spatstat (version 1.28-1)

sumouter: Compute Quadratic Forms

Description

Calculates certain quadratic forms of matrices.

Usage

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

Arguments

x

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

w

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: $$M=\sum _i{w}_i{x}_i{x}_i^{\mathrm{\top }}$$ where the sum is over all rows of x (after removing any rows containing NA or other non-finite values). 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: $$y_i=x_{i}V{x}_i^{\mathrm{\top }}$$ 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

  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)