Simulated high-dimensional functional time series
We generate N populations of functional time series. For each i\in \{1,…, N\}, the ith function at time t\in \{1,…, T\} is given by
X_t^{(i)}(u) = ∑^2_{p=1}β_{p,t}^{(i)}γ_p^{(i)}(u) + θ_t^{(i)}(u),
where θ_t^{(i)}(u) = ∑^{∞}_{p=3}β_{p,t}^{(i)}γ_p^{(i)}(u).
data("hd_data")The coefficients β_{p,t}^{(i)} for all N populations are combined and generated, for all p\in N, by
\bm{β}_{p,t} = = \bm{A}_p\bm{f}_{p,t},
where \bm{β}_{p,t}=\{β_{p,t}^{1},…,β_{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 β_{p,t}^{(i)}s have mean 0 and variance 0 when p>3, so we only construct the coefficients \bm{β}_{p,t} for p\in\{1, 2, 3\}.
The first set of coefficients \bm{β}_{1,t} for N populations are generated with \bm{β}_{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}+ω_t, where ω_t are independent N(0,1) random variables. We generate f_{1,t}^{(i)} for all i\in \{2,…, N\} by f_{1,t}^{(i)}=(1/N) \times g_t^{(i)}, where g_t^{(2)},…,g_t^{(N)} are also AR(1) and follow g_t^{(i)} = 0.2\times g_{t-1}^{(i)}+ω_t. It is then ensured that most of the variance of \bm{β}_{1,t} can be explained by one factor. The second coefficient \bm{β}_{2,t} are constructed the same way as \bm{β}_{1,t}.
We also generate the third functional principal component scores \bm{β}_{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 γ_1^{(i)}(u) = \sin(2π u + π i/2), γ_2^{(i)}(u) = \cos(2π u + π i/2) and γ_3^{(i)}(u) = \sin(4π u + π 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{β}_{1,t}γ_1^{(i)}(u) + \bm{β}_{2,t}γ_2^{(i)}(u) + \bm{β}_{3,t}γ_3^{(i)}(u),
where (\cdot)_i denotes the ith element of the vector.
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.
data(hd_data)
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