Functional time series: dynamic FPCA

Description

Implementation of dynamic functional principle component analysis (FDPCA), simulation of functional AR and functional MA processes and frequency domain tools for funcional data. The package is a wrapper for functionality of the multivariate package freqdom for applying frequency domain on objects from fda. Compared to freqdom some new visualization methods are added – adequate only if data has functional structure.

Details

fda.ts package allows you to analyse functional time series objects in both time and frequency domain. The main feature is dynamic functional principal component analysis. This method allows to transform a stationary functional time series into a vector process with mutually uncorrelated component processes.

There are two key differnces between classical PCA and dynamic PCA:

• Component processes returned by the dynamic procedure are mutually uncorrelated,

• The mapping maximizes the long run variance of compoments, which, in case of stationary functional time series, means that the process reconstructed from and d > 0 first dynamic principal components better approximates the original functional time series process than the first d classic principal components.

For functional data one can conveniently visualize properties of the filters, covariances or the spectral density operator.

For details we refer to the literature below and to help pages of functions fts.dpca for estimating the components, fts.dpca.scores for estimating scores and fts.dpca.KLexpansion for retrieving the signal from components.

The package fda.ts require the package freqdom provides the analogue multivariate toolset.

References

Hormann Siegfried, Kidzinski Lukasz and Hallin Marc. Dynamic functional principal components. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 77.2 (2015): 319-348.

Hormann Siegfried, Kidzinski Lukasz and Kokoszka Piotr. Estimation in functional lagged regression. Journal of Time Series Analysis 36.4 (2015): 541-561.

Hormann Siegfried and Kidzinski Lukasz. A note on estimation in Hilbertian linear models. Scandinavian journal of statistics 42.1 (2015): 43-62.

Estimate autocovariance and cross-covariances operators

Description

This function is used to estimate a collection of cross-covariances operators of two stationary functional series.

Usage

fts.cov.structure(X, Y = X, lags = 0)

Arguments

Details

Let X_1(u),\ldots, X_T(u) and Y_1(u),\ldots, Y_T(u) be two samples of functional data. This function determines empirical lagged covariances between the series (X_t(u)) and (Y_t(u)). More precisely it determines

(\widehat{c}^{XY}_h(u,v)\colon h\in lags ),

where \widehat{c}^{XY}_h(u,v) is the empirical version of the covariance kernel \mathrm{Cov}(X_h(u),Y_0(v)). For a sample of size T we set \hat\mu^X(u)=\frac{1}{T}\sum_{t=1}^T X_t(u) and \hat\mu^Y(v)=\frac{1}{T}\sum_{t=1}^T Y_t(v). Now for h \geq 0

\frac{1}{T}\sum_{t=1}^{T-h} (X_{t+h}(u)-\hat\mu^X(u))(Y_t(v)-\hat\mu^Y(v))

and for h < 0

\frac{1}{T}\sum_{t=|h|+1}^{T} (X_{t+h}(u)-\hat\mu^X(u))(Y_t(v)-\hat\mu^Y(v)).

Since X_t(u)=\boldsymbol{b}_1^\prime(u)\mathbf{x}_t and Y_t(u)=\mathbf{y}_t^\prime \boldsymbol{b}_2(u) we can write

\widehat{c}^{XY}_h(u,v)=\boldsymbol{b}_1^\prime(u)\widehat{C}^{\mathbf{xy}}\boldsymbol{b}_2(v),

where \widehat{C}^{\mathbf{xy}} is defined as for the function “cov.structure” for series of coefficient vectors (\mathbf{x}_t\colon 1\leq t\leq T) and (\mathbf{y}_t\colon 1\leq t\leq T).

Value

An object of class fts.timedom. The list contains the following components:

• operators \quad an array. Element [,,k] contains the covariance matrix of the coefficient vectors of the two time series related to lag \ell_k.

• lags \quad the lags vector from the arguments.

• basisX \quad X$basis, an object of class basis.fd (see create.basis)

• basisY \quad Y$basis, an object of class basis.fd (see create.basis)

See Also

The multivariate equivalent in the freqdom package: cov.structure

Examples

# Generate an autoregressive process
fts = fts.rar(d=3)

# Get covariance at lag 0
fts.cov.structure(fts, lags = 0)

# Get covariance at lag 10
fts.cov.structure(fts, lags = 10)

# Get entire covariance structure between -20 and 20
fts.cov.structure(fts, lags = -20:20)

# Compute covariance with another process
fts0 = fts + fts.rma(d=3)
fts.cov.structure(fts, fts0, lags = -2:2)

Compute Functional Dynamic Principal Components and dynamic Karhunen Loeve extepansion

Description

Functional dynamic principal component analysis (FDPCA) decomposes functional time series to a vector time series with uncorrelated components. Compared to classical functional principal components, FDPCA decomposition outputs components which are uncorrelated in time, allowing simpler modeling of the processes and maximizing long run variance of the projection.

Usage

fts.dpca(X, q = 30, freq = (-1000:1000/1000) * pi, Ndpc = X$basis$nbasis)

Arguments

Details

This convenient function applies the FDPCA methodology and returns filters (fts.dpca.filters), scores (fts.dpca.scores), the spectral density (fts.spectral.density), variances (fts.dpca.var) and Karhunen-Leove expansion (fts.dpca.KLexpansion).

See the example for understanding usage, and help pages for details on individual functions.

Value

A list containing

• scores \quad DPCA scores (fts.dpca.scores)

• filters \quad DPCA filters (fts.dpca.filters)

• spec.density \quad spectral density of X (fts.spectral.density)

• var \quad amount of variance explained by dynamic principal components (fts.dpca.var)

• Xhat \quad Karhunen-Loeve expansion using Ndpc dynamic principal components (fts.dpca.KLexpansion)

References

Hormann, S., Kidzinski, L., and Hallin, M. Dynamic functional principal components. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 77.2 (2015): 319-348.

Brillinger, D. Time Series (2001), SIAM, San Francisco.

Shumway, R., and Stoffer, D. Time series analysis and its applications: with R examples (2010), Springer Science & Business Media

Examples

# Load example PM10 data from Graz, Austria
data(pm10) # loads functional time series pm10 to the environment
X = center.fd(pm10)

# Compute functional dynamic principal components with only one component
res.dpca = fts.dpca(X, Ndpc = 1, freq=(-25:25/25)*pi) # leave default freq for higher precision
plot(res.dpca$Xhat)
fts.plot.filters(res.dpca$filters)

# Compute functional PCA with only one component
res.pca = prcomp(t(X$coefs), center = TRUE)
res.pca$x[,-1] = 0

# Compute empirical variance explained
var.dpca = (1 - sum( (res.dpca$Xhat$coefs - X$coefs)**2 ) / sum(X$coefs**2))*100
var.pca = (1 - sum( (res.pca$x %*% t(res.pca$rotation) - t(X$coefs) )**2 ) / sum(X$coefs**2))*100

cat("Variance explained by PCA (empirical):\t\t",var.pca,"%\n")
cat("Variance explained by PCA (theoretical):\t",
   (1 - (res.pca$sdev[1] / sum(res.pca$sdev)))*100,"%\n")
cat("Variance explained by DPCA (empirical):\t\t",var.dpca,"%\n")
cat("Variance explained by DPCA (theoretical):\t",(res.dpca$var[1])*100,"%\n")

# Plot filters
fts.plot.filters(res.dpca$filters)

# Plot spectral density (note that in case of these data it's concentrated around 0)
fts.plot.operators(res.dpca$spec.density,freq = c(-2,-3:3/30 * pi,2))

# Plot covariance of X
fts.plot.covariance(X)

# Compare values of the first PC scores with the first DPC scores 
plot(res.pca$x[,1],t='l',xlab = "Time",ylab="Score", lwd = 2.5)
lines(res.dpca$scores[,1], col=2, lwd = 2.5)
legend(0,4,c("first PC score","first DPC score"), # puts text in the legend
       lty=c(1,1),lwd=c(2.5,2.5), col=1:2)

Dynamic KL expansion

Description

Computes the dynamic KL expansion up to a given order.

Usage

fts.dpca.KLexpansion(X, dpcs = fts.dpca.filters(fts.spectral.density(X)))

Arguments

Details

This function computes the L-order dynamic functional principal components expansion, defined by

\hat{X}_{t}^L(u):=\sum_{\ell=1}^L\sum_{k\in\mathbf{Z}} Y_{\ell,t+k} \phi_{\ell k}(u),\quad 1\leq L\leq d,

where \phi_{\ell k}(v) and d are explained in fts.dpca.filters and Y_{\ell k} are the dynamic functional PC scores as in fts.dpca.scores. For the sample version the sum extends over the range of lags for which the \phi_{\ell k} are defined.

For more details we refer to Hormann et al. (2015).

Value

An object of class fd.

References

Hormann, S., Kidzinski, L., and Hallin, M. Dynamic functional principal components. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 77.2 (2015): 319-348.

See Also

The multivariate equivalent in the freqdom package: dpca.KLexpansion

Functional dynamic PCA filters

Description

From a given spectral density operator the dynamic principal component filter sequences are computed.

Usage

fts.dpca.filters(F, Ndpc = F$basisX$nbasis, q = 30)

Arguments

Details

Dynamic principal components are linear filters (\phi_{\ell k}(u)\colon k\in \mathbf{Z}), 1\leq\ell\leq d. They are defined as the Fourier coefficients of the dynamic eigenvector \varphi_\ell(\omega)(u) of a spectral density kernel f_\omega(u,v), i.e. \int_0^1 f_\omega(u,v)\varphi_\ell(\omega)(v)dv=\lambda_\ell(\omega)\varphi_\ell(\omega)(u) and

\phi_{\ell k}(u):=\frac{1}{2\pi}\int_{-\pi}^\pi \varphi_\ell(\omega)(u) \exp(-ik\omega) d\omega.

The index \ell is referring to the \ell-th largest dynamic eigenvalue \lambda_\ell(\omega). For a given spectral density operator (provided as on object of class fts.freqdom) the function fts.dpca.filters computes \phi_{\ell k}(u) for |k|\leq q. Filters will be computed for 1\leq \ell\leq \code{Ndpc}.

For more details we refer to Hormann et al. (2015).

Value

An object of class fts.timedom. The list has the following components:

• operators \quad an array. Each matrix in this array has dimension \code{Ndpc}\times d and is assigned to a certain lag. For a given lag k, the rows of the matrix correspond to the coefficient vector of the filter functions.

• lags \quad a vector with the lags of the filter coefficients.

• basisX \quad F$basis, hence an object of class basis.fd (see create.basis).

• correspondence \quad the correspondence matrix: all scalar products between basis functions.

References

Hormann, S., Kidzinski, L., and Hallin, M. Dynamic functional principal components. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 77.2 (2015): 319-348.

Brillinger, D. Time Series (2001), SIAM, San Francisco.

Shumway, R.H., and Stoffer, D.S. Time Series Analysis and Its Applications (2006), Springer, New York.

See Also

The multivariate equivalent in the freqdom package: dpca.filters

Examples

data(pm10)
X = center.fd(pm10)

# Compute the spectral density operator with Bartlett weights
SD = fts.spectral.density(X, freq = (-50:50/50) * pi, q = 2, weight="Bartlett")
filters = fts.dpca.filters(SD, 2, q = 10)

# Plot filters 1 and 2
fts.plot.filters(filters, 2, one.plot = TRUE)

# Recompute with a different estimate of the spectral density (largerg q)
SD = fts.spectral.density(X, freq = (-50:50/50) * pi, q = 5, weight="Bartlett")
filters = fts.dpca.filters(SD, 2, q = 10)

# Plot filters 1 and 2
fts.plot.filters(filters, 2, one.plot = TRUE)

Functional dynamic principal component scores

Description

Computes the dynamic principal component scores of a functional time series.

Usage

fts.dpca.scores(X, dpcs = fts.dpca.filters(spectral.density(X)))

Arguments

Details

The \ell-th dynamic principal components score sequence is defined by

Y_{\ell t}:=\sum_{k\in\mathbf{Z}} \int_0^1 \phi_{\ell k}(v) X_{t-k}(v)dv,\quad 1\leq \ell\leq d,

where \phi_{\ell k}(v) and d are explained in fts.dpca.filters. (The integral is not necessarily restricted to the interval [0,1], this depends on the data.) For the sample version the sum extends over the range of lags for which the \phi_{\ell k} are defined.

For more details we refer to Hormann et al. (2015).

Value

A (T\times \code{Ndpc})-matix with Ndpc = dim(dpcs$operators)[1]. The \ell-th column contains the \ell-th dynamic principal component score sequence.

References

Hormann, S., Kidzinski, L., and Hallin, M. Dynamic functional principal components. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 77.2 (2015): 319-348.

See Also

The multivariate equivalent in the freqdom package: dpca.scores

Proportion of variance explained by dynamic principal components

Description

Computes the proportion and cumulative proportion of variance explained by dynamic principal components.

Usage

fts.dpca.var(F)

Arguments

Details

Consider a spectral density operator \mathcal{F}_\omega and let \lambda_\ell(\omega) by the \ell-th dynamic eigenvalue. The proportion of variance described by the \ell-th dynamic principal component is given as v_\ell:=\int_{-\pi}^\pi \lambda_\ell(\omega)d\omega/\int_{-\pi}^\pi \mathrm{tr}(\mathcal{F}_\omega)d\omega. This function numerically computes the vectors (v_\ell).

For more details we refer to Hormann et al. (2015).

Value

A vector containing the v_\ell.

References

Hormann, S., Kidzinski, L., and Hallin, M. Dynamic functional principal components. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 77.2 (2015): 319-348.

See Also

The multivariate equivalent in the freqdom package: dpca.var

Creates an object of class fts.freqdom.

Description

Creates an object of class fts.freqdom.

Usage

fts.freqdom(F, basisX, basisY = basisX)

Arguments

Details

This class is used to describe a frequency domain operator (for example a spectral density operator) on selected frequencies. Formally we consider an object of class freqdom and add some basis functions. Depending on the context, we have different interpretations for the new object.

(I) In order to define an operator which maps between two functions spaces, the we interpret F$operators as coefficients in the basis function expansion of the kernel of some finite rank operators

\mathcal{F}_k:\mathrm{span}(\code{basisY})+\mathrm{i}\, \mathrm{span}(\code{basisY})\to\mathrm{span}(\code{basisX})+\mathrm{i}\, \mathrm{span}(\code{basisX}).

The kernels are f_k(u,v)=\boldsymbol{b}_1^\prime(u)\, F_k\, \boldsymbol{b}_2(v), where \boldsymbol{b_1}(u)=(b_{11}(u),\ldots, b_{1d_1}(u))^\prime and \boldsymbol{b_2}(u)=(b_{21}(u),\ldots, b_{2d_1}(u))^\prime are the basis functions provided by the arguments basisX and basisY, respectively. Moreover, we consider frequencies \{\omega_1,\ldots, \omega_K\}\subset[-\pi,\pi]. The object this function creates corresponds to the mapping \omega_k \mapsto f_k(u,v).

(II) We may ignore basisX, and represent the linear mapping

\mathcal{F}_k:\mathrm{span}(\code{basisY})+\mathrm{i}\, \mathrm{span}(\code{basisY})\to C^{d_1},

by considering f_k(v):=F_k\,\boldsymbol{b}_2(v) and \mathcal{F}_k(x)=\int f_k(v)x(v)dv.

(III) We may ignore basisY, and represent the linear mapping

\mathcal{F}_k: C^{d_1}\to\mathrm{span}(\code{basisX})+\mathrm{i}\, \mathrm{span}(\code{basisX}),

by considering f_k(u):=\boldsymbol{b}_1^\prime(u)F_k and \mathcal{F}_k(y)=f_k(u)y.

Value

Returns an object of class fts.freqdom. An object of class fts.freqdom is a list containing the following components:

• operators \quad returns the array F$operators.

• basisX \quad returns basisX as given in the argument.

• basisY \quad returns basisY as given in the argument.

• freq \quad returns the vector F$freq.

See Also

The multivariate equivalent in the freqdom package: freqdom

Contour plot for the kernels of cross-covariance operators.

Description

Contour plot for the kernels of cross-covariance operators.

Usage

fts.plot.covariance(X, Y = X, cor = FALSE, res = 200, lags = 0:3, nlevels = 25)

Arguments

Examples

fts = fts.rar(100)

# Plot covariance operators of the time series curves
# We chose resolution equal 150 for better precision 
fts.plot.covariance(fts, lags=0:2, res = 150) 

# Plot correlation operators of the time series curves
fts.plot.covariance(fts, lags=0:2, cor = TRUE, res = 50)

# Make the grid of levels more dense
fts.plot.covariance(fts, lags=0:1, nlevels = 100)

Plot kernels

Description

Plot kernels

Usage

fts.plot.filters(A, Ndpc = 1, lags = -3:3, one.plot = FALSE, ...)

Arguments

Examples

# Load example PM10 data from Graz, Austria
data(pm10) # loads functional time series pm10 to the environment
X = center.fd(pm10)

# Compute functional dynamic principal components with only one component
res.dpca = fts.dpca(X, Ndpc = 1, freq=(-25:25/25)*pi) # leave default freq for higher precision

# Plot Functional Dynamic Principal Component Filters
fts.plot.filters(res.dpca$filters)

Contour plot of operator kernels.

Description

Contour plot of operator kernels.

Usage

fts.plot.operators(A, res = 200, lags = 0, freq = 0, axis = "Re", nlevels = 25)

Arguments

Examples

# Load example PM10 data from Graz, Austria
data(pm10) # loads functional time series pm10 to the environment
X = center.fd(pm10)

# Compute functional dynamic principal components with only one component
res.dpca = fts.dpca(X, Ndpc = 1, freq=(-25:25/25)*pi) # leave default freq for higher precision

# Plot the spectral density operator at frequencies -2, -3:3/30 * pi and 2
fts.plot.operators(res.dpca$spec.density,freq = c(-2,-3:3/30 * pi,2))

Simulate functional autoregressive processes

Description

Generates functional autoregressive process.

Usage

fts.rar(
  n = 100,
  d = 11,
  Psi = NULL,
  op.norms = NULL,
  burnin = 20,
  noise = "mnorm",
  sigma = diag(d:1)/d,
  df = 4
)

Arguments

Details

The purpose is to simulate a functional autoregressive process of the form

X_t(u)=\sum_{k=1}^p \int_0^1\Psi_k(u,v) X_{t-k}(v)dv+\varepsilon_t(u),\quad 1\leq t\leq n.

Here we assume that the observations lie in a finite dimensional subspace of the function space spanned by Fourier basis functions \boldsymbol{b}^\prime(u)=(b_1(u),\ldots, b_d(u)). That is X_t(u)=\boldsymbol{b}^\prime(u)\boldsymbol{X}_t, \varepsilon_t(u)=\boldsymbol{b}^\prime(u)\boldsymbol{\varepsilon}_t and \Psi_k(u,v)=\boldsymbol{b}^\prime(u)\boldsymbol{\Psi}_k \boldsymbol{b}(v). Then it follows that

\boldsymbol{X}_t=\boldsymbol{\Psi}_1\boldsymbol{X}_{t-1}+\cdots+ \boldsymbol{\Psi}_p\boldsymbol{X}_{t-p}+\boldsymbol{\varepsilon}_t.

Hence the dynamic of the functional time series is described by a VAR(p) process.

In this mathematical model the law of \boldsymbol{\varepsilon}_t is determined by noise. The matrices Psi[,,k] correspond to \boldsymbol{\Psi}_k. If op.norms is provided, then the coefficient matrices will be rescaled, such that the Hilbert-Schmidt norms of \boldsymbol{\Psi}_k correspond to the vector.

Value

An object of class fd.

See Also

The multivariate equivalent in the freqdom package: rar

Examples

# Generate a FAR process without burnin (starting from 0)
fts = fts.rar(n = 5, d = 5, burnin = 0)
plot(fts)

# Generate a FAR process with burnin 50 (starting from observations
# already resambling the final distribution)
fts = fts.rar(n = 5, d = 5, burnin = 50)
plot(fts)

# Generate observations with very strong dependance
fts = fts.rar(n = 100, d = 5, burnin = 50, op.norms = 0.999)
plot(fts)

Simulate functional moving average processes

Description

Generate a functional moving average process.

Usage

fts.rma(
  n = 100,
  d = 11,
  Psi = NULL,
  op.norms = NULL,
  noise = "mnorm",
  sigma = diag(d:1)/d,
  df = 4
)

Arguments

Details

The purpose is to simulate a functional autoregressive process of the form

X_t(u)=\sum_{k=1}^p \int_0^1\Psi_k(u,v) X_{t-k}(v)dv+\varepsilon_t(u),\quad 1\leq t\leq n.

Here we assume that the observations lie in a finite dimensional subspace of the function space spanned by Fourier basis functions \boldsymbol{b}^\prime(u)=(b_1(u),\ldots, b_d(u)). That is X_t(u)=\boldsymbol{b}^\prime(u)\boldsymbol{X}_t, \varepsilon_t(u)=\boldsymbol{b}^\prime(u)\boldsymbol{\varepsilon}_t and \Psi_k(u,v)=\boldsymbol{b}^\prime(u)\boldsymbol{\Psi}_k \boldsymbol{b}(v). Then it follows that

\boldsymbol{X}_t=\boldsymbol{\Psi}_1\boldsymbol{X}_{t-1}+\cdots+ \boldsymbol{\Psi}_p\boldsymbol{X}_{t-p}+\boldsymbol{\varepsilon}_t.

Hence the dynamic of the functional time series is described by a VAR(p) process.

In this mathematical model the law of \boldsymbol{\varepsilon}_t is determined by noise. The matrices Psi[,,k] correspond to \boldsymbol{\Psi}_k. If op.norms is provided, then the coefficient matrices will be rescaled, such that the Hilbert-Schmidt norms of \boldsymbol{\Psi}_k correspond to the vector.

Value

An object of class fd.

See Also

The multivariate equivalent in the freqdom package: rma

Examples

# Generate a FMA process without burnin (starting from 0)
fts = fts.rma(n = 5, d = 5)
plot(fts)

# Generate observations with very strong dependance
fts = fts.rma(n = 100, d = 5, op.norms = 0.999)
plot(fts)

# Generate observations with very strong dependance and noise
# from the multivariate t distribution
fts = fts.rma(n = 100, d = 5, op.norms = 0.999, noise = "mt")
plot(fts)

Functional spectral and cross-spectral density operator

Description

Estimates the spectral density operator and cross spectral density operator of functional time series.

Usage

fts.spectral.density(
  X,
  Y = X,
  freq = (-1000:1000/1000) * pi,
  q = ceiling((dim(X$coefs)[2])^{     0.33 }),
  weights = "Bartlett"
)

Arguments

Details

Let X_1(u),\ldots, X_T(u) and Y_1(u),\ldots, Y_T(u) be two samples of functional data. The cross-spectral density kernel between the two time series (X_t(u)) and (Y_t(u)) is defined as

f^{XY}_\omega(u,v)=\sum_{h\in\mathbf{Z}} \mathrm{Cov}(X_h(u),Y_0(v)) e^{-ih\omega}.

The function fts.spectral.density determines the empirical cross-spectral density kernel between the two time series. The estimator is of the form

\widehat{f}^{XY}_\omega(u,v)=\sum_{|h|\leq q} w(|k|/q)\widehat{c}^{XY}_h(u,v)e^{-ih\omega},

with \widehat{c}^{XY}_h(u,v) defined in fts.cov.structure. The other paremeters are as in cov.structure.

Since X_t(u)=\boldsymbol{b}_1^\prime(u)\mathbf{x}_t and Y_t(u)=\mathbf{y}_t^\prime \boldsymbol{b}_2(u) we can write

\widehat{f}^{XY}_\omega(u,v)=\boldsymbol{b}_1^\prime(u)\widehat{\mathcal{F}}^{\mathbf{xy}}(\omega)\boldsymbol{b}_2(v),

where \widehat{\mathcal{F}}^{\mathbf{xy}}(\omega) is defined as for the function spectral.density for series of coefficient vectors (\mathbf{x}_t\colon 1\leq t\leq T) and (\mathbf{y}_t\colon 1\leq t\leq T).

Value

Returns an object of class fts.timedom. The list is containing the following components:

• operators \quad an array. Element [,,k] in the coefficient matrix of the spectral density matrix evaluated at the k-th frequency listed in freq.

• lags \quad returns the lags vector from the arguments.

• basisX \quad returns X$basis, an object of class basis.fd (see create.basis).

• basisY \quad returns Y$basis, an object of class basis.fd (see create.basis)

See Also

The multivariate equivalent in the freqdom package: spectral.density

Examples

data(pm10)
X = center.fd(pm10)

# Compute the spectral density operator with Bartlett weights
SD = fts.spectral.density(X, freq = (-50:50/50) * pi, q = 2, weight="Bartlett")
fts.plot.operators(SD, freq = -2:2)

# Compute the spectral density operator with Tukey weights
SD = fts.spectral.density(X, freq = (-50:50/50) * pi, q = 2, weight="Tukey")
fts.plot.operators(SD, freq = -2:2)
# Note relatively small difference between the two plots

# Now, compute the spectral density operator with Tukey weights and larger q
SD = fts.spectral.density(X, freq = (-50:50/50) * pi, q = 5, weight="Tukey")
fts.plot.operators(SD, freq = -2:2)

Object of class fts.timedom

Description

Creates an object of class fts.timedom. This object corresponds to a sequence of linear operators.

Usage

fts.timedom(A, basisX, basisY = basisX)

Arguments

Details

This class is used to describe a functional linear filter, i.e. a sequence of linear operators, each of which corresponds to a certain lag. Formally we consider an object of class timedom and add some basis functions. Depending on the context, we have different interpretations for the new object.

(I) In order to define operators which maps between two functions spaces, the we interpret A$operators as coefficients in the basis function expansion of the kernel of some finite rank operators

\mathcal{A}_k:\mathrm{span}(\code{basisY})\to\mathrm{span}(\code{basisX}).

The kernels are a_k(u,v)=\boldsymbol{b}_1^\prime(u)\, A_k\, \boldsymbol{b}_2(v), where \boldsymbol{b_1}(u)=(b_{11}(u),\ldots, b_{1d_1}(u))^\prime and \boldsymbol{b_2}(u)=(b_{21}(u),\ldots, b_{2d_1}(u))^\prime are the basis functions provided by the arguments basisX and basisY, respectively. Moreover, we consider lags \ell_1<\ell_2<\cdots<\ell_K. The object this function creates corresponds to the mapping \ell_k \mapsto a_k(u,v).

(II) We may ignore basisX, and represent the linear mapping

\mathcal{A}_k:\mathrm{span}(\code{basisY})\to R^{d_1},

by considering a_k(v):=A_k\,\boldsymbol{b}_2(v) and \mathcal{A}_k(x)=\int a_k(v)x(v)dv.

(III) We may ignore basisY, and represent the linear mapping

\mathcal{A}_k: R^{d_1}\to\mathrm{span}(\code{basisX}),

by considering a_k(u):=\boldsymbol{b}_1^\prime(u)A_k and \mathcal{A}_k(y)=a_k(u)y.

Value

Returns an object of class fts.freqdom. An object of class fts.freqdom is a list containing the following components:

• operators \quad returns the array A$operators.

• basisX \quad returns basisX as given in the argument.

• basisY \quad returns basisY as given in the argument.

• lags \quad returns A$lags.

See Also

The multivariate equivalent in the freqdom package: timedom

Truncate functional timedom object

Description

Truncate functional timedom object

Usage

fts.timedom.trunc(A, lags)

Checks if an object belongs to the class fts.freqdom

Description

Checks if an object belongs to the class fts.freqdom.

Usage

is.fts.freqdom(X)

Arguments

Value

TRUE if X is of type fts.freqdom, FALSE otherwise

See Also

fts.freqdom, fts.timedom, is.fts.timedom

Checks if an object belongs to the class fts.timedom

Description

Checks if an object belongs to the class fts.timedom.

Usage

is.fts.timedom(X)

Arguments

Value

TRUE if X is of type fts.timedom, FALSE otherwise

See Also

fts.freqdom, fts.timedom, is.fts.freqdom

PM10 dataset

Description

Concentration of partical matter of diameter 10 micrometers or smaller (PM10) is one of the most widely adopted measurements for assesment of ambient air quality. In this dataset, we provide 175 measurement of daily temporal distribution of PM10 in Graz, Austria, collected between December 2004 and June 2005.

For the purpose of this R package, raw data of 48 observations per day, was transformed to functional objects in Fourier basis using fda package.

Usage

pm10

Format

175 daily distribution functions in the fd format from the fda package

Source

Styrian Government, FA 17C Technical Environmental Protection and Safety, Air Quality Control Section,

References

Hormann, Siegfried, Brigitte Pfeiler, and Ernst Stadlober. Analysis and prediction of particulate matter PM10 for the winter season in Graz. Austrian Journal of Statistics 34.4 (2005): 307-326.

Examples

data(pm10)
summary(pm10)
plot(pm10)