madness (version 0.2.8)

madness-class: Madness Class.

Description

An S4 class to enable forward differentiation of multivariate computations. Think of ‘madness’ as ‘multivariate automatic differentiation -ness.’ There is also a constructor method for madness objects, and a wrapper method.

Usage

# S4 method for madness
initialize(
  .Object,
  val,
  dvdx,
  xtag = NA_character_,
  vtag = NA_character_,
  varx = matrix(nrow = 0, ncol = 0)
)
madness(val, dvdx = NULL, vtag = NULL, xtag = NULL, varx = NULL)

Value

An object of class madness.

Arguments

.Object

a madness object, or proto-object.

val

an array of some numeric value, of arbitrary dimension.

dvdx

a matrix of the derivative of (the vector of) val with respect to some independent variable, \(X\).

xtag

an optional name for the \(X\) variable.

vtag

an optional name for the \(val\) variable.

varx

an optional variance-covariance matrix of the independent variable, \(X\).

Slots

val

an array of some numeric value. (Note that array includes matrix as a subclass.) The numeric value can have arbitrary dimension.

dvdx

a matrix of the derivative of (the vector of) val with respect to some independent variable, \(X\). A Derivative is indeed a 2-dimensional matrix. Derivatives have all been 'flattened'. See the details. If not given, defaults to the identity matrix, in which case \(val=X\), which is useful to initialization. Note that the derivative is with respect to an 'unrestricted' \(X\).

xtag

an optional name for the \(X\) variable. Operations between two objects of the class with distinct xtag data will result in an error, since they are considered to have different independent variables.

vtag

an optional name for the \(val\) variable. This will be propagated forward.

varx

an optional variance-covariance matrix of the independent variable, \(X\).

Author

Steven E. Pav

Details

A madness object contains a (multidimensional) value, and the derivative of that with respect to some independent variable. The purpose is to simplify computation of multivariate derivatives, especially for use in the Delta method. Towards this usage, one may store the covariance of the independent variable in the object as well, from which the approximate variance-covariance matrix can easily be computed. See vcov.

Note that derivatives are all implicitly 'flattened'. That is, when we talk of the derivative of \(i \times j\) matrix \(Y\) with respect to \(m \times n\) matrix \(X\), we mean the derivative of the \(ij\) vector \(\mathrm{vec}\left(Y\right)\) with respect to the \(mn\) vector \(\mathrm{vec}\left(X\right)\). Moreover, derivatives follow the 'numerator layout' convention: this derivative is a \(ij \times mn\) matrix whose first column is the derivative of \(\mathrm{vec}\left(Y\right)\) with respect to \(X_{1,1}\). Numerator layout feels unnatural because it makes a gradient vector of a scalar-valued function into a row vector. Despite this deficiency, it makes the product rule feel more natural. (2FIX: is this so?)

References

Petersen, Kaare Brandt and Pedersen, Michael Syskind. "The Matrix Cookbook." Technical University of Denmark (2012). http://www2.imm.dtu.dk/pubdb/pubs/3274-full.html

Magnus, Jan R. and Neudecker, H. "Matrix Differential Calculus with Applications in Statistics and Econometrics." 3rd Edition. Wiley Series in Probability and Statistics: Texts and References Section (2007).

Examples

obj <- new("madness", val=matrix(rnorm(10*10),nrow=10), dvdx=diag(100), xtag="foo", vtag="foo")
obj2 <- madness(val=matrix(rnorm(10*10),nrow=10), xtag="foo", vtag="foo^2")