Data2fd: Create a functional data object from data
In fda: Functional Data Analysis
Description
Usage
Arguments
Details
Value
References
See Also
Examples
Description
This function converts an array y of function values plus an
array argvals of argument values into a functional data object.
This function tries to do as much for the user as possible in setting
up a call to function smooth.basis. Be warned that the result
may not be a satisfactory smooth of the data, and consequently that it
may be necessary to use function smooth.basis instead, the help
file for which provides a great deal more information than is provided
here.
Also, function Data2fd can swap the first two arguments,
argvals and y if it appears that they have been included
in reverse order. A warning message is returned if this swap takes place.
Any such automatic decision, though, has the possibility of being wrong,
and the results should be carefully checked. Preferably, the order of
the arguments should be respected: argvals comes first and
y comes second.
Usage
1
2
3
4
Arguments
argvals
a set of argument values. If this is a vector, the same set of
argument values is used for all columns of y. If
argvals is a matrix, the columns correspond to the columns of
y, and contain the argument values for that replicate or
case.
Dimensions for argvals must match the first dimensions of
y, though y can have more dimensions. For example, if
dim(y) = c(9, 5, 2), argvals can be a vector of length 9 or a
matrix of dimensions c(9, 5) or an array of dimensions c(9, 5, 2).
y
an array containing sampled values of curves.
If y is a vector, only one replicate and variable are
assumed. If y is a matrix, rows must correspond to argument
values and columns to replications or cases, and it will be assumed
that there is only one variable per observation. If y is a
three-dimensional array, the first dimension (rows) corresponds to
argument values, the second (columns) to replications, and the third
(layers) to variables within replications. Missing values are
permitted, and the number of values may vary from one replication to
another. If this is the case, the number of rows must equal the
maximum number of argument values, and columns of y having
fewer values must be padded out with NA's.
basisobj
One of the following:
basisfd
a functional basis object (class basisfd).
fd
a functional data object (class fd), from which its
basis component is extracted.
fdPar
a functional parameter object (class fdPar), from which
its basis component is extracted.
integer
an integer giving the order of a B-spline basis,
create.bspline.basis(argvals, norder=basisobj)
numeric vector
specifying the knots for a B-spline basis,
create.bspline.basis(basisobj)
NULL
Defaults to create.bspline.basis(argvals).
nderiv
Smoothing typically specified as an integer order for the derivative
whose square is integrated and weighted by lambda to smooth.
By default, if basisobj[['type']] == 'bspline', the smoothing
operator is int2Lfd(max(0, norder-2)).
A general linear differential operator can also be supplied.
lambda
weight on the smoothing operator specified by nderiv.
fdnames
Either a charater vector of length 3 or a named list of length 3.
In either case, the three elements correspond to the following:
argname
name of the argument, e.g. "time" or "age".
repname
a description of the cases, e.g. "reps" or "weather stations"
value
the name of the observed function value, e.g. "temperature"
If fdnames is a list, the components provide labels for the levels
of the corresponding dimension of y.
covariates
the observed values in y are assumed to be primarily determined
the the height of the curve being estimates, but from time to time
certain values can also be influenced by other known variables. For
example, multi-year sets of climate variables may be also determined by
the presence of absence of an El Nino event, or a volcanic eruption.
One or more of these covariates can be supplied as an n by
p matrix, where p is the number of such covariates. When
such covariates are available, the smoothing is called "semi-parametric."
Matrices or arrays of regression coefficients are then estimated that
define the impacts of each of these covariates for each cueve and each
variable.
method
by default the function uses the usual textbook equations for computing
the coefficients of the basis function expansions. But, as in regression
analysis, a price is paid in terms of rounding error for such
computations since they involved cross-products of basis function
values. Optionally, if method is set equal to the string "qr",
the computation uses an algorithm based on the qr-decomposition which
is more accurate, but will require substantially more computing time
when n is large, meaning more than 500 or so. The default
is "chol", referring the Choleski decomposition of a symmetric positive
definite matrix.
dfscale
the generalized cross-validation or "gcv" criterion that is often used
to determine the size of the smoothing parameter involves the
subtraction of an measue of degrees of freedom from n. Chong
Gu has argued that multiplying this degrees of freedom measure by
a constant slightly greater than 1, such as 1.2, can produce better
decisions about the level of smoothing to be used. The default value
is, however, 1.0.
Details
This function tends to be used in rather simple applications where
there is no need to control the roughness of the resulting curve with
any great finesse. The roughness is essentially controlled by how
many basis functions are used. In more sophisticated applications, it
would be better to use the function smooth.basisPar.
Value
an object of the fd class containing:
coefs
the coefficient array
basis
a basis object
fdnames
a list containing names for the arguments, function values
and variables
References
Ramsay, James O., and Silverman, Bernard W. (2006), Functional
Data Analysis, 2nd ed., Springer, New York.
Ramsay, James O., and Silverman, Bernard W. (2002), Applied
Functional Data Analysis, Springer, New York.
See Also
smooth.basisPar,
smooth.basis,
project.basis,
smooth.fd,
smooth.monotone,
smooth.pos
day.5
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
126
127
128
129
130
131
132
133
134
135
136
##
## Simplest possible example: constant function
##
# 1 basis, order 1 = degree 0 = constant function
b1.1 <- create.bspline.basis(nbasis=1, norder=1)
# data values: 1 and 2, with a mean of 1.5
y12 <- 1:2
# smooth data, giving a constant function with value 1.5
fd1.1 <- Data2fd(y12, basisobj=b1.1)
plot(fd1.1)
# now repeat the analysis with some smoothing, which moves the
# toward 0.
fd1.1.5 <- Data2fd(y12, basisobj=b1.1, lambda=0.5)
# values of the smooth:
# fd1.1.5 = sum(y12)/(n+lambda*integral(over arg=0 to 1 of 1))
# = 3 / (2+0.5) = 1.2
eval.fd(seq(0, 1, .2), fd1.1.5)
##
## step function smoothing
##
sessionInfo()
find("crossprod")
# 2 step basis functions: order 1 = degree 0 = step functions
b1.2 <- create.bspline.basis(nbasis=2, norder=1)
# fit the data without smoothing
fd1.2 <- Data2fd(1:2, basisobj=b1.2)
# plot the result: A step function: 1 to 0.5, then 2
op <- par(mfrow=c(2,1))
plot(b1.2, main='bases')
plot(fd1.2, main='fit')
par(op)
##
## Simple oversmoothing
##
# 3 step basis functions: order 1 = degree 0 = step functions
b1.3 <- create.bspline.basis(nbasis=3, norder=1)
# smooth the data with smoothing
fd1.3.5 <- Data2fd(y12, basisobj=b1.3, lambda=0.5)
# plot the fit along with the points
plot(0:1, c(0, 2), type='n')
points(0:1, y12)
lines(fd1.3.5)
# Fit = penalized least squares with penalty =
# = lambda * integral(0:1 of basis^2),
# which shrinks the points towards 0.
# X1.3 = matrix(c(1,0, 0,0, 0,1), 2)
# XtX = crossprod(X1.3) = diag(c(1, 0, 1))
# penmat = diag(3)/3
# = 3x3 matrix of integral(over arg=0:1 of basis[i]*basis[j])
# Xt.y = crossprod(X1.3, y12) = c(1, 0, 2)
# XtX + lambda*penmat = diag(c(7, 1, 7)/6
# so coef(fd1.3.5) = solve(XtX + lambda*penmat, Xt.y)
# = c(6/7, 0, 12/7)
##
## linear spline fit
##
# 3 bases, order 2 = degree 1
b2.3 <- create.bspline.basis(norder=2, breaks=c(0, .5, 1))
# interpolate the values 0, 2, 1
fd2.3 <- Data2fd(c(0,2,1), basisobj=b2.3, lambda=0)
# display the coefficients
round(fd2.3$coefs, 4)
# plot the results
op <- par(mfrow=c(2,1))
plot(b2.3, main='bases')
plot(fd2.3, main='fit')
par(op)
# apply some smoothing
fd2.3. <- Data2fd(c(0,2,1), basisobj=b2.3, lambda=1)
op <- par(mfrow=c(2,1))
plot(b2.3, main='bases')
plot(fd2.3., main='fit', ylim=c(0,2))
par(op)
##
## quadratic spline fit
##
# 4 bases, order 3 = degree 2 = continuous, bounded, locally quadratic
b3.4 <- create.bspline.basis(norder=3, breaks=c(0, .5, 1))
# fit values c(0,4,2,3) without interpolation
fd3.4 <- Data2fd(c(0,4,2,3), basisobj=b3.4, lambda=0)
round(fd3.4$coefs, 4)
op <- par(mfrow=c(2,1))
plot(b3.4)
plot(fd3.4)
points(c(0,1/3,2/3,1), c(0,4,2,3))
par(op)
# try smoothing
fd3.4. <- Data2fd(c(0,4,2,3), basisobj=b3.4, lambda=1)
round(fd3.4.$coef, 4)
op <- par(mfrow=c(2,1))
plot(b3.4)
plot(fd3.4., ylim=c(0,4))
points(seq(0,1,len=4), c(0,4,2,3))
par(op)
##
## Two simple Fourier examples
##
gaitbasis3 <- create.fourier.basis(nbasis=5)
gaitfd3 <- Data2fd(gait, basisobj=gaitbasis3)
plotfit.fd(gait, seq(0,1,len=20), gaitfd3)
# set up the fourier basis
daybasis <- create.fourier.basis(c(0, 365), nbasis=65)
# Make temperature fd object
# Temperature data are in 12 by 365 matrix tempav
# See analyses of weather data.
tempfd <- Data2fd(CanadianWeather$dailyAv[,,"Temperature.C"],
day.5, daybasis)
# plot the temperature curves
par(mfrow=c(1,1))
plot(tempfd)
##
## argvals of class Date and POSIXct
##
# Date
invasion1 <- as.Date('1775-09-04')
invasion2 <- as.Date('1812-07-12')
earlyUS.Canada <- c(invasion1, invasion2)
BspInvasion <- create.bspline.basis(earlyUS.Canada)
earlyYears <- seq(invasion1, invasion2, length.out=7)
(earlyQuad <- (as.numeric(earlyYears-invasion1)/365.24)^2)
fitQuad <- Data2fd(earlyYears, earlyQuad, BspInvasion)
# POSIXct
AmRev.ct <- as.POSIXct1970(c('1776-07-04', '1789-04-30'))
BspRev.ct <- create.bspline.basis(AmRev.ct)
AmRevYrs.ct <- seq(AmRev.ct[1], AmRev.ct[2], length.out=14)
(AmRevLin.ct <- as.numeric(AmRevYrs.ct-AmRev.ct[1]))
fitLin.ct <- Data2fd(AmRevYrs.ct, AmRevLin.ct, BspRev.ct)
fda documentation built on May 2, 2019, 5:12 p.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 Details Value References See Also Examples
Description
This function converts an array y of function values plus an
array argvals of argument values into a functional data object.
This function tries to do as much for the user as possible in setting
up a call to function smooth.basis. Be warned that the result
may not be a satisfactory smooth of the data, and consequently that it
may be necessary to use function smooth.basis instead, the help
file for which provides a great deal more information than is provided
here.
Also, function Data2fd can swap the first two arguments,
argvals and y if it appears that they have been included
in reverse order. A warning message is returned if this swap takes place.
Any such automatic decision, though, has the possibility of being wrong,
and the results should be carefully checked. Preferably, the order of
the arguments should be respected: argvals comes first and
y comes second.
Usage
1 2 3 4 |
Arguments
argvals |
a set of argument values. If this is a vector, the same set of
argument values is used for all columns of Dimensions for |
y |
an array containing sampled values of curves. If |
basisobj |
One of the following:
|
nderiv |
Smoothing typically specified as an integer order for the derivative
whose square is integrated and weighted by A general linear differential operator can also be supplied. |
lambda |
weight on the smoothing operator specified by |
fdnames |
Either a charater vector of length 3 or a named list of length 3. In either case, the three elements correspond to the following:
If fdnames is a list, the components provide labels for the levels
of the corresponding dimension of |
covariates |
the observed values in |
method |
by default the function uses the usual textbook equations for computing
the coefficients of the basis function expansions. But, as in regression
analysis, a price is paid in terms of rounding error for such
computations since they involved cross-products of basis function
values. Optionally, if |
dfscale |
the generalized cross-validation or "gcv" criterion that is often used
to determine the size of the smoothing parameter involves the
subtraction of an measue of degrees of freedom from |
Details
This function tends to be used in rather simple applications where
there is no need to control the roughness of the resulting curve with
any great finesse. The roughness is essentially controlled by how
many basis functions are used. In more sophisticated applications, it
would be better to use the function smooth.basisPar.
Value
an object of the fd class containing:
coefs |
the coefficient array |
basis |
a basis object |
fdnames |
a list containing names for the arguments, function values and variables |
References
Ramsay, James O., and Silverman, Bernard W. (2006), Functional Data Analysis, 2nd ed., Springer, New York.
Ramsay, James O., and Silverman, Bernard W. (2002), Applied Functional Data Analysis, Springer, New York.
See Also
smooth.basisPar,
smooth.basis,
project.basis,
smooth.fd,
smooth.monotone,
smooth.pos
day.5
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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 | ##
## Simplest possible example: constant function
##
# 1 basis, order 1 = degree 0 = constant function
b1.1 <- create.bspline.basis(nbasis=1, norder=1)
# data values: 1 and 2, with a mean of 1.5
y12 <- 1:2
# smooth data, giving a constant function with value 1.5
fd1.1 <- Data2fd(y12, basisobj=b1.1)
plot(fd1.1)
# now repeat the analysis with some smoothing, which moves the
# toward 0.
fd1.1.5 <- Data2fd(y12, basisobj=b1.1, lambda=0.5)
# values of the smooth:
# fd1.1.5 = sum(y12)/(n+lambda*integral(over arg=0 to 1 of 1))
# = 3 / (2+0.5) = 1.2
eval.fd(seq(0, 1, .2), fd1.1.5)
##
## step function smoothing
##
sessionInfo()
find("crossprod")
# 2 step basis functions: order 1 = degree 0 = step functions
b1.2 <- create.bspline.basis(nbasis=2, norder=1)
# fit the data without smoothing
fd1.2 <- Data2fd(1:2, basisobj=b1.2)
# plot the result: A step function: 1 to 0.5, then 2
op <- par(mfrow=c(2,1))
plot(b1.2, main='bases')
plot(fd1.2, main='fit')
par(op)
##
## Simple oversmoothing
##
# 3 step basis functions: order 1 = degree 0 = step functions
b1.3 <- create.bspline.basis(nbasis=3, norder=1)
# smooth the data with smoothing
fd1.3.5 <- Data2fd(y12, basisobj=b1.3, lambda=0.5)
# plot the fit along with the points
plot(0:1, c(0, 2), type='n')
points(0:1, y12)
lines(fd1.3.5)
# Fit = penalized least squares with penalty =
# = lambda * integral(0:1 of basis^2),
# which shrinks the points towards 0.
# X1.3 = matrix(c(1,0, 0,0, 0,1), 2)
# XtX = crossprod(X1.3) = diag(c(1, 0, 1))
# penmat = diag(3)/3
# = 3x3 matrix of integral(over arg=0:1 of basis[i]*basis[j])
# Xt.y = crossprod(X1.3, y12) = c(1, 0, 2)
# XtX + lambda*penmat = diag(c(7, 1, 7)/6
# so coef(fd1.3.5) = solve(XtX + lambda*penmat, Xt.y)
# = c(6/7, 0, 12/7)
##
## linear spline fit
##
# 3 bases, order 2 = degree 1
b2.3 <- create.bspline.basis(norder=2, breaks=c(0, .5, 1))
# interpolate the values 0, 2, 1
fd2.3 <- Data2fd(c(0,2,1), basisobj=b2.3, lambda=0)
# display the coefficients
round(fd2.3$coefs, 4)
# plot the results
op <- par(mfrow=c(2,1))
plot(b2.3, main='bases')
plot(fd2.3, main='fit')
par(op)
# apply some smoothing
fd2.3. <- Data2fd(c(0,2,1), basisobj=b2.3, lambda=1)
op <- par(mfrow=c(2,1))
plot(b2.3, main='bases')
plot(fd2.3., main='fit', ylim=c(0,2))
par(op)
##
## quadratic spline fit
##
# 4 bases, order 3 = degree 2 = continuous, bounded, locally quadratic
b3.4 <- create.bspline.basis(norder=3, breaks=c(0, .5, 1))
# fit values c(0,4,2,3) without interpolation
fd3.4 <- Data2fd(c(0,4,2,3), basisobj=b3.4, lambda=0)
round(fd3.4$coefs, 4)
op <- par(mfrow=c(2,1))
plot(b3.4)
plot(fd3.4)
points(c(0,1/3,2/3,1), c(0,4,2,3))
par(op)
# try smoothing
fd3.4. <- Data2fd(c(0,4,2,3), basisobj=b3.4, lambda=1)
round(fd3.4.$coef, 4)
op <- par(mfrow=c(2,1))
plot(b3.4)
plot(fd3.4., ylim=c(0,4))
points(seq(0,1,len=4), c(0,4,2,3))
par(op)
##
## Two simple Fourier examples
##
gaitbasis3 <- create.fourier.basis(nbasis=5)
gaitfd3 <- Data2fd(gait, basisobj=gaitbasis3)
plotfit.fd(gait, seq(0,1,len=20), gaitfd3)
# set up the fourier basis
daybasis <- create.fourier.basis(c(0, 365), nbasis=65)
# Make temperature fd object
# Temperature data are in 12 by 365 matrix tempav
# See analyses of weather data.
tempfd <- Data2fd(CanadianWeather$dailyAv[,,"Temperature.C"],
day.5, daybasis)
# plot the temperature curves
par(mfrow=c(1,1))
plot(tempfd)
##
## argvals of class Date and POSIXct
##
# Date
invasion1 <- as.Date('1775-09-04')
invasion2 <- as.Date('1812-07-12')
earlyUS.Canada <- c(invasion1, invasion2)
BspInvasion <- create.bspline.basis(earlyUS.Canada)
earlyYears <- seq(invasion1, invasion2, length.out=7)
(earlyQuad <- (as.numeric(earlyYears-invasion1)/365.24)^2)
fitQuad <- Data2fd(earlyYears, earlyQuad, BspInvasion)
# POSIXct
AmRev.ct <- as.POSIXct1970(c('1776-07-04', '1789-04-30'))
BspRev.ct <- create.bspline.basis(AmRev.ct)
AmRevYrs.ct <- seq(AmRev.ct[1], AmRev.ct[2], length.out=14)
(AmRevLin.ct <- as.numeric(AmRevYrs.ct-AmRev.ct[1]))
fitLin.ct <- Data2fd(AmRevYrs.ct, AmRevLin.ct, BspRev.ct)
|
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.