spatstat (version 1.46-1)

sumouter: Compute Quadratic Forms

Description

Calculates certain quadratic forms of matrices.

Usage

sumouter(x, w=NULL) 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 matrix x will be interpreted as a collection of row vectors. The command sumouter computes the sum of the outer products of these vectors, weighted by the entries of w: $$ M = \sum_i w_i x_i x_i^\top $$ 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 $p * p$ matrix where p = ncol(x). 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 dimensins. For each of the row vectors 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

Run this code
  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)

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