GAEVforecast: Fit a generalized additive extreme value model to the...

In ftsa: Functional Time Series Analysis

Fit a generalized additive extreme value model to the functional data with given basis numbers

Description

One-step-ahead forecast for any given quantile(s) of functional time sereies of extreme values using a generalized additive extreme value (GAEV) model.

Usage

GAEVforecast(data, q, d.loc.max = 10, d.logscale.max = 10)

Arguments

data	a n by p data matrix, where n denotes the number of functional objects and p denotes the number of realizations on each functional object
q	a required scalar or vector of GEV quantiles that are of forecasting interest
d.loc.max	the maximum number of basis functions considered for the location parameter
d.logscale.max	the maximum number of basis functions considered for the (log-)scale parameter

Details

For the functional time seres \{X_t(u),t=1,...,T,u\in \mathcal{I}\}, the GAEV model is given as

X_{t}(u) ~ GEV[\mu_{t}(u),\sigma_t(u),\xi_t],

where

\mu_t(u) = \beta^{(\mu)}_{t,0} + \sum_{i=1}^{d_1}\beta^{(\mu)}_{t,i}b^{(\mu)}_{i}(u),

\ln(\sigma_t(u)) = \beta^{(\sigma)}_{t,0} + \sum_{i=1}^{d_2}\beta^{(\sigma)}_{t,i}b^{(\sigma)}_{i}(u), \xi_t \in [0,\infty),

where d_{j},j=1,2 are positive integers of basis numbers, \{b^{(\mu)}_{i}(u),i=1,\dots,d_{1}\} and \{b^{(\sigma)}_{i}(u),i=1,\dots,d_{2}\} are the cubic regression spline basis functions.

The optimal number of basis functions (d_1,d_2) are chosen by minimizing the Kullback-Leibler divergence on the test set using a leave-one-out cross-validation technique.

The one-step-ahead forecast of the joint coefficients (\widehat{\beta^{(\mu)}}_{T+1,i},\widehat{\beta^{(\sigma)}}_{T+1,j},\widehat{\xi}_{T+1},i=0,...,d_1,j=0,...,d_2) are produced using a vector autoregressive model, whose order is selected via the corrected Akaike information criterion. Then the one-step-ahead forecast of the GEV parameter (\widehat{\mu}_{T+1}(u),\widehat{\sigma}_{T+1}(u),\widehat{\xi}_{T+1}) can be computed accordingly.

The one-step-ahead forecast for the \tau-th quantile of the extreme values \widehat{X}_{T+1}(u) is computed by

Q_{\tau}(u|\widehat{\mu}_{T+1},\widehat{\sigma}_{T+1},\widehat{\xi}_{T+1})

=

\widehat{\mu}_{T+1}(u) + \frac{\widehat{\sigma}_{T+1}(u) \big[(-\ln(\tau))^{-\widehat{\xi}_{T+1}}-1\big]}{\widehat{\xi}_{T+1}}, \xi > 0, \tau\in [0,1);\ \xi < 0, \tau\in (0,1], \\ \widehat{\mu}_{T+1}(u) - \widehat{\sigma}_{T+1}(u) \cdot \ln[-\ln\big(\tau)], \xi=0, \tau \in (0,1).

Value

kdf.location	the optimal number of basis functions considered for the location parameter
kdf.logscale	the optimal number of basis functions considered for the (log-)scale parameter
basis.location	the basis functions for the location parameter
basis.logscale	the basis functions for the (log-)scale parameter
para.location.pred	the predicted location function
para.scale.pred	the predicted scale function
para.shape.pred	the predicted shape parameter
density.pred	the prediced density function(s) for the given quantile(s)

Author(s)

Ruofan Xu and Han Lin Shang

References

Shang, H. L. and Xu, R. (2021) ‘Functional time series forecasting of extreme values’, Communications in Statistics Case Studies Data Analysis and Applications, in press.

Examples

## Not run: 
library(evd)
data = matrix(rgev(1000),ncol=50) 
GAEVforecast(data = data, q = c(0.02,0.7), d.loc.max = 5, d.logscale.max = 5)

## End(Not run)

ftsa documentation built on Sept. 11, 2023, 5:09 p.m.