fts.rma | R Documentation |
Generate a functional moving average process.
fts.rma( n = 100, d = 11, Psi = NULL, op.norms = NULL, noise = "mnorm", sigma = diag(d:1)/d, df = 4 )
n |
number of observations to generate. |
d |
dimension of the underlying multivariate VAR model. |
Psi |
an array of p≥q 1 coefficient matrices (need to be square matrices). |
op.norms |
a vector with non-negative scalar entries to which the |
noise |
|
sigma |
covariance or scale matrix of the coefficients corresponding to functional innovations. The default value
is |
df |
degrees of freqdom if |
The purpose is to simulate a functional autoregressive process of the form
X_t(u)=∑_{k=1}^p \int_0^1Ψ_k(u,v) X_{t-k}(v)dv+\varepsilon_t(u),\quad 1≤q t≤q n.
Here we assume that the observations lie in a finite dimensional subspace of the function space spanned by Fourier basis functions \boldsymbol{b}^\prime(u)=(b_1(u),…, b_d(u)). That is X_t(u)=\boldsymbol{b}^\prime(u)\boldsymbol{X}_t, \varepsilon_t(u)=\boldsymbol{b}^\prime(u)\boldsymbol{\varepsilon}_t and Ψ_k(u,v)=\boldsymbol{b}^\prime(u)\boldsymbol{Ψ}_k \boldsymbol{b}(v). Then it follows that
\boldsymbol{X}_t=\boldsymbol{Ψ}_1\boldsymbol{X}_{t-1}+\cdots+ \boldsymbol{Ψ}_p\boldsymbol{X}_{t-p}+\boldsymbol{\varepsilon}_t.
Hence the dynamic of the functional time series is described by a VAR(p) process.
In this mathematical model the law of \boldsymbol{\varepsilon}_t is determined by noise
. The matrices Psi[,,k]
correspond to \boldsymbol{Ψ}_k. If op.norms
is provided, then the coefficient matrices will be rescaled, such that
the Hilbert-Schmidt norms of \boldsymbol{Ψ}_k correspond to the vector.
An object of class fd
.
The multivariate equivalent in the freqdom
package: rma
# Generate a FMA process without burnin (starting from 0) fts = fts.rma(n = 5, d = 5) plot(fts) # Generate observations with very strong dependance fts = fts.rma(n = 100, d = 5, op.norms = 0.999) plot(fts) # Generate observations with very strong dependance and noise # from the multivariate t distribution fts = fts.rma(n = 100, d = 5, op.norms = 0.999, noise = "mt") plot(fts)
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