/* manual_dwtools.c * * Copyright (C) 1993-2003 David Weenink * * This program is free software; you can redistribute it and/or modify * it under the terms of the GNU General Public License as published by * the Free Software Foundation; either version 2 of the License, or (at * your option) any later version. * * This program is distributed in the hope that it will be useful, but * WITHOUT ANY WARRANTY; without even the implied warranty of * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU * General Public License for more details. * * You should have received a copy of the GNU General Public License * along with this program; if not, write to the Free Software * Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA. */ /* djmw 20020313 GPL djmw 20030317 Latest modification */ #include "ManPagesM.h" #include "TableOfReal_extensions.h" #ifndef _Configuration_h_ #include "Configuration.h" #endif #ifndef _Discriminant_h_ #include "Discriminant.h" #endif static TableOfReal getStandardizedLogFrequencyPolsData (int includeLevels) { TableOfReal me = TableOfReal_createFromPolsData_50males (includeLevels); long i, j; if (! me) return NULL; for (i=1; i <= my numberOfRows; i++) { for (j=1; j <= 3; j++) my data[i][j] = log10 (my data[i][j]); } TableOfReal_standardizeColumns (me); TableOfReal_setColumnLabel (me, 1, "standardized log (%F__1_)"); TableOfReal_setColumnLabel (me, 2, "standardized log (%F__2_)"); TableOfReal_setColumnLabel (me, 3, "standardized log (%F__3_)"); if (includeLevels) { TableOfReal_setColumnLabel (me, 4, "standardized %L__1_"); TableOfReal_setColumnLabel (me, 5, "standardized %L__1_"); TableOfReal_setColumnLabel (me, 6, "standardized %L__3_"); } return me; } static void drawPolsF1F2_log (Graphics g) { TableOfReal me = getStandardizedLogFrequencyPolsData (0); if (me != NULL) { Graphics_setWindow (g, -2.9, 2.9, -2.9, 2.9); TableOfReal_drawScatterPlot (me, g, 1, 2, 0, 0, -2.9, 2.9, -2.9, 2.9, 0, 1); } forget (me); Melder_clearError(); } static void drawPolsF1F2ConcentrationEllipses (Graphics g) { TableOfReal me = getStandardizedLogFrequencyPolsData (0); if (me != NULL) { Discriminant d = TableOfReal_to_Discriminant (me); if (d != NULL) { Discriminant_drawConcentrationEllipses (d, g, 1, 0, 0, 1, 2, -2.9, 2.9, -2.9, 2.9, 12, 1); forget (d); } forget (me); } Melder_clearError(); } static void drawPolsDiscriminantConfiguration (Graphics g) { TableOfReal me = getStandardizedLogFrequencyPolsData (0); if (me != NULL) { Discriminant d = TableOfReal_to_Discriminant (me); if (d != NULL) { Configuration c = Discriminant_and_TableOfReal_to_Configuration (d, me, 2); if (c != NULL) { Configuration_draw (c, g, 1, 2, -2.9, 2.9, -2.9, 2.9, 0, 1, "", 1); forget (c); } forget (d); } forget (me); } Melder_clearError(); } static void drawBoxPlot (Graphics g) { double q25 = 25, q50 = 50, q75 = 60, mean = 45; double hspread = q75 - q25, r = 0.05, w = 0.2; double lowerInnerFence = q25 - 1.5 * hspread; double upperInnerFence = q75 + 1.5 * hspread; double upperOuterFence = q75 + 3.0 * hspread; double lowerWhisker = lowerInnerFence + 0.1 * hspread; double upperWhisker = upperInnerFence - 0.5 * hspread; double ymin = lowerWhisker - 0.1 * hspread, ymax = q75 + 4 * hspread; double x = 0, dx = 0.01, xar = x + 0.7, xtl = xar + dx; double xal1 = x + r + dx, xal2 = x + w + r, y; Graphics_setWindow (g, -1, 2, ymin, ymax); Graphics_setInner (g); Graphics_setTextAlignment (g, Graphics_LEFT, Graphics_HALF); Graphics_line (g, x - r, lowerWhisker, x + r, lowerWhisker); Graphics_line (g, x, lowerWhisker, x, q25); Graphics_line (g, x - w, q25, x + w, q25); Graphics_line (g, x - w, q50, x + w, q50); Graphics_line (g, x - w, q75, x + w, q75); Graphics_line (g, x - w, q25, x - w, q75); Graphics_line (g, x + w, q25, x + w, q75); Graphics_line (g, x, q75, x, upperWhisker); Graphics_line (g, x - r, upperWhisker, x + r, upperWhisker); y = q75 + 2.5 * hspread; Graphics_text (g, x, y, "*"); Graphics_arrow (g, xar, y, xal1, y); Graphics_text (g, xtl, y, "outlier > %upperInnerFence"); y = q75 + 3.5 * hspread; Graphics_text (g, x, y, "o"); Graphics_arrow (g, xar, y, xal1, y); Graphics_text (g, xtl, y, "outlier > %upperOuterFence"); y = upperOuterFence; Graphics_setLineType (g, Graphics_DOTTED); Graphics_line (g, -xtl, y, xtl, y); Graphics_text (g, xtl, y, "%upperOuterFence"); y = upperInnerFence; Graphics_line (g, -xtl, y, xtl, y); Graphics_text (g, xtl, y, "%upperInnerFence"); Graphics_line (g, x - w, mean, x + w, mean); Graphics_setLineType (g, Graphics_DRAWN); y = upperWhisker; Graphics_arrow (g, xar, y, xal2, y); Graphics_text (g, xtl, y, "%upperWhisker"); y = lowerWhisker; Graphics_arrow (g, xar, y, xal2, y); Graphics_text (g, xtl, y, "%lowerWhisker"); y = q75; Graphics_arrow (g, xar, y, xal2, y); Graphics_text (g, xtl, y, "%q75"); y = q25; Graphics_arrow (g, xar, y, xal2, y); Graphics_text (g, xtl, y, "%q25"); y = q50; Graphics_arrow (g, xar, y, xal2, y); Graphics_text (g, xtl, y, "%q50"); y = mean; Graphics_arrow (g, xar, y, xal2, y); Graphics_text (g, xtl, y, "%mean"); Graphics_unsetInner (g); } void manual_dwtools_init (ManPages me); void manual_dwtools_init (ManPages me) { MAN_BEGIN ("AffineTransform", "djmw", 20010927) INTRO ("One of the @@types of objects@ in P\\s{RAAT}.") NORMAL ("An affine transform is a combination of a linear transformation #%A " "and a translation #%t that transforms a vector #%x to a new vector #%y " "in the following way:") FORMULA ("#%y = #%A x + #%t") MAN_END MAN_BEGIN ("AffineTransform: Invert", "djmw", 20011008) INTRO ("Get the inverse of the selected @AffineTransform object.") NORMAL ("The inverse from") FORMULA ("#%y = #%A x + #%t") NORMAL ("is:") FORMULA ("#%x = #%A^^-1^ - #%A^^-1^#%t.") MAN_END MAN_BEGIN ("Band filtering in the frequency domain", "djmw", 20010404) INTRO ("We describe how band filtering in the frequency domain is performed.") NORMAL ("We start with a @Sound and end with a filter bank representation of " "this sound. We assume a standard analysis context: a sound divided into " "frames according to a certain %%window length% and %%time step%. We will " "simulate a filterbank with %N filters.") NORMAL ("The algorithm for each sound frame proceeds in the following way:") LIST_ITEM ("1. Apply a Gaussian window to the sound frame.") LIST_ITEM ("2. Convert the windowed frame into a @Spectrum object.") LIST_ITEM ("3. Convert the spectral amplitudes to %energy values by squaring " "the real and imaginary parts and multiplying by %df, the frequency " "distance between two successive frequency points in the spectrum. " "Since the Spectrum object only contains positive frequencies, " "we have to multiply all energy values, except the first and the last " "frequency, by another factor of 2 to compensate for negative frequencies.") LIST_ITEM ("4. For each of the %N filters in the filter bank: determine the " "inner product of its filter function with the energies as determined in " "the previous step. The result of each inner product is the energy in the " "corresponding filter.") LIST_ITEM ("5. Convert the energies in each filter to power by dividing by " "the %%window length%.") LIST_ITEM ("6. Correct the power, due to the windowing of the frame, by dividing " "by the integral of the %squared windowing function.") LIST_ITEM ("7. Convert all power values to %dB's according to 10 * log10 " "(%power / 4 10^^-10^).") MAN_END MAN_BEGIN ("Bonferroni correction", "djmw", 20011107) NORMAL ("In general, if we have %k independent significance tests " "at the %\\al level, the probability %p that we will get no significant " "differences in all these tests is simply the product of the individual " "probabilities: (1 - %\\al)^^%k^. " "For example, with %\\al = 0.05 and %k = 10 we get %p = 0.95^^10^ = 0.60. " "This means, however, we now have a 40\\% chance that one of these 10 " "tests will turn out significant, despite each individual test only being " "at the 5\\% level. " "In order to guarantee that the overall significance test is still at the " "%\\al level, we have to adapt the significance level %\\al\\'p of the " "individual test. ") NORMAL ("This results in the following relation between the overall and the " "individual significance level:") FORMULA ("(1 - %\\al\\'p)^^%k%^ = 1 - %\\al.") NORMAL ("This equation can easily be solved for %\\al\\'p:") FORMULA ("%\\al\\'p = 1 - (1-%\\al)^^1/%k^,") NORMAL ("which for small %\\al reduces to:") FORMULA ("%\\al\\'p = %\\al / %k") NORMAL ("This is a very simple recipe: If you want an overall significance " "level %\\al and you perform %k individual tests, simply divide %\\al " "by %k to obtain the significance level for the individual tests.") MAN_END MAN_BEGIN ("box plot", "djmw", 20000524) INTRO ("A box plot provides a simple graphical summary of data. These plots " "originate from the work of @@Tukey (1977)@.") ENTRY ("Definitions") NORMAL ("The following figure shows an annotated box plot.") PICTURE (5.0, 5.0, drawBoxPlot) NORMAL ("To understand the box plot we need the following definitions:") LIST_ITEM ("%q25 = lower quartile, 25\\% of the data lie below this value") LIST_ITEM ("%q50 = median, 50\\% of the data lie below this value") LIST_ITEM ("%q75 = upper quartile, 25\\% of the data lie above this value") NORMAL ("The following definitions all depend on these quantiles:") LIST_ITEM ("%hspread = |%q75 \\-- %q25| (50\\% interval)") LIST_ITEM ("%lowerOuterFence = %q25 \\-- 3.0 * %hspread (not in figure)") LIST_ITEM ("%lowerInnerFence = %q25 \\-- 1.5 * %hspread (not in figure)") LIST_ITEM ("%upperInnerFence = %q75 + 1.5 * %hspread") LIST_ITEM ("%upperOuterFence = %q75 + 3.0 * %hspread") LIST_ITEM ("%lowerWhisker = smallest data value larger then %lowerInnerFence") LIST_ITEM ("%upperWhisker = largest data value smaller then %upperInnerFence") NORMAL ("The box plot is a summary of the data in which:") LIST_ITEM ("\\bu the horizontal lines of the rectangle correspond to " "%q25, %q50 and %q75, respectively.") LIST_ITEM ("\\bu the dotted line corresponds to the mean.") LIST_ITEM ("\\bu the outliers outside the %outerFences are drawn with an 'o'.") LIST_ITEM ("\\bu the outliers in the intervals (%lowerOuterFence, %lowerInnerFence) " "and (%upperInnerFence, %upperOuterFence) are drawn with an '*'.") LIST_ITEM ("\\bu the whisker lines outside the rectangle connect %q25 with %lowerWhisker, and, " "%q75 with %upperWhisker, respectively. With no outliers present, the " "whiskers mark minimum and/or maximum of the data.") MAN_END MAN_BEGIN ("BarkFilter", "djmw", 20010404) INTRO ("One of the @@types of objects@ in P\\s{RAAT}.") NORMAL ("An object of type BarkFilter represents an acoustic time-frequency " "representation of a sound: the power spectral density %P(%z, %t), expressed " "in dB's. " "It is sampled into a number of points around equally spaced times %t__%i_ " "and frequencies %z__%j_ (on a Bark scale).") ENTRY ("Inside a BarkFilter") NORMAL ("With @Inspect you will see that this type contains the same " "attributes a @Matrix.") MAN_END MAN_BEGIN ("Categories", "djmw", 19960918) INTRO ("One of the @@types of objects@ in P\\s{RAAT}.") NORMAL ("An object of type Categories represents an ordered collection of categories. Each " "category is a simple text string.") ENTRY ("Categories commands") NORMAL ("Creation:") LIST_ITEM ("\\bu ##Create an empty Categories#") LIST_ITEM ("\\bu @@FFNet & Pattern: To Categories...@") NORMAL ("Viewing and editing:") LIST_ITEM ("\\bu @CategoriesEditor") NORMAL ("Analysis:") LIST_ITEM ("\\bu @@Categories: To Confusion@") LIST_ITEM ("\\bu @@Categories: Difference@") NORMAL ("Synthesis") LIST_ITEM ("\\bu @@Categories: Append@") LIST_ITEM ("\\bu ##Categories: Permute...#") LIST_ITEM ("\\bu ##Categories: To unique Categories#") ENTRY ("Inside a Categories") NORMAL ("With @Inspect you will see the following attributes:") TAG ("%size") DEFINITION ("the number of simple categories.") TAG ("%item[]") DEFINITION ("the categories. Each category is an object of type #SimpleString.") MAN_END MAN_BEGIN ("Categories: Difference", "djmw", 19960918) INTRO ("A command to compute the difference between two selected @Categories objects.") ENTRY ("Behaviour") NORMAL ("Each element in the first object is compared with the corresponding " "object in the second object according to its compare method. " "The number of different %categories will be shown in the @@Info window@.") MAN_END MAN_BEGIN ("Categories: To Confusion", "djmw", 19960918) INTRO ("A command to compute the @Confusion matrix from two selected " "@Categories objects.") ENTRY ("Algorithm") NORMAL ("A confusion matrix is constructed from both #Categories objects in " "the following way: The first Categories object is considered the stimulus " "Categories and its unique (sorted) categories " "form the row indices of the confusion matrix, the unique (sorted) " "categories of the second object form the column indices of this matrix.") NORMAL ("Next, each element in the first #Categories object is compared with " "the corresponding object in the second object and the element in the " "confusion matrix adressed by this pair is incremented by 1.") MAN_END MAN_BEGIN ("Categories: Append", "djmw", 19960918) INTRO ("You can choose this command after selecting 2 objects of type @Categories. " "A new object is created that contains the second object appended after the first.") MAN_END MAN_BEGIN ("Categories: Edit", "djmw", 19960918) INTRO ("You can choose this command after selecting one #Categories. " "A @CategoriesEditor will appear on the screen, with the selected #Categories in it.") MAN_END MAN_BEGIN ("CategoriesEditor", "djmw", 19960918) ENTRY ("An editor for manipulating @Categories.") NORMAL ("To make a selection, use the left mouse button.") NORMAL ("The Ctrl key extends a selection (discontinuously).") NORMAL ("The Shift key extends a selection contiguously.") MAN_END MAN_BEGIN ("CC", "djmw", 20010219) INTRO ("One of the @@types of objects@ in P\\s{RAAT}.") NORMAL ("Any object that represents cepstral coefficients as a function of " "time.") MAN_END MAN_BEGIN ("CCA", "djmw", 20020323) INTRO ("One of the @@types of objects@ in P\\s{RAAT}. ") NORMAL ("An object of type CCA represents the Canonical Correlation " "Analysis of two multivariate datasets.") ENTRY ("Commands") NORMAL ("Creation:") LIST_ITEM ("\\bu @@TableOfReal: To CCA...@") MAN_END MAN_BEGIN ("CCA: Get zero correlation probability...", "djmw", 20020501) INTRO ("Get the probability that for the selected @CCA object the chosen " "canonical correlation coefficient is different from zero.") ENTRY ("Argument") TAG ("%Index") DEFINITION ("is the index of the canonical correlation coefficient that " "you want to test.") ENTRY ("Algorithm") NORMAL ("Wilks' statistic: the probability that coefficient \\ro__%index_ " "differs from zero is ") FORMULA (" %probability = chiSquareQ (\\ci^2, %ndf),") NORMAL ("where the %%number of degrees of freedom% parameter equals") FORMULA ("%ndf = (%n__y_ - %index +1)(%n__x_ - %index +1)") NORMAL ("and the chi-squared parameter is") FORMULA ("\\ci^2 = \\--(%numberOfObservations - (%n__y_ + %n__x_ +3)/2) " "log (\\La__%index_),") NORMAL ("In the formulas above the variables %n__y_ and %n__x_ are the " "dimensions of the dependent and the independent data sets whose " "canonical correlations have been " "obtained, and Wilks' lambda is:") FORMULA ("\\La__index_ = \\Pi__%i=%index..min(%ny,%nx)_ (1 \\-- \\ro__%i_^^2^)") MAN_END MAN_BEGIN ("CCA & Correlation: To TableOfReal (loadings)", "djmw", 20020525) INTRO ("Determine from the selected @CCA and @Correlation objects the correlations " "of the canonical variables with the original variables. These correlations are " "called %%canonical factor loadings%, or also %%structure correlation " "coefficients%.") MAN_END MAN_BEGIN ("CCA & TableOfReal: To TableOfReal (loadings)", "djmw", 20020525) INTRO ("Determine from the selected @CCA and @TableOfReal objects the correlations " "of the canonical variables with the original variables. These correlations are " "called %%canonical factor loadings%, or also %%structure correlation " "coefficients%.") MAN_END MAN_BEGIN ("CCA & TableOfReal: To TableOfReal (scores)...", "djmw", 20020503) INTRO ("Determines the scores on the dependent and the independent canonical " "variates from the selected @CCA and @TableOfReal objects.") ENTRY ("Arguments") TAG ("%%Dimension of dependent variate") DEFINITION ("defines the partition of the table into the dependent and independent " "parts for which separate scores will be determined.") TAG ("%%Number of canonical correlations") DEFINITION ("determines the dimension, i.e., the number of columns of the resulting " "TableOfReal object. The newly created table will have twice this number of " "columns.") ENTRY ("Behaviour") NORMAL ("The scores on the dependent set are determined as #T__%y_ #Y, where #T__%y_") NORMAL ("The scores for the dependent data will be in the lower numbered columns, " "the scores for the independent part will be in the higher numbered columns of " "the newly created object.") MAN_END MAN_BEGIN ("Canonical correlation analysis", "djmw", 20020509) INTRO ("This tutorial will show you how to perform canonical correlation " "analysis with the praat program.") ENTRY ("1. Objective of canonical correlation analysis") NORMAL ("In canonical correlation analysis we try to find the correlations between " "two data sets. One data set is called the %dependent set, the other the " "%independent set. In the %Praat program these two sets must reside into one " "@TableOfReal object. The lower numbered columns of this table will then be " "interpreted as the dependent part, the rest of the columns as the " "independent part. " "The dimension of, i.e., the number of columns in, the dependent part may not " "exceed the dimension of the independent part.") NORMAL ("As an example, we will use the dataset from @@Pols et al. (1972)@ " "with the frequencies and levels of the first three formants from the 12 " "Dutch monophthong vowels as spoken in /h_t/ context by 50 male speakers. " "We will try to find the canonical correlation between formant frequencies " "(the %dependent part) and levels (the %independent part). " "The dimension of both groups of variates is 3. " "In the introduction of the " "@@discriminant analysis@ tutorial you can find how to get these data, " "how to take the logarithm of the formant frequency values and how to " "standardize them. The following script summarizes:") CODE ("Create TableOfReal from Pols data (50 males)... yes") CODE ("Formula... if col < 4 then log10 (self) else self endif") CODE ("Standardize columns") NORMAL ("Before we start with the %canonical correlation analysis we will first have " "a look at the %Pearson correlations of this table and " "calculate the @Correlation matrix. It is given by:") CODE (" F1 F2 F3 L1 L2 L3") CODE ("F1 1 -0.338 0.191 0.384 -0.505 -0.014") CODE ("F2 -0.338 1 0.190 -0.106 0.526 -0.568") CODE ("F3 0.191 0.190 1 0.113 -0.038 0.019") CODE ("L1 0.384 -0.106 0.113 1 -0.038 0.085") CODE ("L2 -0.505 0.526 -0.038 -0.038 1 0.128") CODE ("L3 -0.014 -0.568 0.019 0.085 0.128 1") NORMAL ("The following script summarizes:") CODE ("To Correlation") CODE ("Draw as numbers... 1 0 decimal 3") NORMAL ("The correlation matrix shows that high correlations exist between some " "formant frequencies and some levels. For example, the correlation " "coefficient between F2 and L2 equals 0.526.") NORMAL ("In a canonical correlation analysis of the dataset above, we try " "to find the linear " "combination %u__1_ of %F__1_, %F__2_ and %F__3_ that correlates maximally " "with the linear combination %v__1_ of %L__1_, %L__2_ and %L__3_. " "When we have found these %u__1_ and %v__1_ we next try to find a new " "combination %u__2_ of the formant frequencies and a new combination " "%v__2_ of the levels that have maximum correlation. These %u__2_ and " "%v__2_ must be uncorrelated with %u__1_ and %v__1_. " "When we express the above with formulas we have:") FORMULA ("%u__1_ = %y__11_%F__1_+%y__12_%F__2_ + %y__13_%F__3_") FORMULA ("%v__1_ = %x__11_%L__1_+%x__12_%L__2_ + %x__13_%L__3_") FORMULA ("\\ro(%u__1_, %v__1_) = maximum, \\ro(%u__2_, %v__2_) = submaximum, ") FORMULA ("\\ro(%u__2_, %u__1_) = \\ro (%u__2_, %v__1_) = \\ro (%v__2_, %v__1_) " "= \\ro (%v__2_, %u__1_) = 0,") NORMAL ("where the \\ro(%u__i_, %v__i_) are the correlations between the " "##canonical variates# %u__i_ and %v__i_ and the %y__%ij_'s and %x__%ij_'s are" " the ##canonical coefficients# for the dependent and the independent " "variates, respectively.") ENTRY ("2. How to perform a canonical correlation analysis") NORMAL ("Select the TableOfReal and choose from the dynamic menu the option " "@@TableOfReal: To CCA...|To CCA...@. This command is available in the " "\"Multivariate statistics\" action button. We fill out the form and supply " "3 for %%Dimension of dependent variate%. The resulting CCA object will bear " "the same name as the TableOfReal object. The following script summarizes:") CODE ("select TableOfReal pols_50males") CODE ("To CCA... 3") ENTRY ("3. How to get the canonical correlation coefficients") NORMAL ("You can get the canonical correlation coefficients by queries of the CCA " "object: ##Get correlation...#.") ENTRY ("4. How to obtain canonical scores") NORMAL ("Canonical scores are the linear combinations:") FORMULA ("%u__%i_ = %y__%i1_%F__1_+%y__%i2_%F__2_ + %y__%i3_%F__3_, and,") FORMULA ("%v__%i_ = %x__%i1_%L__1_+%x__%i2_%L__2_ + %x__%i3_%L__3_,") NORMAL ("where the index %i runs from 1 to the number of correlation coefficients.") NORMAL ("You can get the canonical scores by selecting a CCA object together with " "the TableOfReal object and choose ") NORMAL ("@@CCA & TableOfReal: To TableOfReal (scores)...@") ENTRY ("5. How to predict one dataset from the other") NORMAL ("@@CCA & TableOfReal: Predict...@") MAN_END MAN_BEGIN ("CCA & TableOfReal: Predict...", "djmw", 20020503) INTRO ("") MAN_END MAN_BEGIN ("Chebyshev polynomials", "djmw", 19990620) INTRO ("The Chebyshev polynomials %T__%n_(%x) of degree %n are special orthogonal polynomial functions " "defined on the domain [-1, 1].") NORMAL ("Orthogonality:") FORMULA ("__-1_\\in^^1^ %W(%x) %T__%i_(%x) %T__%j_(%x) %dx = \\de__%ij_") FORMULA ("%W(%x) = (1 \\-- %x^^2^)^^\\--1/2^ (-1 < x < 1)") NORMAL ("They obey certain recurrence relations:") FORMULA ("%T__%n_(%x) = 2 %x %T__%n-1_(%x) \\-- %T__%n-2_(%x)") FORMULA ("%T__0_(%x) = 1") FORMULA ("%T__1_(%x) = %x") MAN_END MAN_BEGIN ("ChebyshevSeries", "djmw", 19990620) INTRO ("One of the @@types of objects@ in P\\s{RAAT}.") NORMAL ("An object of type ChebyshevSeries represents a linear combination of @@Chebyshev polynomials@ " "%T__%k_(%x).") FORMULA ("ChebyshevSeries (%x) = \\Si__%k=1..%numberOfCoefficients_ %c__%k_ %T__%k_(%x)") MAN_END MAN_BEGIN ("ChebyshevSeries: To Polynomial", "djmw", 19990620) INTRO ("A command to transform the selected @ChebyshevSeries object into a @@Polynomial@ object.") NORMAL ("We find polynomial coefficients %c__%k_ such that") FORMULA ("\\Si__%k=1..%numberOfCoefficients_ %c__%k_ %x^^%k^ = \\Si__%k=1.." "%numberOfCoefficients_ %l__%k_ %T__%k_(%x)") NORMAL ("We use the recurrence relation for @@Chebyshev polynomials@ to calculate these coefficients.") MAN_END MAN_BEGIN ("ClassificationTable", "djmw", 19990525) INTRO ("One of the @@types of objects@ in P\\s{RAAT}.") NORMAL ("An object of type ClassificationTable represents the result of a classification experiment. " "The numbers in a row show how well a particular input matches the classes represented by the " "column labels. The higher the number the better the match.") MAN_END MAN_BEGIN ("concentration ellipse", "djmw", 20000531) NORMAL ("The percentage of bivariate normally distributed data covered by an ellipse " "whose axes have a length of %numberOfSigmas\\.c\\si can be obtained by integration of the p.d.f. " "over an elliptical area. This results in the following equation as can be " "verified from equation 26.3.21 in @@Abramowitz & Stegun (1970)@:") FORMULA ("%percentage = (1 - exp (-%numberOfSigmas^^2^/2))\\.c 100\\% ,") NORMAL ("where the %numberOfSigmas is the radius of the \"ellipse\":") FORMULA ("(%x/%\\si__x_)^2 + (%y/%\\si__y_)^2 = %numberOfSigmas^2.") NORMAL ("The %numberOfSigmas = 1 ellipse covers 39.3\\% , " "the %numberOfSigmas = 2 ellipse covers 86.5\\% and " "the %numberOfSigmas = 3 ellipse covers 98.9\\% of the data.") MAN_END MAN_BEGIN ("confidence interval", "djmw", 20011105) NORMAL ("The confidence interval gives an estimated range of values which " "is likely to include an unknown population parameter. " "The estimated range is calculated from a given set of observations.") ENTRY ("Examples") NORMAL ("At the \\al level of significance a two sided confidence interval " "for the true mean \\mu for normally distributed data with mean %%mean% and " "known standard deviation %\\si can be constructed as:") FORMULA ("%%mean% - %z__\\al/2_ \\si / \\Vr%N \\<_ \\mu \\<_ " "%%mean% + %z__\\al/2_ \\si / \\Vr%N,") NORMAL ("where %z__\\al/2_ = invGaussQ (\\al/2) and %N is the number of observations.") NORMAL ("If the standard deviation is %not known, we have to estimate its value (%s) " "from the data and the formula above becomes:") FORMULA ("%%mean% - %t__%%\\al/2;N%_ %s / \\Vr%N \\<_ \\mu \\<_ " "%%mean% + %t__%%\\al/2;N%_ %s / \\Vr%N,") NORMAL ("where %t__%%\\al/2;N%_ = invStudentQ (%\\al/2, %N-1).") NORMAL ("For %\\al=0.05 and %N=20 we get %z__0.025_=1.96 and %t__0.025;20_=2.039. " "This shows that when we have to estimate the standard deviation from the data, " "the confidence interval is wider than when the standard deviation is known " "beforehand.") MAN_END MAN_BEGIN ("confidence level", "djmw", 20011105) NORMAL ("The confidence level is the probability value 1-\\al associated " "with a @@confidence interval@, where \\al is the level of significance. " "It can also be expressed as a percentage 100(1-\\al)\\% and is than " "sometimes called the %%confidence coefficient%.") MAN_END MAN_BEGIN ("Confusion", "djmw", 20010501) INTRO ("One of the @@types of objects@ in P\\s{RAAT}.") NORMAL ("An object of type Confusions represents a confusion matrix, with " "stimuli as row indices and responses as column indices. The entry at " "position [%i][%j] represents the number of times that response %j " "was given to the stimulus %i.") ENTRY ("Creating a Confusion from data in a text file") NORMAL ("Suppose you have two objects A and B. " "In one way or another, you have acquired the following " "confusions: %\\de__%AA_ = 6, %\\de__%AB_ = 2 , %\\de__%BA_ = 1, " "and %\\de__%BB_ = 7, where %\\de__%AB_ represents the number of confusions " "between object A and object B.") NORMAL ("You can create a simple text file like the following:") CODE ("\"ooTextFile\" ! The line by which Praat can recognize your file") CODE ("\"Confusion\" ! The line that tells Praat about the contents") CODE ("2 \"A\" \"B\" ! Number of columns, and column labels") CODE ("2 ! Number of rows") CODE ("\"A\" 6 2 ! Row label (A), A-A value, A-B value") CODE ("\"B\" 1 7 ! Row label (B), B-A value, B-B value") NORMAL ("This text file can be read with the @@Read from file...@ command. " "Since a Confusion object has the data structure of a " "TableOfReal, you could also start from an apropriate " "@TableOfReal object and cast it to a Confusion object.") ENTRY ("Commands") NORMAL ("Creation:") LIST_ITEM ("\\bu @@Categories: To Confusion") NORMAL ("Drawing") LIST_ITEM ("\\bu ##Draw as numbers...") LIST_ITEM ("\\bu ##Draw as squares...") NORMAL ("Query") LIST_ITEM ("\\bu ##Get column mean (index)...") LIST_ITEM ("\\bu ##Get column mean (label)...") LIST_ITEM ("\\bu ##Get column stdev (index)...") LIST_ITEM ("\\bu ##Get column stdev (label)...") LIST_ITEM ("\\bu @@Confusion: Get fraction correct|Get fraction correct") NORMAL ("Modification") LIST_ITEM ("\\bu ##Formula...") LIST_ITEM ("\\bu ##Remove column (index)...") LIST_ITEM ("\\bu ##Insert column (index)...") LIST_ITEM ("\\bu ##Set row label (index)...") LIST_ITEM ("\\bu ##Set row label (label)...") LIST_ITEM ("\\bu ##Set column label (index)...") LIST_ITEM ("\\bu ##Set column label (label)...") NORMAL ("Analysis:") LIST_ITEM ("\\bu @@Confusion: To Similarity...@") LIST_ITEM ("\\bu @@Confusion: To Dissimilarity (pdf)...@") ENTRY ("Inside a Confusion") NORMAL ("With @Inspect you will see the following attributes:") TAG ("%numberOfRows") DEFINITION ("the number of stimuli.") TAG ("%numberOfColumns") DEFINITION ("the number of responses.") TAG ("%rowLabels") DEFINITION ("the names of the stimuli.") TAG ("columnLabels") DEFINITION ("the names of the responses.") MAN_END MAN_BEGIN ("Confusion: Condense...", "djmw", 20010705) INTRO ("Groups row and column labels of the selected @Confusion object in " "order to reduce its dimension.") ENTRY ("Arguments") TAG ("%Search") DEFINITION ("the pattern to match") TAG ("%Replace") DEFINITION ("the pattern that replaces the match(es)") TAG ("%%Replace limit%") DEFINITION ("limits the maximum number of times that a match/replace cycle " "may occur within each label.") TAG ("%Search and replace are") DEFINITION ("defines whether the search and replace strings are taken " "litterally or as a @@Regular expressions|regular expression@.") ENTRY ("Behaviour") NORMAL ("First all row and column labels are changed according to the search " "and replace specification. Next all rows or columns that have the same " "labels are summed. ") MAN_END MAN_BEGIN ("Confusion: Get fraction correct", "djmw", 20000225) INTRO ("A @@query@ to ask the selected @Confusion matrix for the fraction of " "correct classifications.") NORMAL ("The \"fraction correct\" is defined as the quotient of the number " "of correct classifications and the total number of entries in the matrix.") NORMAL ("Cells with correct classifications have equal row and column labels.") MAN_END MAN_BEGIN ("Confusion: To TableOfReal (marginals)", "djmw", 20011031) INTRO ("A new @TableOfReal object is created from the selected @Confusion " "object with one extra row and column. ") NORMAL ("The first element of the extra row will contain the sum of the " "confusions in the the first %column, the first element of the extra " "column will contain the sum of the confusions in the the first %row, " "etc... The bottom-right element will contain the sum of all confusions.") MAN_END MAN_BEGIN ("Correlation", "djmw", 19990105) INTRO ("One of the @@types of objects@ in P\\s{RAAT}.") NORMAL ("An object of type Correlation represents the correlation coefficients " "of a multivariate data set.") MAN_END MAN_BEGIN ("Correlation: Confidence intervals...", "djmw", 20011105) INTRO ("Calculates @@confidence interval@s for the correlation coefficients " "from the selected @Correlation object(s) and saves these intervals in a " "new @TableOfReal object.") ENTRY ("Argument") TAG ("%%Confidence level%") DEFINITION ("the @@confidence level@ you want for the confidence intervals.") TAG ("%%Number of tests%") DEFINITION ("determines the @@Bonferroni correction@ for the significance " "level. If the default value (zero) is chosen, it will be set equal " "to the number of correlations involved (a matrix of dimension %n " "has %n\\.c(%n-1)/2 correlations).") TAG ("%%Approximation%") DEFINITION ("defines the approximation that will be used to calculate the " "confidence intervals. It is either Fisher's z transformation or Ruben's " "transformation. According to @@Boomsma (1977)@, Ruben's approximation is " "more accurate than Fisher's.") ENTRY ("Algorithm") NORMAL ("We obtain intervals by the large-sample conservative multiple tests " "with Bonferroni inequality and the Fisher or Ruben transformation. " "We put the upper values of the confidence intervals in the upper " "triangular part of the matrix and the lower values of the confidence " "intervals in lower triangular part of the resulting TableOfReal object.") NORMAL ("In %%Fisher's approximation%, for each element %r__%ij_ of the " "correlation matrix the confidence interval is:") FORMULA ("#[ tanh (%z__%ij_ - %z__%\\al\\'p_ / \\Vr(%N - 3)) , " "tanh (%z__%ij_ + %z__%\\al\\'p_ / \\Vr(%N - 3)) #],") NORMAL ("where %z__%ij_ is the Fisher z-transform of the correlation %r__%ij_:") FORMULA ("%z__%ij_ = 1/2 ln ((1 + %r__%ij_) / (1 - %r__%ij_)), ") NORMAL ("%z__%\\al\\'p_ the Bonferroni corrected %z-value " "%z__%\\al/(2\\.c%numberOfTests)_, ") FORMULA ("%\\al = 1 - %confidenceLevel,") NORMAL ("and %N the number of observations that the correlation matrix is " "based on.") NORMAL ("In %%Ruben's approximation% the confidence interval for element %r__%ij_ " "is:") FORMULA ("#[ %x__1_ / \\Vr(1 - %x__1_^2), %x__2_ / \\Vr(1 - %x__2_^2) #]") NORMAL ("in which %x__1_ and %x__2_ are the smallest and the largest root from") FORMULA ("%a %x^^2^ + %b %x + %c = 0, with") FORMULA ("%a = 2%N - 3 - %z__%\\al\\'p_^^2^") FORMULA ("%b = - 2 %r\\'p \\Vr((2%N - 3)(2%N - 5))") FORMULA ("%c = (2%N - 5 - %z__%\\al\\'p_^^2^) %r\\'p^^2^ - 2%z__%\\al\\'p_^^2^, and") FORMULA ("%r\\'p = %r__%ij_ / \\Vr(1 - %r__%ij_^2),") MAN_END MAN_BEGIN ("Covariance", "djmw", 19990105) INTRO ("One of the @@types of objects@ in P\\s{RAAT}.") NORMAL ("An object of type Covariance represents the sums of squares and cross products of " "a multivariate data set divided by the number of observations.") MAN_END MAN_BEGIN ("Covariance: Difference", "djmw", 19981222) INTRO ("You can choose this command after selecting two objects of type @Covariance. ") NORMAL ("We test the hypothesis that the samples that gave rise to the two " "covariance matrices #%M__1_ and #%M__2_, were drawn from the same " "distribution. The test statistic is %L\\'p which is distributed as " "a \\ci^2 variate with %p(%p+1)/2 degrees of freedom.") FORMULA ("%L\\'p = %L \\.c (1 \\-- (2%p + 1 \\-- 2 / (%p + 1)) / (6 \\.c ( %N \\-- 1))),") NORMAL ("where, ") FORMULA ("%L = (%N \\-- 1) \\.c (ln determinant (#%M__1_) \\-- ln determinant " "(#%M__2_)) + trace (#%M__2_ \\.c #%M__1_^^\\--1^) \\-- %p), ") NORMAL ("%p is dimension of covariance matrix and " "%N is the number of observations underlying the covariance matrix. ") NORMAL ("For more details on this test, see e.g. page 292 of @@Morrison (1990)@.") MAN_END MAN_BEGIN ("Covariance: Get significance of one mean...", "djmw", 20020107) INTRO ("Gets the level of significance for one mean from the selected " "@Covariance object being different from a hypothesized mean.") ENTRY ("Arguments") TAG ("%Index") DEFINITION ("the position of the element of the means vector (centroid).") TAG ("%%Value") DEFINITION ("the hypothesized mean %\\mu (see below).") ENTRY ("Behaviour") NORMAL ("This is the standard test on means when the variance is unknown. " "The test statistic is") FORMULA ("%t = (%mean - %\\mu) \\Vr%(N / %s^2),") NORMAL ("which has the Student %t distribution with %ndf = %N-1 degrees of freedom.") NORMAL ("In the formulas above, %mean is the element of the mean vector at " "position %index, %\\mu is the hypothesized mean, " "%N is the number of observations, %s^2 " "is the variance at position [%index][%index] in the covariance matrix.") NORMAL ("The returned probability %p is the %%two-sided% probability") FORMULA ("%p = 2 * studentQ (%t, %ndf)") NORMAL ("A low probability %p means that the difference is significant.") MAN_END MAN_BEGIN ("Covariance: Get significance of means difference...", "djmw", 20020107) INTRO ("Gets the level of significance for the %difference of two means " "from the selected @Covariance object being different from a hypothesized " "value.") ENTRY ("Arguments") TAG ("%%Index1, Index2") DEFINITION ("the positions of the two elements of the means vector whose " "difference is compared to the hypothesized difference.") TAG ("%%Value") DEFINITION ("the hypothesized difference (%\\mu).") TAG ("%%Paired samples") DEFINITION ("determines whether we treat the two means as being dependent. ") TAG ("%%Equal variances") DEFINITION ("determines whether the distribution of the difference of the means " "is a Student t-distribution (see below).") ENTRY ("Behaviour") NORMAL ("This is Student's t-test for the significance of a difference of means. " "The test statistic is:") FORMULA ("%t = (%mean__1_ - %mean__2_ - %\\mu) \\Vr (%N / %s^2) with %ndf " "degrees of freedom.") NORMAL ("In the formula above %mean__1_ and %mean__2_ are the elements of the " "means vector, %\\mu is the hypothesized difference and %N is the number of " "observations. The value that we use for the (combined) variance %s^2 is:") FORMULA ("%s^2 = %var__1_ + %var__2_ - 2 * %covar__12_,") NORMAL ("when the samples are %paired, and ") FORMULA ("%s^2 = %var__1_ + %var__2_ ") NORMAL ("when they are not.") NORMAL ("The %var__1_ and %var__2_ are the variance components for " "%mean__1_ and %mean__2_, respectively, and %covar__12_ is their covariance." " When we have %%paired samples% we assume that the two variances are " "not independent and their covariance is subtracted, otherwise their " "covariance is not taken into account. Degrees of freedom parameter %ndf " "usually equals 2(%N-1). ") NORMAL ("If the two variances are significantly different, the statistic %t " "above is only %approximately distributed as Student's %t with " "degrees of freedom equal to:") FORMULA ("%ndf = (%N-1) \\.c (%var__1_ + %var__2_)^2 / (%var__1_^2 + " "%var__2_^2).") NORMAL ("The returned probability %p will be the %%two-sided% probability") FORMULA ("%p = 2 * studentQ (%t, %ndf)") NORMAL ("A low probability %p means that the difference is significant.") MAN_END MAN_BEGIN ("Covariance: Get significance of one variance...", "djmw", 20020107) INTRO ("Gets the probability for one variance from the selected " "@Covariance object being different from a hypothesized variance.") ENTRY ("Arguments") TAG ("%Index") DEFINITION ("the position of the variance element.") TAG ("%%Hypothesized variance") DEFINITION ("the hypothesized variance \\si^2") ENTRY ("Behaviour") NORMAL ("The test statistic") FORMULA ("%\\ci^2 = (%N-1)%s^2 / \\si^2,") NORMAL ("is distributed as a chi-squared variate with %ndf = %N-1 degrees of freedom.") NORMAL ("The returned probability %p will be ") FORMULA ("%p = chiSquareQ (%\\ci^2, %ndf)") MAN_END MAN_BEGIN ("Covariance: Get significance of variance ratio...", "djmw", 20020107) INTRO ("Gets the probability for the ratio of two variances " "from the selected @Covariance object being different from a hypothesized " "ratio.") ENTRY ("Arguments") TAG ("%%Index1, index2") DEFINITION ("detemine the variances") TAG ("%%Hypothesized ratio") DEFINITION ("the hypothesized ratio %F") ENTRY ("Behaviour") NORMAL ("The test statistic") FORMULA ("%f = %s__1_^2 / %s__2_^2 / %ratio") NORMAL ("is distributed as Fisher's F distribution with %ndf__1_ = %N-1 and " "%ndf__2_ = %N-1 degrees of freedom for the numerator and denominator terms, " "respectively.") NORMAL ("The returned probability %p will be the %%two-sided% probability") FORMULA ("%p = 2 * fisherQ (%f, %ndf__1_, %ndf__2_)") NORMAL ("If %s__2_^2 > %s__1_^2 we use 1/%f to determine the probability.") MAN_END MAN_BEGIN ("Covariance: To TableOfReal (random sampling)...", "djmw", 20000524) INTRO ("Generate a @TableOfReal object by random sampling from a multi-variate " "normal distribution whose @Covariance matrix is the selected object.") ENTRY ("Argument") TAG ("%%Number of data points") DEFINITION ("determines the number of data points that will be generated. Each " "data point occupies one row in the generated table.") ENTRY ("Algorithm") NORMAL ("The algorithm proceeds as follows:") LIST_ITEM ("1. Calculate the eigenvalues $v__%i_ and " "eigenvectors %#e__%i_ of the %m \\xx %m Covariance matrix. " "In general there will also be %m of these. Let #%E be the %m \\xx %m matrix " "with eigenvector %#e__%j_ in column %j (%j=1..%m).") LIST_ITEM ("2. Generate a vector #x whose elements %x__%k_ equal %x__%k_ = " "randomGauss (0, \\Vr (%v__%k_)). " "Each %x__%k_ is a random deviate drawn from a Gaussian distribution with " "mean zero and standard deviation equal to the square root of the corresponding " "eigenvalue %v__%k_.") LIST_ITEM ("3. Calculate the vector #y = #%E #x, obtained by multiplying the vector " "#x with the matrix #%E.") LIST_ITEM ("4. Add the centroid to #y and copy the elements of #y to the corresponding row of " "the TableOfReal object.") LIST_ITEM ("5. Repeat steps 2, 3 and 4 until the desired number of data points " "has been reached.") LIST_ITEM ("6. Copy the column labels from the Covariance object to the " "TableOfReal object.") MAN_END MAN_BEGIN ("Create ChebyshevSeries...", "djmw", 19990608) INTRO ("A command to create a @ChebyshevSeries from a list of coefficients.") ENTRY ("Arguments") TAG ("%Xmin, %Xmax") DEFINITION ("define the domain of the polynomials.") TAG ("%Coefficients") DEFINITION ("define the coefficients of each @@Chebyshev polynomials|Chebyshev polynomial@. " "The coefficient of the polynomial with the highest degree comes last.") MAN_END MAN_BEGIN ("Create ISpline...", "djmw", 19990608) INTRO ("A command to create an @ISpline from a list of coefficients.") ENTRY ("Arguments") TAG ("%Xmin, %Xmax") DEFINITION ("define the domain of the polynomial @spline.") TAG ("%Degree") DEFINITION ("defines the degree of the polynomial @spline.") TAG ("%Coefficients") DEFINITION ("define the coefficients of the basis polynomials.") TAG ("%%Interior knots") DEFINITION ("define the positions in the domain where continuity conditions are defined.") ENTRY ("Behaviour") NORMAL ("The number of coefficients and the number of interior knots must satisfy " "the following relation:") FORMULA ("%numberOfCoefficients = %numberOfInteriorKnots + %degree") NORMAL ("") MAN_END MAN_BEGIN ("Create MSpline...", "djmw", 19990608) INTRO ("A command to create an @MSpline from a list of coefficients.") ENTRY ("Arguments") TAG ("%Xmin, %Xmax") DEFINITION ("define the domain of the polynomial @spline.") TAG ("%Degree") DEFINITION ("defines the degree of the polynomial @spline.") TAG ("%Coefficients") DEFINITION ("define the coefficients of the basis polynomials.") TAG ("%%Interior knots") DEFINITION ("define the positions in the domain where continuity conditions are defined.") ENTRY ("Behaviour") NORMAL ("The number of coefficients and the number of interior knots must satisfy " "the following relation:") FORMULA ("%numberOfCoefficients = %numberOfInteriorKnots + %degree + 1") NORMAL ("") MAN_END MAN_BEGIN ("Create Polynomial...", "djmw", 19990627) INTRO ("A command to create an @Polynomial from a list of coefficients.") ENTRY ("Arguments") TAG ("%Xmin, %Xmax") DEFINITION ("define the domain of the polynomial.") TAG ("%Degree") DEFINITION ("defines the degree of the basis polynomials.") TAG ("%Coefficients") DEFINITION ("define the coefficients of the polynomial. The coefficient of the " "highest power of %x comes last.") MAN_END MAN_BEGIN ("Create LegendreSeries...", "djmw", 19990608) INTRO ("A command to create a @LegendreSeries from a list of coefficients.") ENTRY ("Arguments") TAG ("%Xmin, %Xmax") DEFINITION ("define the domain of the polynomials.") TAG ("%Coefficients") DEFINITION ("define the coefficients of each @@Legendre polynomials|Legendre polynomial@. " "The coefficient of the polynomial with the highest degree comes last.") MAN_END MAN_BEGIN ("Create Sound from gamma-tone...", "djmw", 19980712) INTRO ("A command to create a @Sound as a @@gamma-tone@.") ENTRY ("Arguments") TAG ("%Name") DEFINITION ("the name of the resulting Sound object.") TAG ("%%Minimum time%, %%Maximum time% (s)") DEFINITION ("the start and end time of the resulting Sound.") TAG ("%%Sample rate% (Hz)") DEFINITION ("the sampling frequency of the resulting Sound.") TAG ("%Gamma") DEFINITION ("the exponent of the polynomial.") TAG ("%Frequency (Hz), %Bandwidth (Hz)") DEFINITION ("determine the frequency and damping of the cosine wave in the gamma-tone.") TAG ("%%Initial phase% (radians)") DEFINITION ("the initial phase of the sine wave.") TAG ("%%Addition factor% (default: 0)") DEFINITION ("determines the degree of asymmetry in the spectrum of the gamma-tone. " "The zero default value gives a gamma-tone. A value unequal to zero results in a " "so called %gamma-chirp. A negative value is used in auditory filter modeling to " "guarantee the usual direction of filter asymmetry, which corresponds to an upward " "glide in instantaneous frequency.") TAG ("%%Scale amplitudes") DEFINITION ("determines whether the amplitudes will be scaled to fit in the range (-1, 1).") ENTRY ("Purpose") NORMAL ("to create a Sound according to the following formula:") FORMULA ("%t^^%\\ga\\--1^ e^^\\--2%\\pi\\.c%bandwidth\\.c%t^ " "cos (2%%\\pi\\.cfrequency\\.ct% + %additionFactor\\.cln(%t) + %initialPhase),") NORMAL ("The %gamma %chirp function has a monotonically frequency-modulated carrier (the chirp) with " "instantaneous frequency ") FORMULA ("%instantaneousFrequency(%t) = %frequency + %additionFactor / (2\\.c\\pi\\.c%t)") NORMAL ("and an envelope that is a gamma distribution function. It is a theoretically optimum " "auditory filter, in the sense that it leads to minimal uncertainty in the joint time and " "scale representation of auditory signal analysis.") NORMAL ("For faithful modeling of the inner ear, " "@@Irino & Patterson (1996)@ conclude that a value of approximately 1.5 * ERB (%frequency) " "is appropriate for %bandwidth. " "ERB stands for @@equivalent rectangular bandwidth@. Their formula for ERB is:") FORMULA ("ERB(%f) = 6.23 10^^\\--6^ %f^2 + 93.39 10^^\\--3^ %f + 28.52.") NORMAL ("To avoid @aliasing in the chirp sound, a sound is only generated during times where the " "instantaneous frequency is greater than zero and smaller than the @@Nyquist frequency@.") MAN_END MAN_BEGIN ("Create Sound from Shepard tone...", "djmw", 19970314) INTRO ("One of the commands that create a @Sound.") ENTRY ("Arguments") TAG ("%Name") DEFINITION ("the name of the resulting Sound object.") TAG ("%%Minimum time%, %%Maximum time% (s)") DEFINITION ("the start and end time of the resulting Sound.") TAG ("%%Sample rate% (Hz)") DEFINITION ("the sampling frequency of the resulting Sound.") TAG ("%%Lowest frequency% (Hz)") DEFINITION ("the frequency of the lowest component.") TAG ("%%Number of components%") DEFINITION ("the number of frequency components.") TAG ("%%Frequency change% (semitones/s)") DEFINITION ("determines how long it takes to change all frequencies by one octave.") TAG ("%%Amplitude range% (dB)") DEFINITION ("determines the relative size of the maximum and the minimum amplitude of components.") ENTRY ("Purpose") NORMAL ("to create a Sound that is a continuous variant of the sound sequences " "used by @@Shepard (1964)@ in his " "experiment about the circularity in judgments of relative pitch.") NORMAL ("The tone consists of many sinusoidal components whose frequencies " "increase exponentially in time. " "All frequencies are always at successive intervals of an octave and sounded simultaneously. " "Thus the frequency of each component above the lowest is at each moment in time exactly twice " "the frequency of the one just below. The amplitudes are large for the components of intermediate " "frequency only, and tapered off gradually to subthreshold levels for the components at the " "highest and lowest extremes of frequency.") NORMAL ("The Sound is generated according to the following specification:") FORMULA ("%s(%t) = \\su__%i=1..%numberOfComponents_ %A__%i_(%t) sin (arg__%i_(%t)), where") FORMULA ("arg__%i_(%t) = \\in 2%\\pi f__%i_(%t) %dt , and") FORMULA ("f__%i_(%t) = %lowestFrequency \\.c 2^^(%i \\-- 1 + %t/(12/%frequencyChange_st)^, with") FORMULA ("%A__%i_(%t) = %L__min_ + (1 \\-- %L__min_) (1 \\-- cos 2%\\pi%\\te__%i_(%t)) / 2, with,") FORMULA ("%\\te__%i_(%t) = ln ((%f__%i_(%t) + 1 Hz) / (%lowestFrequency + 1 Hz)) / " "ln ((%maximumFrequency + 1 Hz) / (%lowestFrequency + 1 Hz))") FORMULA ("%L__min_ = 10^^\\--%amplitudeRange/10^, and,") FORMULA ("%maximumFrequency = %lowestFrequency\\.c2^^%numberOfComponents\\--1^") MAN_END MAN_BEGIN ("Create formant table (Peterson & Barney 1952)", "djmw", 20020620) INTRO ("A command to create a @Table object filled with (two repetitions of) the " "fundamental frequency and the first three formant frequency values from 10 " "American-English monophthong vowels as spoken in /h_d/ context by 76 speakers " "(33 men, 28 women and 15 children).") ENTRY ("Table layout") NORMAL ("The created table will contain 9 columns") TAG ("Column 1, labeled as %Type") DEFINITION ("speaker type: \"m\", \"w\" or \"c\" (for %man, %women or %child).") TAG ("Column 2, labeled as %Sex") DEFINITION ("speaker sex: Either \"m\" or \"f\" (for %male or %female).") TAG ("Column 3, labeled as %Speaker") DEFINITION ("speaker id: a number from 1 to 76.") TAG ("Column 4, labeled as %Vowel") DEFINITION ("the vowel name. The following list gives the vowel in h_d context word " "together with its representation in this column: (%heed, iy), (%hid, ih), " "(%head, eh), (%had, ae), (%hod, ah), (%hawed, aa), (%hood, ao), (%who'd, uh), " "(%hud, uw), (%heard, er).") TAG ("Column 5, labeled as %IPA") DEFINITION ("the IPA-notation for the vowels") TAG ("Column 6, labeled as %F0") DEFINITION ("the fundamental frequency in Hertz") TAG ("Column 7, 8 and 9, labeled as %F1, %F2 and %F3") DEFINITION ("the frequencies in Hertz of the first three formants.") NORMAL ("The data were supplied in printed form by Ignatius Mattingly. " "More details about these data and how they were measured can be found in " "@@Peterson & Barney (1952)@.") MAN_END MAN_BEGIN ("Create formant table (Pols & Van Nierop 1973)", "djmw", 20020620) INTRO ("A command to create a @Table object filled with the frequencies and the levels " "of the first three formants from the 12 Dutch monophthong " "vowels as spoken in /h_t/ context by 50 male and 25 female speakers.") ENTRY ("Table layout") NORMAL ("The created table will contain 10 columns") TAG ("Column 1, labeled as %Sex") DEFINITION ("speaker sex: Either \"m\" or \"f\" (for %male or %female).") TAG ("Column 2, labeled as %Speaker") DEFINITION ("speaker id: a number from 1 to 75.") TAG ("Column 3, labeled as %Vowel") DEFINITION ("the vowel name. The following list gives the vowel in p_t context word " "together with its representation in this column: (%poet, oe), (%paat, aa), " "(%poot, oo), (%pat, a), (%peut, eu), (%piet, ie), (%puut, uu), (%peet, ee), " "(%put, u), (%pet, e), (%pot, o), (%pit, i).") TAG ("Column 4, labeled as %IPA") DEFINITION ("the IPA-notation for the vowels") TAG ("Column 5, 6 and 7, labeled as %F1, %F2 and %F3") DEFINITION ("the frequencies in Hertz of the first three formants.") TAG ("Column 8, 9 and 10, labeled as %L1, %L2 and %L3") DEFINITION ("the levels in decibel below overall SPL of the first three formants.") NORMAL ("More details about these data and how they were measured can be found " "in @@Pols et al. (1972)@ and @@Van Nierop et al. (1973)@.") MAN_END MAN_BEGIN ("Create TableOfReal from Pols data (50 males)...", "djmw", 19990426) INTRO ("A command to create a @TableOfReal filled with the first three formant " "frequency values and (optionally) the levels from the 12 Dutch monophthong " "vowels as spoken in /h_t/ context by 50 male speakers.") NORMAL ("The first three columns will contain the frequencies in Hz, the next three columns " "the levels in decibels below the overall SPL of the measured vowel segment. Each row will " "be labelled with its corresponding vowel symbol.") NORMAL ("More details about these data and how they were measured can be found in the paper of " "@@Pols et al. (1972)@.") MAN_END MAN_BEGIN ("Discriminant", "djmw", 19981103) INTRO ("One of the @@types of objects@ in P\\s{RAAT}.") NORMAL ("An object of type Discriminant represents the discriminant structure of a multivariate " "data set with several groups. This discriminant structure consists of a number of orthogonal " "directions in space, along which maximum separability of the groups can occur.") ENTRY ("Commands") NORMAL ("Creation:") LIST_ITEM ("\\bu @@Discriminant analysis@ tutorial") LIST_ITEM ("\\bu @@TableOfReal: To Discriminant@") NORMAL ("Drawing") LIST_ITEM ("\\bu Draw eigenvalues (scree)...") LIST_ITEM ("\\bu Draw eigenvector...") LIST_ITEM ("\\bu @@Discriminant: Draw sigma ellipses...|Draw sigma ellipses...@") MAN_END MAN_BEGIN ("Discriminant analysis", "djmw", 19990227) INTRO ("This tutorial will show you how to perform discriminant analysis with " "the @@Praat program@.") NORMAL ("As an example, we will use the dataset from @@Pols et al. (1972)@ " "with the frequencies and levels of the first three formants from the 12 " "Dutch monophthong vowels as spoken in /h_t/ context by 50 male speakers. " "This data set has been incorporated into the " "Praat program and can be called into play with the @@Create TableOfReal " "from Pols data (50 males)...@ command that can be found in the \"New / " "TableOfReal\" menu.") NORMAL ("In the list of objects a new TableOfReal object will appear with 6 " "columns and 600 rows " "(50 speakers \\xx 12 vowels). The first three columns contain " "the formant frequencies in Hz, the last three columns contain the levels " "of the first three formants " "given in decibels below the overall sound pressure level of the measured " "vowel segment. Each row is labelled with a vowel label.") NORMAL ("Pols et al. use logarithms of frequency values, we will too. Because " "the measurement units in the first three columns are in Hz and in the last " "three columns in dB, it is probably better to standardize the columns. " "The following script summarizes our achievements up till now:") CODE ("Create TableOfReal from Pols data (50 males)... yes") CODE ("Formula... if col < 4 then log10 (self) else self fi") CODE ("Standardize columns") CODE ("\\# change the column labels too, for nice plot labels.") CODE ("Set column label (index)... 1 standardized log (\\% F\\_ \\_ 1\\_ )") CODE ("Set column label (index)... 2 standardized log (\\% F\\_ \\_ 2\\_ )") CODE ("Set column label (index)... 3 standardized log (\\% F\\_ \\_ 3\\_ )") CODE ("Set column label (index)... 4 standardized \\% L\\_ \\_ 1\\_ ") CODE ("Set column label (index)... 5 standardized \\% L\\_ \\_ 2\\_ ") CODE ("Set column label (index)... 6 standardized \\% L\\_ \\_ 3\\_ ") NORMAL ("To get an indication of what these data look like, we make a scatter " "plot of the " "first standardized log-formant-frequency against the second standardized " "log-formant-frequency. With the next script fragment you can reproduce the " "following picture.") CODE ("Viewport... 0 5 0 5") CODE ("select TableOfReal pols_50males") CODE ("Draw scatter plot... 1 2 0 0 -2.9 2.9 -2.9 2.9 no yes") PICTURE (5, 5, drawPolsF1F2_log) NORMAL ("Apart from a difference in scale this plot is the same as fig. 3 in the " "Pols et al. article.") ENTRY ("1. How to perform a discriminant analysis") NORMAL ("Select the TableOfReal and choose from the dynamic menu the option " "@@TableOfReal: To Discriminant|To Discriminant@. This command is available " "in the \"Multivariate statistics\" action button. The resulting Discriminant " "object will bear the same name as the TableOfReal object. The following " "script summarizes:") CODE ("select TableOfReal pols_50males") CODE ("To Discriminant") ENTRY ("2. How to project data on the discriminant space") NORMAL ("You select a TableOfReal and a Discriminant object together and choose: " "@@Discriminant & TableOfReal: To Configuration...|To Configuration...@. " "One of the options of the newly created Configuration object is to draw it. " "The following picture shows how the data look in the plane spanned by the " "first two dimensions of this Configuration. The directions in this " "configuration are the eigenvectors from the Discriminant.") PICTURE (5, 5, drawPolsDiscriminantConfiguration) NORMAL ("The following script summarizes:") CODE ("select TableOfReal pols_50males") CODE ("plus Discriminant pols_50males") CODE ("To Configuration... 0") CODE ("Viewport... 0 5 0 5") CODE ("Draw... 1 2 -2.9 2.9 -2.9 2.9 yes") NORMAL ("If you are only interested in this projection, there also is a short cut " "without an intermediate Discriminant object: " "select the TableOfReal object and choose @@TableOfReal: To Configuration " "(lda)...|To Configuration (lda)...@.") ENTRY ("3. How to draw concentration ellipses") NORMAL ("Select the Discriminant object and choose @@Discriminant: Draw sigma " "ellipses...|Draw sigma ellipses...@. In the form you can fill out the " "coverage of the ellipse by way of the %%Number of sigmas% parameter. " "You can also select the projection " "plane. The next figure shows the 1-%\\si concentration ellipses in the " "standardized log %F__1_ vs log %F__2_ plane. When the data are multinormally distributed, " "a 1-%\\si ellipse will cover approximately 39.3\\% of the data. " "The following code summarizes:") CODE ("select Discriminant pols_50males") CODE ("Draw sigma ellipses... 1.0 no 1 2 -2.9 2.9 -2.9 2.9 yes") PICTURE (5, 5, drawPolsF1F2ConcentrationEllipses) ENTRY ("4. How to classify") NORMAL ("Select together the Discriminant object (the classifier), and " "a TableOfReal object (the data to be classified). Next you choose " "@@Discriminant & TableOfReal: To ClassificationTable...|To " "ClassificationTable@. " "Normally you will enable the option %%Pool covariance matrices% and " "the pooled covariance matrix will be used for classification. ") NORMAL ("The ClassificationTable can be converted to a @Confusion object " "and its fraction correct can be queried with: " "@@Confusion: Get fraction correct@.") MAN_END MAN_BEGIN ("Discriminant: Draw sigma ellipses...", "djmw", 19990222) INTRO ("A command to draw for each group from the selected @Discriminant an ellipse " "that covers part of the multivariate data.") ENTRY ("Arguments") TAG ("%%Number of sigmas") DEFINITION ("determines the @@concentration ellipse|data coverage@.") TAG ("%%Discriminant plane") DEFINITION ("When on, the selected %X and %Y-dimension will refer to the eigenvectors " "of the discriminant space, and, consequently, the projection of the hyper ellipsoid " "onto the space spanned by these eigenvectors will be drawn. When off, the selected " "%X and Y-dimension will refer to the original dimensions.") TAG ("%Xmin, %Xmax, %Ymin, %Ymax") DEFINITION ("determine the limits of the drawing area.") TAG ("%%Label size") DEFINITION ("determines the size of the labels at the centre of the ellipse. No " "labels will be drawn when a value less than or equal to zero is chosen.") MAN_END MAN_BEGIN ("Discriminant: Extract within-group SSCP...", "djmw", 20020314) INTRO ("Extract the @SSCP for group %%index% from the selected @Discriminant " "object.") MAN_END MAN_BEGIN ("Discriminant: Extract pooled within-groups SSCP", "djmw", 20020314) INTRO ("Extract the pooled within-group @SSCP from the selected @Discriminant " "object.") MAN_END MAN_BEGIN ("Discriminant: Get partial discrimination probability...", "djmw", 19981102) INTRO ("A command to test the selected @Discriminant for the significance of " "discrimination afforded by the remaining %n\\--%k eigenvectors after the acceptance of " "the first %k eigenvectors.") ENTRY ("Details") NORMAL ("The test statistic is:") FORMULA ("%\\ci^2 = \\--(%degreesOfFreedom\\--(%numberOfGroups+%dimension)/2) ln \\La\\'p, where") FORMULA ("%degreesOfFreedom = (%dimension\\--%k)(%numberOfGroups\\--%k\\--1), and, ") FORMULA ("\\La\\'p = \\Pi__%j=%k+1..%numberOfEigenvalues_ 1 / (1 + %%eigenvalue[j]%)") MAN_END MAN_BEGIN ("Discriminant: Get contribution of component...", "djmw", 19981106) INTRO ("A command to ask the selected @Discriminant for the contribution of the %j^^th^ " "discriminant function (component) to the total variance.") ENTRY ("Details") NORMAL ("The contribution is defined as:") FORMULA ("%%eigenvalue[j]% / \\Si__%i=1..%numberOfEigenvalues_ %%eigenvalue[i]%") MAN_END MAN_BEGIN ("Discriminant: Get Wilks' lambda...", "djmw", 19990216) INTRO ("A command to ask the selected @Discriminant for the value of Wilks' lamda (a " "multivariate measure of group differences over several variables).") ENTRY ("Arguments") TAG ("%From") DEFINITION ("the first eigenvalue number from which the value for lambda has to be calculated.") ENTRY ("Details") NORMAL ("Wilks' lambda is defined as:") FORMULA ("%\\La = \\Pi__%i=%from..%numberOfEigenvalues_ 1 / (1 + %eigenvalue[%i])") NORMAL ("Because lambda is a kind of %inverse measure, values of lambda which are near zero " "denote high discrimination between groups.") MAN_END MAN_BEGIN ("Discriminant: Get concentration ellipse area...", "djmw", 20000531) INTRO ("A command to query the @Discriminant object for the area of the concentration " "ellipse of one of its groups.") ENTRY ("Arguments") TAG ("%%Number of sigmas") DEFINITION ("determines the @@concentration ellipse|data coverage@.") TAG ("%%Discriminant plane") DEFINITION ("When on, the selected %X and %Y-dimension will refer to the eigenvectors " "of the discriminant space, and, consequently, the area of the projection of the hyper ellipsoid " "onto the space spanned by these eigenvectors will be calculated. When off, the selected " "%X and Y-dimension will refer to the original dimensions.") ENTRY ("Algorithm") NORMAL ("See @@SSCP: Get sigma ellipse area...") MAN_END MAN_BEGIN ("Discriminant: Get confidence ellipse area...", "djmw", 20000531) INTRO ("A command to query the @Discriminant object for the area of the confidence " "ellipse of one of its groups.") ENTRY ("Arguments") TAG ("%%Discriminant plane") DEFINITION ("When on, the selected %X and %Y-dimension will refer to the eigenvectors " "of the discriminant space, and, consequently, the area of the projection of the hyper ellipsoid " "onto the space spanned by these eigenvectors will be calculated. When off, the selected " "%X and Y-dimension will refer to the original dimensions.") ENTRY ("Algorithm") NORMAL ("See @@SSCP: Get confidence ellipse area...") MAN_END MAN_BEGIN ("Discriminant & SSCP: Project", "djmw", 20020313) INTRO ("A command to project the selected @SSCP object on the eigenspace " "defined by the selected @Discriminant object.") NORMAL ("Further details can be found in @@Eigen & SSCP: Project@") MAN_END MAN_BEGIN ("Discriminant & TableOfReal: To ClassificationTable...", "djmw", 19990506) INTRO ("A command to use the selected @Discriminant to classify each row from the " "selected @TableOfReal. The newly created @ClassificationTable will then contain the posterior " "probabilities of group membership.") ENTRY ("Argument") TAG ("%%Pool covariance matrices") DEFINITION ("when on, all group covariance matrices are pooled and distances will be determined " "on the basis of only this pooled covariance matrix (see below).") ENTRY ("Details") NORMAL ("The posterior probabilities of group membership %p__%j_ for a vector #x are defined as:") FORMULA ("%p__%j_ = %p(%j\\| #%x) = exp (\\--%d__%j_^^2^(#%x) / 2) / " "\\su__%k=1..%numberOfGroups_ exp (\\--%d__%k_^^2^(#%x) / 2),") NORMAL ("where %d__%i_^^2^ is the generalized squared distance function:") FORMULA ("%d__%i_^^2^(#%x) = ((#%x\\--#%\\mu__%i_)\\'p #\\Si__%i_^^-1^ (#%x\\--#%\\mu__%i_) + " "ln determinant (#\\Si__%i_)) / 2 \\-- ln %aprioriProbability__%i_") NORMAL ("that depends on the individual covariance matrix #\\Si__%i_ and the mean " "#%\\mu__%i_ for group %i.") NORMAL ("When the covariances matrices are %pooled, the squared distance function can be reduced to:") FORMULA ("%d__%i_^^2^(#%x) = ((#%x\\--#%\\mu__%i_)\\'p #\\Si^^-1^ (#%x\\--#%\\mu__%i_) " "\\-- ln %aprioriProbability__%i_,") NORMAL ("and #\\Si is now the pooled covariance matrix.") NORMAL ("The apriori probabilities normally will have values that are related to the number of " "%training vectors %n__%i_ in each group:") FORMULA ("%aprioriProbability__%i_ = %n__%i_ / \\Si__%k=1..%numberOfGroups_ %n__%k_") MAN_END MAN_BEGIN ("Discriminant & TableOfReal: To Configuration...", "djmw", 19990429) INTRO ("A command to project each row in the selected @TableOfReal onto " "a space spanned by the eigenvectors of the selected @Discriminant. ") ENTRY ("Arguments") TAG ("%%Number of dimensions") DEFINITION ("specifies the number of eigenvectors taken into account, i.e., determines " "the dimension of the resulting @Configuration. When the default value (0) is " "given the resulting Configuration will have the maximum dimension as allowed by " "the number of eigenvectors in the selected Discriminant.") ENTRY ("Precondition") NORMAL ("The number of columns in the TableOfReal must equal the dimension of the " "eigenvectors in the Discriminant.") NORMAL ("See also @@Eigen & TableOfReal: Project...@.") MAN_END MAN_BEGIN ("DTW", "djmw", 20000525) INTRO ("One of the @@types of objects@ in P\\s{RAAT}.") NORMAL ("An object of type DTW represents the dynamic time warp structure of " "two objects.") ENTRY ("Commands") NORMAL ("Creation:") LIST_ITEM ("\\bu @@CC: To DTW...@ (from 2 objects with cepstral coefficients)") LIST_ITEM ("\\bu ##Spectrogram: To DTW...# (from 2 Spectrogram objects)") NORMAL ("Query:") LIST_ITEM ("\\bu @@DTW: Get time along path...@") MAN_END MAN_BEGIN ("DTW: Get time along path...", "djmw", 20010703) INTRO ("Queries the selected @DTW object for the time along the minimal path " "given the time along the \"x-direction\". ") ENTRY ("Arguments") TAG ("%Time") DEFINITION ("the time along the \"x-direction\".") TAG ("%%In case of ambiguity choose%,") DEFINITION ("determines which time to return when several possibilities exist. " "Ambiguity arises when the path at the choosen time frame has insertions " "(a plot of the path will then show a line parallel to the y-direction in " "this time frame). " "The options %highest and %lowest return the " "highest and lowest value of this parallell line. With no insertions " "present, the returned times will be exactly equal for both options.") ENTRY ("Behaviour") NORMAL ("When the %input time is in the interval [%xmin, %xmax], the %returned " "time will be in the interval [%ymin, %ymax], where [%xmin, %xmax] and " "[%ymin, %ymax] are the domains of the two \"objects\" from which the " "DTW-object was constructed." "For all other input times we assume that the two object are aligned. The " "returned time value will therefor simply equal the input time." ) MAN_END MAN_BEGIN ("Eigen", "djmw", 19981102) INTRO ("One of the @@types of objects@ in P\\s{RAAT}.") NORMAL ("An object of type Eigen represents the eigen structure of " "a matrix whose eigenvalues and eigenvectors are real.") ENTRY ("Inside an Eigen") NORMAL ("With @Inspect you will see the following attributes:") TAG ("%numberOfEigenvalues") DEFINITION ("the number of eigenvalues and eigenvectors") TAG ("%dimension") DEFINITION ("the dimension of an eigenvector.") TAG ("%eigenvalues[1..%numberOfEigenvalues]") DEFINITION ("the real eigenvalues.") TAG ("%eigenvectors[1..%numberOfEigenvalues][1..%dimension]") DEFINITION ("the real eigenvectors, stored by row.") MAN_END MAN_BEGIN ("Eigen: Draw eigenvalues (scree)...", "djmw", 19990111) INTRO ("A command to draw a scree plot of the eigenvalues of the selected @Eigen object(s).") NORMAL ("In a scree plot we plot the pairs (1, eigenvalue[1]), (2, eigenvalue[2]), ... " "(%p, eigenvalue[%p]). The eigenvalues are sorted from large to small.") MAN_END MAN_BEGIN ("Eigen: Draw eigenvector...", "djmw", 19990112) INTRO ("A command to draw an eigenvector from the selected @Eigen.") ENTRY ("Arguments") TAG ("%Eigenvector %number") DEFINITION ("determines the eigenvector to be drawn.") TAG ("%Component %loadings") DEFINITION ("when on, the eigenvector is multiplied with the square root of the corresponding " "eigenvalue. (For @@PCA@-analysis this means that you will draw the so called " "%%component loading vector%. You will be able to compare " "quantitatively the elements in different component loading vectors because " "the %i-th element in the %j-th component loading vector gives the covariance between the %i-th " "original variable and the %j-th principal component.)") TAG ("%First %element, %Last %element") DEFINITION ("determine the first and last element of the vector that must be drawn.") TAG ("%Minimum, %Maximum") DEFINITION ("determine the lower and upper bounds of the plot.") TAG ("%Mark %size, %Mark %string") DEFINITION ("determine size and type of the marks that will be drawn.") TAG ("%Garnish") DEFINITION ("determines whether a bounding box and margins will be drawn.") MAN_END MAN_BEGIN ("Eigen: Get contribution of component...", "djmw", 19981109) INTRO ("A command to ask the selected @Eigen for the contribution of the %j^^th^ " "eigenvalue to the total sum of eigenvalues.") ENTRY ("Details") NORMAL ("The contribution is defined as:") FORMULA ("%%eigenvalue[j]% / \\Si__%i=1..%numberOfEigenvalues_ %%eigenvalue[i]%") MAN_END MAN_BEGIN ("Eigen: Get cumulative contribution of components...", "djmw", 19981109) INTRO ("A command to ask the selected @Eigen for the contribution of the sum of the " "eigenvalues[%from..%to] to the total sum of eigenvalues.") ENTRY ("Details") NORMAL ("The contribution is defined as:") FORMULA ("\\Si__%i=%from..%to_ %%eigenvalue[i]% / \\Si__%i=1..%numberOfEigenvalues_ %%eigenvalue[i]%") MAN_END MAN_BEGIN ("Eigen & Matrix: Project...", "djmw", 20020328) INTRO ("A command to project the columns of the @Matrix object onto the " "eigenspace of the @Eigen object.") ENTRY ("Argument") TAG ("%%Number of dimensions%,") DEFINITION ("defines the dimension, i.e., the number of rows, of the " "resulting object.") ENTRY ("Algorithm") NORMAL ("Project each column of the Matrix on the coordinate " "system given by the eigenvectors of the Eigen object. This can be done " "as follows:") FORMULA ("%y__%ji_ = \\Si__%k=1..%numberOfColums_ %e__jk_ %x__%ki_, where") NORMAL ("%y__%ji_ is the %j-th element of the %i-th column of the resulting " "(matrix) object, %e__%jk_ is the %k-th element of the %j-th eigenvector " "and, %x__%ki_ is the %k-th element of the %i-th column of the selected " "matrix object.") MAN_END MAN_BEGIN ("Eigen & SSCP: Project", "djmw", 20020328) INTRO ("A command to project the @SSCP object onto the eigenspace of " "the @Eigen object.") ENTRY ("Behaviour") NORMAL ("Transform the SSCP object as if it was calculated in a coordinate " "system given by the eigenvectors of the Eigen object. This can be done " "as follows:") FORMULA ("#%S__%t_ = #%E\\'p #%S #%E, where") NORMAL ("where #%E\\'p is the transpose of the matrix with eigenvectors #%E, " "#%S is the square matrix with sums of squares and crossproducts, and " "#%S__%t_ the newly created square matrix. The dimension of #%S__%t_ may " "be smaller than the dimension of #%S.") MAN_END MAN_BEGIN ("Eigen & TableOfReal: Project...", "djmw", 20020328) INTRO ("A command to project the rows of the @TableOfReal object onto the " "eigenspace of the @Eigen object.") ENTRY ("Argument") TAG ("%%Number of dimensions%,") DEFINITION ("defines the dimension, i.e., the number of columns, of the " "resulting object.") ENTRY ("Algorithm") NORMAL ("Project each row of the TableOfReal on the coordinate " "system given by the eigenvectors of the Eigen object. This can be done " "as follows:") FORMULA ("%y__%ij_ = \\Si__%k=1..%numberOfColums_ %e__jk_ %x__%ik_, where") NORMAL ("%e__%jk_ is the %k-th element of the %j-th eigenvector, %x__%ik_ is " "the %k-th element of the %i-th row and %y__%ij_ is the %j-th element at " "the %i-th row of the matrix part of the resulting object.") MAN_END MAN_BEGIN ("equivalent rectangular bandwidth", "djmw", 19980713) INTRO ("The %%equivalent rectangular bandwidth% (ERB) of a filter is defined " "as the width of a rectangular filter whose height equals the peak gain of " "the filter and which passes the same total power as the filter (given a flat " "spectrum input such as white noise or an impulse).") MAN_END MAN_BEGIN ("Excitations", "djmw", 19960918) INTRO ("A collection of objects of type @Excitation. " "You can create an #Excitations by selecting one or more #Excitation's and " "selecting ##To Excitations#. You can add one or more #Excitation's to an " "#Excitations by selecting one #Excitations and one or more " "#Excitation's and selecting ##Add to Excitations# (the #Excitation's will " "be removed from the list of objects).") MAN_END MAN_BEGIN ("Excitations: Append", "djmw", 19960918) INTRO ("You can choose this command after selecting two objects of type @Excitations. ") NORMAL ("A new object is created that contains the second object appended after the first.") MAN_END MAN_BEGIN ("Excitations: To Pattern...", "djmw", 19960918) INTRO ("A command to convert every selected @Excitations to a @Pattern object.") ENTRY ("Argument") TAG ("%Join") DEFINITION ("the number of subsequent @Excitation objects to combine into one row of @Pattern. " "E.g. if an #Excitation has length 26 and %join = 2 then each row of #Pattern " "contains 52 elements. The number of rows in #Pattern wil be %%my size% / 2. " "In the conversion process the elements of an #Excitation will be divided by 100.0 in order " "to guarantee that all patterns have values between 0 and 1.") MAN_END MAN_BEGIN ("FormantFilter", "djmw", 20010404) INTRO ("One of the @@types of objects@ in P\\s{RAAT}.") NORMAL ("An object of type FormantFilter represents an acoustic time-frequency " "representation of a sound: the power spectral density %P(%f, %t), expressed " "in dB's. " "It is sampled into a number of points around equally spaced times %t__%i_ " "and frequencies %f__%j_ (on a linear frequency scale).") ENTRY ("Inside a FormantFilter") NORMAL ("With @Inspect you will see that this type contains the same " "attributes a @Matrix.") MAN_END MAN_BEGIN ("gamma-tone", "djmw", 19980713) INTRO ("A gamma-tone is the product of a rising polynomial, a decaying exponential function, and a " "cosine wave.") NORMAL ("It can be described with the following formula:") FORMULA ("gammaTone (%t) = %a %t^^%\\ga\\--1^ e^^\\--2%\\pi\\.c%bandwidth\\.c%t^ " "cos (2%%\\pi\\.cfrequency\\.ct% + %initialPhase),") NORMAL ("where %\\ga determines the order of the gamma-tone.") NORMAL ("The gammatone function has a monotone carrier (the tone) with an " "envelope that is a gamma distribution function. The amplitude spectrum is essentially " "symmetric on a linear frequency scale. This function is used in some time-domain " "auditory models to simulate the spectral analysis performed by the basilar membrane. " "It was popularized in auditory modeling by @@Johannesma (1972)@. @@Flanagan (1960)@ " "already used it to model basilar membrane motion.") MAN_END MAN_BEGIN ("generalized singular value decomposition", "djmw", 19981007) INTRO ("The %%generalized singular value decomposition% (gsvd) of an %m \\xx %n matrix #%A and " "a %p \\xx %n matrix #%B is given by the pair of factorizations") FORMULA ("#%A = #%U #%\\Si__1_ [#%0, #%R] #%Q\\'p and #%B = #%V #%\\Si__2_ [#%0, #%R] #%Q\\'p") NORMAL ("The matrices in these factorizations have the following properties:") TAG ("\\bu #%U [%m \\xx %m], #%V [%p \\xx %p] and #%Q [%n \\xx %n]") DEFINITION (" are orthogonal matrices. In the reconstruction formula's above we maximally need " "only the first %n columns of matrices #%U and #%V (when %m and/or %p are greater than %n).") TAG ("\\bu #%R [%r \\xx %r],") DEFINITION ("is an upper triangular nonsingular matrix. %r is the rank of [#%A\\'p, #%B\\'p]\\'p " "and %r \\<_ %n. The matrix [#%0, #%R] is %r \\xx %n and its first %n \\xx (%n \\-- %r) part " "is a zero matrix.") TAG ("\\bu #%\\Si__1_ [%m \\xx %r] and #%\\Si__2_ [%p \\xx %r]") DEFINITION ("are real, nonnegative and \"diagonal\".") NORMAL ("In practice, the matrices #%\\Si__1_ and #%\\Si__2_ are never used. Instead a shorter " "representation with numbers %\\al__%i_ and %\\be__%i_ is used. These numbers obey " "0 \\<_ \\al__%i_ \\<_ 1 and \\al__%i_^^2^ + \\be__%i_^^2^ = 1. The following relations exist:") FORMULA ("#%\\Si__1_\\'p #%\\Si__1_ + #%\\Si__2_\\'p #%\\Si__2_ = #%I, ") FORMULA ("#%\\Si__1_\\'p #%\\Si__1_ = #diag (%\\al__1_^^2^, ..., %\\al__%r_^^2^), and, ") FORMULA ("#%\\Si__2_\\'p #%\\Si__2_ = #diag (%\\be__1_^^2^, ..., %\\be__%r_^^2^).") NORMAL ("The ratios \\al__%i_ / \\be__%i_ are called the %%generalized singular values% of the " "pair #%A, #%B. Let %l be the rank of #%B and %k + %l (= %r) the rank of [#%A\\'p, #%B\\'p]\\'p. " "Then the first %k generalized singular values are infinite and the remaining %l are finite. " "(When %#B is of full rank then, of course, %k = 0).") ENTRY ("Special cases") NORMAL ("\\bu If #%B is a square nonsingular matrix, the gsvd of #%A and #%B is equivalent to the " "singular value decomposition of #%A #%B^^\\--1^.") NORMAL ("\\bu The generalized eigenvalues and eigenvectors of #%A\\'p #%A - %\\la #%B\\'p #%B " "can be expressed in terms of the gsvd. The columns of the matrix #%X, constructed as") CODE ("X = Q*( I 0 )") CODE (" ( 0 inv(R) ),") NORMAL ("form the eigenvectors. The important eigenvectors, of course, correspond " "to the positions where the %l eigenvalues are not infinite.") MAN_END MAN_BEGIN ("invFisherQ", "djmw", 20000525) INTRO ("$$invFisherQ$ (%q, %df1, %df2) returns the value %f for which $$@fisherQ (%f, %df1, %df2) = %q.") MAN_END MAN_BEGIN ("fisherQ", "djmw", 20000525) INTRO ("$$fisherQ$ (%f, %df1, %df2) returns the area under Fisher's F-distribution " "from %f to +\\oo.") MAN_END MAN_BEGIN ("ISpline", "djmw", 19990627) INTRO ("One of the @@types of objects@ in P\\s{RAAT}. ") NORMAL ("An object of type ISpline represents a linear combination of basis i@spline functions. Each " "basis %ispline is a monotonically increasing polynomial function of degree %p.") FORMULA ("ISpline (%x) = \\Si__%k=1..%numberOfCoefficients_ %c__%k_ %ispline__%k_(%x)") MAN_END MAN_BEGIN ("Legendre polynomials", "djmw", 19990620) INTRO ("The Legendre polynomials %P__%n_(%x) of degree %n are special orthogonal polynomial functions " "defined on the domain [-1, 1].") NORMAL ("Orthogonality:") FORMULA ("__-1_\\in^^1^ %W(%x) %P__%i_(%x) %P__%j_(%x) %dx = \\de__%ij_") FORMULA ("%W(%x) = 1 (-1 < x < 1)") NORMAL ("They obey certain recurrence relations:") FORMULA ("%n %P__%n_(%x) = (2%n \\-- 1) %x %P__%n-1_(%x) \\-- (%n \\-- 1) %P__%n-2_(%x)") FORMULA ("%P__0_(%x) = 1") FORMULA ("%P__1_(%x) = %x") NORMAL ("We may %change the domain of these polynomials to [%xmin, %xmax] by " "using the following transformation:") FORMULA ("%x\\'p = (2%x \\-- (%xmax + %xmin)) / (%xmax - %xmin).") NORMAL ("We subsequently use %P__%k_(%x\\'p) instead of %P__%k_(%x).") MAN_END MAN_BEGIN ("LegendreSeries", "djmw", 19990620) INTRO ("One of the @@types of objects@ in P\\s{RAAT}.") NORMAL ("An object of type LegendreSeries represents a linear combination of @@Legendre polynomials@ " "%P__%k_(%x).") FORMULA ("LegendreSeries (%x) = \\Si__%k=1..%numberOfCoefficients_ %c__%k_ %P__%k_(%x)") MAN_END MAN_BEGIN ("LegendreSeries: To Polynomial", "djmw", 19990620) INTRO ("A command to transform the selected @LegendreSeries object into a @@Polynomial@ object.") NORMAL ("We find polynomial coefficients %c__%k_ such that") FORMULA ("\\Si__%k=1..%numberOfCoefficients_ %c__%k_ %x^^%k^ = " "\\Si__%k=1..%numberOfCoefficients_ %l__%k_ %P__%k_(%x)") NORMAL ("We use the recurrence relation for @@Legendre polynomials@ to calculate these coefficients.") MAN_END MAN_BEGIN ("Matrix: Solve equation...", "djmw", 19961006) INTRO ("Solve the general matrix equation #A #x = #b for #x.") NORMAL("The matrix #A can be any general %m \\xx %n matrix, #b is a %m-dimensional " "and #x a %n-dimensional vector. The @Matrix contains #A as its first %n columns " "and #b as its last column. The %n-dimensional solution is returned as a #Matrix " "with %n columns.") NORMAL ("When the number of equations (%m) is %greater than the number of unknowns (%n) the " "algorithm gives the best least-squares solution. If on the contrary you " "have %fewer equations than unknowns the solution will not be unique.") ENTRY ("Method") NORMAL ("Singular value decomposition with backsubstitution. " "Zero will be substituted for eigenvalues smaller than %tolerance \\.c " "%%maximum_eigenvalue% (when the user-supplied %tolerance equals 0.0 a " "value of 2.2 \\.c 10^^-16^ \\.c %%number_of_unknowns% " "will be used as %tolerance).") NORMAL ("See for more details: @@Press et al. (1992)@ chapter 2.") MAN_END MAN_BEGIN ("MelFilter", "djmw", 20010404) INTRO ("One of the @@types of objects@ in P\\s{RAAT}.") NORMAL ("An object of type MelFilter represents an acoustic time-frequency " "representation of a sound: the power spectral density %P(%f, %t), " "expressed in dB's. " "It is sampled into a number of points around equally spaced times %t__%i_ " "and frequencies %f__%j_ (on a Mel frequency scale).") ENTRY ("Inside a MelFilter") NORMAL ("With @Inspect you will see that this type contains the same " "attributes a @Matrix.") MAN_END MAN_BEGIN ("MelFilter: To MFCC...", "djmw", 20010404) INTRO ("A command to create a @MFCC object from each selected @MelFilter " "object.") NORMAL ("Mel frequency cepstral coefficients result from the Discrete Cosine " "Transform of the filterbank spectrum (in dB). The following formula " "shows the relation:") FORMULA ("%c__%i_ = \\Si__%j=1_^^%N^ %P__%j_ cos (%i\\pi/%N (%j-0.5))),") NORMAL ("where %N represents the number of filters and %P__%j_ the power in dB " "in the %j^^%th^ filter.") NORMAL ("This transformation was first used by @@Davis & Mermelstein (1980)@.") MAN_END MAN_BEGIN ("MSpline", "djmw", 19990627) INTRO ("One of the @@types of objects@ in P\\s{RAAT}. ") NORMAL ("An object of type MSpline represents a linear combination of basis " "m@spline functions. Each basis %mspline is a positive polynomial function " "of degree %p.") FORMULA ("MSpline (%x) = \\Si__%k=1..%numberOfCoefficients_ %c__%k_ %mspline__%k_(%x)") MAN_END MAN_BEGIN ("Pattern", "djmw", 19960918) INTRO ("One of the @@types of objects@ in P\\s{RAAT}.") INTRO ("An object of type Pattern represents a sequence of patterns that can serve as " "as input patterns for a neural net.") ENTRY ("Pattern commands") NORMAL ("Creation:") LIST_ITEM ("\\bu ##Create Pattern with zeroes...#") NORMAL ("Synthesis:") LIST_ITEM ("\\bu @@FFNet & Pattern: To Categories...@") LIST_ITEM ("\\bu @@Pattern & Categories: To Feedforward Net...@") ENTRY ("inside a Pattern") NORMAL ("With @Inspect you will see that this type contains the same " "attributes a @Matrix.") MAN_END MAN_BEGIN ("PCA", "djmw", 19990323) INTRO ("One of the @@types of objects@ in P\\s{RAAT}. " "See the @@Principal component analysis@ tutorial.") NORMAL ("An object of type PCA represents the principal components analysis " "of a multivariate dataset.") ENTRY ("Commands") NORMAL ("Creation:") LIST_ITEM ("\\bu @@Principal component analysis@ tutorial") LIST_ITEM ("\\bu @@TableOfReal: To PCA@") ENTRY ("Inside a PCA") NORMAL ("With @Inspect you will see that this type contains the same " "attributes as an @Eigen with the following extras:") TAG ("%numberOfObservations") DEFINITION ("the number of observations in the multivariate dataset that originated the PCA, " "usually equal to the dataset's number of rows.") TAG ("%labels[1..%dimension]") DEFINITION ("the label that corresponds to each dimension.") TAG ("%centroid") DEFINITION ("the centroids of the originating multivariate data set.") MAN_END MAN_BEGIN ("PCA: Get fraction variance accounted for...", "djmw", 19990106) INTRO ("A command to ask the selected @PCA for the fraction %%variance " "accounted for% by the selected components.") ENTRY ("Arguments") TAG ("%%From component%, %%To component%") DEFINITION ("define the range of components. By choosing both numbers equal, " "you get the fraction of the \"variance\" explained by one component.") ENTRY ("Details") NORMAL ("The contribution is defined as:") FORMULA ("\\Si__%i=%from..%to_ %%eigenvalue[i]% / \\Si__%i=1..%numberOfEigenvalues_ %%eigenvalue[i]%") MAN_END MAN_BEGIN ("PCA: Get equality of eigenvalues...","djmw", 19981102) INTRO ("A command to get the probability that some of the eigenvalues of the " "selected @PCA object are equal. A low probability means that it is not " "very likely that that these numbers are equal.") NORMAL ("We test the hypothesis %H__0_: %\\la__%from_ = ... = %\\la__%to_ " "that %r (= %to\\--%from+1) of the eigenvalues \\la of the covariance " "matrix are equal. The remaining eigenvalues are unrestricted as to their " "values and multiplicities. The alternative hypothesis to %H__0_ is that " "some of the eigenvalues in the set are distinct.") ENTRY ("Arguments") TAG ("%From, %To") DEFINITION ("define the range of eigenvalues to be tested for equality.") TAG ("%%Conservative test") DEFINITION ("when on, a more conservative estimate for %n is chosen (see below).") ENTRY ("Details") NORMAL ("The test statistic is:") FORMULA ("\\ci^2 = \\--%n \\Si__%j=%from..%to_ ln %eigenvalue[%j] + %n %r " "ln (\\Si__%j=%from..%to_ %eigenvalue[%j] / %r),") NORMAL ("with %r(%r+1)/2 \\--1 degrees of freedom. Here %n = %totalNumberOfCases \\-- 1.") NORMAL ("A special case occurs when the variation in the last %r dimensions is spherical. In a " "slightly more conservative test we may replace %n by %n\\--%from\\--(2%r^2+%r+2)/6%r.") NORMAL ("Also see @@Morrison (1990)@, page 336.") MAN_END MAN_BEGIN ("PCA: Get number of components (VAF)...", "djmw", 19990111) INTRO ("A command to ask the selected @PCA for the minimum number of " "components that are necessary " "to explain the given fraction %%variance accounted for%.") ENTRY ("Arguments") TAG ("%%Variance accounted for% (fraction)") DEFINITION ("the fraction variance accounted for that must be explained.") MAN_END MAN_BEGIN ("PCA: To TableOfReal (reconstruct 1)...", "djmw", 20030108) INTRO ("A command to reconstruct a single data item. The result is stored as " "a @TableOfReal with only one row.") ENTRY ("Argument") TAG ("%Coefficients") DEFINITION ("the weight for the eigenvectors.") NORMAL ("The algorithm is explained in @@PCA & Configuration: To TableOfReal " "(reconstruct)@.") MAN_END MAN_BEGIN ("PCA & Configuration: To TableOfReal (reconstruct)", "djmw", 20030108) INTRO ("A command to reconstruct a @TableOfReal from the selected @Configuration" " and @PCA.") NORMAL ("The TableOfReal is reconstructed from the eigenvectors of the PCA and " "elements of the Configuration are the weight factors: ") FORMULA ("%#t__%i_ = \\Si__%k_ %c__%ik_ #%e__%k_,") NORMAL ("where %#t__%i_ is the %i-th row in the resulting TableOfReal object, %c__%ik_ is " "the element at row %i and column %k in the Configuration object and #%e__%k_ " "the %k-th eigenvector from the PCA object.") MAN_END MAN_BEGIN ("PCA & TableOfReal: To Configuration...", "djmw", 19990111) INTRO ("A command to construct a @Configuration from the selected @TableOfReal" " and @PCA.") NORMAL ("The TableOfReal is projected on the eigenspace of the PCA, i.e., " "each row of the TableOfReal is treated as a vector, and the inner product " "with the eigenvectors of the PCA " "determine its coordinates in the Configuration.") ENTRY ("Argument") TAG ("%%Number of dimensions") DEFINITION ("determines the dimension of the resulting Configuration.") NORMAL ("See also @@Eigen & TableOfReal: Project...@.") MAN_END MAN_BEGIN ("Polynomial", "djmw", 19990608) INTRO ("One of the @@types of objects@ in P\\s{RAAT}.") NORMAL ("An object of type " "Polynomial represents a polynomial function on a domain.") NORMAL ("A polynomial of degree %n is defined as:") FORMULA ("%p(%x) = %c__1_ + %c__2_ %x + %c__3_ %x^^2^ + ... c__%n+1_ %x^^%n^.") NORMAL ("The real numbers %c__%k_ are called the polynomial %coefficients.") ENTRY ("Commands") NORMAL ("Creation") LIST_ITEM ("\\bu @@Create Polynomial...@ (in the ##New menu#)") LIST_ITEM ("\\bu @@LPC: To Polynomial (slice)...@ (from prediction coefficients)") LIST_ITEM ("\\bu @@LegendreSeries: To Polynomial@") LIST_ITEM ("\\bu @@ChebyshevSeries: To Polynomial@") NORMAL ("Drawing") LIST_ITEM ("\\bu ##Draw...#") NORMAL ("Queries") LIST_ITEM ("\\bu @@Polynomial: Get function value...|Get function value...@: get %p(%x)") LIST_ITEM ("\\bu ##Get coefficient value...#: get %c__%i_") LIST_ITEM ("\\bu @@Polynomial: Get minimum...|Get minimum...@: minimum of %p(%x) on an interval") LIST_ITEM ("\\bu @@Polynomial: Get x of minimum...|Get x of minimum...@") LIST_ITEM ("\\bu @@Polynomial: Get maximum...|Get maximum...@: maximum of %p(%x) on an interval") LIST_ITEM ("\\bu @@Polynomial: Get x of maximum...|Get x of maximum...@") LIST_ITEM ("\\bu @@Polynomial: Get area...|Get area...@") NORMAL ("Modification") LIST_ITEM ("\\bu ##Set domain...#: new domain") LIST_ITEM ("\\bu ##Set coefficient value...#: change one coefficient") NORMAL ("Conversion") LIST_ITEM ("\\bu @@Polynomial: To Spectrum...|To Spectrum...@ (evaluation over unit-circle)") LIST_ITEM ("\\bu @@Polynomial: To Polynomial (derivative)|To Polynomial (derivative)@") LIST_ITEM ("\\bu @@Polynomial: To Polynomial (primitive)|To Polynomial (primitive)@") LIST_ITEM ("\\bu @@Polynomial: To Roots|To Roots@: roots of polynomial") MAN_END MAN_BEGIN ("Polynomial: Get area...", "djmw", 19990610) INTRO ("A command to compute the area below the selected @Polynomial object.") ENTRY ("Arguments") TAG ("%Xmin, %Xmax") DEFINITION ("define the interval.") NORMAL ("The area is defined as __%xmin_\\in^^xmax^ %p(%x) %dx.") MAN_END MAN_BEGIN ("Polynomial: Get function value...", "djmw", 19990610) INTRO ("A command to compute %p(%x) for the selected @Polynomial object.") MAN_END MAN_BEGIN ("Polynomial: Get maximum...", "djmw", 19990610) INTRO ("A command to compute, on a specified interval, the maximum value of the selected " "@Polynomial object.") MAN_END MAN_BEGIN ("Polynomial: Get x of maximum...", "djmw", 19990610) INTRO ("A command to compute, on a specified interval, the location of the maximum of the " "selected @Polynomial object.") MAN_END MAN_BEGIN ("Polynomial: Get minimum...", "djmw", 19990610) INTRO ("A command to compute, on a specified interval, the minimum value of the selected " "@Polynomial object.") MAN_END MAN_BEGIN ("Polynomial: Get x of minimum...", "djmw", 19990610) INTRO ("A command to compute, on a specified interval, the location of the minimum of the " "selected @Polynomial object.") MAN_END MAN_BEGIN ("Polynomials: Multiply", "djmw", 19990616) INTRO ("A command to multiply two @@Polynomial|polynomials@ with each other.") NORMAL ("The result of muliplying 1 + 2 %x and 2 \\-- %x^2 will be the polynomial:") FORMULA ("2 + 4 %x \\-- %x^2 \\-- 2 %x^3.") MAN_END MAN_BEGIN ("Polynomial: To Polynomial (derivative)", "djmw", 19990610) INTRO ("A command to compute the derivative of the selected @Polynomial object.") MAN_END MAN_BEGIN ("Polynomial: To Polynomial (primitive)", "djmw", 19990610) INTRO ("A command to compute the primitive of the selected @Polynomial object.") MAN_END MAN_BEGIN ("Polynomial: Scale x...", "djmw", 19990610) INTRO ("A command to transform the selected @Polynomial object to a new domain.") TAG ("%Xmin, %Xmax") DEFINITION ("define the new domain") ENTRY ("Behaviour") NORMAL ("The polynomial is transformed from domain [%x__min_, %x__max_] to " "domain [%Xmin, %Xmax] in such a way that its form stays the same. " "This is accomplished by first calculating:") FORMULA ("%f(%x\\'p) = \\Si__%k=1..%numberOfCoefficients_ %c__%k_ %x\\'p^^%k^, where") FORMULA ("%x\\'p = %a %x + %b,") NORMAL ("and then collecting terms of equal degree. The %a and %b are defined as") FORMULA ("%a = (%x__min_ \\-- %x__max_) / (%Xmin \\-- %Xmax)") FORMULA ("%b = %x__min_ \\-- %a %Xmin") MAN_END MAN_BEGIN ("Polynomial: To Roots", "djmw", 19990608) INTRO ("A command to compute the @@Roots|roots@ of the selected @Polynomial objects.") ENTRY ("Algorithm") NORMAL ("The roots are found from the polished eigenvalues of a special companion matrix. " "For further explanation on these methods see @@Press et al. (1992)@.") MAN_END MAN_BEGIN ("Polynomial: To Spectrum...", "djmw", 19990616) INTRO ("A command to compute the @@Spectrum|spectrum@ of the selected @Polynomial objects.") ENTRY ("Arguments") TAG ("%%Nyquist frequency% (Hz)") DEFINITION ("defines the highest frequency in the spectrum. The lowest frequency of the spectrum " "will be 0 Hz.") TAG ("%%Number of frequencies") DEFINITION ("defines the number of frequencies in the spectrum.") ENTRY ("Algorithm") NORMAL ("We calculate the spectrum by evaluating the polynomial at regularly spaced points %z__%k_ " "on the upper half of a circle with radius %r = 1 in the complex plane. The upperhalf of the " "unit circle, where %k\\.c%\\fi is in the interval [0, %\\pi], will be mapped to frequencies " "[0, @@Nyquist frequency@] in the spectrum. ") NORMAL ("The complex values %z__%k_ (%k=1..%numberOfFrequencies) are defined as:") FORMULA ("%z__%k_ = %r e^^%i %k %\\fi^, where,") FORMULA ("%\\fi = \\pi / (%numberOfFrequencies \\-- 1) and %r = 1.") MAN_END MAN_BEGIN ("Principal component analysis", "djmw", 19990323) INTRO ("This tutorial describes how you can perform principal component " "analysis with the @@Praat program@.") NORMAL ("Principal component analysis (PCA) involves a mathematical procedure " "that transforms a number of (possibly) correlated variables into a " "(smaller) number of uncorrelated variables called %%principal " "components%. The first principal component accounts for as much of the " "variability in the data as possible, and each succeeding component " "accounts for as much of the remaining variability as possible.") NORMAL ("Tradionally, principal component analysis is performed on a square " "symmetric matrix of type @SSCP (pure sums of squares and cross " "products), @Covariance (scaled sums of squares and cross products), or " "@Correlation (sums of squares and cross products from standardized " "data). " "The analysis results for objects of type SSCP and Covariance do not " "differ. A Correlation object has to be used if the variances of " "individual variates differ much, or if the units of measurement of the " "individual variates differ. The result of a principal component analysis " "on such objects will be a new object of type @PCA.") ENTRY ("1. Objectives of principal component analysis") LIST_ITEM ("\\bu To discover or to reduce the dimensionality of the data set.") LIST_ITEM ("\\bu To identify new meaningful underlying variables.") ENTRY ("2. How to start") NORMAL ("We assume that the multi-dimensional data have been collected in a " "@TableOfReal object which " "is a rectangular matrix with (optional) row and column labels.") NORMAL ("If the variances of the individual columns in the TableOfReal " "differ much or the measurement units of the columns differ then you " "should first standardize the data. You can do this by choosing " "@@TableOfReal: Standardize columns|Standardize columns@, since the " "covariance matrix from standardized data equals the correlation matrix.") NORMAL ("Select the TabelOfReal object from the list of objects and choose " "@@TableOfReal: To PCA|To PCA@. This results in a new PCA object in the " "list of objects.") NORMAL ("We can now make a scree plot of the eigenvalues, @@Eigen: Draw " "eigenvalues (scree)...|Draw eigenvalues (scree)...@ " "to get an indication of the importance of each eigenvalue. The exact " "contribution of each eigenvalue (or a range of eigenvalues) to the " "\"explained variance\" can also be queried: " "@@PCA: Get fraction variance accounted for...|Get fraction variance " "accounted for...@. You might also check for the equality of a " "number of eigenvalues: @@PCA: Get equality of eigenvalues...|Get equality " "of eigenvalues...@.") ENTRY ("3. Determining the number of components") NORMAL ("There are two methods to help you to choose the number of components. " "Both methods are based on relations between the eigenvalues.") LIST_ITEM ("\\bu Plot the eigenvalues, @@Eigen: Draw eigenvalues (scree)...|" "Draw eigenvalues (scree)...@") LIST_ITEM1 ("If the points on the graph tend to level out (show an \"elbow\"), " "these eigenvalues are usually close enough to zero that they can be " "ignored.") LIST_ITEM ("\\bu Limit variance accounted for, @@PCA: Get number of components " "(VAF)...|Get number of components (VAF)...@.") ENTRY ("4. Getting the principal components") NORMAL ("Principal components are obtained by projecting the multivariate " "datavectors on the space spanned by the eigenvectors. This can be done " "in two ways:") LIST_ITEM ("1. Directly from the TableOfReal without first forming a " "@PCA object: " "@@TableOfReal: To Configuration (pca)...|To Configuration (pca)...@. " "You can then draw the Configuration or display its numbers. ") LIST_ITEM ("2. Select a PCA and a TableOfReal object together and choose " "@@PCA & TableOfReal: To Configuration...|To Configuration...@. " "In this way you project the TableOfReal onto the PCA's eigenspace.") ENTRY ("5. Mathematical background on principal component analysis") NORMAL ("The mathematical technique used in PCA is called eigen analysis: " "we solve for the eigenvalues and eigenvectors of a square symmetric " "matrix with sums of squares and cross products. " "The eigenvector associated with the largest eigenvalue has the same " "direction as the first principal component. The eigenvector associated " "with the second largest eigenvalue determines the direction of the second " "principal component. " "The sum of the eigenvalues equals the trace of the square matrix and the " "maximum number of eigenvectors equals the number of rows (or columns) of " "this matrix.") ENTRY ("6. Algorithms") NORMAL ("If our matrix happens to be symmetric, with some sort of sums of " "squares and cross products, we solve for the eigenvalue and eigenvectors " "by first performing a Householder reduction to tridiagonal form, followed" " by the QL algorithm with implicit shifts.") NORMAL ("If, conversely, our starting point is a non-square or non-symmetric " "data matrix #%A (that could originate from a TableOfReal), " "we do not have to form explicitly the matrix with sums of squares and " "cross products, #%A\\'p#%A. Instead, we proceed by a numerically more " "stable method, and form the @@singular value decomposition@ " "#%U #%\\Si #%V\\'p of #%A. The matrix #%V then contains the eigenvectors, " "and the squared diagonal elements of #%\\Si contain the eigenvalues.") MAN_END MAN_BEGIN ("Regular expressions", "djmw", 20010706) INTRO ("This tutorial describes the %syntax of regular expressions in the " "@@Praat program@.") ENTRY ("Introduction") NORMAL ("A %%regular expression% is a text string that describes a %set " "of strings. Regular expressions (regex) are useful as a way to search " "for patterns in text strings and, optionally, replace them by another " "pattern.") NORMAL ("Some regex match only one string, i.e., the set they describe has " "only one member. For example, the regex \"ab\" matches the string \"ab\" " "and no others. Other regex match more than one string, i.e., the set " "they describe has more than one member. For example, the regex \"a*\" " "matches the string made up of any number (including zero) of \"a\"s. " "As you can see, some characters match themselves (such as \"a\" and " "\"b\") and these characters are called %ordinary characters. The " "characters that don't match themselves, such as \"*\", are called " "%special characters or %meta characters. Many special characters are only " "special characters in the %search regex and are ordinary characters in " "the substitution regex. ") NORMAL ("You can read the rest of this tutorial sequentially with the help of " "the \"<1\" and \">1\" buttons.") LIST_ITEM ("1. @@Regular expressions 1. Special characters|Special characters@ " "(\\bs \\^ \\$ { } [ ] ( ) . + ? \\| - &)") LIST_ITEM ("2. @@Regular expressions 2. Quantifiers|Quantifiers@ " "(how often do we match).") LIST_ITEM ("3. @@Regular expressions 3. Anchors|Anchors@ (where do we match)") LIST_ITEM ("4. @@Regular expressions 4. Special constructs with parenthesis|" "Special constructs with parenthesis@ (grouping constructs)") LIST_ITEM ("5. @@Regular expressions 5. Special control characters|" "Special control characters@ (difficult-to-type characters like \\bsn)") LIST_ITEM ("6. @@Regular expressions 6. Convenience escape sequences|" "Convenience escape sequences@ " "(\\bsd \\bsD \\bsl \\bsL \\bss \\bsS \\bsw \\bsW \\bsB)") LIST_ITEM ("7. @@Regular expressions 7. Octal and hexadecimal escapes|" "Octal and hexadecimal escapes@ (things like \\bs053 or \\bsX2B)") LIST_ITEM ("8. @@Regular expressions 8. Substitution special characters|" "Substitution special characters@ (\\bs1..\\bs9 \\bsU \\bsu \\bsL \\bsl &)") NORMAL ("More in depth coverage of regular expressions can be found in " "@@Friedl (1997)@.") MAN_END MAN_BEGIN ("Regular expressions 1. Special characters", "djmw", 20010718) INTRO ("The following characters are the %meta characters that give special " "meaning to the regular expression search syntax:") TAG ("#\\bs# the backslash %escape character.") DEFINITION ("The backslash gives special meaning to the character " "following it. For example, the combination \"\\bsn\" stands for the " "%newline, one of the @@Regular expressions 5. Special control characters" "|control characters@. The combination \"\\bsw\" stands for a \"word\" " "character, one of the @@Regular expressions 6. Convenience escape " "sequences|" "convenience escape sequences@ while \"\\bs1\" is one of the @@Regular " "expressions 8. Substitution special characters|substitution special " "characters@.") LIST_ITEM1("Example: The regex \"aa\\bsn\" tries to match two consecutive " "\"a\"s at the end of a line, inclusive the newline character itself.") LIST_ITEM1 ("Example: \"a\\bs+\" matches \"a+\" and not a series of one or " "\"a\"s.") TAG ("##\\^ # the caret is the start of line @@Regular expressions 3. " "Anchors|anchor@ or the negate symbol.") LIST_ITEM1 ("Example: \"\\^ a\" matches \"a\" at the start of a line.") LIST_ITEM1 ("Example: \"[\\^ 0-9]\" matches any non digit.") TAG ("##\\$ # the dollar is the end of line @@Regular expressions 3. " "Anchors|anchor@.") LIST_ITEM1 ("Example: \"b\\$ \" matches a \"b\" at the end of a line.") LIST_ITEM1 ("Example: \"\\^ b\\$ \" matches the empty line.") TAG ("##{ }# the open and close curly bracket are used as range @@Regular " "expressions 2. Quantifiers|quantifiers@.") LIST_ITEM1 ("Example: \"a{2,3}\" matches \"aa\" or \"aaa\".") TAG ("##[ ]# the open and close square bracket define a character class to " "match a %single character.") DEFINITION ("The \"\\^ \" as the first character following the \"[\" negates " "and the match is for the characters %not listed. " "The \"-\" denotes a range of characters. Inside a \"[ ]\" character " "class construction most special characters are interpreted as ordinary " "characters. ") LIST_ITEM1 ("Example: \"[d-f]\" is the same as \"[def]\" and matches \"d\", " "\"e\" or \"f\".") LIST_ITEM1 ("Example: \"[a-z]\" matches any lowercase characters in the " "alfabet.") LIST_ITEM1 ("Example: \"[\\^ 0-9]\" matches any character that is not a digit.") LIST_ITEM1 ("Example: A seach for \"[][()?<>$^.*?^]\" in the string " "\"[]()?<>$^.*?^\" followed by a replace string \"r\" has the result " "\"rrrrrrrrrrrrr\". Here the search string is %one character class and " "all the meta characters are interpreted as ordinary characters without " "the need to escape them.") TAG ("##( )# the open and close parenthesis are used for grouping " "characters (or other regex).") DEFINITION ("The groups can be referenced in " "both the seach and the @@Regular expressions 8. Substitution special " "characters|substitution@ phase. There also exist some @@Regular " "expressions 4. Special constructs with parenthesis|special constructs " "with parenthesis@.") LIST_ITEM1 ("Example: \"(ab)\\bs1\" matches \"abab\".") TAG ("##.# the dot matches any character except the newline.") LIST_ITEM1 ("Example: \".a\" matches two consecutive characters where " "the last one is \"a\".") LIST_ITEM1 ("Example: \".*\\bs.txt\\$ \" matches all strings that end in " "\".txt\".") TAG ("##*# the star is the match-zero-or-more @@Regular expressions 2. " "Quantifiers|quantifier@.") LIST_ITEM1 ("Example: \"\\^ .*\\$ \" matches an entire line. ") TAG ("##+# the plus is the match-one-or-more quantifier.") TAG ("##?# the question mark is the match-zero-or-one " "quantifier. The question mark is also used in " "@@Regular expressions 4. Special constructs with parenthesis|special " "constructs with parenthesis@ and in @@Regular expressions 2. " "Quantifiers|changing match behaviour@.") TAG ("##\\| # the vertical pipe separates a series of alternatives.") LIST_ITEM1 ("Example: \"(a|b|c)a\" matches \"aa\" or \"ba\" or \"ca\".") TAG ("##< ># the smaller and greater signs are @@Regular expressions 3. " "Anchors|anchors@ that specify a left or right word boundary.") TAG ("##-# the minus indicates a range in a character class (when it is " "not at the first position after the \"[\" opening bracket or the last " "position before the \"]\" closing bracket.") LIST_ITEM1 ("Example: \"[A-Z]\" matches any uppercase character.") LIST_ITEM1 ("Example: \"[A-Z-]\" or \"[-A-Z]\" match any uppercase character " "or \"-\".") TAG ("##&# the and is the \"substitute complete match\" symbol.") MAN_END MAN_BEGIN ("Regular expressions 2. Quantifiers", "djmw", 20010708) INTRO ("Quantifiers specify how often the preceding @@Regular expressions|" "regular expression@ should match.") TAG ("##*# Try to match the preceding regular expression zero or more times.") LIST_ITEM1 ("Example: \"(ab)c*\" matches \"ab\" followed by zero or more " "\"c\"s, i.e., \"ab\", \"abc\", \"abcc\", \"abccc\" ...") TAG ("##+# Try to match the preceding regular expression one or more times." "times.") LIST_ITEM1 ("Example: \"(ab)c+\" matches \"ab\" followed by one or more " "\"c\"s, i.e., \"abc\", \"abcc\", \"abccc\" ...") TAG ("##{%m, %n}# Try to match the preceding regular expression between %m " "and %n times.") DEFINITION ("If you leave %m out, it is assumed to be zero. If you leave " "%n out it is assumed to be infinity. I.e., \"{,%n}\" matches from %zero " "to %n times, \"{%m,}\" matches a minimum of %m times, \"{,}\" matches " "the same as \"*\" and \"{n}\" is shorthand for \"{n, n\"} and matches " "exactly %n times.") LIST_ITEM1 ("Example: \"(ab){1,2}\" matches \"ab\" and \"abab\".") TAG ("##?# Try to match zero or one time.") ENTRY ("Changing match behaviour") NORMAL ("Default the quatifiers above try to match as much as possible, they " "are %greedy. " "You can alter greedy behaviour to %lazy behaviour by adding an " "extra \"?\" after the quantifier.") LIST_ITEM1 ("Example: In the string \"cabddde\", the search \"abd{1,2}\" " "matches \"abdd\", while the search for \"abd{1,2}?\" matches \"abd\".") LIST_ITEM1 ("Example: In the string \"cabddde\", the search \"abd+\" " "matches \"abddd\", while the search for \"abd+?\" matches \"abd\".") MAN_END MAN_BEGIN ("Regular expressions 3. Anchors", "djmw", 20010708) INTRO ("Anchors let you specify a very specific position within the search " "text.") TAG ("##\\^ # Try to match the (following) regex at the beginning of a line.") LIST_ITEM1 ("Example: \"\\^ ab\" matches \"ab\" only at the beginning of a " "line and not, for example, in the line \"cab\".") TAG ("##\\$ # Try to match the (following) regex at the end of a line.") TAG ("##<# Try to match the regex at the %start of a word.") DEFINITION ("The character class that defines a %word can be found at the " "@@Regular expressions 6. Convenience escape sequences|convenience escape " "sequences@ page.") TAG ("##># Try to match the regex at the %end of a word.") TAG ("##\\bsB# Not a word boundary") DEFINITION ("") MAN_END MAN_BEGIN ("Regular expressions 4. Special constructs with parenthesis", "djmw", 20010710) INTRO ("Some special constructs exist with parenthesis. ") TAG ("##(?:#%regex#)# is a grouping-only construct.") DEFINITION ("They exist merely for efficiency reasons and facilitate grouping.") TAG ("##(?=#%regex#)# is a positive look-ahead.") DEFINITION ("A match of the regular expression contained in the positive " "look-ahead construct is attempted. If the match succeeds, control is " "passed to the regex following this construct and the text consumed by " "this look-ahead construct is first unmatched. ") TAG ("##(?!#%regex#)# is a negative look-ahead.") DEFINITION ("Functions like a positive look-ahead, only the " "%regex must %not match.") LIST_ITEM ("Example: \"abc(?!.*abc.*)\" searches for the %last " "occurrence of \"abc\" in a string.") TAG ("##(?i#%regex#)# is a case insensitive regex.") TAG ("##(?I#%regex#)# is a case sensitive regex.") DEFINITION ("Default a regex is case sensitive. ") LIST_ITEM1 ("Example: \"(?iaa)\" matches \"aa\", \"aA\", \"Aa\" and \"AA\".") TAG ("##(?n#%regex#)# matches newlines.") TAG ("##(?N#%regex#)# doesn't match newlines.") NORMAL ("All the constructs above do not capture text and can not be " "referenced, i.e., the parenthesis are not counted. However, you " "can make them capture text by surrounding them with %ordinary " "parenthesis.") MAN_END MAN_BEGIN ("Regular expressions 5. Special control characters", "djmw", 20010708) INTRO ("Special control characters in a @@Regular expressions|regular " "expression@ specify characters that are difficult to type.") TAG ("#\\bsa alert (bell).") TAG ("#\\bsb backspace.") TAG ("#\\bse ASCII escape character.") TAG ("#\\bsf form feed (new page).") TAG ("#\\bsn newline.") TAG ("#\\bsr carriage return.") LIST_ITEM1 ("Example : a search for \"\\bsr\\bsn\" followed by a replace " "\"\\bsr\" changes Windows text files to Macintosh text files.") LIST_ITEM1 ("Example : a search for \"\\bsr\" followed by a replace " "\"\\bsn\" changes Macintosh text files to Unix text files.") LIST_ITEM1 ("Example : a search for \"\\bsr\\bsn\" followed by a replace " "\"\\bsn\" changes Windows text files to Unix text files.") TAG ("#\\bst horizontal tab.") TAG ("#\\bsv vertical tab.") MAN_END MAN_BEGIN ("Regular expressions 6. Convenience escape sequences", "djmw", 20010708) INTRO ("Convenience escape sequences in a @@Regular expressions|regular " "expression@ present a shorthand for some character classes.") TAG ("#\\bsd matches a digit: [0-9].") LIST_ITEM1 ("Example: \"-?\\bsd+\" matches any integer.") TAG ("#\\bsD %not a digit: [\\^ 0-9].") TAG ("#\\bsl a letter: [a-zA-Z].") TAG ("#\\bsL %not a letter: [\\^ a-zA-Z].") TAG ("#\\bss whitespace: [ \\bst\\bsn\\bsr\\bsf\\bsv].") TAG ("#\\bsS %not whitespace: [\\^ \\bst\\bsn\\bsr\\bsf\\bsv].") TAG ("#\\bsw \"word\" character: [a-zA-Z0-9\\_ ].") LIST_ITEM1 ("Example: \"\\bsw+\" matches a \"word\", i.e., a string of one " "or more characters that may consist of letters, digits and underscores.") TAG ("#\\bsW %not a \"word\" character: [\\^ a-zA-Z0-9\\_ ].") TAG ("#\\bsB any character that is %not a word-delimiter.") MAN_END MAN_BEGIN ("Regular expressions 7. Octal and hexadecimal escapes", "djmw", 20010709) NORMAL ("An octal number can be represented by the octal escape \"\\bs0\" " "and maximally three digits from the digit class [0-7]. " "The octal number should not exceed \\bs0377. ") NORMAL ("A hexadecimal number can be represented by the octal escape " "\"\\bsx\" or \"\\bsX\"and maximally two characters from the class " "[0-9A-F]. The maximum hexadecimal number should not exceed \\bsxFF. ") LIST_ITEM1 ("Example: \\bs053 and \\bsX2B both specify the \"+\" character.") MAN_END MAN_BEGIN ("Regular expressions 8. Substitution special characters", "djmw", 20010708) INTRO ("The substitution string is mostly interpreted as ordinary text except " "for the @@Regular expressions 5. Special control characters|" "special control characters@, the @@Regular expressions 7. Octal and " "hexadecimal escapes|octal and hexadecimal escapes@ and the following " "character combinations:") TAG ("#\\bs1 ... #\\bs9 are backreferences at sub-expressions 1 ... 9 in the match.") DEFINITION ("Any of the first nine sub-expressions of the match string can " "be inserted into the replacement string by inserting a `\\bs' followed " "by a digit from 1 to 9 that represents the string matched by a " "parenthesized expression within the regular expression. The numbering " "is left to right.") LIST_ITEM1("Example: A search for \"(a)(b)\" in the string \"abc\", " "followed by a replace \"\\bs2\\bs1\" results in \"bac\".") TAG ("#& reference at entire match.") DEFINITION ("The entire string that was matched by the search operation will " "be substituted.") LIST_ITEM1 ("Example: a search for \".\" in the string \"abcd\" followed by " "the replace \"&&\" doubles every character in the result " "\"aabbccdd\".") TAG ("#\\bsU #\\bsu to uppercase.") DEFINITION ("The text inserted by \"&\" or \"\\bs1\" ... \"\\bs9\" is " "converted to %uppercase (\"\\bsu\" only changes the %first character to " "uppercase).") LIST_ITEM1 ("Example: A search for \"(aa)\" in the string \"aabb\", " "followed by a replace \"\\bsU\\bs1bc\" results in the string \"AAbcbb\".") TAG ("#\\bsL #\\bsl to lowercase.") DEFINITION ("The text inserted by \"&\" or \"\\bs1\" ... \"\\bs9\" is " "converted to %lowercase (\"\\bsl\" only changes the %first character to " "lowercase).") LIST_ITEM1 ("Example: A search for \"(AA)\" with a replace \"\\bsl\\bs1bc\" " "in the string \"AAbb\" results in the string \"aAbcbb\".") MAN_END MAN_BEGIN ("Roots", "djmw", 19990608) INTRO ("One of the @@types of objects@ in P\\s{RAAT}.") NORMAL ("An object of type Roots " "represents the (complex) roots of a @@Polynomial|polynomial@ function.") ENTRY ("Commands") NORMAL ("Creation") LIST_ITEM ("\\bu @@Polynomial: To Roots@") NORMAL ("Drawing") LIST_ITEM ("\\bu ##Draw...# (in the complex plane)") NORMAL ("Queries") LIST_ITEM ("\\bu ##Get root...#: get complex root") LIST_ITEM ("\\bu ##Get real part of root...#") LIST_ITEM ("\\bu ##Get imaginary part of root...#") MAN_END MAN_BEGIN ("singular value decomposition", "djmw", 19981007) INTRO ("The %%singular value decomposition% (svd) of a real %m \\xx %n matrix #A is the " "factorization") FORMULA ("#A = #U #\\Si #V\\'p,") NORMAL ("The matrices in this factorization have the following properties:") TAG ("#U [%m \\xx %n] and #V [%n \\xx %n]") DEFINITION ("are othogonal matrices. The columns #u__%i_ of #U =[#u__1_, ..., #u__%n_] " "are the %%left singular vectors%, and the columns #v__%i_ of #V [#v__1_, ..., #v__%n_] " "are the %%right singular vectors%.") TAG ("#\\Si [%n \\xx %n] = diag (%\\si__1_, ..., %\\si__%n_)") DEFINITION ("is a real, nonnegative, and diagonal matrix. Its diagonal contains the so called " "%%singular values% %\\si__%i_, where %\\si__1_ \\>_ ... \\>_ %\\si__%n_ \\>_ 0.") MAN_END MAN_BEGIN ("Sound & Pitch: Change gender...", "djmw", 20030208) INTRO ("A command to create a new Sound object with manipulated characteristics " "from the selected @Sound and @Pitch object.") NORMAL ("With this command you can have finer grained control over the " "pitch than with the @@Sound: Change gender...@ command. " "Accurate pitch measurement determines the quality of the " "@PSOLA synthesis." ) ENTRY ("Parameters") NORMAL ("The parameters are described in @@Sound: Change gender...@ " "(except that we don't need the %%Maximum pitch% here, since it will be " "determined from the selected Pitch object.") MAN_END MAN_BEGIN ("Sound: Change gender...", "djmw", 20030205) INTRO ("A command to create a new @Sound object with manipulated characteristics.") ENTRY ("Parameters") NORMAL ("The arguments that control the pitch measurement are:") TAG ("%%Minimum pitch% (default 75 Hz)") DEFINITION ("pitch candidates below this frequency will not be considered. This " "parameter determines the length of the analysis window.") TAG ("%%Maximum pitch% (default 600 Hz)") DEFINITION ("pitch candidates above this frequency will be ignored.") NORMAL ("The arguments that control the manipulation are:") TAG ("%%Formant shift ratio%") DEFINITION ("determines the frequencies of the formants in the newly created " "sound object. When this ratio equals 1 no frequency shift will occur and " "the formant frequencies will not have changed. A 1.2 ratio will change " "a male voice to a voice with approximately female formant characteristics. " "A 1/1.2 ratio will change a female voice to a voice with approximately male formant " "characteristics.") TAG ("%%New pitch median% (default 0 Hz: same as original)") DEFINITION ("determines what the median pitch of the new sound has to be. " "The pitch values in the newly created sound object will be calculted from the pitch " "values in the selected sound object by multiplying them with a factor " "%%newPitchMedian / oldPitchMedian%. This factor equals 1 when the default " "value for the new pitch median (0) is chosen. ") TAG ("%%Pitch range scale factor% (default 1)") DEFINITION ("determines an %extra% scaling of the new pitches around the %new% " "pitch median. A factor equal to 1 means that no additional pitch modification occurs " "(except the obvious one described above). A factor equal to 0 monotonizes the new " "sound to the new pitch median.") TAG ("%%Length scale factor% (default 1)") DEFINITION ("The factor with which the sound will be lengthened. The default is 1. " "If you take a value less than 1, the resulting sound will be shorter than " "the original. A value larger than 3 will not work.") NORMAL ("If you want more control over the synthesis you can supply your own " "Pitch object and use the @@Sound & Pitch: Change gender...@ command. ") ENTRY ("Algorithm") NORMAL ("The shifting of frequencies is done via manipulation of the sample rate. " "Pitch and length changes are generated with @PSOLA synthesis.") NORMAL ("The new pitches are calculated in a two step process. We first multiply all " "the pitches with the factor %%newPitchMedian / oldPitchMedian% according to:") FORMULA ("%newPitch = %pitch * %newPitchMedian / %oldPitchMedian.") NORMAL ("It follows that when the %newPitchMedian equals the %oldPitchMedian no " "change in pitch value will occur in the first step.") NORMAL ("Subsequently, the pitch range scale factor determines the final pitches " "in the following linear manner:") FORMULA ("%finalPitch = %newPitchMedian + (%newPitch \\-- %newPitchMedian) * %pitchRangeScaleFactor") NORMAL ("Hence, it follows that no further scaling occurs when %pitchRangeScaleFactor " "equals 1.") MAN_END MAN_BEGIN ("Sound: Filter (gammatone)...", "djmw", 19980712) INTRO ("A command to filter a Sound by a fourth order gammatone bandpass filter.") ENTRY ("Arguments") TAG ("%%Centre frequency%, %Bandwidth") DEFINITION ("determine the passband of the filter.") ENTRY ("Algorithm") NORMAL ("The impulse response of the filter is a 4-th order @@gamma-tone@. This " "filter is implemented as a simple 8-th order recursive digital filter with " "4 zeros and 8 poles (these 8 poles consist of one conjugate pole pair to the " "4-th power). In the Z-domain its formula is: ") FORMULA ("%#H (%z) = (1 + \\su__%i=1..4_ %a__%i_%z^^%\\--i^) / " "(1 + \\su__%j=1..8_ %b__%j_%z^^%\\--j^)") NORMAL ("The derivation of the filter coefficients %a__%i_ and %b__%j_ is " "according to @@Slaney (1993)@. " "The gain of the filter is scaled to unity at the centre frequency.") MAN_END MAN_BEGIN ("Sound: To BarkFilter...", "djmw", 20010404) INTRO ("A command that creates a @BarkFilter object from every selected " "@Sound object by @@band filtering in the frequency domain@ with a " "bank of filters.") NORMAL ("The filter functions used are:") FORMULA ("10 log %#H(%z) = 7 - 7.5 * (%z__%c_ - %z - 0.215) - 17.5 * \\Vr " "(0.196 + (%z__%c_ - %z - 0.215)^2)") NORMAL ("where %z__%c_ is the central (resonance) frequency of the filter in Bark. " "The bandwidths of these filters are constant and equal 1 Bark.") MAN_END MAN_BEGIN ("Sound: To FormantFilter...", "djmw", 20010404) INTRO ("A command that creates a @FormantFilter object from every selected " "@Sound object by @@band filtering in the frequency domain@ with a " "bank of filters whose bandwidths depend on the pitch of the signal.") NORMAL ("The analysis proceeds in two steps:") LIST_ITEM ("1. We perform a pitch analysis (see @@Sound: To Pitch...@ for " "details).") LIST_ITEM ("2. We perform a filter bank analysis on a linear frequency scale. " "The bandwidth of the filters depends on the measured pitch (see @@Sound & " "Pitch: To FormantFilter...@ for details).") MAN_END MAN_BEGIN ("Sound & Pitch: To FormantFilter...", "djmw", 20010404) INTRO ("A command that creates a @FormantFilter object from the selected " "@Sound and @Pitch objects by @@band filtering in the frequency domain@ with a " "bank of filters whose bandwidths depend on the Pitch.") NORMAL ("The filter functions used are:") FORMULA ("%#H(%f, %F__0_) = 1 / (((%f__%c_^^2^ - %f^2) /%f\\.c%B(%F__0_)))^2 + 1),") NORMAL ("where %f__%c_ is the central (resonance) frequency of the filter. " "%B(%F__0_) is the bandwidth in Hz and determined as") FORMULA ("%B(%F__0_) = %relativeBandwidth\\.c%F__0_, ") NORMAL ("where %F__0_ is the fundamental frequency as determined from the Pitch " "object. Whenever the value of %F__0_ is undefined, a value of 100 Hz is taken.") MAN_END MAN_BEGIN ("Sound: To MelFilter...", "djmw", 20010404) INTRO ("A command that creates a @MelFilter object from every selected " "@Sound object by @@band filtering in the frequency domain@ with a " "bank of filters.") NORMAL ("The filter functions used are triangular in shape on a %linear " "frequency scale. The filter function depends on three parameters, the " "lower frequency %f__%l_, the central frequency %f__%c_ and the higher " "frequency %f__%h_. " "On a %mel scale, the distances %f__%c_-%f__%l_ and %f__%h_-%f__%c_ " "are the same for each filter and are equal to the distance between the " "%f__%c_'s of successive filters. The filter function is:" ) FORMULA ("%#H(%f) = 0 for %f \\<_ %f__%l_ and %f \\>_ %f__%h_") FORMULA ("%#H(%f) = (%f - %f__%l_) / (%f__%c_ - %f__%l_) for %f__%l_ \\<_ %f \\<_ %f__%c_") FORMULA ("%#H(%f) = (%f__%h_ - %f) / (%f__%h_ - %f__%c_) for %f__%c_ \\<_ %f \\<_ %f__%h_") MAN_END MAN_BEGIN ("Sound: To Pitch (shs)...", "djmw", 19970402) INTRO ("A command that creates a @Pitch object from every selected @Sound object.") ENTRY ("Purpose") NORMAL ("to perform a pitch analysis based on a spectral compression model. " "The concept of this model is that each spectral component not only activates " "those elements of the central pitch processor that are most sensitive to the " "component's frequency, but also elements that have a lower harmonic " "relation with this component. Therefore, when a specific element of the " "central pitch processor is most sensitive at a frequency %f__0_, it receives " "contributions from spectral components in the " "signal at integral multiples of %f__0_.") ENTRY ("Algorithm") NORMAL ("The spectral compression consists of the summation of a sequence of " "harmonically compressed spectra. " "The abscissa of these spectra is compressed by an integral factor, the rank " "of the compression. The maximum of the resulting sum spectrum is the " "estimate of the pitch. Details of the algorithm can be " "found in @@Hermes (1988)@") ENTRY ("Arguments") TAG ("%%Time step% (default 0.01 s)") DEFINITION ("the measurement interval (frame duration), in seconds.") TAG ("%%Minimum pitch% (default 50 Hz)") DEFINITION ("candidates below this frequency will not be recruted. This parameter " "determines the length of the analysis window.") TAG ("%%Max. number of candidates% (default 15)") DEFINITION ("The maximum number of candidates that will be recruted.") TAG ("%%Maximum frequency% (default 1250 Hz)") DEFINITION ("higher frequencies will not be considered.") TAG ("%%Max. number of subharmonics% (default 15)") DEFINITION ("the maximum number of harmonics that add up to the pitch.") TAG ("%%Compression factor% (default 0.84)") DEFINITION ("the factor by which successive compressed spectra are multiplied before the summation.") TAG ("%%Number of points per octave% (default 48)") DEFINITION ("determines the sampling of the logarithmic frequency scale.") TAG ("%Ceiling (default 500 Hz)") DEFINITION ("candidates above this frequency will be ignored.") MAN_END MAN_BEGIN ("SSCP", "djmw", 19981103) INTRO ("One of the @@types of objects@ in P\\s{RAAT}.") NORMAL ("An object of type SSCP represents the sums of squares and cross products of " "a multivariate data set.") ENTRY ("Inside a SSCP") NORMAL ("With @Inspect you will see that this type contains the same " "attributes a @TableOfReal with the following extras:") TAG ("%numberOfObservations") TAG ("%centroid") MAN_END MAN_BEGIN ("SSCP: Draw sigma ellipse...", "djmw", 19990222) INTRO ("A command to draw for the selected @SSCP an ellipse that " "covers a part of the multivariate data.") ENTRY ("Arguments") TAG ("%%Number of sigmas") DEFINITION ("determines the @@concentration ellipse|data coverage@.") MAN_END MAN_BEGIN ("SSCP: Get sigma ellipse area...", "djmw", 20000525) INTRO ("A command to query the selected @SSCP object for the area of a " "sigma ellipse.") ENTRY ("Algorithm") NORMAL ("The algorithm proceeds as follows:") LIST_ITEM ("1. The four array elements in the SSCP-matrix that correspond to the choosen dimensions " "are copied into a two-dimensional matrix #%S (symmetric of course).") LIST_ITEM ("2. The eigenvalues of #%S are determined, call them %s__1_ and %s__2_.") LIST_ITEM ("3. The lengths %l__%i_ of the axes of the ellipse can be obtained as the " "square root of the %s__i_ multiplied by a scale factor: %l__%i_ = %scaleFactor \\.c " "\\Vr (%s__%i_ ), " "where %scaleFactor = %numberOfSigmas / \\Vr(%numberOfObservations \\-- 1).") LIST_ITEM ("4. The area of the ellipse will be %\\pi\\.c%l__1_\\.c%l__2_.") MAN_END MAN_BEGIN ("SSCP: Get confidence ellipse area...", "djmw", 20000525) INTRO ("A command to query the selected @SSCP object for the area of a " "confidence ellipse.") ENTRY ("Algorithm") NORMAL ("The algorithm proceeds as follows:") LIST_ITEM ("1. The four array elements in the SSCP-matrix that correspond to the choosen dimensions " "are copied into a two-dimensional matrix #%S (symmetric of course).") LIST_ITEM ("2. The eigenvalues of #%S are determined, call them %s__1_ and %s__2_.") LIST_ITEM ("3. The lengths %l__%i_ of the axes of the ellipse can be obtained from " "@@Johnson (1998)@, page 410, as the " "square root of the %s__i_ multiplied by a scale factor: %l__%i_ = %scaleFactor \\.c " "\\Vr (%s__%i_ ), where") FORMULA ("%scaleFactor = \\Vr (%f \\.c %p \\.c (%n \\-- 1) / (%n \\.c (%n \\-- %p))), in which,") LIST_ITEM (" %f = $$@@invFisherQ@$ (1 \\-- %confidenceLevel, %p, %n \\-- %p), " "%p is the numberOfRows from the SSCP object and %n the %numberOfObservations.") LIST_ITEM ("4. The area of the ellipse will be %\\pi\\.c%l__1_\\.c%l__2_.") MAN_END MAN_BEGIN ("SSCP: Get diagonality (bartlett)...", "djmw", 20011111) INTRO ("Tests the hypothesis that the selected @SSCP matrix object is " "diagonal.") ENTRY ("Argument") TAG ("%%Number of constraints") DEFINITION ("modifies the number of independent observations. " "The default value is 1.") ENTRY ("Algorithm") NORMAL ("The test statistic is |#R|^^N/2^, the N/2-th power of the determinant" " of the correlation matrix. @@Bartlett (1954)@ developed the following " "approximation to the limiting distribution:") FORMULA ("\\ci^2 = -(%N - %numberOfConstraints - (2%p + 5) /6) ln |#R|") NORMAL ("In the formula's above, %p is the dimension of the correlation " "matrix, %N-%numberOfConstraints is the number of independent " "observations. Normally %numberOfConstraints would " "equal 1, however, if the matrix has been computed in some other way, " "e.g., from within-group sums of squares and cross-products of %k " "independent groups, %numberOfConstraints would equal %k.") NORMAL ("We return the probability %\\al as ") FORMULA ("%\\al = chiSquareQ (\\ci^2 , %p(%p-1)/2).") NORMAL ("A very low %\\al indicates that it is very improbable that the " "matrix is diagonal.") MAN_END MAN_BEGIN ("SSCP: To Covariance...", "djmw", 19990524) INTRO ("A command that creates a @Covariance object from each selected @SSCP object.") ENTRY ("Arguments") TAG ("%%Number of constraints") DEFINITION ("determines the factor by which each entry in the " "SSCP-matrix is scaled to obtain the Covariance matrix.") ENTRY ("Details") NORMAL ("The relation between the numbers %c__%ij_ in the covariance matrix and the numbers %s__%ij_ in " "the originating SSCP matrix is:") FORMULA ("%c__%ij_ = %s__%ij_ / (%numberOfObservations - %numberOfConstraints)") NORMAL ("Normally %numberOfConstraints will equal 1. However, when the originating SSCP was the " "result of summing %g SSCP objects, as is, for example, the case when you obtained the total " "within-groups SSCP from the individual group SSCP's, %numberOfConstraints will equal %g.") MAN_END MAN_BEGIN ("T-test", "djmw", 20020117) INTRO ("A test on the mean of a normal variate when the variance is unknown.") NORMAL ("In Praat, the t-test is used to query a @Covariance object and:") LIST_ITEM ("1. get the significance of one mean. See @@Covariance: Get " "significance of one mean...@.") LIST_ITEM ("2. get the significance of the %difference between two means. " "See @@Covariance: Get significance of means difference...@.") NORMAL ("You should use a t-test when you want to test a hypothesis about " "the mean of one column in a @TableOfReal object, or, if you want to test " "a hypothesis about the difference between the means of two columns in " "this object.") NORMAL ("You can perform these t-tests in Praat by first transforming the " "TableOfReal object into a Covariance object (see @@TableOfReal: To " "Covariance@) and then choosing the appropriate query method on the " "latter object.") MAN_END MAN_BEGIN ("TableOfReal: Change row labels...", "djmw", 20010822) INTRO ("Changes the row labels of the selected @TableOfReal object according " "to the specification in the search and replace fields.") NORMAL ("Both search and replace fields may contain @@regular expressions|" "Regular expressions@. The %%Replace limit% parameter limits the number of " "replaces that may occur within each label.") MAN_END MAN_BEGIN ("TableOfReal: Change column labels...", "djmw", 20010822) INTRO ("Changes the column labels of the selected @TableOfReal object according " "to the specification in the search and replace fields.") NORMAL ("Both search and replace fields may contain @@regular expressions|" "Regular expressions@. The %%Replace limit% parameter limits the number of " "replaces that may occur within each label.") MAN_END MAN_BEGIN ("TableOfReal: Draw biplot...", "djmw", 20020603) INTRO ("A command to draw a biplot for each column in the selected " "@TableOfReal object.") ENTRY ("Arguments") TAG ("%%Xmin%, %%Xmax%, %%Ymin%, %%Ymax%") DEFINITION ("determine the drawing boundaries.") TAG ("%Split factor") DEFINITION ("determines the weighing of the row and column structure " "(see below).") ENTRY ("Behaviour") LIST_ITEM ("1. Get the @@singular value decomposition@ #U #\\Si #V\\'p of the " "table.") LIST_ITEM ("2. Calculate weighing factors %\\la for row and columns") FORMULA ("%\\la__r,1_ = %\\si__1_^^%splitFactor^") FORMULA ("%\\la__c,1_ = %\\si__1_^^1-%splitFactor^") FORMULA ("%\\la__r,2_ = %\\si__2_^^%splitFactor^") FORMULA ("%\\la__c,2_ = %\\si__2_^^1-%splitFactor^") DEFINITION ("where %\\si__1_ and %\\si__2_ are the first and the second singular values") LIST_ITEM ("3. For the rows (%i from 1..%numberOfRows) form:") FORMULA ("%xr__%i_ = %U__%i1_ %\\la__%r,1_") FORMULA ("%yr__%i_ = %U__%i2_ %\\la__%r,2_") LIST_ITEM ("4. For the columnss (%i from 1..%numberOfColumns) form:") FORMULA ("%xc__%i_ = %V__%i1_ %\\la__%c,1_") FORMULA ("%yc__%i_ = %V__%i2_ %\\la__%c,2_") LIST_ITEM ("5. Plot the points (%xr__%i_, yr__%i_) and (%xc__%i_, yc__%i_) in the " "same figure with the corresponding row and column labels.") MAN_END MAN_BEGIN ("TableOfReal: Draw box plots...", "djmw", 20000523) INTRO ("A command to draw a @@box plot@ for each column in the selected " "@TableOfReal object.") ENTRY ("Arguments") TAG ("%%From row%, %%To row%, %%From column%, %%To column%") DEFINITION ("determine the part of the table that you want to analyse.") TAG ("%Ymin and %Ymax") DEFINITION ("determine the drawing boundaries.") MAN_END MAN_BEGIN ("TableOfReal: Select columns where row...", "djmw", 20020502) INTRO ("Copy columns from the selected @TableOfReal object to a new " "TableOfReal object.") ENTRY ("Arguments") TAG ("%Columns") DEFINITION ("defines the indices of the columns to be selected. Ranges can be " "defined with a colon \":\". Columns will be selected in the specified " "order.") TAG ("%%Row condition") DEFINITION ("specifies a condition for the selection of rows. If the " "condition evaluates as %true for a particular row, the selected elements " "in this row will be copied. See @@Matrix: Formula...@ for the kind of " "expressions that can be used here.") ENTRY ("Examples") CODE ("Select columns where row... \"1 2 3\" 1") CODE ("Select columns where row... \"1 : 3\" 1") NORMAL ("Two alternative expressions to copy the first three columns to a new table " "with the same number of rows.") CODE ("Select columns where row... \"3 : 1\" 1") NORMAL ("Copy the first three columns to a new table with the same number of " "rows. The new table will have the 3 columns reversed.") CODE ("Select columns where row... \"1:6 9:11\" self[row,8]>0") NORMAL ("Copy the first six columns and columns 9, 10, and 11 to a new table. " "Copy only elements from rows where the element in column 8 is greater " "than zero.") MAN_END MAN_BEGIN ("TableOfReal: Standardize columns", "djmw", 19990428) INTRO ("Standardizes each column of the selected @TableOfReal.") NORMAL ("The entries %x__%ij_ in the TableOfReal will change to:") FORMULA ("(%x__%ij_ \\-- %\\mu__%j_) / %\\si__%j_, ") NORMAL ("where %\\mu__%j_ and %\\si__%j_ are the mean and the standard deviation as calculated " "from the %j^^th^ column, respectively. After standardization all column means will equal zero " "and all column standard deviations will equal one.") MAN_END MAN_BEGIN ("TableOfReal: To Configuration (lda)...", "djmw", 19981103) INTRO ("Calculates a @Configuration based on the @Discriminant scores obtained " "from the selected @TableOfReal. Row labels in the table indicate group membership.") ENTRY ("Parameters") TAG ("%%Number of dimensions") DEFINITION ("determines the number of dimensions of the resulting Configuration.") ENTRY ("Algorithm") NORMAL ("First we calculate the Discriminant from the data in the TableOfReal. " "See @@TableOfReal: To Discriminant@ for details.") NORMAL ("The eigenvectors of the Discriminant determine the directions that " "the data in the TableOfReal will be projected unto.") MAN_END MAN_BEGIN ("TableOfReal: To Configuration (pca)...", "djmw", 19980909) INTRO ("Calculates a @Configuration based on the principal components from the " "selected @TableOfReal.") ENTRY ("Parameters") TAG ("%%Number of dimensions") DEFINITION ("determines the number of dimensions of the resulting Configuration.") ENTRY ("Algorithm") NORMAL ("We form principal components without explicitely calculating the covariance matrix " "#C = #M\\'p\\.c#M, where #M is the matrix part of the TableOfReal. ") LIST_ITEM ("1. Make the singular value decomposition of #M. This results in " "#M = #U\\.c#d\\.c#V\\'p.") LIST_ITEM ("2. Sort singular values #d and corresponding row vectors in #V (descending).") LIST_ITEM ("3. The principalComponent__%ij_ = \\su__%k=1..%numberOfColumns_ %M__%ik_ \\.c %V__%jk_.") ENTRY ("Remark") NORMAL ("The resulting configuration is unique up to reflections along the new principal directions.") MAN_END MAN_BEGIN ("TableOfReal: To Correlation", "djmw", 20020105) INTRO ("A command that creates a (%Pearson) @Correlation object from every " "selected @TableOfReal object. The correlations are calculated between " "columns.") ENTRY ("Algorithm") NORMAL ("The linear correlation coefficient %r__%ij_ (also called the %%product" " moment correlation coefficient% or %%Pearson's correlation coefficient%) " " between the elements of columns %i and %j is calculated as:") FORMULA ("%r__%ij_ = \\Si__%k_ (%x__%ki_ - %mean__%i_)(%x__%kj_ - %mean__%j_)/" "(\\Vr (\\Si__%k_(%x__%ki_ - %mean__%i_)^2) \\Vr (\\Si__%k_(%x__%kj_ -" " %mean__%j_)^2)),") NORMAL ("where %x__%mn_ is the element %m in column %n, and %mean__%n_ " "is the mean of column %n.") MAN_END MAN_BEGIN ("TableOfReal: To Correlation (rank)", "djmw", 20020105) INTRO ("A command that creates a (%%Spearman rank-order%) @Correlation object " "from every selected @TableOfReal object. The correlations are calculated " "between columns.") ENTRY ("Algorithm") NORMAL ("The Spearman rank-order correlation coefficient %r__%ij_ between " "the elements of columns %i and %j is calculated as the linear correlation" " of the ranks:") FORMULA ("%r__%ij_ = \\Si__%k_ (%R__%ki_ - %Rmean__%i_) " "(%R__%kj_ - %Rmean__%j_) / (\\Vr (\\Si__%k_(%R__%ki_ - %Rmean__%i_)^2) " "\\Vr (\\Si__%k_(%R__%kj_ - %Rmean__%j_)^2)),") NORMAL ("where %R__%mn_ is the rank of element %m in column %n, " "and %Rmean__%n_ is the mean of the ranks in column %n.") MAN_END MAN_BEGIN ("TableOfReal: To Covariance", "djmw", 20020117) INTRO ("A command that creates a @Covariance object from every " "selected @TableOfReal object. The covariances are calculated between " "columns.") ENTRY ("Algorithm") NORMAL ("The covariance coefficients %s__%ij_ " " between the elements of columns %i and %j are defined as:") FORMULA ("%s__%ij_ = \\Si__%k_ (%x__%ki_ - %mean__%i_)(%x__%kj_ - %mean__%j_)/" "(%numberOfObservations - %numberOfConstraints),") NORMAL ("where %x__%ki_ is the element %k in column %i, %mean__%i_ " "is the mean of column %i, %numberOfObservations equals the number of rows in " "the table, and %numberOfConstraints equals 1.") NORMAL ("The actual calculation goes as follows") LIST_ITEM ("1. Centralize each column (subtract the mean).") LIST_ITEM ("2. Get its @@singular value decomposition@ #U #\\Si #V\\'p.") LIST_ITEM ("3. Form #S = #V #\\Si #V\\'p.") LIST_ITEM ("4. Divide all elements in #S by (%numberOfObservations - 1).") MAN_END MAN_BEGIN ("TableOfReal: To Discriminant", "djmw", 19990104) INTRO ("A command that creates a @Discriminant object from every selected " "@TableOfReal object. Row labels in the table indicate group membership.") ENTRY ("Algorithm") NORMAL ("We solve for directions #x that are eigenvectors of the generalized " "eigenvalue equation:") FORMULA ("#%B #x - %\\la #%W #x = 0,") NORMAL ("where #%B and #%W are the between-groups and the within-groups sums of " "squares and cross-products matrices, respectively. Both #%B and #%W are symmetric " "matrices. Standard formula show that both matrices can also " "be written as a matrix product. The formula above then transforms to:") FORMULA ("#%B__1_\\'p#%B__1_ #x - %\\la #%W__1_\\'p#%W__1_ #x = 0") NORMAL ("The equation can be solved with the @@generalized singular value decomposition@. " "This procedure is numerically very stable and can even cope with cases when both " "matrices are singular.") NORMAL ("The apriori probabilities in the Discriminant will be calculated from the number of " "%training vectors %n__%i_ in each group:") FORMULA ("%aprioriProbability__%i_ = %n__%i_ / \\Si__%k=1..%numberOfGroups_ %n__%k_") MAN_END MAN_BEGIN ("TableOfReal: To PCA", "djmw", 19980106) INTRO ("A command that creates a @PCA object from every selected " "@TableOfReal object.") MAN_END MAN_BEGIN ("TableOfReal: To SSCP...", "djmw", 19990218) INTRO ("Calculates Sums of Squares and Cross Products (@SSCP) from the selected @TableOfReal.") ENTRY ("Algorithm") NORMAL ("The sums of squares and cross products %s__%ij_ " " between the elements of columns %i and %j are calculated as:") FORMULA ("%s__%ij_ = \\Si__%k_ (%x__%ki_ - %mean__%i_)(%x__%kj_ - %mean__%j_),") NORMAL ("where %x__%mn_ is the element %m in column %n and %mean__%n_ " "is the mean of column %n.") MAN_END MAN_BEGIN ("TableOfReal: Split into Pattern and Categories...", "djmw", 20000229) INTRO ("Splits the selected @TableOfReal into a @Pattern and a @Categories object.") NORMAL ("The selected rows and columns in the (sub)matrix part of the TableOfReal " "are copied into the Pattern object and " "the corresponding row labels are copied into a Categories object.") MAN_END MAN_BEGIN ("TableOfReal: To CCA...", "djmw", 20020424) INTRO ("A command that creates a @CCA object from the selected " "@TableOfReal object.") ENTRY ("Argument") TAG ("%%Dimension of dependent variate (ny)") DEFINITION ("defines the partition of the table into the two parts whose " "correlations will be determined. The first %ny columns must be the " "dependent part, the rest of the columns will be interpreted as the " "independent part (%nx columns). In general %nx must be larger than or " "equal to %ny.") ENTRY ("Behaviour") NORMAL ("Calculates canonical correlations between the %dependent and the " "%independent parts of the table. The corresponding " "canonical coefficients are also determined.") ENTRY ("Algorithm") NORMAL ("The canonical correlation equations for two data sets #T__%y_ " "[%n \\xx %p] and #T__%x_ [n \\xx %q] are:") FORMULA ("(1) (#S__%yx_ #S__%xx_^^-1^ #S__%yx_\\'p -\\la #S__%yy_)#y = #0") FORMULA ("(2) (#S__%yx_\\'p #S__%yy_^^-1^ #S__%yx_ -\\la #S__%xx_)#x = #0") NORMAL ("where #S__%yy_ [%p \\xx %p] and #S__%xx_ [%q \\xx %q] are the " "covariance matrices of data sets #T__%y_ and #T__%x_, respectively, " "#S__%yx_ [%p \\xx %q] is the matrix of covariances between data sets " "#T__%y_ and #T__%x_, and the vectors #y and #x are the %%canonical " "weights% or the %%canonical function coefficients% for the dependent and " "the independent data, respectively. " "In terms of the (dependent) data set #T__%y_ and the (independent) data set " "#T__%x_, these covariances can be written as:") FORMULA ("#S__%yy_ = #T__%y_\\'p #T__%y_, #S__%yx_ = #T__%y_\\'p #T__%x_ and " "#S__%xx_ = #T__%x_\\'p #T__%x_.") NORMAL ("The following @@singular value decomposition@s ") FORMULA ("#T__%y_ = #U__%y_ #D__%y_ #V__%y_\\'p and #T__%x_ = #U__%x_ #D__%x_ " "#V__%x_\\'p ") NORMAL ("transform equation (1) above into:") FORMULA ("(3) (#V__%y_ #D__%y_ #U__%y_\\'p#U__%x_ #U__%x_\\'p #U__%y_ #D__%y_ " "#V__%y_\\'p - \\la #V__%y_ #D__%y_ #D__%y_ #V__%y_\\'p)#y = 0 ") NORMAL ("where we used the fact that:") FORMULA ("#S__%xx_^^-1^ = #V__%x_ #D__%x_^^-2^ #V__%x_\\'p.") NORMAL ("Equation (3) can be simplified by multiplication from the left by " "#D__%y_^^-1^ #V__%y_' to:") FORMULA (" (4) ((#U__%x_\\'p #U__%y_)\\'p (#U__%x_\\'p #U__%y_) - \\la #I)#D__%y_ " "#V__%y_\\'p #y = #0") NORMAL ("This equation can, finally, be solved by a substitution of the s.v.d " "of #U__%x_\\'p #U__%y_ = #U #D #V\\'p into (4). This results in") FORMULA ("(5) (#D^^2^ - \\la #I) #V\\'p #D__%y_ #V__%y_\\'p #y = #0") NORMAL ("In an analogous way we can reduce eigenequation (2) to:") FORMULA ("(6) (#D^^2^ - \\la #I) #U #D__%x_ #V__%x_\\'p #x = #0") NORMAL ("From (5) and (6) we deduce that the eigenvalues in both equations " "are equal to the squared singular values of the product matrix " "#U__%x_\\'p#U__%y_. " "These singular values are also called %%canonical " "correlation coefficients%. The eigenvectors #y and #x can be obtained " "from the columns of the following matrices #Y and #X:") FORMULA ("#Y = #V__%y_ #D__%y_^^-1^ #V") FORMULA ("#X = #V__%x_ #D__%x_^^-1^ #U") NORMAL ("For example, when the vector #y equals the first column of #Y and " "the vector #x equals " "the first column of #X, then the vectors #u = #T__%y_#y and #v = #T__%x_#x " "are the linear combinations from #T__%y_ and #T__%x_ that have maximum " "correlation. Their correlation coefficient equals the first canonical " "correlation coefficient.") MAN_END MAN_BEGIN ("TextGrid: Extend time...", "djmw", 20020702) INTRO ("Extends the domain of the selected @TextGrid object.") ENTRY ("Arguments") TAG ("%%Extend domain by") DEFINITION ("defines the amount of time by which the domain will be extended.") TAG ("%At") DEFINITION ("defines whether starting times or finishing times will be " "modified.") ENTRY ("Behaviour") NORMAL ("We add an extra (empty) interval into each %%interval tier%. " "This is necessary to keep original intervals intact. According to the " "value of the second argument, the new interval will be added at the " "beginning or at the end of the tier.") NORMAL ("For %%point tiers% only the domain will be changed.") MAN_END MAN_BEGIN ("TIMIT acoustic-phonetic speech corpus", "djmw", 19970320) INTRO ("A large American-English speech corpus that resulted from the joint efforts " "of several American research sites.") NORMAL ("The TIMIT corpus contains a total of 6300 sentences, 10 sentences spoken by " "630 speakers selected from 8 major dialect regions of the USA. 70\\% of " "the speakers are male, 30\\% are female.") NORMAL ("The text corpus design was done by the Massachusetts Institute of " "Technology (MIT), Stanford Research Institute and Texas Instruments (TI). " "The speech was recorded at TI, transcribed at MIT, and has been maintained, " "verified and prepared for CDROM production by the American National Institute " "of Standards and Technology (NIST) (@@Lamel et al. (1986)@).") MAN_END MAN_BEGIN ("Sequence", "djmw", 19970509) INTRO ("One of the @@types of objects@ in P\\s{RAAT}.") NORMAL ("An object of type Sequence holds a particular ordening of integer numbers.") MAN_END /********************* References **************************************/ MAN_BEGIN ("Bai & Demmel (1993)", "djmw", 19981007) NORMAL ("Z. Bai & J. Demmel (1993), \"Computing the generalized singular value " "decomposition\", %%SIAM J. Sci. Comput.% #14, 1464-1486.") MAN_END MAN_BEGIN ("Bartlett (1954)", "djmw", 20011111) NORMAL ("M.S. Bartlett(1954), \"A note on multiplying factors for various " "chi-squared approximations\", %%Joural of the Royal Statistical Society, " "Series B%, vol. 16, 296-298") MAN_END MAN_BEGIN ("Boomsma (1977)", "djmw", 20020524) NORMAL ("A. Boomsma (1977), \"Comparing approximations of confidence intervals " "for the product-moment correlation coefficient\", %%Statistica Neerlandica% " "#31, 179-186.") MAN_END MAN_BEGIN ("Davis & Mermelstein (1980)", "djmw", 20010419) NORMAL ("S.B. Davis & P. Mermelstein (1980), \"Comparison of parametric " "representations for monosyllabic word recognition in continuously " "spoken sentences\", " "%%IEEE Trans. on ASSP% #28, 357-366.") MAN_END MAN_BEGIN ("Flanagan (1960)", "djmw", 19980713) NORMAL ("J.L. Flanagan (1960), \"Models for approximating basilar membrane " "displacement\", %%Bell Sys. Tech. J.% #39, 1163-1191.") MAN_END MAN_BEGIN ("Friedl (1997)", "djmw", 20010710) NORMAL ("J.E.F. Friedl (1997), %%Mastering Regular Expressions%, " "O'Reilly & Associates.") MAN_END MAN_BEGIN ("Heath et al. (1986)", "djmw", 19981007) NORMAL ("M.T. Heath, J.A. Laub, C.C. Paige & R.C. Ward (1986), \"Computing the " "singular value decomposition of a product of two matrices\", " "%%SIAM J. Sci. Statist. Comput.% #7, 1147-1159.") MAN_END MAN_BEGIN ("Hermes (1988)", "djmw", 19980123) NORMAL ("D.J. Hermes (1988), \"Measurement of pitch by subharmonic " "summation\", %%J.Acoust.Soc.Am.% #83, 257-264.") MAN_END MAN_BEGIN ("Irino & Patterson (1996)", "djmw", 19980123) NORMAL ("T. Irino & R.D. Patterson (1996), \"A time-domain, level-dependent " "auditory filter: The gammachirp\", %%J.Acoust.Soc.Am.% #101, 412-419.") MAN_END MAN_BEGIN ("Johannesma (1972)", "djmw", 19980123) NORMAL ("P.I.M. Johannesma (1972): \"The pre-response stimulus ensemble of " "neurons in the cochlear nucleus\", in %%Symposium on Hearing Theory% " "(IPO, Eindhoven, Holland), 58-69.") MAN_END MAN_BEGIN ("Johnson (1998)", "djmw", 20000525) NORMAL ("D.E. Johnson (1998), %%Applied Multivariate methods%") MAN_END MAN_BEGIN ("Lamel et al. (1986)", "djmw", 19980123) NORMAL ("L.F. Lamel, R.H. Kassel & S. Sennef (1986): \"Speech Database " "Development: Design and Analysis of the Acoustic-Phonetic Corpus\", " "%%Proc. DARPA Speech Recognition Workshop%\", Report No. SAIC-86/1546, " "100-19.") MAN_END MAN_BEGIN ("Morrison (1990)", "djmw", 19980123) NORMAL ("D.F. Morrison (1990), %%Multivariate Statistical Methods%, " "McGraw-Hill, New York.") MAN_END MAN_BEGIN ("Peterson & Barney (1952)", "djmw", 20020620) NORMAL ("G.E. Peterson & H.L. Barney (1952), \"Control methods used in a study " "of the vowels\", %%J.Acoust.Soc.Am.% #24, 175-184") MAN_END MAN_BEGIN ("Pols et al. (1972)", "djmw", 19990426) NORMAL ("L.C.W. Pols, H.R.C. Tromp & R. Plomp (1972), " "\"Frequency analysis of Dutch vowels from 50 male speakers\", " "%%J.Acoust.Soc.Am.% #53, 1093-1101.") MAN_END MAN_BEGIN ("Press et al. (1992)", "djmw", 19980114) NORMAL ("W.H. Press, S.A. Teukolsky, W.T. Vetterling & B.P. Flannery (1992), " "%%Numerical Recipes in C: the art of scientific computing%, " "Second Edition, Cambridge University Press.") MAN_END MAN_BEGIN ("Shepard (1964)", "djmw", 19980114) NORMAL ("R.N. Shepard (1964), \"Circularity in Judgments of Relative Pitch\", " "%%J.Acoust.Soc.Am.% #36, 2346-2353.") MAN_END MAN_BEGIN ("Slaney (1993)", "djmw", 19980712) NORMAL ("M. Slaney (1993), \"An efficient implementation of the " "Patterson-Holdsworth auditory filterbank\", " "%%Apple Computer Technical Report% #35, 41 pages.") MAN_END MAN_BEGIN ("Tribolet et al. (1979)", "djmw", 20010114) NORMAL ("J.M. Tribolet & T.F. Quatieri (1979), \"Computation of the Complex " "Cepstrum\", in: %%Programs for Digital Signal Processing%, " "Digital Signal Processing Committee (eds.), IEEE Press.") MAN_END MAN_BEGIN ("Tukey (1977)", "djmw", 20000524) NORMAL ("J.W. Tukey (1977), %%Exploratory data analysis%, " "Addison-Wesley, Reading, Mass.") MAN_END MAN_BEGIN ("Van Nierop et al. (1973)", "djmw", 20020620) NORMAL ("D.J.P.J. Van Nierop, L.C.W. Pols & R. Plomp (1973), \"Frequency " "analysis of Dutch vowels from 25 female speakers\", %%Acustica% #29, 110-118") MAN_END MAN_BEGIN ("Weenink (1999)", "djmw", 20000412) NORMAL ("D.J.M. Weenink (1999), \"Accurate algorithms for performing " "principal component analysis and discriminant analysis\", " "%%Proceedings of the Institute of Phonetic Sciences of the " "University of Amsterdam% #23, 77-89.") MAN_END } /* End of file manual_dwtools.c */