hd_data {ftsa}R Documentation

Simulated high-dimensional functional time series

Description

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

Xt(i)(u)=p=12βp,t(i)γp(i)(u)+θt(i)(u),X_t^{(i)}(u) = \sum^2_{p=1}\beta_{p,t}^{(i)}\gamma_p^{(i)}(u) + \theta_t^{(i)}(u),

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

Usage

data("hd_data")

Details

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

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

where βp,t={βp,t1,,βp,tN}\bm{\beta}_{p,t}=\{\beta_{p,t}^{1},\dots,\beta_{p,t}^N\}. Here, Ap\bm{A}_p is an N×NN\times N matrix, and fp,t\bm{f}_{p,t} is an N×1N\times 1 vector. Furthermore, we assume that the βp,t(i)\beta_{p,t}^{(i)}s have mean 0 and variance 0 when p>3p>3, so we only construct the coefficients βp,t\bm{\beta}_{p,t} for p{1,2,3}p\in\{1, 2, 3\}.

The first set of coefficients β1,t\bm{\beta}_{1,t} for NN populations are generated with β1,t=A1f1,t\bm{\beta}_{1,t}=\bm{A}_1\bm{f}_{1,t}. Each element in the matrix A1\bm{A}_1 is generated by aij=N1/4×bija_{ij}=N^{-1/4}\times b_{ij}, where bijN(2,4)b_{ij}\sim N(2,4).

The factors f1,t\bm{f}_{1,t} are generated using an autoregressive model of order 1, i.e., AR(1). Define the iith element in vector f1,t\bm{f}_{1,t} as f1,t(i)f_{1,t}^{(i)}. Then, f1,t1f_{1,t}^{1} is generated by f1,t1=0.5×f1,t11+ωtf_{1,t}^{1}=0.5\times f_{1,t-1}^{1}+\omega_t, where ωt\omega_t are independent N(0,1)N(0,1) random variables. We generate f1,t(i)f_{1,t}^{(i)} for all i{2,,N}i\in \{2,\dots, N\} by f1,t(i)=(1/N)×gt(i)f_{1,t}^{(i)}=(1/N) \times g_t^{(i)}, where gt(2),,gt(N)g_t^{(2)},\dots,g_t^{(N)} are also AR(1) and follow gt(i)=0.2×gt1(i)+ωtg_t^{(i)} = 0.2\times g_{t-1}^{(i)}+\omega_t. It is then ensured that most of the variance of β1,t\bm{\beta}_{1,t} can be explained by one factor. The second coefficient β2,t\bm{\beta}_{2,t} are constructed the same way as β1,t\bm{\beta}_{1,t}.

We also generate the third functional principal component scores β3,t\bm{\beta}_{3,t} but with small values. Moreover, A3\bm{A}_3 is generated by aij=N1/4×bija_{ij}=N^{-1/4}\times b_{ij}, where bijN(0,0.04)b_{ij}\sim N(0, 0.04). The factors bmf3,tbm{f}_{3,t} are generated as f1,t\bm{f}_{1,t}.

The three basis functions are constructed by γ1(i)(u)=sin(2πu+πi/2)\gamma_1^{(i)}(u) = \sin(2\pi u + \pi i/2), γ2(i)(u)=cos(2πu+πi/2)\gamma_2^{(i)}(u) = \cos(2\pi u + \pi i/2) and γ3(i)(u)=sin(4πu+πi/2)\gamma_3^{(i)}(u) = \sin(4\pi u + \pi i/2), where u[0,1]u\in [0,1]. Finally, the functional time series for the iith population is constructed by

Xt(i)(u)=β1,tγ1(i)(u)+β2,tγ2(i)(u)+β3,tγ3(i)(u),\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 ()i(\cdot)_i denotes the iith 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.

See Also

hdfpca, forecast.hdfpca

Examples

data(hd_data)

[Package ftsa version 6.4 Index]