Description Usage Details References See Also Examples

We generate *N* populations of functional time series. For each *i\in \{1,…, N\}*, the *i*th 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)*.

1 | ```
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 *i*th 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 *i*th 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 *i*th 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**.

1 |

```
Loading required package: forecast
Loading required package: rainbow
Loading required package: MASS
Loading required package: pcaPP
Loading required package: sde
Loading required package: stats4
Loading required package: fda
Loading required package: splines
Loading required package: Matrix
Attaching package: 'fda'
The following object is masked from 'package:forecast':
fourier
The following object is masked from 'package:graphics':
matplot
Loading required package: zoo
Attaching package: 'zoo'
The following objects are masked from 'package:base':
as.Date, as.Date.numeric
sde 2.0.15
Companion package to the book
'Simulation and Inference for Stochastic Differential Equations With R Examples'
Iacus, Springer NY, (2008)
To check the errata corrige of the book, type vignette("sde.errata")
Attaching package: 'ftsa'
The following objects are masked from 'package:stats':
sd, var
```

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