Data2LD: Evaluate the mean of squared errors data-fitting criterion...
In Data2LD: Functional Data Analysis with Linear Differential Equations
Description
Usage
Arguments
Value
References
See Also
Examples
Description
Data2LD ... stands for "Data to Linear Dynamics."
The function approximates the data in argument YLIST by one or smooth
functions x_i, i=1,…,d.. This approximation is defined by a set
of d linear differential or algebraic equations defined by a set of
parameters some of which require estimation from the data.
The approximation minimizes the mean of squared residuals, expressed as
follows:
H(θ) = ∑_i w_i ∑_j^{n_i}
∑_{\ell}^N x [y_{ij \ell} - x_i(t_j)]^2
where:
-
i \in (1,…,d) indexes equations in a system differential and
algebraic equations.
-
j = 1,…,n_i indexes times of observation of a variable.
-
\ell = 1,…,N indexes replications of observations.
-
w_i is a positive fixed real number weighting the contribution of
a variable to the fitting criterion.
Only a subset of the variables may be observed, so that not all values of index i are
actually used. The number and location of times of observation t_j can
vary from one observed variable to another.
The fitting functions x_i(t) in turn, here assumed to be defined over
the interval [0,T], are defined by basis function expansions:
x_i(t_j) = ∑_k^{K_i} c_{ik}(θ|ρ) φ_{ik}(t_j)
where φ_{ik}(t) is the kth function in a system of
K_i basis functions used to approximate the ith variable, and
c_{ik} is the coefficientin the expansion of φ_{ik}(t).
The number of K_i basis functions and the type of basis function system
can vary from one variable to another. This information is contained in argument
XbasisList described below.
The coefficients c_{ik}(θ|ρ) defining the smoothing functions
are functions of the unknown parameters in vector θ that define
the differential and algebraic equations and that require estimation
from the data. The smoothing parameter ρ is a value in the interval
[0,1). The coefficient functions c_{ik}(θ|ρ) minimize the
inner least squares criterion, expressed here for simplicity for a single
variable or equation:
J(c|θ) = (1-ρ) ∑_j^n [y_j - c_k^K φ_k(t_j)]^2/n +
ρ \int_0^T {L(θ)x(t)]^2 dt/T}
Linear differential operator L(θ) is a linear differential or
algebraic equation rewritten as an operator by subtracting the right
side of the equation from the left, so that a solution x of the
equation satisfies Lx = 0.
Each L in a system of equation depends on one or more parameters that
need to be estimated and are contained in parameter vector θ.
The linear differential equation is of the form
D^m x_i = sum_k^d sum_j^{m_k} beta_{kj}(t) D^{j-1} x_k +
sum_f^{F_i} α_{fi}(t) u_{f,i},
i=1,...,d, f=1,...,F_i
where each coefficient is expanded in terms of its own number of
B-spline basis functions:
β_{ij}(t) = ∑_k^{K_{ij}} b_{ijk} φ_{ijk}(t),
α_{fi}(t) = ∑_k^{K_{fi}} a_{fik} ψ_{fik}(t)
As smoothing parameter ρ increases toward its upper limit of 1,
the second roughness penalty term in J is more and more emphasized
and the fit to the data less and less emphasized, so that x_i is
required to approach a solution to its respective equation.
The highest order of derivative can vary from one equation to another,
and in particular can be larger than 1.
Each right side may contain contributions from all variables in the
equation. Morever, these contributions may be from all permissible
derivatives of each equation, wher "permissible" means up to one less
than the highest order in the variable.
In addition, each equation may be forced by a number of forcing terms
that can vary from equation to equation.
This version approximates the integrals in the penalty terms by using
inprod_basis to compute the cross-product matrices for the
β-coefficient basis functions and the corresponding derivative of
the x-basis functions,and the cross-product matrices for the
α-coefficients and the corresponding U functions.
These are computed upon the first call to D2LD, and then retained
for subsequent calls by using the persistent command. This technique
for speeding up computaton is called memoization.
The structure of the model is defined in list modelList, which is
described below.
This version disassociates coefficient functions from equation
definitions to allow some coefficients to be used repeatedly and for
both homogeneous and forcing terms. It requires an extra argument
COEFLIST that contains the coefficients and the position of their
coefficient vectors in vector θ.
Usage
1
2
Arguments
yList
A list of length NVAR. Each list contains a list object with these
fields:
argvals: a vector of length n_i of observation times
y: a matrix with n_i rows and N columns.
The number of columns must be the same for all variables,
except that, if a list is empty, that variable is taken to be not observed.
XbasisList
A list array of length d. Each member contains in turn
a functional data object or a functional basis object.
modelList
A list each member of which specifies one of linear differential
equations in the system. These can be constructed using function make.Variable.
See the help file for that function for further details.
rhoVec
A real number in the half-open interval [0,T)].
Its value determines the relative emphasis on fitting the data versus being a
solution of the differential equation. The smaller the value, the more the emphasis
on data fitting.
summary
A logical constant. If TRUE, a variety of summeary measures are
computed, including degrees of freedom (df), the generalized cross-validation
index (gcv) and an estimate of the sampling variance-covariance matrix.
However, these can in some circumstances be computationally demanding, and
summary=FALSE limits the output to the fitting criterion, gradient and
hessian matrix, which can greatly speed up computation. Function Data2LD is
called in this way within function Data2LD.opt, for example.
Value
A named list object containing these results of the analysis:
MSE
The weighted mean squared errors computed over the variables with data.
DpMSE
The gradient of the objective function MSE with respect to the estimated
parameters.
D2ppMSE
A square symmetric matrx of that contains the second partial
derivatives of the objective function with respect to the parameter being estimated.
XfdParList
A list of length d containing functional parameter
objects for the estimated functions x_i(t).
df
An equivalent degrees of freedom value
df = trace(2 YM - YM*YM) where YM is the matrix y2cMap
described below.
gcv
The generalized cross-validation measure GCV. The value of
ρ corresponding to the minimim of GCV across values of
smoothing parameter ρ is often chose for an automatic
data-driven level of smoothing.
ISE
The sum across variables of the integrated squared value of
the differential operator. This value multiplied by ρ
and divided by T, the width of the domain, is the second
term the objective function.
Var.theta
A square symmetric matrix containing estimates of the sampling
variances and covariances for the estimated parameter values.
Rmat
A square symmetric matrix associated with the homogeneous portions of
the equations that has many useful applications, including constructions of more
compact and meaningful basis function expansions for the x_i's.
Smat
A matrix containing information on the covariation of the basis functions
and the forcing functions.
References
J. O. Ramsay and G. Hooker (2017) Dynamic Data Analysis. Springer.
See Also
checkModel,
make.Variable,
make.Xterm,
make.Fterm
checkModel
Examples
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
#
# Example 1: The refinery data
#
N <- 194
TimeData <- RefineryData[,1]
TrayData <- RefineryData[,2]
ValvData <- RefineryData[,3]
# Define the List array containing the tray data
TrayList <- list(argvals=RefineryData[,1], y=RefineryData[,2])
TrayDataList <- vector("list",1)
TrayDataList[[1]] <- TrayList
# construct the basis object for tray variable
# Order 5 spline basis with four knots at the 67
# to allow discontinuity in the first derivative
# and 15 knots between 67 and 193
Traynorder <- 5
Traybreaks <- c(0, rep(67,3), seq(67, 193, len=15))
Traynbasis <- 22
TrayBasis <- create.bspline.basis(c(0,193), Traynbasis, Traynorder,
Traybreaks)
# Set up the basis list for the tray variable
TrayBasisList <- vector("list",1)
TrayBasisList[[1]] <- TrayBasis
# Both alpha and beta coefficient functions will be constant.
# Define the constant basis
conbasis <- create.constant.basis(c(0,193))
betafdPar <- fdPar(conbasis)
alphafdPar <- fdPar(conbasis)
# Construct a step basis (order 1) and valve level
Valvbreaks <- c(0,67,193)
Valvnbasis <- 2
Valvnorder <- 1
Valvbasis <- create.bspline.basis(c(0,193), Valvnbasis, Valvnorder, Valvbreaks)
# smooth the valve data
Valvfd <- smooth.basis(TimeData, ValvData, Valvbasis)$fd
# Set up the model list for the tray variable
# Define single homogeneous term
# Xterm Fields: funobj parvec estimate variable deriv. factor
XTerm <- make.Xterm(betafdPar, 0.04, TRUE, 1, 0, -1)
XList <- vector("list", 1)
XList[[1]] <- XTerm
# Define the single forcing term
# Fterm Fields: funobj parvec estimate Ufd factor
FTerm <- make.Fterm(alphafdPar, 1.0, TRUE, Valvfd, 1)
FList <- vector("list", 1)
FList[[1]] <- FTerm
# Define the single differential equation in the model
TrayVariable <- make.Variable(XList=XList, FList=FList,
name="Tray level", order=1)
# Set up the model List array
TrayModelList <- vector("list",1)
TrayModelList[[1]] <- TrayVariable
# Run a check on TrayVariableList and make the modelList object
checkModelList <- checkModel(TrayBasisList, TrayModelList, summarywrd=TRUE)
TrayModelList <- checkModelList$model
nparam <- checkModelList$nparam
# Evaluate the fit to the data given the initial parameter estimates (0 and 0)
# This also initializes the four-way tensors so that they are not re-computed
# for subsquent analyses.
rhoVec <- 0.5
Data2LDList <- Data2LD(TrayDataList, TrayBasisList, TrayModelList, rhoVec)
MSE <- Data2LDList$MSE # Mean squared error for fit to data
DMSE <- Data2LDList$DpMSE # gradient with respect to parameter values
#
# Example 2: The head impact data
#
ImpactTime <- HeadImpactData[,2] # time in milliseconds
ImpactData <- HeadImpactData[,3]*100 # acceleration in millimeters/millisecond^2
ImpactRng <- c(0,60) # Define range time for estimated model
# Define the List array containing the data
ImpactDataList1 <- list(argvals=ImpactTime, y=ImpactData)
ImpactDataList <- vector("list",1)
ImpactDataList[[1]] <- ImpactDataList1
# Define a unit pulse function located at times 14-15
Pulsebasis <- create.bspline.basis(ImpactRng, 3, 1, c(0,14,15,60))
Pulsefd <- fd(matrix(c(0,1,0),3,1),Pulsebasis)
# Construct basis for output x(t),
# multiple knots at times 14 and 15.
# Order 6 spline basis, with three knots at the impact point and
# three knots at impact + delta to permit discontinuous first
# derivatives at these points
knots <- c(0,14,14,14,15,15,seq(15,60,len=11))
norder <- 6
nbasis <- 21
ImpactBasis <- create.bspline.basis(ImpactRng,nbasis,norder,knots)
# plot the basis
ImpactTimefine <- seq(0,60,len=201)
ImpactBasisfine <- eval.basis(ImpactTimefine, ImpactBasis)
# set up basis list
ImpactBasisList <- vector("list",1)
ImpactBasisList[[1]] <- ImpactBasis
# Set up the constant basis, make three coefficients and check them
conbasis <- create.constant.basis(ImpactRng)
# Define the terms in the second order linear equation
# Define the two terms in the homogeneous part of the equation
# Xterm Fields: funobj parvec estimate variable deriv. factor
Xterm1 <- make.Xterm(conbasis, 1, TRUE, 1, 0, -1)
Xterm2 <- make.Xterm(conbasis, 1, TRUE, 1, 1, -1)
# Set up the XList vector of length two
XList <- vector("list",2)
XList[[1]] <- Xterm1
XList[[2]] <- Xterm2
# Define the forcing term
# Fterm Fields: funobj parvec estimate Ufd factor
Fterm <- make.Fterm(conbasis, 1.0, TRUE, Pulsefd, 1)
# Set up the forcing term list of length one
FList <- vector("list",1)
FList[[1]] <- Fterm
# Define the single differential equation in the model
ImpactVariable <- make.Variable(name="Impact", order=2, XList=XList, FList=FList)
# Define list of length one containing the equation definition
ImpactModelList=vector("list",1)
ImpactModelList[[1]] <- ImpactVariable
# Check the object for internal consistency and
# set up the model object
# Run a check on TrayVariableList and make the modelList object
checkModelList <- checkModel(ImpactBasisList, ImpactModelList, summarywrd=TRUE)
ImpactModelList <- checkModelList$model
nparam <- checkModelList$nparam
# An evaluation of the criterion at the initial values
rhoVec <- 0.5 # light smoothing
Data2LDResult <- Data2LD(ImpactDataList, ImpactBasisList, ImpactModelList, rhoVec)
MSE <- Data2LDResult$MSE # Mean squared error for fit to data
DpMSE <- Data2LDResult$DpMSE # gradient with respect to parameter values
D2ppMSE <- Data2LDResult$D2ppMSE # Hessian matrix
Data2LD documentation built on Aug. 6, 2020, 1:08 a.m.
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
Description Usage Arguments Value References See Also Examples
Description
Data2LD ... stands for "Data to Linear Dynamics." The function approximates the data in argument YLIST by one or smooth functions x_i, i=1,…,d.. This approximation is defined by a set of d linear differential or algebraic equations defined by a set of parameters some of which require estimation from the data. The approximation minimizes the mean of squared residuals, expressed as follows:
H(θ) = ∑_i w_i ∑_j^{n_i} ∑_{\ell}^N x [y_{ij \ell} - x_i(t_j)]^2
where:
-
i \in (1,…,d) indexes equations in a system differential and algebraic equations.
-
j = 1,…,n_i indexes times of observation of a variable.
-
\ell = 1,…,N indexes replications of observations.
-
w_i is a positive fixed real number weighting the contribution of a variable to the fitting criterion.
Only a subset of the variables may be observed, so that not all values of index i are actually used. The number and location of times of observation t_j can vary from one observed variable to another. The fitting functions x_i(t) in turn, here assumed to be defined over the interval [0,T], are defined by basis function expansions:
x_i(t_j) = ∑_k^{K_i} c_{ik}(θ|ρ) φ_{ik}(t_j)
where φ_{ik}(t) is the kth function in a system of
K_i basis functions used to approximate the ith variable, and
c_{ik} is the coefficientin the expansion of φ_{ik}(t).
The number of K_i basis functions and the type of basis function system
can vary from one variable to another. This information is contained in argument
XbasisList described below.
The coefficients c_{ik}(θ|ρ) defining the smoothing functions
are functions of the unknown parameters in vector θ that define
the differential and algebraic equations and that require estimation
from the data. The smoothing parameter ρ is a value in the interval
[0,1). The coefficient functions c_{ik}(θ|ρ) minimize the
inner least squares criterion, expressed here for simplicity for a single
variable or equation:
J(c|θ) = (1-ρ) ∑_j^n [y_j - c_k^K φ_k(t_j)]^2/n + ρ \int_0^T {L(θ)x(t)]^2 dt/T}
Linear differential operator L(θ) is a linear differential or algebraic equation rewritten as an operator by subtracting the right side of the equation from the left, so that a solution x of the equation satisfies Lx = 0. Each L in a system of equation depends on one or more parameters that need to be estimated and are contained in parameter vector θ. The linear differential equation is of the form
D^m x_i = sum_k^d sum_j^{m_k} beta_{kj}(t) D^{j-1} x_k + sum_f^{F_i} α_{fi}(t) u_{f,i}, i=1,...,d, f=1,...,F_i
where each coefficient is expanded in terms of its own number of B-spline basis functions:
β_{ij}(t) = ∑_k^{K_{ij}} b_{ijk} φ_{ijk}(t),
α_{fi}(t) = ∑_k^{K_{fi}} a_{fik} ψ_{fik}(t)
As smoothing parameter ρ increases toward its upper limit of 1,
the second roughness penalty term in J is more and more emphasized
and the fit to the data less and less emphasized, so that x_i is
required to approach a solution to its respective equation.
The highest order of derivative can vary from one equation to another,
and in particular can be larger than 1.
Each right side may contain contributions from all variables in the
equation. Morever, these contributions may be from all permissible
derivatives of each equation, wher "permissible" means up to one less
than the highest order in the variable.
In addition, each equation may be forced by a number of forcing terms
that can vary from equation to equation.
This version approximates the integrals in the penalty terms by using
inprod_basis to compute the cross-product matrices for the
β-coefficient basis functions and the corresponding derivative of
the x-basis functions,and the cross-product matrices for the
α-coefficients and the corresponding U functions.
These are computed upon the first call to D2LD, and then retained
for subsequent calls by using the persistent command. This technique
for speeding up computaton is called memoization.
The structure of the model is defined in list modelList, which is
described below.
This version disassociates coefficient functions from equation
definitions to allow some coefficients to be used repeatedly and for
both homogeneous and forcing terms. It requires an extra argument
COEFLIST that contains the coefficients and the position of their
coefficient vectors in vector θ.
Usage
1 2 |
Arguments
yList |
A list of length NVAR. Each list contains a list object with these fields:
|
XbasisList |
A list array of length d. Each member contains in turn a functional data object or a functional basis object. |
modelList |
A list each member of which specifies one of linear differential
equations in the system. These can be constructed using function |
rhoVec |
A real number in the half-open interval [0,T)]. Its value determines the relative emphasis on fitting the data versus being a solution of the differential equation. The smaller the value, the more the emphasis on data fitting. |
summary |
A logical constant. If TRUE, a variety of summeary measures are
computed, including degrees of freedom ( |
Value
A named list object containing these results of the analysis:
MSE |
The weighted mean squared errors computed over the variables with data. |
DpMSE |
The gradient of the objective function MSE with respect to the estimated parameters. |
D2ppMSE |
A square symmetric matrx of that contains the second partial derivatives of the objective function with respect to the parameter being estimated. |
XfdParList |
A list of length d containing functional parameter objects for the estimated functions x_i(t). |
df |
An equivalent degrees of freedom value
df = trace(2 YM - YM*YM) where YM is the matrix |
gcv |
The generalized cross-validation measure GCV. The value of ρ corresponding to the minimim of GCV across values of smoothing parameter ρ is often chose for an automatic data-driven level of smoothing. |
ISE |
The sum across variables of the integrated squared value of the differential operator. This value multiplied by ρ and divided by T, the width of the domain, is the second term the objective function. |
Var.theta |
A square symmetric matrix containing estimates of the sampling variances and covariances for the estimated parameter values. |
Rmat |
A square symmetric matrix associated with the homogeneous portions of the equations that has many useful applications, including constructions of more compact and meaningful basis function expansions for the x_i's. |
Smat |
A matrix containing information on the covariation of the basis functions and the forcing functions. |
References
J. O. Ramsay and G. Hooker (2017) Dynamic Data Analysis. Springer.
See Also
checkModel,
make.Variable,
make.Xterm,
make.Fterm
checkModel
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 | #
# Example 1: The refinery data
#
N <- 194
TimeData <- RefineryData[,1]
TrayData <- RefineryData[,2]
ValvData <- RefineryData[,3]
# Define the List array containing the tray data
TrayList <- list(argvals=RefineryData[,1], y=RefineryData[,2])
TrayDataList <- vector("list",1)
TrayDataList[[1]] <- TrayList
# construct the basis object for tray variable
# Order 5 spline basis with four knots at the 67
# to allow discontinuity in the first derivative
# and 15 knots between 67 and 193
Traynorder <- 5
Traybreaks <- c(0, rep(67,3), seq(67, 193, len=15))
Traynbasis <- 22
TrayBasis <- create.bspline.basis(c(0,193), Traynbasis, Traynorder,
Traybreaks)
# Set up the basis list for the tray variable
TrayBasisList <- vector("list",1)
TrayBasisList[[1]] <- TrayBasis
# Both alpha and beta coefficient functions will be constant.
# Define the constant basis
conbasis <- create.constant.basis(c(0,193))
betafdPar <- fdPar(conbasis)
alphafdPar <- fdPar(conbasis)
# Construct a step basis (order 1) and valve level
Valvbreaks <- c(0,67,193)
Valvnbasis <- 2
Valvnorder <- 1
Valvbasis <- create.bspline.basis(c(0,193), Valvnbasis, Valvnorder, Valvbreaks)
# smooth the valve data
Valvfd <- smooth.basis(TimeData, ValvData, Valvbasis)$fd
# Set up the model list for the tray variable
# Define single homogeneous term
# Xterm Fields: funobj parvec estimate variable deriv. factor
XTerm <- make.Xterm(betafdPar, 0.04, TRUE, 1, 0, -1)
XList <- vector("list", 1)
XList[[1]] <- XTerm
# Define the single forcing term
# Fterm Fields: funobj parvec estimate Ufd factor
FTerm <- make.Fterm(alphafdPar, 1.0, TRUE, Valvfd, 1)
FList <- vector("list", 1)
FList[[1]] <- FTerm
# Define the single differential equation in the model
TrayVariable <- make.Variable(XList=XList, FList=FList,
name="Tray level", order=1)
# Set up the model List array
TrayModelList <- vector("list",1)
TrayModelList[[1]] <- TrayVariable
# Run a check on TrayVariableList and make the modelList object
checkModelList <- checkModel(TrayBasisList, TrayModelList, summarywrd=TRUE)
TrayModelList <- checkModelList$model
nparam <- checkModelList$nparam
# Evaluate the fit to the data given the initial parameter estimates (0 and 0)
# This also initializes the four-way tensors so that they are not re-computed
# for subsquent analyses.
rhoVec <- 0.5
Data2LDList <- Data2LD(TrayDataList, TrayBasisList, TrayModelList, rhoVec)
MSE <- Data2LDList$MSE # Mean squared error for fit to data
DMSE <- Data2LDList$DpMSE # gradient with respect to parameter values
#
# Example 2: The head impact data
#
ImpactTime <- HeadImpactData[,2] # time in milliseconds
ImpactData <- HeadImpactData[,3]*100 # acceleration in millimeters/millisecond^2
ImpactRng <- c(0,60) # Define range time for estimated model
# Define the List array containing the data
ImpactDataList1 <- list(argvals=ImpactTime, y=ImpactData)
ImpactDataList <- vector("list",1)
ImpactDataList[[1]] <- ImpactDataList1
# Define a unit pulse function located at times 14-15
Pulsebasis <- create.bspline.basis(ImpactRng, 3, 1, c(0,14,15,60))
Pulsefd <- fd(matrix(c(0,1,0),3,1),Pulsebasis)
# Construct basis for output x(t),
# multiple knots at times 14 and 15.
# Order 6 spline basis, with three knots at the impact point and
# three knots at impact + delta to permit discontinuous first
# derivatives at these points
knots <- c(0,14,14,14,15,15,seq(15,60,len=11))
norder <- 6
nbasis <- 21
ImpactBasis <- create.bspline.basis(ImpactRng,nbasis,norder,knots)
# plot the basis
ImpactTimefine <- seq(0,60,len=201)
ImpactBasisfine <- eval.basis(ImpactTimefine, ImpactBasis)
# set up basis list
ImpactBasisList <- vector("list",1)
ImpactBasisList[[1]] <- ImpactBasis
# Set up the constant basis, make three coefficients and check them
conbasis <- create.constant.basis(ImpactRng)
# Define the terms in the second order linear equation
# Define the two terms in the homogeneous part of the equation
# Xterm Fields: funobj parvec estimate variable deriv. factor
Xterm1 <- make.Xterm(conbasis, 1, TRUE, 1, 0, -1)
Xterm2 <- make.Xterm(conbasis, 1, TRUE, 1, 1, -1)
# Set up the XList vector of length two
XList <- vector("list",2)
XList[[1]] <- Xterm1
XList[[2]] <- Xterm2
# Define the forcing term
# Fterm Fields: funobj parvec estimate Ufd factor
Fterm <- make.Fterm(conbasis, 1.0, TRUE, Pulsefd, 1)
# Set up the forcing term list of length one
FList <- vector("list",1)
FList[[1]] <- Fterm
# Define the single differential equation in the model
ImpactVariable <- make.Variable(name="Impact", order=2, XList=XList, FList=FList)
# Define list of length one containing the equation definition
ImpactModelList=vector("list",1)
ImpactModelList[[1]] <- ImpactVariable
# Check the object for internal consistency and
# set up the model object
# Run a check on TrayVariableList and make the modelList object
checkModelList <- checkModel(ImpactBasisList, ImpactModelList, summarywrd=TRUE)
ImpactModelList <- checkModelList$model
nparam <- checkModelList$nparam
# An evaluation of the criterion at the initial values
rhoVec <- 0.5 # light smoothing
Data2LDResult <- Data2LD(ImpactDataList, ImpactBasisList, ImpactModelList, rhoVec)
MSE <- Data2LDResult$MSE # Mean squared error for fit to data
DpMSE <- Data2LDResult$DpMSE # gradient with respect to parameter values
D2ppMSE <- Data2LDResult$D2ppMSE # Hessian matrix
|
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
Embedding an R snippet on your website
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.