acfpacf {perARMA}

Plotting usual ACF and PACF

Description

Plots values of usual ACF and PACF functions with confidence intervals. Function acfpacf uses procedures acfpacf.acf and acfpacf.pacf, which computes values of ACF and PACF function, respectively.

Usage

acfpacf(x, nac, npac, datastr,...)
acfpacf.acf(x, normflg)
acfpacf.pacf(x, m)

Arguments

x

input time series, missing values are not permitted.

nac

number of ACF values to return (typically much less than length of x).

npac

number of PACF values to return (typically much less than length of x).

datastr

string name of data to be printed on the plot.

normflg

computing parameter for ACF values. These values are returned as a series acf(k) for k = 0, ..., nr, where nr is length of x. Parameter normflg can be equal to:
0 - acf(k) values will be normalized by nr,
1 - acf(k) values will be normalized by nr multiplied by sample variance (to obtain correlations),
2 - acf(k) values will be divided by nr-k, thus normalized by number of samples at each lag,
3 - acf(k) values will be divided by nr-k multiplied by sample variance.
In acfpacf procedure normflg=3 is used.

m

maximum lag to compute PACF values. Value for the first lag (pacf(1)) is equal to acf(2), which is the lag 1 acf value, and then for computing values for k = 2, ..., m the Toeplitz structure of the projection equations is used (see Brockwell, P. J., Davis, R. A., 1991, Time Series: Theory and Methods, example 4.4.2).

...

other arguments:
plfg, acalpha, pacacalpha, valcol, thrcol, thrmhcol, where
plfg is plotting flag, this parameter should be positive number to plot computed acfpacf values,
acalpha and pacalpha are p-values (or alpha is I type error) thresholds for ACF and PACF plots based on independent normal values,
valcol,thrcol,thrmhcol are colors of function values, confidence interval markers on the ACF/PACF and confidence interval markers on the ACF/PACF for multiple hypothesis alpha correction on the plot.
By default parameters are fixed to plfg=1, acalpha=.05, pacacalpha=.05, valcol="red", thrcol="green", thrmhcol="blue".

Details

Function acfpacf returns plot of ACF and PACF values with two types of thresholds for input acalpha and pacalpha, respectively. The first one denoted by thr is given for ACF values by Pr[acf(j)>thr] = \alpha/2 and Pr[acf(j)<-thr] = \alpha/2 where acf(k) is the ACF value at lag k. This threshold corresponds to type I error for null hypothesis that acf(k) = 0. The plot allows to check if any of the ACF values are significantly non-zero. Actual threshold calculations are based on the following asymptotic result: if X_t is IID (0,\sigma^2), then for large n, \hat{\rho}(k) for k << n are IID N(0,1/n) (see Brockwell, P. J., Davis, R. A., 1991, Time Series: Theory and Methods, example 7.2.1, p. 222). Thus, under the null hypothesis, the plots for thr = qnorm(1-acalpha/2,0,1/sqrt(nr)) should exhibit a proportion of roughly acalpha points that lie outside the interval [-thr, thr]. Threshold for PACF is based on the same results.
On the other hand we can also interpret the plots as a multiple hypothesis testing problem to compute second threshold thrm. Suppose, we decided to plot some number of nonzero lags (equal to nac) of the ACF function. If the estimated acf values estimates are IID then we have nac independent tests of acf(k) = 0. The probability that at least one of values lies outside the interval [-thr, thr] is equal to 1-Pr[all lie inside], which is [1-(1-acalpha)]^nac. Finally, the last expression is approximately equal to nac*acalpha. To get a threshold thrmh such that 1-Pr[all lie inside] = acalpha we take the threshold as
thrmh = qnorm (1-(acalpha/2)/nac, 0, 1/sqrt(nr)) (for more details check the Bonferroni correction).
For the PACF, the threshold thrm calculation is based on Theorem 8.1.2 of Time Series: Theory and Methods, p. 241, which states that the PACF values for an AR sequence are asymptotically normal.

Value

No return value, called for side effects

Note

Procedure acfpacf is an implementation of the usual (meaning not periodic) ACF and PACF functions. It happens that for PC time series the usual ACF and PACF are still useful in the identification of model orders and in some cases can be more sensitive than the perodic versions. The ACF and PACF values inform about the correlations of the random part. It is possible to run acfpacf on original data which include the peridic mean as a deterministic component. But typically the periodic mean can distort our understanding (or view) of the random fluctuations, thus running acfpacf additionally on the data after removing periodic mean is suggested. It is possible to use also acfpacf.acf, acfpacf.pacf procedures to obtain values of ACF and PACF functions, respectively. When subjected to a truly PC sequence, the usual ACF and PACF produce an average of the instantaneous (time indexed) values produced by periodic ACF and periodic PACF. Depending therefore on correlations between these averaged quantities, the usual ACF and PACF may be more or less sensitive that the instantaneous ones.

Author(s)

Harry Hurd

References

Box, G. E. P., Jenkins, G. M., Reinsel, G. (1994), Time Series Analysis, 3rd Ed., Prentice-Hall, Englewood Cliffs, NJ.

Brockwell, P. J., Davis, R. A. (1991), Time Series: Theory and Methods, 2nd Ed., Springer: New York.

Bretz, F., Hothorn, T., Westfall, P. (2010), Multiple Comparisons Using R, CRC Press.

Westfall, P. H., Young, S. S. (1993), Resampling-Based Multiple Testing: Examples and Methods for p-Value Adjustment, Wiley Series in Probability and Statistics.

See Also

peracf, perpacf

Examples

data(volumes)
# for original data
dev.set(which=1)
acfpacf(volumes,24,24,'volumes')

# for data after removing periodic mean
pmean_out<-permest(t(volumes),24, 0.05, NaN,'volumes',pp=0)
xd=pmean_out$xd
dev.set(which=1)
acfpacf(xd,24,24,'volumes')