spatstat (version 1.48-0)

sumouter: Compute Quadratic Forms

Description

Calculates certain quadratic forms of matrices.

Usage

sumouter(x, w=NULL, y=x) quadform(x, v) bilinearform(x, v, y)

Arguments

x,y

A matrix, whose rows are the vectors in the quadratic form.

w

Optional vector of weights

v

Matrix determining the quadratic form

Value

A vector or matrix.

Details

The matrices x and y will be interpreted as collections of row vectors. They must have the same number of rows.

The command sumouter computes the sum of the outer products of corresponding row vectors, weighted by the entries of w: $$ M = \sum_i w_i x_i y_i^\top $$ where the sum is over all rows of x (after removing any rows containing NA or other non-finite values). If w is missing, the weights will be taken as 1. The result is a $p * q$ matrix where p = ncol(x) and q = ncol(y). The command quadform evaluates the quadratic form, defined by the matrix v, for each of the row vectors of x: $$ y_i = x_i V x_i^\top $$ The result y is a numeric vector of length n where n = nrow(x). If x[i,] contains NA or other non-finite values, then y[i] = NA.

The command bilinearform evaluates the more general bilinear form defined by the matrix v. Here x and y must be matrices of the same dimensions. For each row vector of x and corresponding row vector of y, the bilinear form is $$ z_i = x_i V y_i^\top $$ The result z is a numeric vector of length n where n = nrow(x). If x[i,] or y[i,] contains NA or other non-finite values, then z[i] = NA.

Examples

  x <- matrix(1:12, 4, 3)
  dimnames(x) <- list(c("Wilma", "Fred", "Barney", "Betty"), letters[1:3])
  x

  sumouter(x)

  w <- 4:1
  sumouter(x, w)
  v <- matrix(1, 3, 3)
  quadform(x, v)

  # should be the same as quadform(x, v)
  bilinearform(x, v, x)

  # See what happens with NA's
  x[3,2] <- NA
  sumouter(x, w)
  quadform(x, v)