/* manual_MDS.c * * Copyright (C) 1993-2005 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 20020422 GPL + removed "C syntax" part of manpage djmw 20030317 Latest modification. djmw 20030825 Spelling corrections. djmw 20040118 Procrustes modifications. djmw 20040214 Removed last link to @praat program@. djmw 20040407 Change Arguments -> Settings. djmw 20040513 Spelling. djmw 20050406 Latest modification. */ #include "ManPagesM.h" #include "MDS.h" static void drawLetterRConfigurationExample (Graphics g) { Configuration me = Configuration_createLetterRExample (1); if (me == NULL) return; Graphics_setWindow (g, -6, 4, -7, 5); Configuration_draw (me, g, 1, 2, -6, 4, -7, 5, 0, 1, "", 1); forget (me); } static void drawLetterRConfigurationExample2 (Graphics g) { Configuration me = Configuration_createLetterRExample (2); if (me == NULL) return; Configuration_draw (me, g, 1, 2, -0.8, 1.2, -0.8, 0.7, 0, 1, "", 1); forget (me); } static void drawLetterRShepard (Graphics g) { Dissimilarity d = Dissimilarity_createLetterRExample (32.5); if (d != NULL) { Configuration c = Configuration_createLetterRExample (2); if (c != NULL) { Dissimilarity_Configuration_drawShepardDiagram (d, c, g, 0, 200, 0, 2.2, 1, "+", 1); forget (c); } forget (d); } } static void drawLetterRRegression (Graphics g) { Dissimilarity d = Dissimilarity_createLetterRExample (32.5); if (d != NULL) { Configuration c = Configuration_createLetterRExample (2); if (c != NULL) { Dissimilarity_Configuration_drawMonotoneRegression (d, c, g, MDS_PRIMARY_APPROACH, 0, 200, 0, 2.2, 1, "+", 1); forget (c); } forget (d); } } static void drawCarrollWishConfigurationExample (Graphics g) { Configuration me = Configuration_createCarrollWishExample (); if (me == NULL) return; Graphics_setWindow (g, -2, 2, -2, 2); Configuration_draw (me, g, 1, 2, -2, 2, -2, 2, 0, 1, "", 1); forget (me); } static void drawCarrollWishSalienceExample (Graphics g) { Salience me = Salience_createCarrollWishExample (); if (me == NULL) return; Salience_draw (me, g, 1, 2, 1); forget (me); } static void drawMsplineExample (Graphics g) { drawSplines (g, 0, 1, 0, 10, 1, 3, "0.3 0.5 0.6", 1); } static void drawIsplineExample (Graphics g) { drawSplines (g, 0, 1, 0, 1.5, 2, 3, "0.3 0.5 0.6", 1); } void manual_MDS_init (ManPages me); void manual_MDS_init (ManPages me) { MAN_BEGIN ("CANDECOMP", "djmw", 19971201) ENTRY ("An algorithm to solve the INDSCAL problem.") NORMAL ("In the analysis of the INDSCAL three-way data matrix (%numberOfPoints " "\\xx %numberOfDimensions \\xx %numberOfSources) we seek to minimize the " "function: ") FORMULA ("%f(%X, %W__1_,..., %W__%numberOfSources_) = " "\\su__%i=1..%numberOfSources_ | %S__%i_ \\-- %X%W__%i_%X\\'p |^2") NORMAL ("where %S__%i_ is a known symmetric %numberOfPoints \\xx %numberOfPoints " "matrix with scalar products of distances for source %i, %X is the unknown configuration " "%numberOfPoints \\xx %numberOfDimensions matrix, %X\\'p its transpose, and, %W__%i_ is " "the diagonal %numberOfDimensions \\xx %numberOfDimensions weight matrix for source %i. The function " "above has no analytical solution for %X and the %W__%i_. It can be solved, however, " "by an iterative procedure which Carroll & Chang have christened CANDECOMP " "(CANonical DECOMPosition). This method minimizes, instead of the function " "given above, the following function:") LIST_ITEM ("%g(%X, %Y, %W__1_,..., %W__%numberOfSources_) = \\su__%i=1..%numberOfSources_ " "| %S__%i_ \\-- %X%W__%i_%Y\\'p |^2") NORMAL ("where %X and %Y are both %numberOfPoints \\xx %numberOfDimensions configuration matrices.") NORMAL ("The algorithm proceeds as follows:") NORMAL ("1. Initialize the $W matrices and the configuration matrix %X. This can for example be " "done according to a procedure given in @@Young, Takane & Lewyckyj (1978)@.") NORMAL ("2. An alternating least squares minimization process is started as described that " "sequentially updates %Y, %X an %W (@@Carroll & Chang (1970)@):") LIST_ITEM ("2.1. Solve for a new %Y given %X and the %W__%i_") LIST_ITEM ("2.2. Solve for a new %X given the %W__%i_ and the new %Y.") LIST_ITEM ("2.3. Solve for the %W__%i_ given the new %X and %Y.") NORMAL ("Evaluate the goodness-of-fit criterion and either repeat the minimization sequence " "(2.1\\--2.3) or continue.") NORMAL ("3. Done: make %Y equal to %X and solve a last time for the %W__%i_.") NORMAL ("Note: during the minimization the following constraints are effective:") LIST_ITEM ("The configuration must be centered.") LIST_ITEM ("The sum of squared coordinates in the configuration space is one for " "each dimension, i.e., the configuration always has unit variance in each dimension.") MAN_END MAN_BEGIN ("Configuration", "djmw", 19980909) INTRO ("One of the @@types of objects@ in P\\s{RAAT}.") NORMAL ("An object of type Configuration represents the positions of a number " "of labelled points in a multidimensional space.") ENTRY ("How to create a Configuration") NORMAL ("From the New menu:") LIST_ITEM (" \\bu @@Create Configuration...") NORMAL ("By multidimensional scaling:") LIST_ITEM (" \\bu @@Dissimilarity: To Configuration (monotone mds)...") LIST_ITEM (" \\bu @@Dissimilarity: To Configuration (i-spline mds)...") LIST_ITEM (" \\bu @@Dissimilarity: To Configuration (interval mds)...") LIST_ITEM (" \\bu @@Dissimilarity: To Configuration (ratio mds)...") LIST_ITEM (" \\bu @@Dissimilarity: To Configuration (absolute mds)...") NORMAL ("By multidimensional scaling with weights (@@Dissimilarity & Weight: To Configuration...@):") LIST_ITEM (" \\bu ##Dissimilarity & Weight: To Configuration (monotone mds)...") LIST_ITEM (" \\bu ##Dissimilarity & Weight: To Configuration (i-spline mds)...") LIST_ITEM (" \\bu ##Dissimilarity & Weight: To Configuration (interval mds)...") LIST_ITEM (" \\bu ##Dissimilarity & Weight: To Configuration (ratio mds)...") LIST_ITEM (" \\bu ##Dissimilarity & Weight: To Configuration (absolute mds)...") NORMAL ("By multidimensional scaling with a start Configuration:") LIST_ITEM (" \\bu @@Dissimilarity & Configuration: To Configuration (monotone mds)...") LIST_ITEM (" \\bu @@Dissimilarity & Configuration: To Configuration (i-spline mds)...") LIST_ITEM (" \\bu @@Dissimilarity & Configuration: To Configuration (interval mds)...") LIST_ITEM (" \\bu @@Dissimilarity & Configuration: To Configuration (ratio mds)...") LIST_ITEM (" \\bu @@Dissimilarity & Configuration: To Configuration (absolute mds)...") NORMAL ("By transforming an existing Configuration:") LIST_ITEM (" \\bu @@Configuration: To Configuration (varimax)...") LIST_ITEM (" \\bu @@Configuration & AffineTransform: To Configuration") LIST_ITEM (" \\bu @@Configuration & Procrustes: To Configuration") NORMAL ("From @@Principal component analysis@:") LIST_ITEM (" \\bu @@TableOfReal: To Configuration (pca)...") LIST_ITEM (" \\bu @@PCA & TableOfReal: To Configuration...") NORMAL ("From @@Discriminant analysis@:") LIST_ITEM ("@@TableOfReal: To Configuration (lda)...") LIST_ITEM (" \\bu @@Discriminant & TableOfReal: To Configuration...") ENTRY ("How to draw a Configuration") LIST_ITEM ("\\bu @@Configuration: Draw...") LIST_ITEM ("\\bu ##Configuration: Draw as numbers...") LIST_ITEM ("\\bu ##Configuration: Draw as squares...") ENTRY ("How to modify a Configuration") LIST_ITEM ("\\bu @@Configuration: Randomize") LIST_ITEM ("\\bu @@Configuration: Rotate (pc)@ (to principal directions)") LIST_ITEM ("\\bu @@Configuration: Rotate...@ (in a plane around the origin)") LIST_ITEM ("\\bu @@Configuration: Invert dimension...") LIST_ITEM ("\\bu @@Configuration: Normalize...") ENTRY ("Inside a Configuration") NORMAL ("With @Inspect you will see the following attributes:") TAG ("%numberOfRows") DEFINITION ("the number of points (%numberOfPoints\\>_1).") TAG ("%numberOfColumns") DEFINITION ("the dimension of the space (%numberOfDimensions\\>_1).") TAG ("%rowLabels") DEFINITION ("the names associated with the points.") TAG ("%columnLabels") DEFINITION ("the names for the dimensions.") TAG ("%data [1..%numberOfPoints][1..%numberOfDimensions]") DEFINITION ("the coordinates of the points.") TAG ("%metric") DEFINITION ("determines the way distances between points are measured. In general " "the distance between points #x__%i_ and #x__%j_ is:") FORMULA ("%d__%ij_ = " "(\\su__%k=1..%numberOfDimensions_ %w__%k_ |%x__%ik_ \\-- " "%x__%jk_|^^%metric^))^^1/%metric^") DEFINITION ("For Euclidean distances %metric is 2.") TAG ("%w [1..%numberOfDimensions]") DEFINITION ("weight given to each dimension in the distance calculation.") MAN_END MAN_BEGIN ("Configuration: Centralize", "djmw", 19980413) INTRO ("Makes the centre of the selected @Configuration equal to the origin.") NORMAL ("") MAN_END MAN_BEGIN ("Configuration: Draw...", "djmw", 20040407) INTRO ("Draws a projection of the selected @Configuration on a coordinate plane.") ENTRY ("Settings") TAG ("%%X-coordinate%, %%Y-coordinate%") DEFINITION ("control the dimensions that will show in the plot.") TAG ("%xmin, %xmax; %ymin, %ymax") DEFINITION ("range for horizontal and vertical axes, respectively.") TAG ("%garnish") DEFINITION ("when on, draws a bounding box with decoration.") MAN_END MAN_BEGIN ("Configuration: Invert dimension...", "djmw", 20040407) INTRO ("Inverts one dimension of a @Configuration.") NORMAL ("Settings") TAG ("%%Dimension") DEFINITION ("the dimensions that has to be inverted.") ENTRY ("Behaviour") NORMAL ("For all points %i=1..%numberOfPoints: if %j == %dimension then %x__%ij_ " "= \\--%x__%ij_.") MAN_END MAN_BEGIN ("Configuration: Normalize...", "djmw", 20040407) INTRO ("Normalizes the selected @Configuration.") ENTRY ("Settings") TAG ("%%Sum of squares% (default: 0.0)") DEFINITION ("The desired value for the variance.") TAG ("%%Each dimension separately") DEFINITION ("When on, the sum of squares in each dimension (column) will be scaled to %sumOfSquares " "When off, the sum of squares of all the matrix elements will equal %sumOfSquares.") NORMAL ("With the default value (0.0) for %sumOfSquares, and %eachDimensionSeparately chosen, " "an INDSCAL-like normalization is applied: the sum of squares for each column is scaled to " "equal 1.0. When %eachDimensionSeparately is off, a Kruskal-like normalization is applied: " "the sum of squares of the whole matrix is scaled equal to %numberOfRows.") ENTRY ("Behaviour") NORMAL ("Before the normalization will be applied, however, we first translate the centre of the " "configuration to the origin by subtracting the mean for each dimension. " "The sum of squares than equals variance.") MAN_END MAN_BEGIN ("Configuration: Randomize", "djmw", 19971201) INTRO ("Changes all coordinates of the points #%x__%i_ in the @Configuration according to:") LIST_ITEM ("%x__%ij_ = randomUniform (-1, 1)") MAN_END MAN_BEGIN ("Configuration: Rotate...", "djmw", 20040407) INTRO ("Rotates the @Configuration in a plane around the origin.") NORMAL ("Settings") TAG ("%%Dimension 1%, %%Dimension 2%") DEFINITION ("the dimensions that span the plane.") TAG ("%Angle") DEFINITION ("the clockwise rotation angle in degrees w.r.t. the direction vector " "of the lowest of the two dimensions.") MAN_END MAN_BEGIN ("Configuration: Rotate (pc)", "djmw", 19971201) INTRO ("Rotates the @Configuration to principal directions. The principal directions " "correspond to the principal components.") MAN_END MAN_BEGIN ("Configuration: To Configuration (procrustes)", "djmw", 19971219) INTRO ("A command that transforms the second selected @Configuration object " "to match the first selected Configuration object as closely as possible." "This problem of fitting one configuration (testee) to another (target) " "as closely as possible is called the Procrustes problem. We use a " "special @@Procrustes transform@ algorithm that does not " "mutilate or distort the testee configuration.") NORMAL ("Both Configuration objects must have the same dimensions.") MAN_END MAN_BEGIN ("Configuration: To Configuration (varimax)...", "djmw", 20040407) INTRO ("A command that rotates the selected @Configuration object to a new " "Configuration object whose coordinates have maximum %squared variance. ") ENTRY ("Settings") TAG ("%%Normalize rows") DEFINITION ("when selected, the distances of all points to the origin will " "be made equal before iteration starts. We remember these scale factors " "and restore the original distances after the iteration process has " "stopped.") TAG ("%Quartimax") DEFINITION ("when selected, the sum of fourth powers, normalized or raw, " "will be maximized. ") TAG ("%%Maximum number of iterations") DEFINITION ("sets a limit to the number of iterations. One iteration consists " "of %numberOfDimensions\\.c (%numberOfDimensions\\--1)/2 planar rotations " "of all pairs of dimensions.") TAG ("%%Tolerance") DEFINITION ("also determines when the iteration stops. This happens if " "|%v__%i_\\--%v__%i+1_| < %Tolerance \\.c %v__%i_, where %v__%i_ is the " "squared variance for the %i^^th^ iteration.") NORMAL ("The iteration process stops when either the %%maximum number of " "iterations% is reached or the %tolerance criterion is met, which ever " "one is first.") ENTRY ("Algorithm") NORMAL ("The Varimax rotation procedure was first proposed by @@Kaiser " "(1958)@. Given a %numberOfPoints \\xx %numberOfDimensions configuration " "#A, the procedure tries to find an orthonormal rotation matrix #T such " "that the sum of variances of the columns of #B*#B is a maximum, where #B " "= #A#T and * is the element wise (Hadamard) product of matrices. A direct " "solution for the optimal #T is not available, except for the case when " "%numberOfDimensions equals two. Kaiser suggested an iterative " "algorithm based on planar rotations, i.e., alternate rotations of all " "pairs of columns of #A.") NORMAL ("However, this procedure is not without problems: the varimax function " "may have stationary points that are not even local maxima. We have " "incorporated an algorithm of @@Ten Berge (1995)@ " "that prevents this unpleasant situation from happening.") MAN_END MAN_BEGIN ("Configuration: To Distance", "djmw", 19971207) INTRO ("A command that computes a @Distance object for each selected " "@Configuration.") ENTRY ("Algorithm") NORMAL ("The distance %d__%ij_ between objects %i and %j is calculated as:") FORMULA ("%d__%ij_ = %d__%ji_ = (\\su__%k=1..%numberOfDimensions_ |%x__%ik_ " "\\-- %x__%jk_|^2)^^1/2^") MAN_END MAN_BEGIN ("Configuration: To Similarity (cc)", "djmw", 19980130) INTRO ("A command that create one @Similarity object from the selected " "@Configuration objects.") NORMAL ("In the Similarity object entry %s__%ij_ equals the @@congruence " "coefficient@ for the %i-th and %j-th selected Configuration object.") NORMAL ("All Configuration objects must have the same number of points and " "the same dimensions.") MAN_END MAN_BEGIN ("Configuration & AffineTransform: To Configuration", "djmw", 20011008) INTRO ("A command that transforms the selected @Configuration to a new " "Configuration object according to the specifications in the selected " "@AffineTransform object.") MAN_END MAN_BEGIN ("Configuration & Procrustes: To Configuration", "djmw", 20011008) INTRO ("A command that transforms the selected @Configuration to a new " "Configuration object according to the specifications in the selected " "@Procrustes object.") MAN_END MAN_BEGIN ("Configurations: To AffineTransform (congruence)...", "djmw", 20040407) INTRO ("A command that creates an @AffineTransform object from two selected " "@Configuration objects.") NORMAL ("We calculate the affine transform that transforms the second " "selected Configuration object to match the first selected Configuration " "object as closely as possible. " "The degree of proportionality is the congruence between corresponding " "dimensions.") ENTRY ("Settings") TAG ("%%Maximum number of iterations") DEFINITION ("sets a limit to the number of iterations.") TAG ("%%Tolerance") DEFINITION ("also determines when the iteration stops. This happens if " "|%f(#T__%i_)\\--%f(#T__%i+1_)| < %Tolerance \\.c %f(#T__%i_), where " "%f(#T__%i_) is the sum of the congruences for the %i^^th^ " "iteration (see below).") NORMAL ("The iteration process stops when either the %%maximum number of " "iterations% is reached or the %tolerance criterion is met, which ever " "one is first.") ENTRY ("Algorithm") NORMAL ("Sometimes the criterion used in a @@Procrustes transform@ is too " "restrictive for comparing two configurations. This criterion is only " "zero when the positions in the rotated configuration (#A#T) equal the " "positions in the other configuration (#B). @@Brokken (1983)@ proposed an " "algorithm to maximize instead the sum of congruences between " "corresponding dimensions of #AT and #B. " "Specifically he proposed to maximize") FORMULA ("%f(#T) = \\su__%i=1..%numberOfDimensions_ #t\\'p__%i_#A\\'p#b__%i_ /" " ((#t\\'p__%i_#A\\'p#A#t__%i_)^^1/2^(#b\\'p__%i_#b__%i_)^^1/2^),") NORMAL ("where #t\\'p__%i_ and #b\\'p__%i_ are the %i^^th^ column of #T and " "#B, respectively. A direct solution for #T is not available, it can only " "be obtained by an iterative procedure. The implemented algorithm is from " "@@Kiers & Groenen (1996)@ and shows excellent convergence properties.") MAN_END MAN_BEGIN ("Configuration & Configuration: To Procrustes...", "djmw", 20011008) INTRO ("A command that creates a @Procrustes object from two selected " "@Configuration objects.") ENTRY ("Argument") TAG ("%%Orthogonal transform%") DEFINITION ("determines whether or not a translation and a scaling are allowed in the transform.") NORMAL ("We calculate the @@Procrustes transform@ that transforms the second " "selected Configuration object to match the first selected Configuration " "object as closely as possible.") MAN_END MAN_BEGIN ("Confusion: To Dissimilarity...", "djmw", 20040407) INTRO ("A command that creates a @Dissimilarity from every selected " "@Confusion.") ENTRY ("Settings") TAG ("%Normalize") DEFINITION ("when on, normalize rows by dividing each row element by the row " "sum. In this way you correct for unequal stimulus numbers.") TAG ("%%No symmetrization%, %Average, %Houtgast") DEFINITION ("determine the symmetrization procedure. See " "@@Confusion: To Similarity...") TAG ("%%Maximum dissimilarity") DEFINITION ("determines the maximum dissimilarity possible. When the default " "value, 0.0, is chosen, %maximumDissimilarity " "is calculated as the maximum element in the Similarity object.") ENTRY ("Algorithm") NORMAL ("We first transform the Confusion to a Similarity. See " "@@Confusion: To Similarity...") NORMAL ("To obtain dissimilarities from similarities we \"reverse\" the " "latter:") FORMULA ("%%dissimilarity__%ij_ = %maximumDissimilarity \\-- %similarity__%ij_") MAN_END MAN_BEGIN ("Confusion: To Dissimilarity (pdf)...", "djmw", 20040407) INTRO ("A command that creates a @Dissimilarity from every selected " "@Confusion.") ENTRY ("Settings") TAG ("%%Symmetrize first") DEFINITION ("when on, the confusion matrix is symmetrized before we calculate " "dissimilarities.") TAG ("%%Maximum dissimilarity% (units of sigma)") DEFINITION ("specifies the dissimilarity from confusion matrix elements that " "are zero.") ENTRY ("Algorithm") TAG ("1. Normalize rows by dividing each row element by the row sum (optional).") TAG ("2. Symmetrize the matrix by averaging %f__%ij_ and %f__%ji_.") TAG ("3. Transformation of the confusion measure which is a sort of " "%similarity measure to the %dissimilarity measure.") NORMAL ("Similarity and dissimilarity have an inverse relationship: the " "greater the similarity, the smaller the dissimilarity and vice versa. " "Both have a monotonic relationship with distance. " "The most simple way to transform the similarities %f__%ij_ into " "dissimilarities is:") FORMULA ("%dissimilarity__%ij_ = %maximumSimilarity \\-- %similarity__%ij_") NORMAL ("For ordinal analyses like Kruskal this transformation is fine because " "only order relations are important in this analysis. However, for " "metrical analyses like INDSCAL this is not optimal. " "In INDSCAL, distance is a linear function of dissimilarity. This means " "that, with the transformation " "above, you ultimately fit an INDSCAL model in which the distance " "between object %i and %j will be linearly related to the confusion " "between %i and %j.") NORMAL ("For the relation between confusion and dissimilarity, the model " "implemented here, makes the assumption that the amount of confusion " "between objects %i and %j is related to the amount that their " "probability density functions, pdf's, overlap. Because we do not know " "these pdf's we make the assumption that both are normal, have equal " "%sigma and are one-dimensional. The parameter to be determined is the " "distance between the centres of both pdf's. " "According to formula 26.2.23 in @@Abramowitz & Stegun (1970)@, for each " "fraction %f__%ij_, we have to find an %x that solves:") FORMULA ("%f__%ij_ = 1 / \\Vr(2%\\pi) \\in__%x_^^\\oo^ e^^-%t\\.c%t/2^ %dt") NORMAL ("This %x will be used as the dissimilarity between %i and %j. The " "relation between %x and %f__%ij_ is monotonic. This means that the " "results for a Kruskal analysis will not change much. For INDSCAL, in " "general, you will note a significantly better fit.") MAN_END MAN_BEGIN ("Confusion: To Similarity...", "djmw", 20040407) INTRO ("A command that creates a @Similarity from every selected @Confusion.") ENTRY ("Settings") TAG ("%Normalize") DEFINITION ("when on, normalize rows by dividing each row element by the row " "sum. In this way you correct for unequal stimulus numbers.") TAG ("%%No symmetrization%, %Average, %Houtgast") DEFINITION ("determine the symmetrization procedure.") ENTRY ("Algorithm") NORMAL ("The %Average procedure averages:") FORMULA ("%similarity__%ij_= %similarity__%ji_ = (%confusion__%ij_ + " "%confusion__%ji_) / 2") NORMAL ( "The %Houtgast procedure as described in the paper by @@Klein, Plomp " "& Pols (1970)@, expresses similarity between stimuli %i and %j by the " "number of times that stimulus %i and %j have " "resulted in the same response, summated over all response categories.") NORMAL ("We use the following formula to calculate the %Houtgast " "dissimilarities:") FORMULA ("%similarity__%ij_ = \\su__%k=1..%numberOfColumns_ min " "(%confusion__%ik_, %confusion__%jk_)") NORMAL ("which is equivalent to the formula in the Klein et al. paper:") FORMULA ("%similarity__%ij_ = \\su__%k=1..%numberOfColumns_ (%confusion__%ik_ " "+ %confusion__%jk_ \\-- |%confusion__%ik_ \\-- %confusion__%jk_|)") MAN_END MAN_BEGIN ("congruence coefficient", "djmw", 20040407) INTRO ("The %%congruence coefficient% is a measure of similarity between two " "@@Configuration@s.") NORMAL ("The congruence coefficient %c(#X, #Y) for the configurations #X and " "#Y is defined as:") FORMULA ("%c(%X, %Y) = \\su__%i<%j_ %w__%ij_ %d__%ij_(#X) %d__%ij_(#Y) / " "([\\su__%i<%j_ %w__%ij_ %d^2__%ij_(#X)]^^1/2^ [\\su__%i<%j_ %w__%ij_ " "%d^2__%ij_(#Y)]^^1/2^),") NORMAL ("where %d__%ij_(#X) is the distance between the points %i and %j in " "configuration #X and %w__%ij_ are nonnegative weights (default: %w__%ij_" " = 1).") NORMAL ("Since distances are nonnegative, the congruence coefficient has a " "value between 0 and 1.") NORMAL ("The %%congruence coefficient% is a better measure of the similarity " "between configurations than the %%correlation coefficient% of the " "distances. @@Borg & Groenen (1997)@ give a simple example where things " "go wrong with correlation coefficients: two configurations #X and #Y with three points each, have " "distances %d__12_(#X) = 1, %d__13_(#X) = 2, %d__23_(#X) = 3 and " "%d__12_(#Y) = 2, %d__13_(#Y) = 3, %d__23_(#Y) = 4. " "These distances have a correlation coefficient of 1. " "However, in #X the three points lie on a straight line and in #Y the " "points form a triangle. This unwanted situation occurs because " "in the calculation of the correlation coefficient the mean is subtracted " "from the distances and the resulting values are no longer distances " "(they may become negative). In calculating the correlation " "between the distances we should not subtract the mean. " "In fact, the congruence coefficient is exactly this correlation " "coefficient calculated with respect to the origin and " "not with respect to the centroid position (the \"mean\").") NORMAL ("For further information on how well one number can assess the " "similarity between two configurations see @@Borg & Groenen (1997)@ " "section 19.7.") MAN_END MAN_BEGIN ("ContingencyTable", "djmw", 19971216) INTRO ("One of the @@types of objects@ in P\\s{RAAT}.") NORMAL ("In a two-way contingency table, cell %f__%ij_ contains the frequency " "with which row category %i co-occurs with column category %j. " "Necessarily, all %f__%ij_ \\>_ 0.") ENTRY ("Commands") NORMAL ("Creation") LIST_ITEM ("\\bu ##TableOfReal: To ##ContingencyTable") NORMAL ("Query") LIST_ITEM ("\\bu ##ContingencyTable: Get chi squared probability") LIST_ITEM ("\\bu ##ContingencyTable: Get Cramer's statistic") LIST_ITEM ("\\bu ##ContingencyTable: Get contingency coefficient") NORMAL ("Analysis") LIST_ITEM ("\\bu @@ContingencyTable: To Configuration (ca)...") MAN_END MAN_BEGIN ("ContingencyTable: To Configuration (ca)...", "djmw", 20040407) INTRO ("A command that creates a @Configuration object from the selected " "@ContingencyTable object by means of @@Correspondence analysis@.") ENTRY ("Settings") TAG ("%%Number of dimensions") DEFINITION ("The dimensionality of the Configuration.") TAG ("%%Scaling of the final configuration") DEFINITION ("determines whether row points are in the centre of gravity of " "column points, or, column points are in the centre of gravity of row " "points, or, whether roes and columns are treated symmetrically.") ENTRY ("Algorithm") NORMAL ("1. We start with the following transformation of the entries " "%f__%ij_:") FORMULA ("%h__%ij_ = %f__%ij_ / \\Vr (%f__%i+_%f__+%j_) - \\Vr " "(%f__%i+_%f__+%j_) / %N,") NORMAL ("where %h__%ij_ is the entry for a cell in the matrix #H with " "transformed data, %f__%i+_ " "is the total count for row %i, %f__+%j_ is the total count for column %j " "and %N is the grand total. " "This can be written in matrix form as:") FORMULA ("#H = #R^^\\--1/2^#F#C^^\\--1/2^ \\-- #R^^1/2^#uu\\'p#C^^1/2^ / %N,") NORMAL ("where #R and #C are diagonal matrices with the row and column totals, " "respectively and #u a column vector with all elements equal to 1. ") NORMAL ("2. Next the singular value decomposition of matrix #H is performed:") FORMULA ("#H = #K #\\La #L\\'p,") NORMAL ("where #K\\'p#K = #I, #L\\'p#L = #I, and #\\La is a diagonal matrix " "with singular values.") NORMAL ("3. Now the row (#X) and column points (#Y) can be determined. " "Three normalizations are possible:") TAG ("\\bu Scale row points in the centre of gravity of column points") DEFINITION ("#X = \\Vr%N #R^^\\--1/2^#K#\\La") DEFINITION ("#Y = \\Vr%N #C^^\\--1/2^#L") TAG ("\\bu Scale column points in the centre of gravity of row points") DEFINITION ("#X = \\Vr%N #R^^\\--1/2^#K") DEFINITION ("#Y = \\Vr%N #C^^\\--1/2^#L#\\La") TAG ("\\bu Treat row points and column points symmetrically") DEFINITION ("#X = \\Vr%N #R^^\\--1/2^#K#\\La^^\\--1/2^") DEFINITION ("#Y = \\Vr%N #C^^\\--1/2^#L\\La^^\\--1/2^") NORMAL ("For more details see @@Gifi (1990)@, chapter 8.") MAN_END MAN_BEGIN ("Correspondence analysis", "djmw", 19971216) INTRO ("Correspondence analysis provides a method for representing data in " "an Euclidean space so that the results can be visually examined for " "structure. For data in a typical two-way @ContingencyTable both the row " "variables and the column variables are represented in the same space. " "This means that one can examine relations not only among row " "or column variables but also between row and column variables.") NORMAL ("In correspondence analysis the data matrix is first transformed by " "dividing each cell by the square root of the corresponding row and column " "totals. " "The transformed matrix is then decomposed with singular value " "decomposition resulting in the singular values (which in this case are " "canonical correlations) and a set of row vectors and column vectors. " "Next the row and column vectors are rescaled with the original total " "frequencies to obtain optimal scores. " "These optimal scores are weighted by the square root of the singular " "values and become the coordinates of the points in the @Configuration.") NORMAL ("Examples can be found in the books by @@Weller & Romney (1990)@ and " "@@Gifi (1990)@.") MAN_END MAN_BEGIN ("Create Configuration...", "djmw", 19980413) INTRO ("A command to create a @Configuration with the specified number of " "points and number of dimensions. The location of the points will be " "determined by the formula (see @@Formulas@ for more " "information about possible formulas).") MAN_END MAN_BEGIN ("Create INDSCAL Carroll & Wish example...", "djmw", 19971201) INTRO ("Creates eight @Dissimilarity objects that bear names \"1\" ... \"8\".") NORMAL ("These objects contain the interpoint distances for a twodimensional " "3\\xx3 @Configuration of points, labelled A, B, C, ... I. " "All Dissimilarity objects are based on the following underlying configuration.") PICTURE (4.0, 4.0, drawCarrollWishConfigurationExample) NORMAL ("The eight sources weigh this configuration in the following manner:") PICTURE (4.0, 4.0, drawCarrollWishSalienceExample) NORMAL ("For each source, the distances were subjected to the transformation: ") FORMULA ("%dissimilarity__%ij_ = %distance__%ij_ + %noiseRange \\.c #u, ") NORMAL ("where #u is a uniform random variable between 0 and 1.") NORMAL ("Now you can do the following for example:") TAG ("Select all the Dissimilarity objects and choose @@Dissimilarity: To Distance...|" "To Distance...@.") DEFINITION ("Uncheck scale (add \"additive constant\").") TAG ("Select all the Distance objects and choose @@Distance: To Configuration (indscal)...|" "To Configuration (indscal)...@.") DEFINITION ("and an @@INDSCAL analysis@ will be performed. In order to reproduce the saliences, " "you have to uncheck the \"Normalize scalar products\" option.") NORMAL ("This example was adapted from @@Carroll & Wish (1974)@.") MAN_END MAN_BEGIN ("Create letter R example...", "djmw", 19971201) INTRO ("Creates a @Dissimilarity object that bears the name %R. The " "dissimilarities in this object were chosen to be a monotone " "transformation of the distances between the 32 two-dimensional points " "that make up the capital letter #R.") PICTURE (4.0, 4.0, drawLetterRConfigurationExample) NORMAL ("All 32 \\.c (32-1)/2 interpoint distances were subjected to the " "transformation: ") FORMULA ("%dissimilarity__%ij_^ = %distance__%ij_^2 + 5 + %noiseRange \\.c #u, ") NORMAL ("where #u is a uniform random variable between 0 and 1.") NORMAL ("This example was chosen from @@Green, Carmone & Smith (1989)@.") MAN_END MAN_BEGIN ("disparities", "djmw", 19980111) INTRO ("The numbers %d\\'p__%ij_ that result from applying an admissible " "transformation %f on the dissimilarities %\\de__%ij_, i.e., %d\\'p__%ij_ " "= %f (%\\de__%ij_). Disparities have the same dimension as distances. " "Other names for disparities are %%pseudo distances% and %%target " "distances%.") MAN_END MAN_BEGIN ("Dissimilarity", "djmw", 20010327) INTRO ("One of the @@types of objects@ in P\\s{RAAT}.") NORMAL ("It represents a one-way table with " "dissimilarities between \"objects\".") ENTRY ("Creating a Dissimilarity from data in a text file") NORMAL ("Suppose you have three objects A, B and C. " "In one way or another, you have acquired the following (symmetric) " "dissimilarities: %\\de__%AB_ = 2 (= %\\de__%BA_) , %\\de__%AC_ = 1 " "(= %\\de__%CA_), and %\\de__%BC_ = 1.4 (= %\\de__CB_), where %\\de__%AB_" " represents the dissimilarity 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 ("\"Dissimilarity\" ! The line that tells Praat about the contents") CODE ("3 \"A\" \"B\" \"C\" ! Number of columns, and column labels") CODE ("3 ! Number of rows") CODE ("\"A\" 0 2 1 ! Row label (A), A-B value, A-C value") CODE ("\"B\" 2 0 1.4 ! Row label (B), B-A value, B-C value") CODE ("\"C\" 1 1.4 0 ! Row label (C), C-A value, C-B value") NORMAL ("Notice that:") LIST_ITEM ("\\bu the row and column labels are identical.") LIST_ITEM ("\\bu the matrix elements on the diagonal are zero.") LIST_ITEM ("\\bu the matrix is symmetrical.") NORMAL ("This text file can be read with the @@Read from file...@ command. " "Since a Dissimilarity object has the data structure of a square " "symmetrical TableOfReal, you could also start from an appropriate " "@TableOfReal object and cast it to a Dissimilarity object.") ENTRY ("Commands") NORMAL ("Creation") LIST_ITEM ("\\bu @@Confusion: To Dissimilarity...") 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 @@Dissimilarity: Get additive constant") NORMAL ("Modification") LIST_ITEM ("\\bu @@Formula...") LIST_ITEM ("\\bu ##Set value...") 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 ("Multidimensional scaling analysis") LIST_ITEM ("\\bu @@Dissimilarity: To Configuration (monotone mds)...") LIST_ITEM ("\\bu @@Dissimilarity: To Configuration (i-spline mds)...") LIST_ITEM ("\\bu @@Dissimilarity: To Configuration (interval mds)...") LIST_ITEM ("\\bu @@Dissimilarity: To Configuration (ratio mds)...") LIST_ITEM ("\\bu @@Dissimilarity: To Configuration (absolute mds)...") LIST_ITEM ("\\bu @@Dissimilarity: To Configuration (kruskal)...") LIST_ITEM ("Transformations") LIST_ITEM ("\\bu @@Dissimilarity: To Distance...") LIST_ITEM ("\\bu @@Dissimilarity: To Weight") MAN_END MAN_BEGIN ("Dissimilarity: Get additive constant", "djmw", 19971201) INTRO ("A command that calculates the \"additive constant\" from the selected @Dissimilarity.") NORMAL ("Distances %d__%ij_ will be obtained from dissimilarities %\\de__%ij_ according to:") FORMULA (" %distance__%ij_ = %dissimilarity__%ij_ + %additiveConstant") NORMAL ("We use a procedure by @@Cailliez (1983)@ to solve the \"additive constant problem\", i.e. " "find the smallest %additiveConstant such that all %distance__%ij_ in the above equation " " have a Euclidean representation.") MAN_END MAN_BEGIN ("Dissimilarity: To Configuration (absolute mds)...", "djmw", 19980105) INTRO ("A command that creates a @Configuration object from a @Dissimilarity object.") NORMAL ("The @disparities %d\\'p__%ij_ will be obtained from dissimilarities %\\de__%ij_ according to:") FORMULA ("%d\\'p__%ij_ = %\\de__%ij_") MAN_END MAN_BEGIN ("Dissimilarity: To Configuration (interval mds)...", "djmw", 19980105) INTRO ("A command that creates a @Configuration object from a @Dissimilarity object.") NORMAL ("The @disparities %d\\'p__%ij_ will be obtained from dissimilarities %\\de__%ij_ according to:") FORMULA ("%d\\'p__%ij_ = %a + %b \\.c %\\de__%ij_") MAN_END MAN_BEGIN ("Dissimilarity: To Configuration (i-spline mds)...", "djmw", 20040407) INTRO ("A command that creates a @Configuration object from a @Dissimilarity object.") NORMAL ("Dissimilarities %\\de__%ij_ and @disparities %d\\'p__%ij_ will be related by a @spline function:") FORMULA ("%d\\'p__%ij_ = \\su__%k=1..(%%numberOfInteriorKnots%+%order)_ spline__%k_ (%knots, %order, %\\de__%ij_),") NORMAL ("where spline__%k_ (\\.c) is the value of the %k^^th^ I-spline of order %order and knot sequence " "%knot evaluated at %\\de__%ij_.") ENTRY ("Settings") TAG ("%%Number of dimensions") DEFINITION ("determines the dimensionality of the configuration.") TAG ("%%Number of interior knots%") DEFINITION ("determines the number of segment boundaries. Each interior knot " "is the boundary between two segments. The splines in each segment will " "be joined as continuously as possible.") TAG ("%%Order of I-spline") DEFINITION ("The order of the polynomial basis of the I-spline.") NORMAL ("Finding the optimal Configuration involves a minimization process:") TAG ("%Tolerance") DEFINITION ("When successive values for the stress differ by less than " "%Tolerance, the minimization process stops.") TAG ("%%Maximum number of iterations%") DEFINITION ("Minimization stops after this number of iterations has been " "reached.") TAG ("%%Number of repetitions") DEFINITION ("If chosen larger than 1, the minimization process will be " "repeated, each time with another random start configuration. " "The configuration that results in minimum stress, will be saved.") ENTRY ("Hints") NORMAL ("If %numberOfInteriorKnots is zero, polynomial regression will be " "performed. Therefore , the combination %numberOfInteriorKnots = 0 and " "%order = 1 also gives interval " "scaling (in fact, it is the implementation in this program).") NORMAL ("In the limit when %order = 0 and %numberOfInteriorKnots = " "%numberOfDissimilarities, monotone regression is performed.") MAN_END MAN_BEGIN ("Dissimilarity: To Configuration (kruskal)...", "djmw", 20040407) INTRO ("A command that creates a @Configuration object from a @Dissimilarity " "object.") ENTRY ("Settings") TAG ("%%Number of dimensions% (default: 2)") DEFINITION ("The dimensionality of the Configuration.") TAG ("%%Distance metric% (default: 2, i.e. Euclidean)") DEFINITION ("the general distance between points #x__%i_ and #x__%j_ (%i,%j " "= 1..%numberOfPoints) is:") DEFINITION ("(\\su__%k=1..%numberOfDimensions_ |%x__%ik_ \\--%x__%jk_|" "^^%metric^)^^1/%metric^") TAG ("%%Sort distances") DEFINITION ("determines the handling of ties in the data. When off, whenever " "two or more dissimilarities are equal we do not care whether the fitted " "distances are equal or not. " "Consequently, no constraints are imposed on the fitted distances. " "When on, however, we impose the constaint that the fitted distances be " "equal whenever the dissimilarities are equal.") NORMAL ("For the calculation of stress:") TAG ("%Formula1 (default)") FORMULA ("%stress = \\Vr(\\su(%distance__%k_ \\-- %fittedDistance__%k_)^2 / " "\\su %distance__%k_^2)") TAG ("%%Formula2%") FORMULA ("%stress = \\Vr(\\su(%distance__%k_ \\-- %fittedDistance__%k_)^2 / " "\\su (%distance__%k_ \\-- %averageDistance)^2)") DEFINITION ("Note that values of stress 2 are generally more than double those " "of stress 1 for the same degree of fit.") NORMAL ("Finding the optimal Configuration involves a minimization process:") TAG ("%Tolerance") DEFINITION ("When successive values for the stress differ less than %Tolerance " "the minimization process stops.") TAG ("%%Maximum number of iterations%") DEFINITION ("Minimization stops after this number of iterations has been " "reached.") TAG ("%%Number of repetitions") DEFINITION ("When chosen larger than 1, the minimalization process will be " "repeated, each time with another random start configuration. " "The configuration that results in minimum stress will be saved.") ENTRY ("Precautions") NORMAL ("When there are few objects it is impossible to recover many " "dimensions. A rough rule of thumb is that there should be at least twice " "as many number of observations, i.e. the %numberOfPoints \\.c " "(%numberOfPoints - 1) / 2 (dis)similarities, than parameters " "to be estimated, i.e. the %numberOfPoints \\.c %numberOfDimensions " "position coordinates. A practical guide is:") LIST_ITEM ("for %numberOfDimensions = 1 you need \\>_ 5 objects") LIST_ITEM ("for %numberOfDimensions = 2 you need \\>_ 9 objects") LIST_ITEM ("for %numberOfDimensions = 3 you need \\>_ 13 objects") NORMAL ("There is no feasible way to be certain that you have found the " "true global minimum. However, by using a great number of different " "random starting configurations to scale the same data it is often " "possible to obtain practical certainty. " "Although the procedure for obtaining an initial configuration is based " "on a %linear relation between distance and (dis)similarity, it gives a " "very good approximation of the optimal #Configuration and " "the #Minimizer practically always finds the global minimum from it " "(I guess...). A way to find out is to try the %numberOfRepetitions " "parameter which gives you the possibility to fit many times and each " "time start with another random initial configuration.") ENTRY ("Algorithm") LIST_ITEM ("1. The Dissimilarity object is converted to a Distance object in " "the same way as in @@Dissimilarity: To Distance...@.)") LIST_ITEM ("2. From the Distance object an initial Configuration is found by " "first transforming the Distance object to a matrix with scalar products " "of distances and subsequently solving for the first %numberOfDimensions " "eigenvectors of this matrix.") LIST_ITEM ("3. A minimalization algorithm is started that tries to minimize a " "function. In this function:") LIST_ITEM ("\\bu 3.1 We normalize the current Configuration from the minimizer") LIST_ITEM ("\\bu 3.2 Calculate a new Distance object from the configuration") LIST_ITEM ("\\bu 3.3 Do a monotone regression of this Distance on the " "Dissimilarity. This results in a new Distance object.") LIST_ITEM ("\\bu 3.4 Calculate stress from this Distance and the Distance " "obtained from Dissimilarity.") NORMAL ("The optimization process is ccontrolledby a conjugate gradient " "minimization algorithm that tries to minimize the %stress function. " "In @@Kruskal (1964)@, a steepest descent " "algorithm is used wwhichis less efficient. ") MAN_END MAN_BEGIN ("Dissimilarity: To Configuration (monotone mds)...", "djmw", 20040407) INTRO ("A command that creates a @Configuration object from a @Dissimilarity " "object.") NORMAL ("Dissimilarities %\\de__%ij_ and @disparities %d\\'p__%ij_ are " "related by:") FORMULA ("%d\\'p__%ij_ \\<_ %d\\'p__%kl_ if %\\de__%ij_ \\<_ %\\de__%kl_") ENTRY ("Settings") TAG ("%%Number of dimensions") DEFINITION ("determines the dimension of the configuration.") TAG ("%Primary or %secondary% approach to ties") DEFINITION ("When dissimilarities are equal, i.e., %\\de__%ij_ = %\\de__%kl_, " "the primary approach imposes no conditions on the corresponding " "@disparities %d\\'p__%ij_ and %d\\'p__%kl_, while the %secondary " "approach demands that also %d\\'p__%ij_ = %d\\'p__%kl_.") NORMAL ("Finding the optimal Configuration involves a minimization process:") TAG ("%Tolerance") DEFINITION ("When successive values for the stress differ less than %Tolerance " "the minimization process stops.") TAG ("%%Maximum number of iterations%") DEFINITION ("Minimization stops after this number of iterations has been reached.") TAG ("%%Number of repetitions") DEFINITION ("When chosen larger than 1, the minimalization process will be " "repeated, each time with another random start configuration. " "The configuration that results in minimum stress will be saved.") MAN_END MAN_BEGIN ("Dissimilarity: To Configuration (ratio mds)...", "djmw", 19980105) INTRO ("A command that creates a @Configuration object from a @Dissimilarity " "object.") NORMAL ("The @disparities %d\\'p__%ij_ will be obtained from dissimilarities " "%\\de__%ij_ according to:") FORMULA ("%d\\'p__%ij_ = %b \\.c %\\de__%ij_") MAN_END MAN_BEGIN ("Dissimilarity: To Distance...", "djmw", 20040407) INTRO ("A command that creates a @Distance object from a selected " "@Dissimilarity object.") ENTRY ("Settings") TAG ("%Scale") DEFINITION ("when on, the @@Dissimilarity: Get additive constant|" "additiveConstant@ is determined, when off the %additiveConstant = 0.") NORMAL ("dissimilarities are transformed to distances according to:") FORMULA (" %distance__%ij_ = %dissimilarity__%ij_ + %additiveConstant.") MAN_END MAN_BEGIN ("Dissimilarity: To Weight", "djmw", 19980108) INTRO ("Creates an object of type @Weight for each selected @Dissimilarity " "object.") NORMAL ("The values in the weight matrix will be:") LIST_ITEM ("%w__%ii_ = 0") LIST_ITEM ("%w__%ij_ = 1 if %\\de__%ij_ > 0") MAN_END MAN_BEGIN ("Dissimilarity & Configuration: Draw regression (absolute mds)...", "djmw", 20040407) INTRO ("Draws a scatterplot of the dissimilarities %\\de__%ij_ from the " "selected @Dissimilarity object versus @disparities %d\\'p__%ij_ obtained " "from the \"regression\" of distances %d__%ij_ " "from @Configuration on the dissimilarities %\\de__%ij_.") FORMULA ("%d\\'p__%ij_ = %\\de__%ij_") ENTRY ("Settings") TAG ("%%Minimum proximity%, %%Maximum proximity%") DEFINITION ("minimum and maximum values for the proximities (horizontal axis).") TAG ("%%Minimum distance%, %%Maximum distance%") DEFINITION ("minimum and maximum values for the distances (vertical axis).") TAG ("%%Mark size (mm)%, %%Mark string%") DEFINITION ("size and kind of the marks in the plot.") TAG ("%garnish") DEFINITION ("when on, draws a bounding box with decoration.") MAN_END MAN_BEGIN ("Dissimilarity & Configuration: Draw regression (interval mds)...", "djmw", 20040407) INTRO ("Draws a scatterplot of the dissimilarities %\\de__%ij_ from the " "selected @Dissimilarity versus @disparities %d\\'p__%ij_ obtained " "from the regression of distances %d__%ij_ " "from @Configuration on the dissimilarities %\\de__%ij_.") FORMULA ("%d\\'p__%ij_ = %a + %b \\.c %\\de__%ij_,") NORMAL ("where the values of %a and %b are determined by regression.") ENTRY ("Settings") TAG ("%%Minimum proximity%, %%Maximum proximity%") DEFINITION ("minimum and maximum values for the proximities (horizontal axis).") TAG ("%%Minimum distance%, %%Maximum distance%") DEFINITION ("minimum and maximum values for the distances (vertical axis).") TAG ("%%Mark size (mm)%, %%Mark string%") DEFINITION ("size and kind of the marks in the plot.") TAG ("%garnish") DEFINITION ("when on, draws a bounding box with decoration.") MAN_END MAN_BEGIN ("Dissimilarity & Configuration: Draw regression (i-spline mds)...", "djmw", 20040407) INTRO ("Draws a scatterplot of the dissimilarities %\\de__%ij_ from the " "selected @Dissimilarity versus @disparities %d\\'p__%ij_ obtained " "from the regression of distances %d__%ij_ from @Configuration on the " "@spline transformed dissimilarities %\\de__%ij_.") ENTRY ("Settings") TAG ("%%Number of interior knots") DEFINITION ("determines the number of segments.") TAG ("Order of I-spline") DEFINITION ("The order of the polynomial basis of the I-spline.") TAG ("%%Minimum proximity%, %%Maximum proximity%") DEFINITION ("minimum and maximum values for the proximities (horizontal axis).") TAG ("%%Minimum distance%, %%Maximum distance%") DEFINITION ("minimum and maximum values for the distances (vertical axis).") TAG ("%%Mark size (mm)%, %%Mark string%") DEFINITION ("size and kind of the marks in the plot.") TAG ("%garnish") DEFINITION ("when on, draws a bounding box with decoration.") MAN_END MAN_BEGIN ("Dissimilarity & Configuration: Draw regression (monotone mds)...", "djmw", 20040407) INTRO ("Draws a scatterplot of the dissimilarities %\\de__%ij_ from the " "selected @Dissimilarity versus @disparities %d\\'p__%ij_ obtained " "from the monotone regression of distances %d__%ij_ " "from @Configuration on the dissimilarities %\\de__%ij_.") ENTRY ("Settings") TAG ("%Primary or %secondary% approach to ties") DEFINITION ("When dissimilarities are equal, i.e., %\\de__%ij_ = %\\de__%kl_ " "the primary approach imposes no conditions on the corresponding distances " "%d__%ij_ and %d__%kl_, while the %secondary approach demands that also " "%d__%ij_ = %d__%kl_.") TAG ("%%Minimum proximity%, %%Maximum proximity%") DEFINITION ("minimum and maximum values for the proximities (horizontal axis).") TAG ("%%Minimum distance%, %%Maximum distance%") DEFINITION ("minimum and maximum values for the distances (vertical axis).") TAG ("%%Mark size (mm)%, %%Mark string%") DEFINITION ("size and kind of the marks in the plot.") TAG ("%garnish") DEFINITION ("when on, draws a bounding box with decoration.") MAN_END MAN_BEGIN ("Dissimilarity & Configuration: Draw regression (ratio mds)...", "djmw", 20040407) INTRO ("Draws a scatterplot of the dissimilarities %\\de__%ij_ from the " "selected @Dissimilarity versus @disparities %d\\'p__%ij_ obtained " "from the \"regression\" of distances %d__%ij_ " "from @Configuration on the dissimilarities %\\de__%ij_.") FORMULA ("%d\\'p__%ij_ = %b \\.c %\\de__%ij_,") NORMAL ("where the value of %b is determined by regression.") ENTRY ("Settings") TAG ("%%Minimum proximity%, %%Maximum proximity%") DEFINITION ("minimum and maximum values for the proximities (horizontal axis).") TAG ("%%Minimum distance%, %%Maximum distance%") DEFINITION ("minimum and maximum values for the distances (vertical axis).") TAG ("%%Mark size (mm)%, %%Mark string%") DEFINITION ("size and kind of the marks in the plot.") TAG ("%garnish") DEFINITION ("when on, draws a bounding box with decoration.") MAN_END MAN_BEGIN ("Dissimilarity & Configuration: Draw Shepard diagram...", "djmw", 20040407) INTRO ("Draws the Shepard diagram. This is a scatterplot of the " "dissimilarities from the @Dissimilarity object versus distances (as " "calculated from the @Configuration).") ENTRY ("Settings") TAG ("%%Minimum proximity%, %%Maximum proximity%") DEFINITION ("minimum and maximum values for the proximities (horizontal axis).") TAG ("%%Minimum distance%, %%Maximum distance%") DEFINITION ("minimum and maximum values for the distances (vertical axis).") TAG ("%%Mark size (mm)%, %%Mark string%") DEFINITION ("size and kind of the marks in the plot.") TAG ("%garnish") DEFINITION ("when on, draws a bounding box with decoration.") MAN_END MAN_BEGIN ("Dissimilarity & Configuration: Get stress (absolute mds)...", "djmw", 19980119) INTRO ("A command to obtain the @stress value for the selected @Dissimilarity " "and @Configuration object.") ENTRY ("Behaviour") NORMAL ("Stress formula's are #dependent of the scale of the Configuration: " "you will get #another stress value if you had pre-multiplied the " "selected Configuration with any number greater than zero.") MAN_END MAN_BEGIN ("Dissimilarity & Configuration: Get stress (interval mds)...", "djmw", 19980119) INTRO ("A command to obtain the @stress value for the selected @Dissimilarity " "and @Configuration object.") ENTRY ("Behaviour") NORMAL ("We use stress formula's that are independent of the scale of the " "Configuration: you would have got the same stress value if you had " "pre-multiplied the selected Configuration with any number greater " "than zero.") MAN_END MAN_BEGIN ("Dissimilarity & Configuration: Get stress (i-spline mds)...", "djmw", 19980119) INTRO ("A command to obtain the @stress value for the selected @Dissimilarity " "and @Configuration object.") ENTRY ("Behaviour") NORMAL ("We use stress formula's that are independent of the scale " "of the Configuration: you would have got the same stress value if " "you had pre-multiplied the selected Configuration with any number " "greater than zero.") MAN_END MAN_BEGIN ("Dissimilarity & Configuration: Get stress (monotone mds)...", "djmw", 19980119) INTRO ("A command to obtain the @stress value for the selected @Dissimilarity " "and @Configuration object.") ENTRY ("Behaviour") NORMAL ("We use stress formula's that are independent of the scale " "of the Configuration: you would have got the same stress value if " "you had pre-multiplied the selected Configuration with any number " "greater than zero.") MAN_END MAN_BEGIN ("Dissimilarity & Configuration: Get stress (ratio mds)...", "djmw", 19980119) INTRO ("A command to obtain the @stress value for the selected @Dissimilarity " "and @Configuration object.") ENTRY ("Behaviour") NORMAL ("We use stress formula's that are independent of the scale " "of the Configuration: you would have got the same stress value if " "you had pre-multiplied the selected Configuration with any number " "greater than zero.") MAN_END MAN_BEGIN ("Dissimilarity & Configuration: To Configuration (absolute mds)...", "djmw", 19980119) INTRO ("A command that creates a @Configuration object from a @Dissimilarity " "object. The selected Configuration object serves as a starting " "configuration for the minimization process.") MAN_END MAN_BEGIN ("Dissimilarity & Configuration: To Configuration (interval mds)...", "djmw", 19980119) INTRO ("A command that creates a @Configuration object from a @Dissimilarity " "object. The selected Configuration object serves as a starting " "configuration for the minimization process.") MAN_END MAN_BEGIN ("Dissimilarity & Configuration: To Configuration (i-spline mds)...", "djmw", 19980119) INTRO ("A command that creates a @Configuration object from a @Dissimilarity " "object. The selected Configuration object serves as a starting " "configuration for the minimization process.") MAN_END MAN_BEGIN ("Dissimilarity & Configuration: To Configuration (kruskal)...", "djmw", 19971201) INTRO ("A command to fit an optimal @Configuration for the selected " "@Dissimilarity object. The selected @Configuration will be used as the " "starting configuration in the kruskal analysis.") MAN_END MAN_BEGIN ("Dissimilarity & Configuration: To Configuration (monotone mds)...", "djmw", 19980119) INTRO ("A command that creates a @Configuration object from a @Dissimilarity " "object. The selected Configuration object serves as a starting " "configuration for the minimization process.") MAN_END MAN_BEGIN ("Dissimilarity & Configuration: To Configuration (ratio mds)...", "djmw", 19980119) INTRO ("A command that creates a @Configuration object from a @Dissimilarity " "object. The selected Configuration object serves as a starting " "configuration for the minimization process.") MAN_END MAN_BEGIN ("Dissimilarity & Configuration & Weight: Get stress...", "djmw", 20040407) INTRO ("A command that calculates the @stress between distances %d__%ij_ " "derived from the selected @Configuration object and @disparities " "%d\\'p__%ij_ derived from the selected @Dissimilarity object. " "With the selected @Weight object the evaluation of the influence " "of each dissimilarity on stress can be influenced.") ENTRY ("Settings") LIST_ITEM ("%%Normalized stress%, %%Kruskal's stress-1%, %%Kruskal's " "stress-2% or %Raw stress%") ENTRY ("Behaviour") NORMAL ("Except for %absolute %mds, we us stress formula's that are " "independent of the scale of the Configuration (see @stress): you would " "have got the same stress value if you had pre-multiplied the selected " "Configuration with any number greater than zero.") MAN_END MAN_BEGIN ("Dissimilarity & Configuration & Weight: To Configuration...", "djmw", 20040407) INTRO ("A command that creates a @Configuration object from a @Dissimilarity " "object. With the selected @Weight object the influence of each " "dissimilarity on @stress can be influenced. The selected Configuration " "object serves as a starting configuration for the minimization process.") ENTRY ("Settings") LIST_ITEM ("\\bu @@Dissimilarity: To Configuration (monotone mds)...") LIST_ITEM ("\\bu @@Dissimilarity: To Configuration (i-spline mds)...") LIST_ITEM ("\\bu @@Dissimilarity: To Configuration (interval mds)...") LIST_ITEM ("\\bu @@Dissimilarity: To Configuration (ratio mds)...") LIST_ITEM ("\\bu @@Dissimilarity: To Configuration (absolute mds)...") MAN_END MAN_BEGIN ("Dissimilarity & Weight: To Configuration...", "djmw", 20040407) INTRO ("A command that creates a @Configuration object from a @Dissimilarity " "object. With the selected @Weight object the influence of each " "dissimilarity on @stress can be influenced.") ENTRY ("Settings") NORMAL ("May be different and depend on the representation function, i.e. " "the scale of measurement.") LIST_ITEM ("\\bu @@Dissimilarity: To Configuration (monotone mds)...") LIST_ITEM ("\\bu @@Dissimilarity: To Configuration (i-spline mds)...") LIST_ITEM ("\\bu @@Dissimilarity: To Configuration (interval mds)...") LIST_ITEM ("\\bu @@Dissimilarity: To Configuration (ratio mds)...") LIST_ITEM ("\\bu @@Dissimilarity: To Configuration (absolute mds)...") MAN_END MAN_BEGIN ("Dissimilarity & Weight: To Configuration...", "djmw", 20040407) INTRO ("A command that creates a @Configuration object from a @Dissimilarity " "object. With the selected @Weight object the influence of each " "dissimilarity on @stress can be influenced.") ENTRY ("Settings") NORMAL ("May be different and depend on the representation function, i.e. the scale of measurement.") LIST_ITEM ("\\bu @@Dissimilarity: To Configuration (monotone mds)...") LIST_ITEM ("\\bu @@Dissimilarity: To Configuration (i-spline mds)...") LIST_ITEM ("\\bu @@Dissimilarity: To Configuration (interval mds)...") LIST_ITEM ("\\bu @@Dissimilarity: To Configuration (ratio mds)...") LIST_ITEM ("\\bu @@Dissimilarity: To Configuration (absolute mds)...") MAN_END MAN_BEGIN ("Distance", "djmw", 19971124) INTRO ("One of the @@types of objects@ in P\\s{RAAT}.") NORMAL ("An object of type Distance represents distances between objects in a metrical space.") NORMAL ("Creation") LIST_ITEM ("\\bu @@Confusion: To Dissimilarity (pdf)...") LIST_ITEM ("\\bu @@Dissimilarity: To Distance...") MAN_END MAN_BEGIN ("Distance: To Configuration (indscal)...", "djmw", 19971124) INTRO ("Perform an @@INDSCAL analysis@ on the selected object(s) of type @Distance that " "results in a @Configuration and a @Salience object.") MAN_END MAN_BEGIN ("Distance: To Configuration (ytl)...", "djmw", 19971124) INTRO ("A command that creates one @Configuration and one @Salience object " "from a collection of one or more @Distance objects.") NORMAL ("This Configuration and Salience object normally serve as starting points " "for an @@individual difference scaling@ such as an @@INDSCAL analysis@.") NORMAL ("The algorithm is ddescribedin @@Young, Takane & Lewyckyj (1978)@.") MAN_END MAN_BEGIN ("Distance: To ScalarProduct...", "djmw", 20040407) INTRO ("A command that creates a @ScalarProduct for each selected @Distance.") ENTRY ("Settings") LIST_ITEM ("%%Make sum of squares equal 1.0") DEFINITION ("when selected, the elements in the resulting matrix part will be scaled such that " "the sum of all the squared elements in the matrix equals 1.0.") ENTRY ("Algorithm") NORMAL ("ScalarProduct entries %b__%ij_ are created from distances %d__%ij_ bij double centering " "the matrix with elements \\--1/2 %d__%ij_^2, i.e.,") FORMULA ("%b__%ij_= \\--1/2(%d__%ij_^2 \\-- %d__%\\.cj_^2 \\-- %d__%i\\.c_^2 + %d__%\\.c\\.c_^2),") NORMAL ("where the dot (\\.c) means averaging over that dimension.") MAN_END MAN_BEGIN ("Distance & Configuration: Draw scatter diagram...", "djmw", 19971201) MAN_END MAN_BEGIN ("Distance & Configuration: Get VAF...", "djmw", 19971201) INTRO ("Calculates the \"%%variance accounted for%\" from the selected collection of @Distance objects and the " "selected @Configuration. The optimal @Salience necessary for the calculation " "will be inferred from the selected Distance and Configuration objects.") MAN_END MAN_BEGIN ("Distance & Configuration: To Configuration (indscal)...", "djmw", 19971201) INTRO ("Performs an @@INDSCAL analysis@ on the selected objects of type @Distance and calculates " "a @Configuration from them. Uses the selected Configuration object as the initial Configuration in " "the iteration process.") MAN_END MAN_BEGIN ("Distance & Configuration & Salience: Get VAF...", "djmw", 19971201) INTRO ("Calculates the \"variance accounted for\" from the selected collection " "of @Distance objects, the " "selected @Configuration and the selected #Salience.") MAN_END MAN_BEGIN ("Distance & Configuration & Salience: To Configuration " "(indscal)...", "djmw", 19971201) INTRO ("A command that creates a new @Configuration from the selected " "collection of @Distance objects, the selected @Configuration and the " "selected #Salience. The selected Configuration and Salience " "serve as start values for the @@INDSCAL analysis@.") MAN_END MAN_BEGIN ("individual difference scaling", "djmw", 19970502) INTRO ("The purpose of individual difference scaling is to represent objects, " "whose dissimilarities are given, as points in a metrical space. " "The distances in the space should be in accordance with the " "dissimilarities as well as is possible. Besides the configuration a " "@Salience matrix is calculated.") NORMAL ("The basic Euclidean model is:") LIST_ITEM ("%\\de__%ijk_ \\~~ (\\su__%s=1..%r_ %w__%ks_(%x__%is_ \\-- " "%x__%js_)^2)^^1/2^") NORMAL ("Here \\de__%ijk_ is the (known) dissimilarity between %objects %i and " "%j, as measured on %data %source %k. The %x's are the %coordinates of " "the objects in an %r-dimensional space and the %w's are weights or " "saliences. Because straight minimization of the expression above " "is difficult, one applies transformations on this expression. " "Squaring both sides gives the model:") LIST_ITEM ("%\\de^2__%ijk_ \\~~ \\su__%s=1..%r_ %w__%ks_(%x__%is_ \\-- " "%x__%js_)^2") NORMAL ("and the corresponding least squares loss function:") LIST_ITEM ("\\su__%k=1..%numberOfSources_ \\su__%i=1..%numberOfPoints_ " "\\su__%j=1..%numberOfPoints_ (%\\de^2__%ijk_ \\-- %d^2__%ijk_)^2") NORMAL ("This loss function is minimized in the (ratio scale option of the) " "#ALSCAL program of @@Takane, Young & de Leeuw (1976)@.") NORMAL ("The transformation used by @@Carroll & Chang (1970)@ in the INDSCAL " "model, transforms the data from each source into scalar products " "of vectors. For the dissimilarities:") LIST_ITEM ("%\\be__%ijk_ = \\--{ %\\de^2__%ijk_ \\-- %\\de^2__%i.%k_ \\-- " "%\\de^2__.%jk_ + %\\de^2__..%k_ } / 2,") NORMAL ("where dots replacing indices indicate averaging over the range of " "that index. In the same way for the distances:") LIST_ITEM ("%z__%ijk_ = \\--{ %d^2__%ijk_ \\-- %d^2__%i.%k_ \\-- %d" "^2__%.%jk_ + %d^2__%..%k_ } / 2.") LIST_ITEM ("%\\be__%ijk_ \\~~ %z__%ijk_ = \\su__%s=1..%numberOfDimensions_ " "%w__%ks_ %x__%is_ %x__%js_") NORMAL ("Translated into matrix algebra, the equation above translates to:") LIST_ITEM ("%B__%k_ \\~~ %Z__%k_ = %X %W__%k_ %X\\'p,") NORMAL ("where %X is a %numberOfPoints \\xx %numberOfDimensions configuration " "matrix, %W__%k_, a non-negative %numberOfDimensions \\xx " "%numberOfDimensions matrix with weights, and %B__%k_ " "the %k^^th^ slab of %\\be__%ijk_.") NORMAL ("This translates to the following INDSCAL loss function:") FORMULA ("%f(%X, %W__1_,..., %W__%numberOfSources_) = " "\\su__%k=1..%numberOfSources_ | %B__%k_ \\-- %X%W__%k_%X\\'p |^2") NORMAL ("") MAN_END MAN_BEGIN ("INDSCAL analysis", "djmw", 19970424) INTRO ("A method for @@individual difference scaling@ analysis in P\\s{RAAT}.") NORMAL ("An INDSCAL analysis can be performed on objects of type Distance.") NORMAL ("If you start with @Dissimilarity objects you first have to transform " "them to Distance objects.") LIST_ITEM ("\\bu @@Dissimilarity: To Distance...@") NORMAL ("If you start with a @Confusion you can use:") LIST_ITEM ("\\bu @@Confusion: To Dissimilarity (pdf)...@") ENTRY ("Examples") LIST_ITEM ("\\bu @@Distance: To Configuration (indscal)...@") DEFINITION ("Perform an INDSCAL analysis on one or more objects of type " "@Distance to calculate a @Configuration.") LIST_ITEM ("\\bu @@Distance & Configuration: To Configuration (indscal)...@") DEFINITION ("Perform an INDSCAL analysis on one or more objects of type " "@Distance to calculate a @Configuration. Use the selected Configuration " "object as the initial Configuration in the iteration process.") ENTRY ("Algorithm") NORMAL ("The function to be minimized in INDSCAL is the following:") FORMULA ("%f(%X, %W__1_,..., %W__%numberOfSources_) = " "\\su__%i=1..%numberOfSources_ |%S__%i_ \\-- %XW__%i_%X\\'p|^2") NORMAL ("where %X an unknown %numberOfPoints x %numberOfDimensions " "configuration matrix, the %W__%i_ are %numberOfSources unknown " "diagonal %numberOfDimensions x %numberOfDimensions matrices with weights, " "often called saliences, and the %S__%i_ are known symmetric " "matrices with scalar products of dimension %numberOfPoints x " "%numberOfPoints.") NORMAL ("In the absence of an algorithm that minimizes %f, @@Carroll & " "Chang (1970)@ resorted to the @CANDECOMP algorithm, which instead of the " "function given above minimizes the following function:") FORMULA ("%g(%X, %Y, %W__1_,..., %W__%numberOfSources_) = " "\\su__%i=1..%numberOfSources_ |%S__%i_ \\-- %XW__%i_%Y\\'p|^2.") NORMAL ("Carroll & Chang claimed that for most practical circumstances %X " "and %Y converge to matrices that will be columnwise proportional. " "However, INDSCAL does not only require symmetry of the solution, " "but also non-negativity of the weights. Both these aspects can not be " "guaranteed with the CANDECOMP algorithm.") NORMAL ("@@Ten Berge, Kiers & Krijnen (1993)@ describe an algorithm that " "automatically satisfies symmetry because it solves %f directly, and, " "also, can guarantee non-negativity of the weights. " "This algorithm proceeds as follows:") NORMAL ("Let #%x__%h_ be the %h-th column of %X. We then write the function %f above as:") FORMULA ("%f(#%x__%h_, %w__1%h_, ..., %w__%numberOfSources %h_) = \\su__%i=1.." "%numberOfSources_ |%S__%ih_ \\-- #%x__%h_%w__%ih_#%x\\'p__%h_|^2,") NORMAL ("with %S__%ih_ defined as:") FORMULA ("%S__%ih_ = (%S__%i_ - \\su__%j\\=/%h, %j=1..%numberOfDimensions_ " "#%x__%j_%w__%ij_#%x\\'p__%j_).") NORMAL ("Without loss of generality we may require that") FORMULA ("#%x\\'p__%h_#%x__%h_ = 1") NORMAL ("Minimizing %f over #%x__%h_ is equivalent to minimizing") FORMULA ("\\su__%i=1..%numberOfSources_ |%S__%ih_|^2 \\-- 2tr \\su " "%S__%ih_#%x__%h_%w__%ih_#%x\\'p__%h_ + \\su %w^2__%ih_") NORMAL ("This amounts to maximizing") FORMULA ("%g(#%x__%h_) = #%x\\'p__%h_(\\su %w__%ih_%S__%ih_)#%x__%h_") NORMAL ("subject to #%x\\'p__%h_#%x__%h_ = 1. The solution for #%x__%h_ is " "the dominant eigenvector of (\\su %w__%ih_%S__%ih_), " "which can be determined with the power method (see @@Golub & van Loan " "(1996)@). The optimal value " "for the %w__%ih_, given that all other parameters are fixed:") FORMULA ("%w__%ih_ = #%x\\'p__%h_%S__%ih_#%x__%h_") NORMAL ("In an alternating least squares procedure we may update columns of " "%X and the diagonals of the %W matrices in any sensible order.") MAN_END MAN_BEGIN ("Kruskal analysis", "djmw", 19971201) INTRO ("One of the @@MDS models@ in P\\s{RAAT}.") NORMAL ("You can perform a Kruskal-type multidimensional scaling only on " "objects of type @Dissimilarity. Objects of other types first have to " "be converted to objects of Dissimilarity type.") ENTRY ("Example") NORMAL ("Convert a @Dissimilarity object into a @Configuration object.") LIST_ITEM ("\\bu @@Dissimilarity: To Configuration (monotone mds)...@") DEFINITION ("choose appropriate parameters") LIST_ITEM ("\\bu @@Dissimilarity & Configuration: Get stress (monotone mds)...@") DEFINITION ("choose stress-1 to obtain the value for the @stress according " "to Kruskal.") ENTRY ("How to get started") NORMAL ("You can create an example @Dissimilarity object with the @@Create " "letter R example...@ button which you can find under the " "##Multidimensional scaling# option in the #New menu.") MAN_END MAN_BEGIN ("MDS models", "djmw", 19980108) INTRO ("Multidimensional scaling (MDS) models are defined by specifying " "how given @Dissimilarity data, %\\de__%ij_, are " "mapped into distances of an %m-dimensional MDS @Configuration %#X. " "The mapping is specified by a %%representation function%, %f : " "%\\de__%ij_ \\-> %d__%ij_(#X), which specifies how " "dissimilarities should be related to the distances. The MDS analysis " "tries to find the configuration (in a given dimensionality) whose " "distances satisfy %f as closely as possible. " "This closeness is quantified by a badness-of-fit measure which is often " "called @stress.") ENTRY ("Representation functions") NORMAL ("In the application of MDS we try to find a configuration #X such that " "the following relations are sasatisfieds well as possible:") FORMULA ("%f(%\\de__%ij_) \\~~ %d__%ij_(#X)") NORMAL ("The numbers that result from applying %f on %\\de__%ij_ are sometimes " "called @disparities %d\\d'__%ij_. In most applications of MDS, besides " "the configuration #X, also the function %f is not " "completely specified, i.e., the exact parameters of %f are unknown and " "must also be estimated during the analysis. If no particular %f can be " "derived from a theoretical model, one often restricts %f to a particular " "class of functions on the basis of the scale level of the dissimilarity " "data. If the disparities are related to the proximities by a specific " "parametric function we speak of %metric MDS otherwise we speak of " "%ordinal or %%non-metric% MDS.") LIST_ITEM ("\\bu %absolute mds: %d\\'p__%ij_ = \\de__%ij_") DEFINITION ("No parameters need to be estimated.") LIST_ITEM ("\\bu %ratio mds: %d\\'p__%ij_ = %b \\.c \\de__%ij_,") DEFINITION ("where the value of %b can be estimated by a linear regression of " "%d__%ij_ on %\\de__%ij_.") LIST_ITEM ("\\bu %interval mds: %d\\'p__%ij_ = %a + %b \\.c %\\de__%ij_,") DEFINITION ("where the values of %a and %b can be estimated by a linear " "regression of %d__%ij_ on %\\de__%ij_.") LIST_ITEM ("\\bu i-spline mds: %d\\'p__%ij_ = %i-spline(%\\de__%ij_),") DEFINITION ("where %i-spline(\\.c) is a smooth monotonically increasing " "@spline curve. The conceptual idea is that it is not possible to map " "all dissimilarities into disparities by one simple function.") LIST_ITEM ("\\bu %monotone mds: %d\\'p__%ij_ = %monotone(\\de__%ij_),") DEFINITION ("where %monotone(\\.c) is restricted to be a monotonic function " "that preserves the order of the dissimilarities:") FORMULA ("if %\\de__%ij_ < %\\de__%kl_, then %d__%ij_(#X) < %d__%kl_(#X)") DEFINITION ("If %\\de__%ij_ = %\\de__%kl_ and no particular constraint is involved for %d__%ij_(#X) " "and %d__%kl_(#X) this is referred to as the %%primary approach% to ties. The %%secondary " "approach% to ties requires that if %\\de__%ij_ = %\\de__%kl_, then also %d__%ij_(#X) = %d__%kl_(#X).") NORMAL ("More information on all aspects of multidimensional scaling can be found in: " "@@Borg & Groenen (1997)@ and @@Ramsay (1988)@.") NORMAL ("The most important types and the conversions between them are shown in the " "following figure.") PICTURE (6, 6, drawMDSClassRelations) MAN_END MAN_BEGIN ("Measurement levels", "djmw", 19970502) INTRO ("According to the measurement theory of @@Stevens (1951)@, there are four measurement levels, namely " "#Nominal, #Ordinal, #Interval and #Ratio. In the light of multidimensional scaling, the first " "two levels, Nominal and Ordinal, are often called %non-%metric. The last two are %metric.") MAN_END MAN_BEGIN ("Multidimensional scaling", "djmw", 20010409) INTRO ("This tutorial describes how you can use P\\s{RAAT} to " "perform ##M#ulti##D#imensional ##S#caling (MDS) analysis.") NORMAL ("MDS helps us to represent %dissimilarities between objects as " "%distances in a %%Euclidean space%. In effect, the more dissimilar two " "objects are, the larger the distance between the objects in the Euclidean " "space should be. The data types in P\\s{RAAT} that " "incorporate these notions are @Dissimilarity, @Distance and " "@Configuration.") NORMAL ("In essence, an MDS-analysis is performed when you select a " "Dissimilarity object and choose one of the ##To Configuration (xxx)...# " "commands to obtain a Configuration object. In the above, method (xxx) " "represents on of the possible @@MDS models|multidimensional scaling " "models@.") ENTRY ("MDS-analysis") NORMAL ("Let us first create a Dissimilarity object. You can for example " "@@Dissimilarity|create a Dissimilarity object from a file@. Here we " "will the use the Dissimilarity object from @@Create letter R example...|" "the letter R example@. We have chchosenhe default value (32.5) for the " "(uniform) noise range. Note that this may result in substantial " "distortions between the dissimilarities and the distances.") NORMAL ("Now you can do the following, for example:") NORMAL ("Select the Dissimilarity and choose @@Dissimilarity: To Configuration " "(monotone mds)...|To Configuration (monotone mds)...@, and you perform " "a @@Kruskal analysis|kruskal@-like multidimensional scaling which " "results in a new " "Configuration object. (This Configuration could subsequently be used as " "the starting Configuration for a new MDS-analysis!).") NORMAL ("Select the Configuration and choose @@Configuration: Draw...|Draw...@ " "and the following picture will result") PICTURE (4.0, 4.0, drawLetterRConfigurationExample2) NORMAL ("The following script summarizes:") CODE ("Create letter R example... 32.5") CODE ("select Dissimilarity R") CODE ("To Configuration (monotone mds)... 2 \"Primary approach\" 0.00001 50 1") CODE ("Draw... 1 2 -0.8 1.2 -0.8 0.7 yes") ENTRY ("Obtaining the stress value") NORMAL ("Select the Dissimilarity and the Configuration together and query for " "the @stress value with: " "@@Dissimilarity & Configuration: Get stress (monotone mds)...|" "Get stress (monotone mds)...@. ") NORMAL ("The following script summarizes:") CODE ("select Dissimilarity R") CODE ("plus Configuration R_monotone") CODE ("Get stress (monotone mds)... \"Primary approach\" \"Kruskals's " "stress-1\")") ENTRY ("The Shepard diagram") NORMAL ("Select the Dissimilarity and the Configuration together to " "@@Dissimilarity & Configuration: Draw Shepard diagram...|" "draw the Shepard diagram@.") PICTURE (4.0, 4.0, drawLetterRShepard) NORMAL ("The following script summarizes:") CODE ("select Dissimilarity R") CODE ("plus Configuration R_monotone") CODE ("Draw Shepard diagram...0 200 0 2.2 1 + yes") ENTRY ("The (monotone) regression") NORMAL ("Select the Dissimilarity and the Configuration together to " "@@Dissimilarity & Configuration: Draw regression (monotone mds)...|" "draw the monotone regression@ of distances on dissimilarities.") PICTURE (4.0, 4.0, drawLetterRRegression) NORMAL ("The following script summarizes:") CODE ("select Dissimilarity R") CODE ("plus Configuration R_monotone") CODE ("Draw monotone regresion... \"Primary approach\" 0 200 0 2.2 1 + yes") NORMAL ("When you enter %noiseRange = 0 in the form for the letter #R, perfect " "reconstruction is possible. The Shepard diagram then will show " "a perfectly smooth monotonically increasing function.") ENTRY ("Weighing the dissimilarities") NORMAL ("When you can't have equal confidence in all the number in the " "Dissimilarity object, you can give different weights to these numbers by " "associating a @Weight object with the Dissimilarity object. " "An easy way to do this is to select the Dissimilarity object and first " "choose @@Dissimilarity: To Weight|To Weight@. Then you might change the " "individual weights in the Weight object with the @@TableOfReal: Set " "value...| Set value...@ command (remember: make %w__%ij_ = %w__%ji_).") NORMAL ("The following script summarizes:") CODE ("select Dissimilarity R") CODE ("To Weight") CODE ("! Change [i][j] and [j][i] cells in the Weight object") CODE ("Set value... i j val") CODE ("Set value... j i val") CODE ("...") CODE ("! now we can do a weighed analysis.") CODE ("select Dissimilarity R") CODE ("plus Weight R") CODE ("To Configuration (monotone mds)... 2 \"Primary approach\" 0.00001 50 1") NORMAL ("You can also query the @stress values with three objects selected. " "The following script summarizes:") CODE ("select Dissimilarity R") CODE ("plus Weight R") CODE ("plus Configuration R_s_monotone") CODE ("Get stress (monotone mds)... \"Primary approach\" \"Kruskals's " "stress-1\")") ENTRY ("Using a start Configuration") NORMAL ("You could also use a Configuration object as a starting " "configuration in the minimization process. " "Lets assume that you are not satisfied with the stress value from the " "Configuration object that you obtained in the previous analysis. " "You can than use this Configuration object as a " "starting point for further analysis:") NORMAL ("The following script summarizes:") CODE ("select Dissimilarity R") CODE ("plus Configuration R_monotone") CODE ("plus Weight R") CODE ("To Configuration (monotone mds)... 2 \"Primary approach\" 0.00001 50 1") ENTRY ("Multiple Dissimilarity's (INDSCAL)") NORMAL ("When you have multiple Dissimilarity objects you can also perform " "@@individual difference scaling@ (often called @@INDSCAL analysis@). ") NORMAL ("As an example we can use an @@Create INDSCAL Carroll & Wish " "example...| example taken from Carrol & Wish@. " "Because INDSCAL only works on metrical data, we can not use Dissimilarity " "objects directly. We have to transform them first @@Dissimilarity: To " "Distance...|to Distance@ objects.") NORMAL ("This type of analysis on multiple objects results in two new objects: " "a Configuration and a @Salience.") MAN_END MAN_BEGIN ("Procrustes", "djmw", 20010927) INTRO ("One of the @@types of objects@ in P\\s{RAAT}.") NORMAL ("An object of type Procrustes represents the special @@AffineTransform|" "affine transform@ that consists of a " "combination of a translation, a shape preserving transformation and a scaling (this scaling is often called %dilation). " "Because the transformation has to be shape preserving, only a combination of a rotation and a reflection is allowed. " "A configuration matrix #%X is transformed in the following way to a new configuration matrix #%Y: ") FORMULA ("#%Y = %s #%X #%T+ #1#%t',") NORMAL ("where %s is the scaling factor, #%T is the shape preserving transformation matrix, #%t is the translation vector, " "and #1 is the vector with only ones as its elements.") NORMAL ("For more information about the Procrustes transform and its algorithm " "see chapter 19 in @@Borg & Groenen (1997)@.") MAN_END MAN_BEGIN ("Procrustes transform", "djmw", 19980119) INTRO ("A transformation that only uses a combination of a translation, " "a scaling and a rigid transformation to transform one Configuration such that it " "matches as closely as possible another Configuration. ") NORMAL ("We speak of %%orthogonal Procrustes transform% when only the rigid " "transformation is allowed but no scaling or translation.") NORMAL ("For more information about the Procrustes transform and its algorithm " "see chapter 19 in @@Borg & Groenen (1997)@.") MAN_END MAN_BEGIN ("Proximity", "djmw", 19961008) INTRO ("One of the @@types of objects@ in P\\s{RAAT}.") NORMAL ("An object of type #Proximity represents proximities between objects.") ENTRY ("Inside a Proximity") NORMAL ("With @Inspect you will see the following attributes:") TAG ("%numberOfRows, %numberOfColumns") DEFINITION ("the number of objects (%numberOfRows and %numberOfColumns are " "equal and \\>_1).") TAG ("%rowLabels, %columnLabels") DEFINITION ("the names associated with the objects (%rowLabels and " "%columnLabels are equal.") TAG ("%data [1..%numberOfRows][1..%numberOfColumns]") DEFINITION ("the proximities between the objects.") MAN_END MAN_BEGIN ("Salience", "djmw", 19980112) INTRO ("One of the @@types of objects@ in P\\s{RAAT}.") NORMAL ("Elements %s__%ij_ in the " "Salience matrix represent the importance of dimension %j (in the " "@Configuration) for data source %i.") ENTRY ("Commands") NORMAL ("Creation, as a by-product of:") LIST_ITEM ("\\bu @@Distance: To Configuration (indscal)...") LIST_ITEM ("\\bu @@Distance: To Configuration (ytl)...") MAN_END MAN_BEGIN ("ScalarProduct", "djmw", 19980125) INTRO ("One of the @@types of objects@ in P\\s{RAAT}.") NORMAL ("An object of type ScalarProduct represents scalar products %b__%ij_ " "between objects %i and %j in a metrical space.") FORMULA ("%b__%ij_ = \\su__%k=1..%numberOfDimensions_ %x__%ik_%x__%jk_,") NORMAL ("where %x__%ik_ and %x__%jk_ are the coordinates of the %k-th " "dimension of points %i and %j, respectively. From this definition one " "can see that scalar products, in contrast to distances, " "do change when the origin is shifted.") NORMAL ("Creation") LIST_ITEM ("\\bu @@Distance: To ScalarProduct...@") MAN_END MAN_BEGIN ("Similarity", "djmw", 19961008) INTRO ("One of the @@types of objects@ in P\\s{RAAT}.") NORMAL ("An object of type Similarity represent a one-way " "table of similarities between \"objects\".") ENTRY ("Commands") NORMAL ("Creation") LIST_ITEM ("\\bu @@Confusion: To Similarity...") 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)...") 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 @@Similarity: To Dissimilarity...") MAN_END MAN_BEGIN ("Similarity: To Dissimilarity...", "djmw", 20040407) INTRO ("A command that creates a @Dissimilarity from every selected " "@Similarity.") ENTRY ("Settings") TAG ("%%Maximum dissimilarity") DEFINITION ("determines the maximum dissimilarity possible. When the default " "value, 0.0, is chchosen%maximumDissimilarity " "is calculated as the maximum element in the Similarity object.") ENTRY ("Algorithm") NORMAL ("To obtain dissimilarities we 'reverse' similarities:") FORMULA ("%%dissimilarity__%ij_ = %maximumDissimilarity \\-- %similarity__%ij_") NORMAL ("In this way the order of dissimilarities is the reverse of the order " "of the similarities.") MAN_END MAN_BEGIN ("smacof", "djmw", 19980119) INTRO ("Scaling by Majorizing a Complicated Function, the iterative algorithm " "to find an optimal Configuration.") LIST_ITEM ("1. Initialize") LIST_ITEM (" 1.a. Get initial Configuration #Z") LIST_ITEM (" 1.b. Set stress %\\si__%n_^^[0]^ to a very large value.") LIST_ITEM (" 1.c. Set iteration counter %k = 0") LIST_ITEM ("2. Increase iteration counter by one: %k = %k + 1") LIST_ITEM ("3. Calculate distances %d__%ij_(#Z).") LIST_ITEM ("4. Transform dissimilarities %\\de__%ij_ into disparities " "%d\\'p__%ij_.") LIST_ITEM ("5. Standardize the disparities so that %\\et__%d\\'p_^2 = " "%n(%n\\--1)/2.") LIST_ITEM ("6. Compute the Guttman transform #X^^[%k]^ of #Z.") LIST_ITEM ("7. Compute new distances %d__%ij_(#X^^[%k]^).") LIST_ITEM ("8. Compute normalized stress %\\si__%n_ (#d\\'p, #X^^[%k]^)") LIST_ITEM ("9. If |%\\si__%n_^^[%k]^ \\-- %\\si__%n_^^[%k\\--1]^| / %\\si__%n_" "^^[%k\\--1]^ < %\\ep or %k > %maximumNumberOfIterations, then stop") LIST_ITEM ("10. Set #Z = #X^^[%k]^, and go to 2.") NORMAL ("This algorithm goes back to @@De Leeuw (1977)@.") MAN_END MAN_BEGIN ("spline", "djmw", 19980112) INTRO ("A spline function %f is a piecewise polynomial function defined on " "an interval [%x__%min_, %x__%max_] " "with specified continuity constraints, i.e., when the interval [%x__%min_," " %x__%max_] is subdivided by points \\xi__%i_ such that %x__%min_ = " "\\xi__%1_ < ... < \\xi__%q_ = %%x__%max_, then within each subinterval " "[\\xi__%j_, \\xi__%j+1_) the function is a polynomial %P__%j_ of " "specified order %k.") NORMAL ("A %%knot sequence% %t = {%t__1_, ..., %t__%n+%k_}, where %n is the " "number of free parameters that specify the spline function, is derived " "from the \\xi__%i_ by placing knots at the boundary values \\xi__%i_ " "according to the order of continuity at that boundary. The most common " "continuity chcharacteristicsmposed on %f request that for adjacent " "polynomials the derivatives up to order %k\\--2 match. For example, " "the knot sequence of a spline of order %k for a partition of [%x__%min_," " %%x__%max_] into three intervals (%q = 4) will be %t__1_ = ... = " "%t__%k_ = %x__%min_ (=\\xi__1_), %t__%k+1_ = \\xi__2_, %t__%k+2_ = " "\\xi__3_ , %t__%k+3_ = ... = %t__%k+%k+2_ = %x__%max_ (= \\xi__4_). " "This is called a %simple knot sequence, because all interior knots are " "simple. The number of free parameters %n for this case obeys a simple " "formula:") FORMULA ("%n = %numberOfInteriorKnots + %order.") NORMAL ("With suitable basis functions, for example, the M-spline family " "%M__%i_(%x|%k, %t), %i=1..%n, we can write any spline %f in the form:") FORMULA ("%f = \\su__%i=1..%n_ %a__%i_%M__%i_,") NORMAL ("where the %M__%i_ are defined by the following recursive formulas:") FORMULA ("%M__%i_(%x|1,%t) = 1 / (%t__%i+1_ \\-- %t__%i_), %t__%i_ " "\\<_ %x < %t__%i+1_, 0 otherwise") FORMULA ("%M__%i_(%x|%k,%t) = %k [(%x\\--%t__%i_)%M__%i_(%x|%k\\--1,%t) + " "(%t__%i+%k_\\--%x)%M__%i+1_(%x|%k\\--1,%t)] / " "((%k\\--1)(%t__%i+%k_\\--%t__%i_))") NORMAL ("These %M__%i_ are localized because %M__%i_(%x|%k,%t) > 0 only when " "%t__%i_ \\<_ %x < %t__%i+%k_ and zero otherwise. Also, we have \\in" " M__%i_(%x)%dx = 1. Because of this localization a change in coefficient " "%a__%i_ will only effect %f within this interval.") NORMAL ("The following picture shows an M-spline of order 3 on the interval " "[0, 1], with three interior knots at 0.3, 0.5 and 0.6.") PICTURE (5.0, 5.0, drawMsplineExample) NORMAL ("Because the M-splines are nonnegative, %monotone splines% can be " "derived from them by %integration:") FORMULA ("%I__%i_(%x|%k,%t) = \\in__%xmin_^%x %M__%i_(%u|%k,%t) %du") NORMAL ("Because each %M__%i_(%x|%k, %t) is a piecewise polynomial of degree" " %k\\--1, each %I__%i_ will be of degree %k. Now we can write:") FORMULA ("%f = \\su__%i=1..%n_ %b__%i_%I__%i_(%x|%k,%t)") NORMAL ("We can use an M-spline of order %k+1 with a simple knot sequence %t, " "for which %t__%j_ \\<_ x < %t__%j+1_, to put " "the I-spline of order %k into a more convenient form:") FORMULA ("%I__%i_(%x|%k,%t) = 0, %i > %j") FORMULA ("%I__%i_(%x|%k,%t) = \\su__%m=%i+1..%j_ (%t__%m+%k+1_\\--" "%t__%m_)%M__%m_(%x|%k+1,%t)/(%k+1), %j\\--%k \\<_ %i \\<_ %j") FORMULA ("%I__%i_(%x|%k,%t) = 1, %i < %j\\--%k") NORMAL ("The following figure shows the I-splines that were derived from " "the M-splines above.") PICTURE (5.0, 5.0, drawIsplineExample) NORMAL ("These spline formula's were taken from @@Ramsay (1988)@ and the " "errors in his I-spline formulas were corrected.") MAN_END MAN_BEGIN ("stress", "djmw", 19980108) INTRO ("A badness-of-fit measure for the entire MDS representation.") NORMAL ("Several measures exist.") ENTRY ("Raw stress") FORMULA ("%\\si__%r_ (#d\\'p, #X) = \\su__%i<%j_ %w__%ij_(%d\\'p__%ij_ \\-- " "%d__%ij_(#X))^2") FORMULA ("= \\su__%i<%j_ %w__%ij_%d\\'p__%ij_^2 + \\su__%i<%j_ " "%w__%ij_%d__%ij_^2(#X) \\-- " "2 \\su__%i<%j_ %w__%ij_%d\\'p__%ij_%d__%ij_(#X)") FORMULA ("= %\\et__%d\\'p_^2 + %\\et^2(#X) \\-- 2%\\ro(#d\\'p, #X)") NORMAL ("where the %d\\'p__%ij_ are the @disparities that are the result " "from the transformation of the dissimilarities, i.e., %f(%\\de__%ij_). " "Raw stress can be misleading because it is dependent on the normalization " "of the disparities. The following measure tries to circumvent this " "inconvenience.") ENTRY ("Normalized stress") FORMULA ("%\\si__%n_ = \\si__%r_ / %\\et__%d\\'p_^2") NORMAL ("This is the stress function that we minimize by iterative " "majorization. It goes back to @@De Leeuw (1977)@.") ENTRY ("Kruskal's stress-1") FORMULA ("%\\si__1_ = \\Vr (\\su__%i<%j_ %w__%ij_(%d\\'p__%ij_ \\-- " "%d__%ij_(#X))^2 / \\su__%i<%j_ %w__%ij_%d__%ij_^2(#X))^^1/2^") NORMAL ("In this measure, which is due to @@Kruskal (1964)@, stress is " "expressed in relation to the size of #X.") ENTRY ("Kruskal's stress-2") FORMULA ("%\\si__2_ = \\Vr (\\su__%i<%j_ %w__%ij_(%d\\'p__%ij_ \\-- " "%d__%ij_(#X))^2 / \\su__%i<%j_ %w__%ij_(%d__%ij_(#X) - " "%averageDistance)^2)^^1/2^.") NORMAL ("In general, this measure results in a stress value that is " "approximately twice the value for stress-1.") ENTRY ("Relation between %\\si__1_ and %\\si__n_") NORMAL ("When we have calculated %\\si__%n_ for Configuration #X, " "disparities #d\\'p and Weight #W we cannot " "directly use #X, #d\\'p and #W to calculate %\\si__1_ because the " "scale of #X is not necessarily optimal " "for %\\si__1_. We allow therefore a scale factor %b > 0 and " "try to calculate %\\si__1_ (#d\\'p, %b #X). We minimize the resulting " "expression for %b and substitute " "the result back into the formula for stress, i.e.,") FORMULA ("%\\si__1_^2 (#d\\'p, %b #X) = (%\\et__%d\\'p_^2 + %b^2 %\\et^2(#X) " "\\-- 2 %b %\\ro(#d\\'p, #X)) / %b^2 %\\et^2(#X)") FORMULA ("d%\\si__1_^2 (%b) / d%b == 0, gives") FORMULA ("%b = %\\et__%d\\'p_^2 / %\\ro") FORMULA ("%\\si__1_^2 = (1 - %\\ro^2 / (%\\et__%d\\'p_^2\\.c%\\et^2(#X)))") NORMAL ("This means that %\\si__1_ = \\Vr %\\si__%n_.") ENTRY ("Relation between %\\si__2_ and %\\si__n_") NORMAL ("We can do the same trick as before for %\\si__2_:") FORMULA ("%\\si__2_^2 (#d\\'p, %b #X) = (%\\et__%d\\'p_^2 + %b^2 %\\et^2(#X) " "\\-- 2 %b %\\ro(#d\\'p, #X)) / " "(%b^2 \\su__%i<%j_ %w__%ij_(%d__%ij_(#X) - %averageDistance)^2)") NORMAL ("From which we derive:") FORMULA ("%\\si__2_ = \\Vr ((%\\et__%d\\'p_^2 \\.c %\\et^2(#X) - " "%\\ro^2(#d\\'p, #X)) / (%\\et__%d\\'p_^2 \\.c \\su__%i<%j_ " "%w__%ij_(%d__%ij_(#X) - %averageDistance)^2))") MAN_END MAN_BEGIN ("TableOfReal: Centre columns", "djmw", 19980422) INTRO ("A command that centres the columns in the selected @TableOfReal " "objects.") ENTRY ("Algorithm") NORMAL ("The new values in the table, %x\\'p__%ij_, will be:") FORMULA ("%x\\'p__%ij_ = %x__%ij_ \\-- %x__\\.c%j_,") NORMAL ("where") FORMULA ("%x__\\.c%j_ = \\su__%i=1..%numberOfRows_ %x__%ij_ / %numberOfRows,") NORMAL ("the average of column %j.") MAN_END MAN_BEGIN ("TableOfReal: Centre rows", "djmw", 19980422) INTRO ("A command that centres the rows in the selected @TableOfReal objects.") ENTRY ("Algorithm") NORMAL ("The new values in the table, %x\\'p__%ij_, will be:") FORMULA ("%x\\'p__%ij_ = %x__%ij_ \\-- %x__%i\\.c_,") NORMAL ("where") FORMULA ("%x__%i\\.c_ = \\su__%j=1..%numberOfColumns_ %x__%ij_ / " "%numberOfColumns,") NORMAL ("the average of row %i.") MAN_END MAN_BEGIN ("TableOfReal: Get table norm", "djmw", 19980422) INTRO ("A command that returns the norm of the selected @TableOfReal object.") ENTRY ("Algorithm") NORMAL ("Returns: sqrt (\\su__%i=1..%numberOfRows_ \\su__%j=1..%numberOfColumns" "_ %x__%ij_^2).") MAN_END MAN_BEGIN ("TableOfReal: Normalize columns...", "djmw", 19980422) INTRO ("A command that normalizes the columns in the selected @TableOfReal " "objects.") ENTRY ("Argument") TAG ("%Norm") DEFINITION ("determines the sum of the squared elements in each column after " "normalization.") ENTRY ("Algorithm") NORMAL ("All elements %x__%ij_ in each column %j=1..%numberOfColumns will be " "multiplied by sqrt (%norm / \\su__%i=1..%numberOfRows_ %x__%ij_^2).") MAN_END MAN_BEGIN ("TableOfReal: Normalize rows...", "djmw", 19980422) INTRO ("A command that normalizes the rows in the selected @TableOfReal " "objects.") ENTRY ("Argument") TAG ("%Norm") DEFINITION ("determines the sum of the squared elements in each row after " "normalization.") ENTRY ("Algorithm") NORMAL ("All elements %x__%ij_ in each row %i=1..%numberOfRows will be " "multiplied by sqrt (%norm / \\su__%j=1..%numberOfColumns_ %x__%ij_^2).") MAN_END MAN_BEGIN ("TableOfReal: Normalize table...", "djmw", 19980422) INTRO ("A command that normalizes the elements in the selected @TableOfReal " "objects.") ENTRY ("Argument") TAG ("%Norm") DEFINITION ("determines the sum of the squared elements after normalization.") ENTRY ("Algorithm") NORMAL ("All elements %x__%ij_ will be multiplied by " "sqrt (%norm / \\su__%i=1..%numberOfRows_ \\su__%j=1..%numberOfColumns_" " %x__%ij_^2.") MAN_END MAN_BEGIN ("Weight", "djmw", 19980108) INTRO ("One of the @@types of objects@ in P\\s{RAAT}.") NORMAL ("An object of type Weight represents a matrix " "with weights %w__%ij_.") NORMAL ("An object of type Weight selected together with an onject of type" " @Dissimilarity can be used to make distinctions in the importance of the" " contribution of each individual dissimilarity %\\de__%ij_ to " "@stress and therefore to the final configuration. Weights can be " "used for instance to code for missing values, i.e., take %w__%ij_ = 0 if " "dissimilarity %\\de__%ij_ is missing and %w__%ij_ = 1 if %\\de__%ij_ is" " known.") ENTRY ("Commands") NORMAL ("Creation") LIST_ITEM ("\\bu @@Dissimilarity: To Weight") NORMAL ("Analysis") NORMAL ("See @@Dissimilarity & Weight: To Configuration...@ for help on the " "following analysis items: ") LIST_ITEM ("\\bu ##Dissimilarity & Weight: To Configuration (monotone mds)...") LIST_ITEM ("\\bu ##Dissimilarity & Weight: To Configuration (i-spline mds)...") LIST_ITEM ("\\bu ##Dissimilarity & Weight: To Configuration (interval mds)...") LIST_ITEM ("\\bu ##Dissimilarity & Weight: To Configuration (ratio mds)...") LIST_ITEM ("\\bu ##Dissimilarity & Weight: To Configuration (absolute mds)...") NORMAL ("Query") NORMAL ("See @@Dissimilarity & Configuration & Weight: Get stress...@ for help " "on the following query items: ") LIST_ITEM ("\\bu ##Dissimilarity & Configuration & Weight: Get stress " "(monotone mds)...") LIST_ITEM ("\\bu ##Dissimilarity & Configuration & Weight: Get stress " "(i-spline mds)...") LIST_ITEM ("\\bu ##Dissimilarity & Configuration & Weight: Get stress " "(interval mds)...") LIST_ITEM ("\\bu ##Dissimilarity & Configuration & Weight: Get stress " "(ratio mds)...") LIST_ITEM ("\\bu ##Dissimilarity & Configuration & Weight: Get stress " "(absolute mds)...") MAN_END /************ references ***********************************************/ MAN_BEGIN ("Abramowitz & Stegun (1970)", "djmw", 19971201) NORMAL ("M. Abramowitz & I. Stegun (1970), %%Handbook of Mathematical " "Functions%, Dover Publications, Inc., New York.") MAN_END MAN_BEGIN ("Borg & Groenen (1997)", "djmw", 19971219) NORMAL ("I. Borg & P. Groenen (1997), %%Modern Multidimensional Scaling: " "Theory and Applications%, Springer.") MAN_END MAN_BEGIN ("Brokken (1983)", "djmw", 19980406) NORMAL (" F.B. Brokken (1983), \"Orthogonal Procrustes rotation maximizing " "congruence\", %Psychometrika #48, 343-352.") MAN_END MAN_BEGIN ("Cailliez (1983)", "djmw", 19971201) NORMAL (" F. Cailliez (1983), \"The analytical solution of the additive " "constant problem\", %Psychometrika #48, 305-308.") MAN_END MAN_BEGIN ("Carroll & Chang (1970)", "djmw", 19971201) NORMAL ("J.D. Carroll & J.-J. Chang, (1970): \"Analysis of Individual " "Differences in Multidimensional scaling via an N-way generalization of " "\"Eckart-Young\" Decomposition\", %Psychometrika #35, 283-319.") MAN_END MAN_BEGIN ("Carroll & Wish (1974)", "djmw", 19971201) NORMAL ("J.D. Carroll & M. Wish, (1974): \"Models and methods for three-way " "multidimensional scaling\", in D.H. Krantz, R.C. Atkinson, R.D. Luce & " "P. Suppes (Eds.), %%Contemporary developments in mathematical psychology:" " Vol. 2 Measurement, psychophysics, and neural " "information processing%, 283-319, New York, Academic Press.") MAN_END MAN_BEGIN ("De Leeuw (1977)", "djmw", 19971201) NORMAL ("J. de Leeuw (1977), \"Applications of convex analysis to " "multidimensional scaling\", In J.R. Barra, F. Brodeau, G. Romier & " "B. van Cutsem (eds.), %% Recent developments in statistics%, Amsterdam, " "The Netherlands: North-Holland, 133-145.") MAN_END MAN_BEGIN ("De Leeuw & Pruzansky (1978)", "djmw", 19971201) NORMAL ("J. de Leeuw & S. Pruzansky (1978), \"A new computational method to " "fit the weighted Euclidean distance model\", %Psychometrika , 479-490.") MAN_END MAN_BEGIN ("Gifi (1990)", "djmw", 19971207) NORMAL ("A. Gifi (1990), %%Nonlinear Multivariate Analysis%, John Wiley & " "Sons Ltd., reprint 1996") MAN_END MAN_BEGIN ("Golub & van Loan (1996)", "djmw", 19971207) NORMAL ("G. Golub & C. van Loan (1996), %%Matrix computations%, third edition, " "The Johns Hopkins University Press, London") MAN_END MAN_BEGIN ("Green, Carmone & Smith (1989)", "djmw", 19971201) NORMAL ("P. Green, F. Carmone, S. Smith (1989): " "%%Multidimensional scaling: concepts and applications%, section 3. " "Allyn and Bacon.") MAN_END MAN_BEGIN ("Kaiser (1958)", "djmw", 19980404) NORMAL (" H.F. Kaiser (1958), \"The varimax criterion for analytic rotation " "in factor analysis\", %Psychometrika #23, 187-200.") MAN_END MAN_BEGIN ("Kiers & Groenen (1996)", "djmw", 19971219) NORMAL ("H.A.L. Kiers & P. Groenen (1996), \"A monotonically convergent " "algorithm for orthogonal congruence rotation\", %Psychometrika #61, " "375-389.") MAN_END MAN_BEGIN ("Klein, Plomp & Pols (1970)", "djmw", 19971201) NORMAL (" W. Klein, R. Plomp, & L.C.W. Pols (1970), \"Vowel Spectra, " "Vowel Spaces, and Vowel Identification\", JASA 48, 999-1009.") MAN_END MAN_BEGIN ("Kruskal (1964)", "djmw", 19971201) NORMAL ("J.B. Kruskal (1964), \"Nonmetric multidimensional scaling: a " "numerical method\", %Psychometrika #29, 115-129.") MAN_END MAN_BEGIN ("Ramsay (1988)", "djmw", 19980106) NORMAL ("J.O. Ramsay (1988), \"Monotone Regression Splines in Action\", " "%%Statistical Science% #3, 425-461.") MAN_END MAN_BEGIN ("Stevens (1951)", "djmw", 19971201) NORMAL ("S.S. Stevens (1951), \"Mathematics, measurement, and psychophysics\"," " in S.S. Stevens (Ed.), %%Handbook of experimental Psychology%, New York:" " Wiley.") MAN_END MAN_BEGIN ("Takane, Young & de Leeuw (1976)", "djmw", 19971201) NORMAL ("Y. Takane, F. Young, J. de Leeuw (1976), \"Non-metric individual " "differences multidimensional scaling: an alternating least squares method " "with optimal scaling features\", %Psychometrika #42, 7-67.") MAN_END MAN_BEGIN ("Ten Berge (1995)", "djmw", 19980404) NORMAL ("J.M.F. ten Berge (1995), \"Suppressing permutations or rigid planar " "rotations: A remedy against nonoptimal varimax rotations\", " "%Psychometrika #60, 437-446.") MAN_END MAN_BEGIN ("Ten Berge, Kiers & Krijnen (1993)", "djmw", 19971207) NORMAL ("J.M.F. ten Berge, H.A.L. Kiers & W.P. Krijnen (1993), \"Computational " "Solutions for the Problem of Negative Saliences and Nonsymmetry in " "INDSCAL\", %Journal of Classification #10, 115-124.") MAN_END MAN_BEGIN ("Torgerson (1958)", "djmw", 19971201) NORMAL ("W.S. Torgerson (1958), %%Theory and methods of scaling%, " "New York: Wiley.") MAN_END MAN_BEGIN ("Young, Takane & Lewyckyj (1978)", "djmw", 19971201) NORMAL ("F.W. Young, Y. Takane & R. Lewyckyj (1978), " "\"Three notes on ALSCAL\", %Psychometrika #43, 433-435.") MAN_END MAN_BEGIN ("Weller & Romney (1990)", "djmw", 19971216) NORMAL ("S.C. Weller & A.K. Romney (1990) %%Metric Scaling: " "Correspondence Analysis%, Sage University Paper Series on Quantitative " "Applications in the Social Sciences 07-075, Newbury Park, CA: Sage.") MAN_END } /* End of file manual_MDS.c */