fts.dpca | R Documentation |
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.
fts.dpca(X, q = 30, freq = (-1000:1000/1000) * pi, Ndpc = X$basis$nbasis)
X |
a functional time series as a |
q |
window size for the kernel estimator, i.e. a positive integer. |
freq |
a vector containing frequencies in [-π, π] on which the spectral density should be evaluated. |
Ndpc |
is the number of principal component filters to compute as in |
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.
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
)
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
# 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)
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