JSL Syntax Reference > JSL Functions > Matrix Functions
Publication date: 08/13/2020

# Matrix Functions

All(A, ...)

Returns

1 if all matrix arguments are nonzero; 0 otherwise.

Any(A, ...)

Returns

1 if one or more elements of one or more matrices are nonzero; 0 otherwise.

B Spline Coef(x, Internal Knot Grid, <degree=3>, <Knot End Points=min(x)||max(x)>)

Description

Finds the matrix of B-spline coefficients for the data in the x argument.

Returns

The matrix of B-spline basis coefficients. This matrix can be used as a design matrix in a linear model. The first column of the matrix contains an intercept term.

Arguments

x

A row or column vector that contains the data.

Internal Knot Grid

Either a single number that designates the number of desired knot points based on percentiles of x or a vector of values that designate the internal knot points. The number of internal knots must be greater than zero and less than or equal to the number of unique elements in x minus two.

degree

A number that indicates the degree of the B-splines. Defaults to 3.

Knot End Points

A 2x1 matrix that designates the lower and upper knot points. If this argument is not specified, the default lower and upper knot points are the minimum and maximum values of x, respectively.

Notes

This function is used in column formulas created by the Functional Data Explorer platform.

CDF(Y)

Description

Returns values of the empirical cumulative probability distribution function for Y, which can be a vector or a list. Cumulative probability is the proportion of data values less than or equal to the value of QuantVec.

Syntax

`{QuantVec, CumProbVec} = CDF(YVec)`

Chol Update(L, V, C)

Description

If L is the Cholesky root of an nxn matrix A, then after calling cholUpdate L is replaced with the Cholesky root of A+V*C*V' where C is an mxm symmetric matrix and V is an n*m matrix.

Cholesky(A)

Description

Finds the lower Cholesky root (L) of a positive semi-definitive matrix, L*L' = A.

Returns

L (the Cholesky root).

Arguments

A

a symmetric matrix.

Correlation(matrix, < <<"Pairwise">, < <<"Shrink">, < <<Freq(vector)>, < <<Weight(vector)>)

Description

Calculates the correlation matrix of the data in the matrix argument.

Returns

The correlation matrix for the specified matrix.

Argument

matrix

A matrix that contains the data. If the data has m rows and n columns, the result is an m-by-m matrix.

"Pairwise"

Uses the pairwise method for missing values rather than the row-wise method.

"Shrink"

Performs the Schafer-Strimmer shrinkage estimate.

<<Freq(vector)

A vector that specifies frequencies for the rows of the matrix argument.

<<Weight(vector)

A vector that specifies weights for the rows of the matrix argument.

Notes

By default, rows are discarded if they contain missing values. If the "Pairwise" option is specified, all pairs of nonmissing values are used in the correlation matrix calculation.

This function uses multithreading if available, so it is recommended for large problems with many rows.

When a column is constant, the correlations for it are 0, and the diagonal element is also 0.

Covariance(matrix, < <<"Pairwise">, < <<"Shrink">, < <<Freq(vector)>, < <<Weight(vector)>)

Description

Calculates the covariance matrix of the data in the matrix argument.

Returns

The covariance matrix for the specified matrix.

Argument

matrix

A matrix that contains the data. If the data has m rows and n columns, the result is an m-by-n matrix.

"Pairwise"

Uses the pairwise method for missing values rather than the row-wise method.

"Shrink"

Performs the Schafer-Strimmer shrinkage estimate.

<<Freq(vector)

A vector that specifies frequencies for the rows of the matrix argument.

<<Weight(vector)

A vector that specifies weights for the rows of the matrix argument.

Notes

By default, rows are discarded if they contain missing values. If the "Pairwise" option is specified, all pairs of nonmissing values are used in the covariance matrix calculation.

This function uses multithreading if available, so it is recommended for large problems with many rows.

Design(vector, < levelsList | <<levels, <<ElseMissing >)

Description

Creates a design matrix that contains a column of 1s and 0s for each unique value of a vector of values.

Returns

A design matrix with a column of 1s and 0s for each unique value of the argument or a list that contains the design matrix and a list of levels.

Argument

vector

A vector.

levelsList

An optional list or matrix argument that specifies the levels in the returned matrix.

<<levels

An optional argument that changes the return value to a list that contains the design matrix and a list of the levels.

<<ElseMissing

An optional argument that changes the handling of values in the vector argument that do not appear in the levelsList argument. If this argument is specified, missing values are placed in the design matrix. Otherwise, 0s are placed in the design matrix.

Note

Missing values in the levelsList argument are not ignored. For example:

`Show( Design ( ., [. 0 1] ),`
`Design( 0, [. 0 1] ),`
`Design( 1, [. 0 1] ),`
`Design( [0 0 1 . 1], [. 0 1] ),`
`Design( {0, 0, 1, ., 1}, [. 0 1] ) );`

Design(., [. 0 1]) = [1 0 0];

Design(0, [. 0 1]) = [0 1 0];

Design(1, [. 0 1]) = [0 0 1];

Design([0 0 1 . 1], [. 0 1]) =

[ 0 1 0,

0 1 0,

0 0 1,

1 0 0,

0 0 1];

Design({0, 0, 1, ., 1}, [. 0 1]) =

[ 0 1 0,

0 1 0,

0 0 1,

1 0 0,

0 0 1];

Design Last(vector, < levelsList, <<ElseMissing >)

Description

Creates a design matrix that contains a column of 1s and 0s for all but the last of the unique values of the argument. The last level is coded as a row of 0s.

Returns

A full-rank design matrix or a list that contains the design matrix and a list of levels.

Arguments

vector

A vector.

levelsList

An optional list or matrix argument that specifies the levels in the returned matrix. If this argument is specified, the last level in this list or matrix is treated as the last level in the design matrix. Otherwise, the last level is defined as the largest value in the vector argument.

<<ElseMissing

An optional argument that changes the handling of values in the vector argument that do not appear in the levelsList argument. If this argument is specified, missing values are placed in the design matrix. Otherwise, 0s are placed in the design matrix.

Design Nom(vector, < levelsList | <<levels, <<ElseMissing >)

DesignF(vector, < levelsList | <<levels, <<ElseMissing >)

Description

Creates a design matrix that contains a column of 1s and 0s for all but the last of the unique values of the argument. The last level is coded as a row of -1s.

Returns

A full-rank design matrix or a list that contains the design matrix and a list of levels.

Argument

vector

A vector.

levelsList

An optional list or matrix argument that specifies the levels in the returned matrix. If this argument is specified, the last level in this list or matrix is treated as the last level in the design matrix. Otherwise, the last level is defined as the largest value in the vector argument.

<<levels

An optional argument that changes the return value to a list that contains the design matrix and a list of levels.

<<ElseMissing

An optional argument that changes the handling of values in the vector argument that do not appear in the levelsList argument. If this argument is specified, missing values are placed in the design matrix. Otherwise, 0s are placed in the design matrix.

Note

Missing values in the levelsList argument are not ignored. For example:

`Show( Design Nom( ., [. 0 1] ),`
`Design Nom( 0, [. 0 1] ),`
`Design Nom( 1, [. 0 1] ),`
`Design Nom( [0 0 1 . 1], [. 0 1] ),`
`Design Nom( {0, 0, 1, ., 1}, [. 0 1] ) );`

Design Nom(., [. 0 1]) = [1 0];

Design Nom(0, [. 0 1]) = [0 1];

Design Nom(1, [. 0 1]) = [-1 -1];

Design Nom([0 0 1 . 1], [. 0 1]) = [0 1, 0 1, -1 -1, 1 0, -1 -1];

Design Nom({0, 0, 1, ., 1}, [. 0 1]) = [0 1, 0 1, -1 -1, 1 0, -1 -1];

Design Ord(vector, < levelsList | <<levels, <<ElseMissing >)

Description

Creates a design matrix that contains a column for all but the last of the unique values of the argument. The first level is coded as a row of 0s. Each subsequent (nth) level in the levelsList argument is coded as a row of (n-1) 1s and the rest 0s.

Returns

A full-rank design matrix or a list that contains the design matrix and a list of levels.

Argument

vector

A vector.

levelsList

An optional list or matrix argument that specifies the levels in the returned matrix.

<<levels

An optional argument that changes the return value to a list that contains the design matrix and a list of levels.

<<ElseMissing

An optional argument that changes the handling of values in the vector argument that do not appear in the levelsList argument. If this argument is specified, missing values are placed in the design matrix. Otherwise, 0s are placed in the design matrix.

Note

Missing values in the levelsList argument are not ignored. For example:

`Show( Design Ord( ., [. 0 1] ),`
`Design Ord( 0, [. 0 1] ),`
`Design Ord( 1, [. 0 1] ),`
`Design Ord( [0 0 1 . 1], [. 0 1] ),`
`Design Ord( {0, 0, 1, ., 1}, [. 0 1] ) );`

Design Ord(., [. 0 1]) = [0 0];

Design Ord(0, [. 0 1]) = [1 0];

Design Ord(1, [. 0 1]) = [1 1];

Design Ord([0 0 1 . 1], [. 0 1]) = [1 0, 1 0, 1 1, 0 0, 1 1];

Design Ord({0, 0, 1, ., 1}, [. 0 1]) = [1 0, 1 0, 1 1, 0 0, 1 1];

Det(A)

Description

Determinant of a square matrix.

Returns

The determinant.

Argument

A

A square matrix.

Diag(A, <B>)

Description

Creates a diagonal matrix from a square matrix or a vector. If two matrices are provided, concatenates the matrices diagonally.

Returns

The matrix.

Argument

A

a matrix or a vector.

Direct Product(A, B)

Description

Direct (Kronecker) product of square matrices or scalars A[i,j]*B.

Returns

The product.

Arguments

A, B

Square matrices or scalars.

Distance(x1, x2, <scales>, <powers>)

Description

Produces a matrix of distances between rows of x1 and rows of x2.

Returns

A matrix.

Arguments

x1, x2

Two matrices.

scales

Optional argument to customize the scaling of the matrix.

powers

Optional argument to customize the powers of the matrix.

E Div(A, B)

A:/B

Description

Element-by-element division of two matrices.

Returns

The resulting matrix.

Arguments

A, B

Two matrices.

E Mult(A, B)

A:*B

Description

Element-by-element multiplication of two matrices.

Returns

The resulting matrix.

Arguments

A, B

Two matrices.

Eigen(A)

Description

Eigenvalue decomposition.

Returns

A list {M, E} such that E * Diag(M) * E = A'.

Argument

A

A symmetric matrix.

Estimate Factor Score(dataRow, Covariance, ManMeans, LatMeans)

Description

Estimates factor scores from a structural equation model (SEM). This function is used in the Save Factor Scores option in an Structural Equation Model report. See Model Options in Multivariate Methods.

Returns

A row vector of estimated factor scores based on the structural equation model.

Arguments

dataRow

A row vector of data values.

Covariance

A model-implied variance-covariance matrix.

ManMeans

A vector of model-implied manifest variable means.

LatMeans

A vector of model-implied latent variable means.

Fourier Basis Coef(x, Number Pairs, <Period=max(x)-min(x)+1>)

Description

Finds the matrix of Fourier basis coefficients for the data in the x argument.

Returns

The matrix of Fourier basis coefficients. This can be used as a design matrix in a linear model. The first column of the matrix contains an intercept term. The remaining columns contain pairs of basis coefficients, where pair i is defined as the sin() and cos() of i * (2 * π / Period) * x.

Arguments

x

A row or column vector that contains the data.

Number Pairs

The number of sin() and cos() pairs for the Fourier basis.

Period

The period for trigonometric functions that make up the Fourier basis.

Notes

This function is used in column formulas created by the Functional Data Explorer platform.

G Inverse(A)

Description

Generalized (Moore-Penrose) matrix inverse.

H Direct Product(A, B)

Description

Horizontal direct product of two square matrices of the same dimension or scalars.

Description

Takes a matrix of intensities and transforms it in a way that is useful for finding streaks in the matrix. Produces a matrix containing the Hough Line Transform with angles as columns and radiuses as rows.

Argument

matrix

A matrix that can be derived from the intensities of an image, but is more likely from a semiconductor wafer that may have defects across in a streak due to planarization machines.

NAngle(number)

Enter the number of the angle to obtain a different sized transform. The default is 180 degrees.

Enter the number of the radius to obtain a different sized transform. The default is sqrt(NRow*nRow+nCol*nCol).

Identity(n)

Description

Creates an n-by-n identity matrix with ones on the diagonal and zeros elsewhere.

Returns

The matrix.

Argument

n

An integer.

Index(i, j, <increment>)

i::j

Description

Creates a column matrix whose values range from i to j.

Returns

The matrix.

Arguments

i, j

Integers that define the range: i is the beginning of the range, j is the end.

increment

Optional argument to change the default increment, which is +1.

Inv()

See Inverse(A).

Inv Update(A, X, 1|-1)

Description

Efficiently update an X´X matrix.

Arguments

A

The matrix to be updated.

X

One or more rows to be added to or deleted from the matrix A.

1|-1

The third argument controls whether the row or rows defined in the second argument, X, are added to or deleted from the matrix A. 1 means to add the row or rows and -1 means to delete the row or rows.

Inverse(A)

Inv(A)

Description

Returns the matrix inverse. The matrix must be square non-singular.

Is Matrix(x)

Description

Returns 1 if the evaluated argument is a matrix, or 0 otherwise.

J(nrows, <ncols>, <value>)

Description

Creates a matrix of identical values.

Returns

The matrix.

Arguments

nrows

Number of rows in matrix. If ncols is not specified, nrows is also used as ncols.

ncols

Number of columns in matrix.

value

The value used to populate the matrix. If value is not specified, 1 is used.

KDTable(matrix)

Description

Returns a table to efficiently look up near neighbors.

Returns

A KDTable object.

Argument

matrix

A matrix of k-dimensional points. The number of dimensions or points is not limited. Each column in the matrix represents a dimension to the data, and each row represents a data point.

Messages

<<Distance between rows(row1, row2)

Returns the distance between two the two specified rows in the KDTable. The distance applies to removed and inserted rows as well.

<<K nearest rows(stop, <position>)

Returns a matrix. Position is a point that is described as a row vector for the coordinate of a row, or as the number of a row. If position is not specified, returns the n nearest rows and distances to all rows. If position is specified, returns the n nearest rows and distances to either a point or a row. Stop is either n or {n, limit}. The limit parameter limits the number of rows that will be found. It can be specified one of two ways: a number, like 5, means return the 5 nearest rows. A list, like {5,10}, means return up to 5 nearest rows, stopping when the distance of 10 is exceeded. In the second case, the last row may have a distance greater than 10. Since the command continues until it finds the closest row beyond the stop radius, this point is also returned. This can be especially useful if there are no rows within the radius.

<<Remove rows(number | vector)

Remove either the row specified by number or the rows specified by vector. Returns the number of rows that were removed. Rows that were already removed are ignored.

<<Insert rows(number | vector)

Re-insert either the row specified by number or the rows specified by vector. Returns the number of rows that were inserted. Rows that were already inserted are ignored.

Note

When rows are removed or inserted, the row indices do not change. You can remove and re-insert only rows that are in the KDTable object. If you need different rows, construct a new KDTable.

Least Squares Solve(y, X, < <<noIntercept, <<weights(OptionalWeightVector), <<method("Sweep"|"GInv")>)

Description

Computes least squares regression estimates for the assumed model y = X * beta + error.

Returns

A list that contains the matrix Beta=Inverse(X'X)X'y and the estimated variance matrix of Beta.

Optional Named Arguments

<<noIntercept

Specifies a no-intercept model.

<<weights(optional weight vector)

Specifies a vector of weights to perform weighted least squares.

<<method("Sweep" | "GInv")

Specifies the method for solving the normal equations. The default Sweep method is more computationally efficient, but you can also specify the generalized inverse ("GInv") method, which is more numerically stable.

Linear Regression(y, X, < <<noIntercept, <<printToLog, <<weight(OptionalWeightVector), <<freq(OptionalFreqVector)>)

Description

Fits a linear regression for the assumed model y = X * beta + error.

Returns

A list that contains a vector of the estimates, a vector of the standard error, and a list of diagnostics. The list of diagnostics contains vectors of the t statistics and p-values for the estimates, as well as the R-Square and adjusted R-Square values for the regression fit.

Optional Named Arguments

<<noIntercept

Excludes the intercept.

<<printToLog

Prints a summary of the fit to the log.

<<weight(vector)

Specifies a vector of weights to perform weighted least squares.

<<freq(vector)

Specifies a vector of frequencies for each row of y and X.

Example

`n = 10;`
`x = J( n, 1, Random Normal() );`
`y = 1 + x * 3 + J( n, 1, Random Normal() );`
`{Estimates, Std_Error, Diagnostics} = Linear Regression( y, x, <<printToLog );`
`As Table( y || x );`
`Bivariate( Y( :Col1 ), X( :Col2 ), Fit Line( 1 ) );`

Loc(A)

Loc(A, item)

Description

Returns a matrix of subscript positions where A is nonzero and nonmissing. For the two-argument function, Loc returns a matrix of positions where item is found within A. If the first argument is a list, the second argument is required.

Argument

A

a matrix or a list

item

the item to be found within the matrix or list A

Loc Max(A)

Description

Returns the position of the maximum element in a matrix.

Returns

An integer that is the specified position.

Argument

A

a matrix

Loc Min(A)

Description

Returns the position of the minimum element in a matrix.

Returns

An integer that is the specified position.

Argument

A

a matrix

Loc NonMissing(matrix, ..., {list}, ...)

Description

Returns indices of nonmissing rows in matrices or lists. In lists, the function can also return indices of nonempty characters.

Returns

The new matrix or list.

Loc Sorted(A, B)

Description

Returns a column vector of subscript positions where the values of A have values less than or equal to the values in B based on a binary search. A must be a matrix sorted in ascending order without missing values.

Returns

The new matrix, which has the same dimensions as B. If a value in B is less than the first value in A, the returned subscript position for that value is 1.

Argument

A, B

matrices

Matrix({{x11, ..., x1m}, {x21, ..., 2m}, {...}, {xn1, ..., xnm}})

Matrix({x1, ..., xn})

Matrix(n, m)

Description

Constructs an n-by-m matrix. The following specification methods are available:

If you specify a list of n lists that each contain m row values, the matrix is formed by vertically concatenating the evaluated lists. The list items must evaluate to numeric values or row vectors, and the dimensions of the items must be conformable.

If you specify a single list of n items, the return value is an n-by-1 column vector. The items of the evaluated list must evaluate to numeric values.

If you specify two integer arguments, the return value is a matrix of zeros that contains n rows and m columns.

Examples

`Matrix({{1, 2, 3}, {4, [5 6]}, {7, 8, 9}});`

[1 2 3, 4 5 6, 7 8 9]

`Matrix({{[1 2 3], 4, 5, 6, 7, 8, 9}});`

[1 2 3 4 5 6 7 8 9]

`Matrix({2,3+7});`

[2, 10]

`Matrix(2,3);`

[0 0 0, 0 0 0]

Matrix Mult(A, B)

C=A*B, ...

Description

Matrix multiplication.

Arguments

A, B, ...

Two or more matrices, which must be conformable (all matrices after the first one listed must have the same number of rows as the number of columns in the first matrix).

Note

Matrix Mult() allows only two arguments, while using the * operator enables you to multiply several matrices.

Matrix Rank(A)

Description

Returns the rank of the matrix A.

Mode({list } or matrix)

Description

Selects the most frequent item from a numeric or character list or a numeric matrix. In the event of a tie, the lower value is selected. If multiple arguments are specified, a combination of numeric values and character strings is acceptable.

Arguments

Specify either a list or a matrix.

Multivariate Normal Impute(yVec, meanYvec, symCovMat, colMin, colMax)

Description

Imputes missing values in yVec based on the mean and covariance.

Arguments

yVec

The vector of responses.

meanYvec

The vector of response means.

symCovMat

A symmetric matrix containing the response covariances. If the covariance matrix is not specified, then JMP imputes with means.

colMin

A vector of column minimums. Provides lower bounds for the imputations.

colMax

A vector of column maximums. Provides upper bounds for the imputations.

NChooseK Matrix(n, k)

Description

Returns a matrix of n things taken k at a time (n select k).

N Col(x)

N Cols(x)

Description

Returns the number of columns in either a data table or a matrix.

Argument

x

Can be a data table or a matrix.

Ortho(A, <Centered(0)>, <Scaled(1)>)

Description

Orthonormalizes the columns of matrix A using the Gram Schmidt method. Centered(0) makes the columns to sum to zero. Scaled(1) makes them unit length.

Ortho Poly(vector, order)

Description

Returns orthogonal polynomials for a vector of indices representing spacings up to the order given.

P Spline Coef(x, Internal Knot Grid, <degree=3>)

Description

Finds the matrix of penalized basis spline (P-spline) coefficients for the data in the x argument. This function is used in column formulas created by the Functional Data Explorer platform.

Returns

The matrix of P-spline basis coefficients, which is the truncated power basis of the specified degree. The truncated power basis of degree p with knots k1 through kK is defined as follows:

where (x - k1)+ is the positive part of (x - k1) and is set to zero for negative values of (x - k1).

Arguments

x

A row or column vector that contains the data.

Internal Knot Grid

Either a single number that designates the number of desired knot points based on percentiles of x or a vector of values that designate the internal knot points.

degree

A number that indicates the degree of the P-splines. Defaults to 3.

Notes

This function is used in column formulas created by the Functional Data Explorer platform.

Parallel Assign({thread_local_var = global_var, ...}, matrix[a, b] = expression using a and b)

Description

Uses multiple threads to assign values to the matrix. Enables you to take advantage of multiple cores on a computer. The function has two arguments.

The first argument is a list of assignment statements that copies global variables into each thread’s local variable list.

The second argument is an assignment expression with a left-hand-side that is a matrix with one or two prototype indexes and a right hand side that can be any JSL expression using those indexes and the local variables from the list (and in JMP global: variables).

Example

`a = 42;`
`x = [1 2 3, 4 5 6, 7 8 9];`
`Show( Parallel Assign( {}, x[i, j] = global:a ) );`
`Show( x );`

Parallel Assign({}, x[i,j] = global:a) = 1;

x =

[ 42 42 42,

42 42 42,

42 42 42];

Print Matrix(M, <named arguments>)

Description

Returns a string that contains a well-formatted matrix. You can use the function, for example, to print the matrix to the log.

Argument

M

A matrix.

Optional Named Arguments

<<ignore locale(Boolean)

Set to false (0) to use the decimal separator for your locale. Set to true (1) to always use a period (.) as a separator. The default value is false (0).

<<decimal digits(n)

An integer that specifies the number of digits after the decimal separator to print.

<<style("style name")

Use one of three available styles: Parseable is a reformatted JSL matrix expression. Latex is formatted for LaTex. If you specify Other, you must define the following three arguments.

<<separate("character")

Define the separator for concatenated entries.

<<line begin("character")

Define the beginning line character.

<<line end("character")

Define the ending line character.

QR(A)

Description

Returns the QR decomposition of A. Typical usage is {Q, R} = QR(A).

Rank Index(vector)

Rank(vector)

Description

Returns a vector of indices that, used as a subscript to the original vector, sorts the vector by rank. Excludes missing values. Lists of numbers or strings are supported in addition to matrices.

Ranking(vector)

Description

Returns a vector of ranks of the values of vector, low to high as 1 to n, ties arbitrary. Lists of numbers or strings are supported in addition to matrices.

Ranking Tie(vector)

Description

Returns a vector of ranks of the values of vector, but ranks for ties are averaged. Lists of numbers or strings are supported in addition to matrices.

Scoring Impute(rowWithMissing, VMat, colMeanVec, colStdDevVec)

Description

Provides streaming functionality for the Automated Data Imputation (ADI) algorithm.

Returns

Returns the row vector with the missing values imputed using the standard least squares estimation.

Arguments

rowwithMissing

A row vector that contains missing values.

VMat

colMeanVec

A vector of the column means ignoring missing cells.

colStdDevVec

a vector of the column standard deviations ignoring missing cells.

Shape(A, nrow, <ncol>, < <<bycol>)

Description

Reshapes the matrix A across rows to the specified dimensions. Each value from the matrix A is placed into the reshaped matrix. By default, the values are placed row-by-row.

Returns

The reshaped matrix.

Arguments

A

A matrix.

nrow

The number of rows that the new matrix should have.

ncol

(Optional) The number of columns the new matrix should have.

<<bycol

(Optional) Specifies that the values be placed into the reshaped matrix column-by-column, instead of row-by-row.

Notes

If ncol is not specified, the number of columns is whatever is necessary to fit all of the original values of the matrix into the reshaped matrix.

If a missing value is specified for nrow, the number of rows is whatever is necessary to fit all of the original values of the matrix into the reshaped matrix.

If the new matrix is smaller than the original matrix, the extra values are discarded.

If the new matrix is larger than the original matrix, the values are repeated to fill the new matrix.

Examples

`a = Matrix({ {1, 2, 3}, {4, 5, 6}, {7, 8, 9} });`

[ 1 2 3,

4 5 6,

7 8 9]

` `
`Shape(a, 2);`

[ 1 2 3 4 5,

6 7 8 9 1]

` `
`Shape(a, 2, 2);`

[ 1 2,

3 4]

` `
`Shape(a, 4, 4);`

[ 1 2 3 4,

5 6 7 8,

9 1 2 3,

4 5 6 7]

` `
`Shape(a, 4, 4, <<bycol);`

[ 1 5 9 4,

2 6 1 5,

3 7 2 6,

4 8 3 7]

Solve(A, b)

Description

Solves a linear system. In other words, x=inverse(A)*b.

Sort Ascending(source)

Description

Returns a copy of a list or matrix source with the items in ascending order.

Sort Descending(source)

Description

Returns a copy of a list or matrix source with the items in descending order.

Sparse SVD(X, <nSingularValues=min(nRow, nCol)>,<tolerance=1e-10>)

Description

Computes the singular value decomposition of matrix X using the implicitly restarted, partially reorthogonalized Lanczos method for sparse matrices.

Returns

Returns a list (U, M, V) such that U*diag(M)*V is equal to x.

Spline Coef(x, y, lambda)

Description

Returns a five column matrix of the form knots||a||b||c||d where knots is the unique values in x.

x is a vector of regressor variables, y is the vector of response variables, and lambda is the smoothing argument. Larger values for lambda result in smoother splines.

Spline Eval(x, coef)

Description

Evaluates the spline predictions using the coef matrix in the same form as returned by SplineCoef(), in other words, knots||a||b||c||d. The x argument can be a scalar or a matrix of values to predict. The number of columns of coef can be any number greater than 1 and each is used for the next higher power. The powers of x are centered at the knot values. For example, the calculation for coef of knots||a||b||c||d is j is such that knots[j] is the largest knot smaller than x.

xx = x-knots[j] is the centered x value:

result = a[j] + xx * (b[j] + xx * (c[j] + xx * d[j]))

The following line is equivalent:

result = a[j] + b[j] * xx + c[j] * xx ^ 2 + d[j] * xx ^ 3

Spline Smooth(x, y, lambda)

Description

Returns the smoothed predicted values from a spline fit.

x is a vector of regressor variables, y is the vector of response variables, and lambda is the smoothing argument. Larger values for lambda result in smoother splines.

SVD(A)

Description

Singular value decomposition.

Sweep(A, <indices>)

Description

Sweeps, or inverts a matrix a partition at a time.

Trace(A)

Description

The trace, or the sum of the diagonal elements of a square matrix.

Transpose(A)

Description

Transposes the rows and columns of the matrix A.

Returns

The transposed matrix.

Arguments

A

A matrix.

Equivalent Expression

`A‘`

V Concat(A, B, ...)

Description

Vertical concatenation of two or more matrices.

Returns

The new matrix.

Arguments

Two or more matrices.

V Concat To(A, B, ...)

Description

Vertical concatenation in place. This is an assignment operator.

Returns

The new matrix.

Arguments

Two or more matrices.

V Max(matrix)

Description

Returns a row vector containing the maximum of each column of matrix.

V Mean(matrix)

Description

Returns a row vector containing the mean of each column of matrix.

V Median(matrix)

Description

Returns a row vector containing the median of each column of matrix.

V Min(matrix)

Description

Returns a row vector containing the minimum of each column of matrix.

V Quantile(matrix, p)

Description

Returns a row vector containing the pth quantile of each column of matrix.

V Standardize(matrix)

Description

Returns a matrix column-standardized to mean = 0 and standard deviation = 1.

V Std(matrix)

Description

Returns a row vector containing the standard deviations of each column of matrix.

V Sum(matrix)

Description

Returns a row vector containing the sum of each column of matrix.

Varimax(matrix, <norm=1>)

Description

Performs a varimax rotation.

Returns

A list that contains the rotated matrix and the orthogonal rotation matrix.

Arguments

matrix

A matrix to be rotated.

norm

Specify 1 to perform a normalized rotation, and specify 0 to perform a non-normalized rotation. The default value is 1.

Vec Diag(A)

Description

Creates a vector from the diagonals of a square matrix A.

Returns

The new matrix.

Arguments

A

A square matrix.

Note

Using a matrix that is not square results in an error.

Description

Constructs an n-by-m matrix. Used in calculation of hat values.

Returns

The new matrix.

Arguments

Two matrices. The first must be symmetric.

Equivalent Expression

`Vec Diag(X*Sym*X‘)`