
A numeric vector is an array of numbers, regarded as a single object. For instance, the squares of the first five integers can be collected in the vector { 1, 4, 9, 16, 25 }. In a Praat script, you can put a vector into a variable whose name ends in a number sign ("#"):
squares# = { 1, 4, 9, 16, 25 }
After this, the variable squares# contains the value { 1, 4, 9, 16, 25 }. We say that the vector squares# has five dimensions, i.e. it contains five numbers.
Whereas in Scripting 3.2. Numeric variables we talked about a numeric variable as being analogous to a house where somebody (the numeric value) could live, a numeric vector with five dimensions can be seen as a street that contains five houses, which are numbered with the indexes 1, 2, 3, 4 and 5, each house containing a numeric value. Thus, the street squares# contains the following five houses: squares# [1], squares# [2], squares# [3], squares# [4] and squares# [5]. Their values (the numbers that currently live in these houses) are 1, 4, 9, 16 and 25, respectively.
To list the five values with a loop, you could do:
writeInfoLine: "Some squares:"
for i from 1 to size (squares#)
appendInfoLine: "The square of ", i, " is ", squares# [i]
endfor
Instead of the above procedure to get the vector squares#, with a precomputed list of five squares, you could compute the five values with a formula, as in the example of Scripting 5.6. Arrays and dictionaries. However, in order to put a value into an element of the vector, you have to create the vector first (i.e., you have to build the whole street before you can put something in a house), so we start by creating a vector with five zeroes in it:
squares# = zero# (5)
After this, squares# is the vector { 0, 0, 0, 0, 0 }, i.e., the value of each element is zero. Now that the vector (street) exists, we can put values into (populate) the five elements (houses):
for i from 1 to size (squares#)
squares# [i] = i * i
endfor
After this, the variable squares#
has the value { 1, 4, 9, 16, 25 }, as before, but now we had the computer compute the squares.
You can create a vector in many ways. The first way we saw was with a vector literal, i.e. a series of numbers (or numeric formulas) between braces:
lengths# = { 1.83, 1.795, 1.76 }
The second way we saw was to create a series of zeroes. To create a vector consisting of 10,000 zeroes, you do
zero# (10000)
Another important type of vector is a series of random numbers. To create a vector consisting of 10,000 values drawn from a Gaussian distribution with true mean 0.0 and true standard deviation 1.0, you could do
noise# = randomGauss# (10000, 0.0, 1.0)
To create a vector consisting of 10,000 values drawn from a uniform distribution of real numbers with true minimum 0.0 and true maximum 1.0, you use
randomUniform# (10000, 0.0, 1.0)
To create a vector consisting of 10,000 values drawn from a uniform distribution of integer numbers with true minimum 1 and true maximum 10, you use
randomInteger# (10000, 1, 10)
Vectors can also be created by some menu commands. For instance, to get vectors representing the times and pitch frequencies of the frames in a Pitch object, you can do
selectObject: myPitch
times# = List all frame times
pitches# = List values in all frames: "Hertz"
For the vector defined above, you can compute the sum of the five values as
sum (squares#)
which gives 55. You compute the average of the five values as
mean (squares#)
which gives 11. You compute the standard deviation of the values as
stdev (squares#)
which gives 9.669539802906858 (the standard deviation is undefined for vectors with fewer than 2 elements). The center of gravity of the distribution defined by regarding the five values as relative frequencies as a function of the index from 1 to 5 is computed by
center (squares#)
which gives 4.090909090909091 (for a vector with five elements, the result will always be a number between 1.0 and 5.0). You compute the inner product of two equally long vectors as follows:
other# = { 2, 1.5, 1, 0.5, 0 }
result = inner (squares#, other#)
which gives 1*2 + 4*1.5 + 9*1 + 16*0.5 + 25*0 = 25. The formula for this is ∑_{i=1}^{5} squares[i] * other[i], so that an alternative piece of code could be
result = sumOver (i to 5, squares# [i] * other# [i])
a# = squares# + 5 ; adding a number to each element of a vector
causes a# to become the vector { 6, 9, 14, 21, 30 }.
b# = a# + { 3.14, 2.72, 3.16, 1, 7.5 } ; adding two vectors of the same length
causes b# to become the vector { 9.14, 11.72, 17.16, 20, 37.5 }.
c# = b# / 2 ; dividing each element of a vector
causes c# to become the vector { 4.57, 5.86, 8.58, 10, 18.75 }.
d# = b# * c# ; elementwise multiplication
causes d# to become the vector { 41.7698, 68.6792, 147.2328, 200, 703.125 }.
A vector can also be given to a menu command that returns another vector. For instance, to get a vector representing the pitch frequencies at 0.01second intervals in a Pitch object, you can do
selectObject: myPitch
tmin = Get start time
tmax = Get end time
times# = between_by# (tmin, tmax, 0.01)
pitches# = List values at times: times#, "hertz", "linear"
A numeric matrix is a twoindexed array of numbers, regarded as a single object. In a Praat script, you can put a matrix into a variable whose name ends in two number signs ("##"):
confusion## = {{ 3, 6, 2 }, { 8, 2, 1 }}
After this, the variable confusion## contains the value {{ 3, 6, 2 }, { 8, 2, 1 }}. We say that the matrix confusion# has two rows and three columns, i.e. it contains six numbers.
Whereas a numeric vector with five dimensions could be seen (see above) as a street that contains five houses, the matrix confusion## can be seen as a city district with two avenues crossed by three streets.
You can create a matrix in many ways. The first way we saw was with a matrix literal, i.e. a series of series of numbers (or numeric formulas) between nested braces.
The second way is as a matrix of zeroes. To create a matrix consisting of 100 rows of 10,000 zeroes, you do
a## = zero## (100, 10000)
After this,
numberOfRows (a##)
is 100, and
numberOfColumns (a##)
is 10000.
Another important type of matrix is one filled with random numbers. To create a matrix consisting of 100 rows of 10,000 values drawn from a Gaussian distribution with true mean 0.0 and true standard deviation 1.0, you can do
noise## = randomGauss## (100, 10000, 0.0, 1.0)
You can create a matrix as the outer product of two vectors:
m## = outer## (u#, v#)
which is the same as
m## = zeros## (size (u#), size (v#))
for irow to size (u#)
for icol to size (v#)
m## [irow, icol] = u# [irow] * v# [icol]
endfor
endfor
or in mathematical notation
m_{ij} = u_{i} v_{j} (i = 1..M, j = 1..N) 
where M is the number of rows and N is the number of columns.
You can add matrices:
c## = a## + b##
Elementwise multiplication:
c## = a## * b##
which does
c_{ij} = a_{ij} b_{ij} (i = 1..M, j = 1..N) 
Matrix multiplication:
c## = mul## (a##, b##)
which does
m_{ij} = ∑_{k=1}^{K} a_{ik} b_{kj} (i = 1..M, j = 1..N) 
where M is the number of rows of a, N is the number of columns of b, and K is the number of columns of a, which has to be equal to the number if rows of b.
Matrixbyvector multiplication:
v# = mul# (m##, u#)
which does
v_{i} = ∑_{j=1}^{N} m_{ij} u_{j} (i = 1..M) 
where M is the number of rows of m, and N is the number of columns of m, which has to be equal to the dimension of u. Also
v# = mul# (u#, m##)
which does
v_{j} = ∑_{i=1}^{M} u_{i} m_{ij} (j = 1..N) 
where M is the number of rows of m, which has to be equal to the dimension of u, and N is the number of columns of m.
© ppgb, August 25, 2019