spatstat (version 1.31-3)

sumouter: Compute Quadratic Forms

Description

Calculates certain quadratic forms of matrices.

Usage

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

Arguments

x

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 $p \times p$ 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.

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)

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