Partial Autocorrelations Research Articles

Bounds on partial correlations of sequences

Two approaches to bounding the partial auto- and crosscorrelations of binary sequences are considered. The first approach uses the discrete Fourier transform and bounds for character sums to obtain bounds on partial autocorrelations of m-sequences and on the partial auto- and crosscorrelations for the small Kasami sets and dual-BCH families of sequences. The second approach applies to binary sequences obtained by interleaving m-sequences. A bound on the peak partial correlation of such sequences is derived in terms of the peak partial autocorrelation of the underlying m-sequences. The bound is applied to GMW, No (1987), and other families of sequences for particular parameters. A comparison of the two approaches shows that the elementary method gives generally weaker results but is more widely applicable. On the other hand, both methods show that well-known sequence families can have favorable partial correlation characteristics, making them useful in certain spread-spectrum applications.

Limited distribution of sample partial autocorrelations: A matrix approach

We develop a technique for derivation of the asymptotic joint distribution of the sample partial autocorrelations of a process, given the corresponding distribution of sample autocorrelations. No assumption of asymptotic normality is needed. The underlying process need not be stationary. The technique is demonstrated through a detailed study of ARMA (1,1)-like processes, but is applicable to other models. The results extend those of Mills and Seneta (1989) for the AR(1)-like case. The study is motivated by the known relationships and properties, especially is the classical AR( p) case, of population and sample partial autocorrelations.

Further results on the finite-sample distribution of Monti's portmanteau test for the adequacy of an ARMA (p,q) model

This note investigates the finite-sample performance of Monti's test, paying special attention to its estimated sizes and empirical powers. Our simulation results indicate that (i) the test size can be affected by the choice of the number of residual partial autocorrelations, m, and (ii) the empirical powers of the Monti and the Ljung-Box tests are similar in the cases of both seasonal and non-seasonal data if m is properly chosen.

Genetic and environmental variation of female reproductive traits in rainbow trout ( Oncorhynchus mykiss)

The variation and covariation of reproductive traits were studied by analyzing data from 2020 females divided into 377 full sib families and covering five generations of three lines—a random mated control line (C), an egg size (+) line (E) and a body weight (+) line (Y), in a selection experiment with rainbow trout conducted at Davis, California. Variance components were estimated from a single trait animal model and covariance components from a two-trait animal model by using a derivative-free restricted maximum likelihood algorithm. Pooled over the three lines, the estimates of heritability were 0.65 for spawning date, 0.14 for spawning body weight, 0.60 for egg size, 0.55 for egg number, 0.52 for egg volume and 0.13 for fertility-hatchability. The estimates in line C were lower for spawning date, spawning body weight and egg number, and higher for egg size and fertility-hatchability than those in line E and line Y. Full-sib family effects caused by factors other than additive genetic effects were considerable for spawning body weight but small for other traits. Genetic correlations were estimated from data pooled over the three lines. Spawning date had significant genetic correlations with spawning body weight, egg size and egg volume (0.51–0.73) as well as with egg number (0.25). Significant genetic correlations were also found for spawning body weight with egg size, egg number and egg volume (0.47–0.67); and for egg size with fertility-hatchability (0.35). As expected, the genetic correlations between egg number and egg volume and between egg size and egg volume (0.81 and 0.48) were strong and significant due to the partial auto-correlations originating from commonality among biological components and methods of measuring these traits. The estimated genetic correlations between spawning body weight and egg production traits (egg size, egg number and egg volume) were positive, so that direct selection for growth rate or egg production traits should result in favorable correlated responses. The low heritability of body weight, the moderately high heritability of egg volume, and the strong genetic correlation between spawning body weight and egg volume, suggest that combined selection for body weight and egg volume could be an effective alternative selection method for improving growth rate and also reproductive capacity in rainbow trout.

Iterated logarithm law for sample generalized partial autocorrelations

The so-called generalized partial autocorrelations for a regular stationary process xt are the array of real coefficients ϕλ,λ(ν) that are defined by the equation E((xt + ϕλ,1(ν)xt−1 + … + ϕλ,λ(ν)xt−λ)xt−v−j) = 0; j = 1, …, λ. If the xt process is an ARMA(p,q) and if the ϕλ,j(ν)'s are usual estimates of ϕλ,j(ν), such as the extended Yule-Walker estimates, then under the weak assumption that the noise in the xt process is a martingale difference sequence, an iterated logarithm law is obtained for (ϕλ,j(q) − ϕλ,j(q)), which applied to the sample generalized partial autocorrelations ϕλ,λ(q), yields lim supnw(n)−1|ϕλ,λ(q)| ⩽ K almost surely, for λ ⩾ p + 1, where w(n) = (2n−1log log n)12 and the constant K depends only on the MA parameters of the xt process. For stationary AR(p) models, the following finer result is also obtained: For λ ⩾ p + 1, almost surely, lim supnw(n)−1ϕλ,λ(0) = 1 and lim infn w(n)−1ϕλ,λ(0) = −1.

Is Human Behavior Autocorrelated? An Empirical Analysis

The current review and analysis investigated the presence of serial dependency (or autocorrelation) in single-subject applied behavior-analytic research. While well researched, few studies have controlled for the number of data points that appeared in the time-series and, thus, the negative bias of the r coefficient, and the power to detect true serial dependency effects. Therefore, all baseline graphs that appeared in the Journal of Applied Behavior Analysis (JABA) between 1968 and 1993 that provided more than 30 data points were examined for the presence of serial dependency (N = 103). Results indicated that 12% of the baseline graphs provided a significant lag-1 autocorrelation, and over 83% of them had coefficient values less than or equal to (±.25). The distribution of the lag-1 autocorrelation coefficients had a mean of .10. Subsequent distributions of partial autocorrelations at lags two through seven had smaller means indicating that as the distance between observations increases (i.e., the lag), serial dependency decreased. Although serial dependency did not appear to be a common property of the single-subject behavioral experiments, it is recommended that, whenever statistical analyses are contemplated, its presence should always be examined. Alternatives for coping with the presence of significant levels of serial dependency were discussed in terms of: (a) using alternative statistical procedures (e.g., ARIMA models, randomization tests, the Shewhart quality-control charts); (b) correcting statistics of traditional parametric procedures (e.g., t, F); or (c) using the autocorrelation coefficient as an indicator and estimate of reliable intervention effects.

Quenouille-type theorem on autocorrelations

The central result is a limit theorem for not necessarily stationary processes resembling AR (p). Assumption of a vector limit distribution for standardized sample autocorrelations leads to the convergence of a vector limit distribution for ordinary sample partial autocorrelations, and to a clear relationship between the two limit distributions. The motivation is the study of the case p=1 by Mills and Seneta (1989, Stochastic Process Appl., 33, 151–161). The central result is used to explain the nature of the relationship between the two results of Quenouille in the classical stationary AR (p) setting.

Bayesian estimation of an autoregressive model using Markov chain Monte Carlo

We present a complete Bayesian treatment of autoregressive model estimation incorporating choice of autoregressive order, enforcement of stationarity, treatment of outliers, and allowance for missing values and multiplicative seasonality. The paper makes three distinct contributions. First, we enforce the stationarity conditions using a very efficient Metropolis-within-Gibbs algorithm to generate the partial autocorrelations. Second we show how to carry out the Gibbs sampler when the autoregressive order is unknown. Third, we show how to combine the various aspects of fitting an autoregressive model giving a more comprehensive and efficient treatment than previous work. We illustrate our methodology with a real example.

More effective time-series analysis and forecasting

Our aim is to suggest ways of improving time-domain modelling, for the purpose of more effective forecasting, by better interpretation of the sample autocorrelations and partial autocorrelations obtained from raw time-series data. For this objective, we assume no specialist knowledge, as we start by surveying all those standard ideas of univariate analysis which are needed for the subsequent development of our thesis.

A Proposal for a Residual Autocorrelation Test in Linear Models

This note proposes a test of goodness of fit for time series models based on the sum of the squared residual partial autocorrelations. The test statistic is asymptotically X2. Its small-sample performance is studied through a Monte Carlo experiment. It appears sensitive to erroneous specifications especially when the fitted model understates the order of the moving average component.

Partial period autocorrelations of geometric sequences

Partial period autocorrelations of geometric sequences

A proposal for a residual autocorrelation test in linear models

This note proposes a test of goodness of fit for time series models based on the sum of the squared residual partial autocorrelations. The test statistic is asymptotically X2. Its small-sample performance is studied through a Monte Carlo experiment. It appears sensitive to erroneous specifications especially when the fitted model understates the order of the moving average component.

Approximate moments to O( n−2 for the sampled partial autocorrelations from a white noise process

We obtain first approximations to all the moments of the low-lag sampled partial autocorrelations, for finite-lengthed realisations from a Gaussian white noise process; and indicate how closer approximations could be achieved. The method is to write products, or powers, of the partial correlations as expansions of the serial correlations, cutting off an expansion at a sufficiently small order term. Formulae for the moments of the partials then depend on knowledge of the expectations for those products, and powers, of the serial correlations which appear in the truncated expansions. We give a small simulation example of how our main result, a first-order approximation for the expectation of a white noise partial correlation, could prove useful in significance testing; and suggest that this rough result, for the mean, might be robust to quite substantial departures from normality. The paper concludes with large scale simulations, for the first four moments of the sampled partial autocorrelations, from length-10 realisations for each of Gaussian and negative exponential white noise. As an addendum, simulated first moments are also reported for some smaller simulations of longer series ( n = 20 and 50) from these two types of white noise.

Bootstrap estimates of the sample bivariate autocorrelation and partial autocorrelation distributions

This paper shows how the bootstrap method can be used to estimate the joint distribution of sample autocorrelations and partial autocorrelations. The exact joint distribution of sample autocorrelations is mathematically intractable and attempts at workable approximations are difficult and rely on special assumptions. The bootstrap offers an accurate solution to this problem without requiring special assumptions and in a way that avoids theoretical difficulties. The bootstrap-estimated joint distributions of the autocorrelations and partial autocorrelations of time series are shown to lead to better ARMA model identification. This is demonstrated using simulated series.

A note on time series model specification in the presence of outliers

We investigate the usefulness of sample autocorrelations and partial autocorrelations as model specification tools when the observed time series is contaminated by an outlier. The results indicate that the specification power of these statistics could be significantly jeopardized by an additive outlier. On the other hand, an innovational outlier seems to cause no harm to them.

Independence of partial autocorrelations for a classical immigration branching process

It is shown that for a data set from a branching process with immigration, where the offspring distribution is Bernoulli and the immigration distribution is Poisson, the normed sample partial autocorrelations are asymptotically independent. This makes possible a goodness-of-fit test of known (Quenouille) form. The underlying process is a classical model in statistical mechanics.

On the partial autocorrelations for an explosive process

On the partial autocorrelations for an explosive process

On the partial autocorrelations for an explosive process

We obtain approximate formulae for the sampled autocovariances, and sampled serial and partial autocorrelations, arising from sufficiently long (but finite) series realisations of explosive first-order autoregressive processes. The correlation results are shown to agree closely with those derived for a length-10 simulation from the literature.

Goodness-of-fit for a branching process with immigration using sample partial autocorrelations

A limit theorem is developed for sample partial autocorrelations, when the vector {N 1 2 (R(k)−m k), k=1,…,H} converges in distribution, the R( k) being sample autocorrelations from a not-necessarily stationary process. The result is used to develop a Quenouille-type goodness-of-fit test based on sample partial autocorrelations for the simple branching process with immigration. This is compared with a test of Venkataraman (1982); and both are applied to historical data.

Randomly Choosing Parameters from the Stationarity and Invertibility Region of Autoregressive-Moving Average Models

SUMMARY Choice of appropriate parameter configurations for time series simulations is not always easy. One possible approach when simulating from autoregressive-moving average models is to choose parameter values from a uniform distribution on the stationarity and invertibility region associated with such models. In this paper, well-known time series results are applied to this problem to give a neat method which comprises generating partial autocorrelations independently distributed as appropriate beta variates and applying a standard transformation to obtain the parameters from these.