Matrix Functions

1 if all matrix arguments are nonzero; 0 otherwise.

Any(A, ...)

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

CDF(Y)

{QuantVec, CumProbVec} = CDF(YVec)

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

Chol Update(L, V, C)

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)

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

L (the Cholesky root).

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

The correlation matrix for the specified matrix.

Rows are discarded if they contain missing values.

matrix

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

Notes

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">

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

The covariance matrix for the specified matrix.

Performs pairwise covariances rather than row-wise covariances.

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"

"Shrink"

Performs the Schafer-Strimmer shrinkage estimate.

Notes

Rows are discarded if they contain missing values.

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

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

Creates design columns for a vector of values.

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.

An optional argument that is a list or matrix specifying 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.

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

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

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]) =

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

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

DesignF(vector, < <<levels >)

A version of Design for making full-rank versions of design matrices for nominal effects.

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

An optional argument that is a list or matrix specifying 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.

Show( Design Nom( ., [. 0 1] ),

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

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 >)

A version of Design for making full-rank versions of design matrices for ordinal effects.

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

An optional argument that is a list or matrix specifying 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.

Show(Design Ord( ., [. 0 1] ),

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

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];

DesignF()

See Design Nom(vector, < levelsList | <<levels >).

Det(A)

Determinant of a square matrix.

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

The matrix.

a matrix or a vector.

Direct Product(A, B)

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

Square matrices or scalars.

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

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

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

Element-by-element division of two matrices.

Element-by-element multiplication of two matrices.

The resulting matrix.

Eigenvalue decomposition.

The resulting matrix.

Generalized (Moore-Penrose) matrix inverse.

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

H Direct Product(A, B)

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

Hough Line Transform(matrix, <NAngle(number)>, <NRadius(number)>)

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.

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.

matrix

NAngle(number)

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

NRadius(number)

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

Identity(n)

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

Index(i, j, <increment>)

i::j

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

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

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

increment

Inv()

See Inverse(A).

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

Efficiently update an X´X matrix.

The matrix to be updated.

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)

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

Is Matrix(x)

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

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

Creates a matrix of identical values.

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)

Returns a table to efficiently look up near neighbors.

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. 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}. Position is a point that is described as a row vector for the coordinate of a row, or as the number of a row.

<<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.

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.

Loc(A)

Loc(list, item)

the item to be found within the list

Creates 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 list.

Loc Max(A)

Returns the position of the minimum element in a matrix.

An integer that is the specified position.

Returns the position of the minimum element in a matrix.

An integer that is the specified position.

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

a matrix

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

The new matrix or list.

Loc Sorted(A, B)

Matrix({{x11, x12, ..., x1m}, {x21, x22, ..., 2m}, {...}, {xn1, xn2, ..., xnm}})

Creates a matrix of subscript positions where the values of A have values less than or equal to the values in B. A must be a matrix sorted in ascending order.

Constructs an n-by-m matrix from a list of n lists of m row values or from the number of rows and columns.

The matrix.

A list of lists in which each list forms a row with the specified values.

Example

mymatrix = matrix({{1, 2, 3}, {4, 5, 6}, {7, 8, 9}, {10, 11, 12}});

Equivalent Expression

Matrix multiplication.

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).

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

Mode(list or matrix)

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.

Specify either a list or a matrix.

Multivariate Normal Impute(yVec, meanYvec, symCovMat)

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

The vector of response means.

yVec

The vector of responses.

meanYvec

symCovMat

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

NChooseK Matrix(n, k)

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

N Col(x)

N Cols(x)

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

Can be a data table or a matrix.

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

Ortho Poly(vector, order)

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.

Print Matrix(M, <named arguments>)

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

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

Note that the following named arguments are all optional.

A matrix.

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)

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

Rank Index(vector)

Rank(vector)

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)

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)

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.

Shape(A, nrow, <ncol>)