hd_data {ftsa} | R Documentation |
Simulated high-dimensional functional time series
Description
We generate N
populations of functional time series. For each i∈{1,…,N}
, the i
th function at time t∈{1,…,T}
is given by
Xt(i)(u)=∑p=12βp,t(i)γp(i)(u)+θt(i)(u),
where θt(i)(u)=∑p=3∞βp,t(i)γp(i)(u)
.
Usage
data("hd_data")
Details
The coefficients βp,t(i)
for all N
populations are combined and generated, for all p∈N
, by
βp,t=Apfp,t,
where βp,t={βp,t1,…,βp,tN}
. Here, Ap
is an N×N
matrix, and fp,t
is an N×1
vector. Furthermore, we assume that the βp,t(i)
s have mean 0 and variance 0 when p>3
, so we only construct the coefficients βp,t
for p∈{1,2,3}
.
The first set of coefficients β1,t
for N
populations are generated with β1,t=A1f1,t
. Each element in the matrix A1
is generated by aij=N−1/4×bij
, where bij∼N(2,4)
.
The factors f1,t
are generated using an autoregressive model of order 1, i.e., AR(1). Define the i
th element in vector f1,t
as f1,t(i)
. Then, f1,t1
is generated by f1,t1=0.5×f1,t−11+ωt
, where ωt
are independent N(0,1)
random variables. We generate f1,t(i)
for all i∈{2,…,N}
by f1,t(i)=(1/N)×gt(i)
, where gt(2),…,gt(N)
are also AR(1) and follow gt(i)=0.2×gt−1(i)+ωt
. It is then ensured that most of the variance of β1,t
can be explained by one factor. The second coefficient β2,t
are constructed the same way as β1,t
.
We also generate the third functional principal component scores β3,t
but with small values. Moreover, A3
is generated by aij=N−1/4×bij
, where bij∼N(0,0.04)
. The factors bmf3,t
are generated as f1,t
.
The three basis functions are constructed by γ1(i)(u)=sin(2πu+πi/2)
, γ2(i)(u)=cos(2πu+πi/2)
and γ3(i)(u)=sin(4πu+πi/2)
, where u∈[0,1]
. Finally, the functional time series for the i
th population is constructed by
Xt(i)(u)=β1,tγ1(i)(u)+β2,tγ2(i)(u)+β3,tγ3(i)(u),
where (⋅)i
denotes the i
th 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]