pda.fd | Language Reference for FDA Library
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pda.fd(xfdlist, bwtlist=NULL, awtlist=NULL, ufdlist=NULL, nfine=501)
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 T for the
estimate
slot, and the first
and third coefficient functions would have value 0.
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.
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.
See analyses of daily weather data for examples.