pda.fd {fda}R Documentation

Principal Differential Analysis

Description

Principal differential analysis (PDA) estimates a system of linear differential equations that define functions that fit the data and their derivatives.

Usage

pda.fd(xfdlist, bwtlist=NULL,
       awtlist=NULL, ufdlist=NULL, nfine=501)

Arguments

xfdlist a list whose members are functional data objects. Each of these objects contain one or more functions that are variables to be represented by a differential equation. The length of the list is the size of the system of differential equations. The number of replications must be the same for each member functional data object.
bwtlist a list array with the first two dimensions are equal to the number of variables in the system (the length of list xfdlist) and the last dimension equal to the order of each equation. The order of the equations is assumed to be the same for each equation. If only a single differential equation is involved, this argument can be an ordinary list with length equal to the order of the equation.
Each member of bwtlist defines a weight function in the system of linear differential equation. For any equation, there can be a term in the equation for each order of derivative from 0 to m-1, where m is the order of the equation. These derivatives can be those of each variable in the system, moreover. Thus, a two-variable system of order 3 can have, for each of the two equations, contributions from derivatives of order 0, 1 and 2 for each of the variables, or six terms in total.
The first dimension of bwtlist corresponds to equations, and the second dimension to contributions of variables for a fixed derivative within the equation defined by the first index. The third index indicates the order of derivative of the contribution, which is one less than the index value.
Each member of bwtlist is a either a functional parameter object or a functional data object. Functional data objects are converted to functional parameters objects with no smoothing. Functional parameter objects permit selected weight functions to be held fixed and not estimated. These are often constant functions with the value 0, corresponding to variables and derivatives that make no contribution.
For example, the harmonic acceleration differential equation would be of the form $D^3 x(t) = -b Dx(t)$ so that bwtlist would be an ordinary list of length 3, each functional parameter object being defined with a constant basis, but only the second object would have a value of TRUE for the estimate slot, and the first and third coefficient functions would have value 0.
awtlist a list containing weight functions for forcing functions.
In addition to terms in each of the equations involving terms corresponding to each derivative of each variable in the system, each equation can also have a contribution from one or more exogenous variables, often called forcing functions.
This argument defines the weights multiplying these forcing functions, and is a list array with first dimension equal to the number of variables or equations, and second dimension equal to the number of forcing functions. It is assumed that the number of forcing functions is the same for all equations. If only one forcing function is involved, or there is only one equation, then awtlist can be an ordinary list.
Each member is a functional parameter object (or a functional data object) defining a weighting function for a forcing function. As with bwtlist, each of these weighting functions may be estimated or held fixed. If the number of forcing functions actually varies from equation to equation, one can still use all forcing functions for all variables, but just weight unwanted ones in a particular equation by a fixed zero function.
Each member's functional data object has only a single replicate.
ufdlist a list containing forcing functions. This is a list array of the same size as awtlist and each member is a functional data object corresponding to a forcing function. The number of replicates must be equal to that of the variables themselves, which is assumed to be the same for all variables.
nfine a number of values for a fine mesh. The estimation of the differential equation involves discrete numerical quadrature estimates of integrals, and these require that functions be evaluated at a fine mesh of values of the argument. This argument defines the number to use. If not constant, this number is set to five times the largest number of basis functions used to represent any variable in the system.

Value

a named list of length 3 with components:

bwtlist a list array of the same dimensions as the corresponding argument, containing the estimated or fixed weight functions defining the system of linear differential equations.
resfdlist a list of length equal to the number of variables or equations. Each members is a functional data object giving the residual functions or forcing functions defined as the left side of the equation (the derivative of order m of a variable) minus the linear fit on the right side. The number of replicates for each residual functional data object is the same as that for the variables.
awtlist a list of the same dimensions as the corresponding argument. Each member is an estimated or fixed weighting function for a forcing function.

See Also

pca.fd, cca.fd

Examples

#See analyses of daily weather data for examples.

[Package fda version 1.2.3 Index]