Frequency Response Function

Compute the frequency response function (also called transfer function) associated to a VARMA or state space model.

Usage

freqresp(obj, n.f, ...)

# S3 method for armamod
freqresp(obj, n.f = 128, ...)

# S3 method for stspmod
freqresp(obj, n.f = 128, ...)

# S3 method for impresp
freqresp(obj, n.f = 128, ...)

Arguments

obj

armamod(), stspmod() or impresp() object. Note that for an impulse response object the result is only an approximation of the "true" frequency response due to the finite number of coefficients.

n.f

number of frequencies.

...

not used.

Value

freqresp object, i.e. a list with slots

frr

rationalmatrices::zvalues() object.

sigma_L

(n,n)-dimensional matrix which contains a left square root of the noise covariance matrix \(\Sigma\).

names

(m)-dimensional character vector or NULL. This optional slot stores the names for the components of the time series/process.

label

character string or NULL.

Details

The frequency response function (or transfer function) associated to an ARMA or state space model is $$ K(\lambda) = \sum_{j=0}^{\infty} k_j e^{-i\lambda j} $$ where \((k_j \,|\, j\geq 0)\) is the impulse response of the model. See also impresp().

For an ARMA model the frequency response is equal to $$ K(\lambda) = (a_0 + a_1 e^{-i\lambda} + \cdots + a_p e^{-i\lambda p})^{-1} (b_0 + b_1 e^{-i\lambda} + \cdots + b_q e^{-i\lambda q}) $$ and for a state space model we have $$ K(\lambda) = C(e^{i\lambda}I_s - A)^{-1}B+D $$ Note that \(K()\) is the discrete-time Fourier transform (DTFT) of the impulse response. If the impulse response is absolutely summable then the coefficents \(k_j\) may be reconstructed from the frequency response via the inverse DTFT $$ k_j = \frac{1}{2\pi} \int_{-\pi}^{\pi} K(\lambda) e^{i\lambda j} d\lambda $$

The S3 methods freqresp.* evaluate the function on a grid of angular frequencies \(\lambda_j = 2\pi j/N\), \(j=0,\ldots,N-1\) and store the result (together with sigma_L) in a freqresp object.

Examples

set.seed(3451) # set seed in order to get reproducible results

### generate random bivariate ARMA(1,1) model
# "bpoles = 1.1" implies that the poles have moduli larger than 1.1
# and therefore the impulse response coefficients decay with a rate (1.1)^k
arma_model = test_armamod(dim = c(2,2), degrees = c(1,1), bpoles = 1.1)
# frequency response
frr = freqresp(arma_model)
# compute the frequency response via the impulse response
irf = impresp(arma_model, lag.max = 100)
frr1 = freqresp(irf)
# since the impulse response quickly decays
# the "truncated" frequency response should be close to the true frequency response
all.equal(frr, frr1)
#> [1] TRUE
# create an equivalent state space model
stsp_model = as.stspmod(arma_model)
# of course the state space model has the same frequency response
# as the original ARMA model
frr1 = freqresp(stsp_model)
all.equal(frr, frr1)
#> [1] TRUE

# we can also reconstruct the impulse response from the
# frequency response, provided the frequency grid is "fine enough"
n.f = 2^6
frr = freqresp(arma_model, n.f = n.f)
# compute the impulse response via the inverse DTFT
K = unclass(frr$frr)
k1 = Re(apply(K, MARGIN = c(1,2), FUN = fft, inverse = TRUE)) / n.f
k1 = aperm(k1, c(2,3,1))
# impulse response
irf = impresp(arma_model, lag.max = n.f-1)
k = unclass(irf$irf)
# compare
all.equal(k, k1)
#> [1] TRUE

set.seed(NULL) # reset seed