One-step-ahead forecast for any given quantile(s) of functional time sereies of extreme values using a generalized additive extreme value (GAEV) model.
GAEVforecast(data, q, d.loc.max = 10, d.logscale.max = 10)the optimal number of basis functions considered for the location parameter
the optimal number of basis functions considered for the (log-)scale parameter
the basis functions for the location parameter
the basis functions for the (log-)scale parameter
the predicted location function
the predicted scale function
the predicted shape parameter
the prediced density function(s) for the given quantile(s)
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
a required scalar or vector of GEV quantiles that are of forecasting interest
the maximum number of basis functions considered for the location parameter
the maximum number of basis functions considered for the (log-)scale parameter
Ruofan Xu and Han Lin Shang
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 $$ Q_{\tau}(u|\widehat{\mu}_{T+1},\widehat{\sigma}_{T+1},\widehat{\xi}_{T+1})$$ = $$ \widehat{\mu}_{T+1}(u) + \frac{\widehat{\sigma}_{T+1}(u) \big[(-\ln(\tau))^{-\widehat{\xi}_{T+1}}-1\big]}{\widehat{\xi}_{T+1}}, \xi > 0, \tau\in [0,1);\ \xi < 0, \tau\in (0,1], \\ \widehat{\mu}_{T+1}(u) - \widehat{\sigma}_{T+1}(u) \cdot \ln[-\ln\big(\tau)], \xi=0, \tau \in (0,1). $$
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.
if (FALSE) {
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)
}
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