Fits a functional partial least squares (PLSR) model using nonlinear partial least squares (NIPALS) algorithm or simple partial least squares (SIMPLS) algorithm.
fplsr(data, order = 6, type = c("simpls", "nipals"), unit.weights =
TRUE, weight = FALSE, beta = 0.1, interval = FALSE, method =
c("delta", "boota"), alpha = 0.05, B = 100, adjust = FALSE,
backh = 10)A list containing the following components is returned.
\((p \times m)\) matrix containing the regression coefficients. \(p\) is the number of variables in the predictors and \(m\) is the number of variables in the responses.
\((p \times order)\) matrix containing the predictor loadings.
\((m \times order)\) matrix containing the response loadings.
(ncol(data$y)-1) x order matrix containing the predictor scores.
\((p\times order)\) matrix containing the weights used to construct the latent components of predictors.
(ncol(data$y)-1) x order matrix containing the response scores.
\((p\times order)\) projection matrix used to convert predictors to predictor scores.
An object of class fts containing the column means of predictors.
An object of class fts containing the column means of responses.
An object of class fts containing the 1-step-ahead predicted values of the responses.
An object of class fts containing the fitted values.
An object of class fts containing the regression residuals.
A vector with the amount of predictor variance explained by each number of component.
Total variance in predictors.
When weight = TRUE, a set of geometrically decaying weights is given. When weight = FALSE, weights are all equal 1.
Time period of a fts object, which can be obtained from colnames(data$y).
Variables of a fts object, which can be obtained from data$x.
Returns the original functional predictors.
Returns the original functional responses.
Bootstrapped point forecasts.
Lower bound of prediction intervals.
Upper bound of prediction intervals.
Adjusted lower bound of prediction intervals.
Adjusted upper bound of prediction intervals.
Adjusted lower bound factor, which lies generally between 0.9 and 1.1.
Adjusted upper bound factor, which lies generally between 0.9 and 1.1.
An object of class fts.
Number of principal components to fit.
When type = "nipals", uses the NIPALS algorithm; when type = "simpls", uses the SIMPLS algorithm.
Constrains predictor loading weights to have unit norm.
When weight = TRUE, a set of geometrically decaying weights is applied to the decentralized data.
When weight = TRUE, the speed of geometric decay is governed by a weight parameter.
When interval = TRUE, produces distributional forecasts.
Method used for computing prediction intervals.
1-alpha gives the nominal coverage probability.
Number of replications.
When adjust = TRUE, an adjustment is performed.
When adjust = TRUE, an adjustment is performed by evaluating the difference between
predicted and actual values in a testing set. backh specifies the testing set.
Han Lin Shang
Point forecasts:
The NIPALS function implements the orthogonal scores algorithm, as described in Martens and Naes (1989). This is one of the two classical PLSR algorthms, the other is the simple partial least squares regression in DeJong (1993). The difference between these two approaches is that the NIPALS deflates the original predictors and responses, while the SIMPLS deflates the covariance matrix of original predictors and responses. Thus, SIMPLS is more computationally efficient than NIPALS.
In a functional data set, the functional PLSR can be performed by setting the functional responses to be 1 lag ahead of the functional predictors. This idea has been adopted from the Autoregressive Hilbertian processes of order 1 (ARH(1)) of Bosq (2000).
Distributional forecasts:
Parametric method:
Influenced by the works of Denham (1997) and Phatak et al. (1993), one way of constructing prediction intervals in the PLSR is via a local linearization method (also known as the Delta method). It can be easily understood as the first two terms in a Taylor series expansion. The variance of coefficient estimators can be approximated, from which an analytic-formula based prediction intervals are constructed.
Nonparametric method:
After discretizing and decentralizing functional data \(f_t(x)\) and \(g_s(y)\), a PLSR model with \(K\) latent components is built. Then, the fit residuals \(o_s(y_i)\) between \(g_s(y_i)\) and \(\hat{g}_s(y_i)\) are calculated as $$o_s(y_i)=g_s(y_i)-\hat{g}_s(y_i), i=1,...,p.$$
The next step is to generate \(B\) bootstrap samples \(o_s^b(y_i)\) by randomly sampling with replacement
from \([o_1(y_i),...,o_n(y_i)]\). Adding bootstrapped residuals to the original
response variables in order to generate new bootstrap responses,
$$g_s^b(y_i)=g_s(y_i)+o_s^b(y_i).$$
Then, the PLSR models are constructed using the centered and discretized predictors and bootstrapped responses
to obtain the boostrapped regression coefficients and point forecasts, from which the empirical prediction intervals and kernel density plots are constructed.
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M. Febrero-Bande, P. Galeano, W. Gonz\'alez-Manteiga (2017), "Functional principal component regression and functional partial least-squares regression: An overview and a comparative study", International Statistical Review, 85(1), 61-83.
ftsm, forecast.ftsm, plot.fm,
summary.fm, residuals.fm, plot.fmres
# When weight = FALSE, all observations are assigned equally.
# When weight = TRUE, all observations are assigned geometrically decaying weights.
fplsr(data = ElNino_ERSST_region_1and2, order = 6, type = "nipals")
fplsr(data = ElNino_ERSST_region_1and2, order = 6)
fplsr(data = ElNino_ERSST_region_1and2, weight = TRUE)
fplsr(data = ElNino_ERSST_region_1and2, unit.weights = FALSE)
fplsr(data = ElNino_ERSST_region_1and2, unit.weights = FALSE, weight = TRUE)
# The prediction intervals are calculated numerically.
fplsr(data = ElNino_ERSST_region_1and2, interval = TRUE, method = "delta")
# The prediction intervals are calculated by bootstrap method.
fplsr(data = ElNino_ERSST_region_1and2, interval = TRUE, method = "boota")
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