perpacf function

Periodic PACF function

The function perpacf, given an input time series, a specified period T and a lag p, computes the periodic sample correlation coefficients $\pi(t,n)$ and returns their values as a matrix ppa of size $T \times (p+1)$.

The ppfcoeffab procedure transforms the output of perpacf into Fourier form, i.e. into Fourier coeficients, so we can represent $\pi(t,n)$ by its Fourier coefficients.

Function ppfplot plots perpacf coefficients returned by perpacf as function of n for each specified lag $t=1, 2,\ldots, T$.

perpacf(x, T_t, p, missval)
ppfcoeffab(ppf, nsamp, printflg, datastr)
ppfplot(ppf, nsamp, alpha, datastr)

Arguments

• x: input time series.
• T_t: period of PC-T structure.
• p: maximum lag used in computation.
• missval: notation for missing values.
• ppf: matrix of periodic PACF values (of size $T \times (p+1)$) returned by perpacf function.
• nsamp: number of samples (periods) used to compute ppf.
• printflg: parameter should be positive to return messages.
• alpha: parameter for thresolds are displayed along with the Bonferroni corrected thresholds.
• datastr: string name of data for printing.

Details

Procedure perpacf returns ppa matrix, where for each separation n=0,1,...,p, ppa[,n] is the value of $\hat{\pi}(t,n)$ for t=1,2,...,T. Further, since T is assumed to be the period of the underlying PC process, $\pi(t,n)$ is periodic in t with period T. So we can represent $\pi(t,n)$ by its Fourier coefficients. Further, if the variation in time of $\pi(t,n)$ is really smooth over the period, then looking at these Fourier coefficients (the output of ppfcoeffab) may be a more sensitive detector of linear dependence of $x_{t+1}$ on the preceding n samples (think of n as fixed here) than looking at $\pi(t,n)$ for individual times. The ppfcoeffab procedure also needs the sample size nsamp used by perpacf in computing the $\pi(t,n)$

in order to compute p-values for the pkab coefficients. The p-values are computed assuming that for each t, $\pi(t,n)$ is N(0,1/sqrt(nsamp)) under the null. The procedure ppfcoeffab is called in parma_ident.

Function ppfplot plots values of $\pi(t,n+1)$ and computes p-values for testing if $\pi(n_0+1,t)=0$ for all t = 1, ..., T and fix $n_0$ (p-values in column labelled $n_0=n$) and if $\pi(n+1,t)=0$ for all t = 1, ..., T and $n_0 \leq n \leq nmax$ (p-values in column labelled $n_0 \leq n \leq nmax$). perpacf is plotted as function of n for each specified lag $t=1, 2,\ldots, T$.

Returns

The function perpacf returns two matrixes: - ppa: matrix of size $T \times (p+1)$ with perpacf coefficients.

• nsamp: matrix of size $T \times (p+1)$ with numbers of samples used in estimation of sample correlation.

The function ppfcoeffab returns table of values: - pihat_k: Fourier coefficients pkab of ppf values.

• pv: Bonferroni corrected p-values.

The function ppfplot returns plot of $\pi(t,n+1)$ coefficients and table of p-vaules for provided tests. Note that there are two plots; the first plot presents values of $\pi(t,n+1)$ for all considered $t$ and $n$, whereas the second plot presents separate charts for particular $t$ values.

References

Hurd, H. L., Miamee, A. G., (2007), Periodically Correlated Random Sequences: Spectral Theory and Practice, Wiley InterScience.

Author(s)

Harry Hurd

See Also

peracf

Examples

data(volumes)
 perpacf_out<-perpacf(t(volumes),24,12,NaN)
 ppa=perpacf_out$ppa
 nsamp=perpacf_out$nsamp
 ppfcoeffab(ppa,nsamp,1,'volumes')
 ppfplot(ppa,41,.05,'volumes')