hd_data: Simulated high-dimensional functional time series
In ftsa: Functional Time Series Analysis

hd_data

R Documentation

Simulated high-dimensional functional time series

Description

We generate N populations of functional time series. For each i\in \{1,\dots, N\}, the ith function at time t\in \{1,\dots, T\} is given by

X_t^{(i)}(u) = \sum^2_{p=1}\beta_{p,t}^{(i)}\gamma_p^{(i)}(u) + \theta_t^{(i)}(u),

where \theta_t^{(i)}(u) = \sum^{\infty}_{p=3}\beta_{p,t}^{(i)}\gamma_p^{(i)}(u).

Usage

data("hd_data")

Details

The coefficients \beta_{p,t}^{(i)} for all N populations are combined and generated, for all p\in N, by

\bm{\beta}_{p,t} = \bm{A}_p\bm{f}_{p,t},

where \bm{\beta}_{p,t}=\{\beta_{p,t}^{1},\dots,\beta_{p,t}^N\}. Here, \bm{A}_p is an N\times N matrix, and \bm{f}_{p,t} is an N\times 1 vector. Furthermore, we assume that the \beta_{p,t}^{(i)}s have mean 0 and variance 0 when p>3, so we only construct the coefficients \bm{\beta}_{p,t} for p\in\{1, 2, 3\}.

The first set of coefficients \bm{\beta}_{1,t} for N populations are generated with \bm{\beta}_{1,t}=\bm{A}_1\bm{f}_{1,t}. Each element in the matrix \bm{A}_1 is generated by a_{ij}=N^{-1/4}\times b_{ij}, where b_{ij}\sim N(2,4).

The factors \bm{f}_{1,t} are generated using an autoregressive model of order 1, i.e., AR(1). Define the ith element in vector \bm{f}_{1,t} as f_{1,t}^{(i)}. Then, f_{1,t}^{1} is generated by f_{1,t}^{1}=0.5\times f_{1,t-1}^{1}+\omega_t, where \omega_t are independent N(0,1) random variables. We generate f_{1,t}^{(i)} for all i\in \{2,\dots, N\} by f_{1,t}^{(i)}=(1/N) \times g_t^{(i)}, where g_t^{(2)},\dots,g_t^{(N)} are also AR(1) and follow g_t^{(i)} = 0.2\times g_{t-1}^{(i)}+\omega_t. It is then ensured that most of the variance of \bm{\beta}_{1,t} can be explained by one factor. The second coefficient \bm{\beta}_{2,t} are constructed the same way as \bm{\beta}_{1,t}.

We also generate the third functional principal component scores \bm{\beta}_{3,t} but with small values. Moreover, \bm{A}_3 is generated by a_{ij}=N^{-1/4}\times b_{ij}, where b_{ij}\sim N(0, 0.04). The factors bm{f}_{3,t} are generated as \bm{f}_{1,t}.

The three basis functions are constructed by \gamma_1^{(i)}(u) = \sin(2\pi u + \pi i/2), \gamma_2^{(i)}(u) = \cos(2\pi u + \pi i/2) and \gamma_3^{(i)}(u) = \sin(4\pi u + \pi i/2), where u\in [0,1]. Finally, the functional time series for the ith population is constructed by

\bm{X}_t^{(i)}(u) = \bm{\beta}_{1,t}\gamma_1^{(i)}(u) + \bm{\beta}_{2,t}\gamma_2^{(i)}(u) + \bm{\beta}_{3,t}\gamma_3^{(i)}(u),

where (\cdot)_i denotes the ith element of the vector.

References

Y. Gao, H. L. Shang and Y. Yang (2018) High-dimensional functional time series forecasting: An application to age-specific mortality rates, Journal of Multivariate Analysis, forthcoming.