This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations .(September 2011) |

In time series analysis, the **partial autocorrelation function** (**PACF**) gives the partial correlation of a stationary time series with its own lagged values, regressed the values of the time series at all shorter lags. It contrasts with the autocorrelation function, which does not control for other lags.

This function plays an important role in data analysis aimed at identifying the extent of the lag in an autoregressive model. The use of this function was introduced as part of the Box–Jenkins approach to time series modelling, whereby plotting the partial autocorrelative functions one could determine the appropriate lags **p** in an AR (**p**) model or in an extended ARIMA (**p**,**d**,**q**) model.

Given a time series , the partial autocorrelation of lag *k*, denoted , is the autocorrelation between and with the linear dependence of on through removed; equivalently, it is the autocorrelation between and that is not accounted for by lags through , inclusive.

where is the surjective operator of orthogonal projection of onto the linear subspace of Hilbert space spanned by .

There are algorithms for estimating the partial autocorrelation based on the sample autocorrelations (Box, Jenkins, and Reinsel 2008 and Brockwell and Davis, 2009). These algorithms derive from the exact theoretical relation between the partial autocorrelation function and the autocorrelation function.

Partial autocorrelation plots (Box and Jenkins, Chapter 3.2, 2008) are a commonly used tool for identifying the order of an autoregressive model. The partial autocorrelation of an AR(*p*) process is zero at lag *p* + 1 and greater. If the sample autocorrelation plot indicates that an AR model may be appropriate, then the sample partial autocorrelation plot is examined to help identify the order. One looks for the point on the plot where the partial autocorrelations for all higher lags are essentially zero. Placing on the plot an indication of the sampling uncertainty of the sample PACF is helpful for this purpose: this is usually constructed on the basis that the true value of the PACF, at any given positive lag, is zero. This can be formalised as described below.

An approximate test that a given partial correlation is zero (at a 5% significance level) is given by comparing the sample partial autocorrelations against the critical region with upper and lower limits given by , where *n* is the record length (number of points) of the time-series being analysed. This approximation relies on the assumption that the record length is at least moderately large (say *n*>30) and that the underlying process has finite second moment.

**Autocorrelation**, sometimes known as **serial correlation** in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations as a function of the time lag between them. The analysis of autocorrelation is a mathematical tool for finding repeating patterns, such as the presence of a periodic signal obscured by noise, or identifying the missing fundamental frequency in a signal implied by its harmonic frequencies. It is often used in signal processing for analyzing functions or series of values, such as time domain signals.

In probability theory and statistics, the **beta distribution** is a family of continuous probability distributions defined on the interval [0, 1] parameterized by two positive shape parameters, denoted by *α* and *β*, that appear as exponents of the random variable and control the shape of the distribution. The generalization to multiple variables is called a Dirichlet distribution.

In statistics, the **Pearson correlation coefficient** ― also known as **Pearson's r**, the

In econometrics, the **autoregressive conditional heteroskedasticity** (**ARCH**) model is a statistical model for time series data that describes the variance of the current error term or innovation as a function of the actual sizes of the previous time periods' error terms; often the variance is related to the squares of the previous innovations. The ARCH model is appropriate when the error variance in a time series follows an autoregressive (AR) model; if an autoregressive moving average (ARMA) model is assumed for the error variance, the model is a **generalized autoregressive conditional heteroskedasticity** (**GARCH**) model.

In the statistical analysis of time series, **autoregressive–moving-average** (**ARMA**) **models** provide a parsimonious description of a (weakly) stationary stochastic process in terms of two polynomials, one for the autoregression (AR) and the second for the moving average (MA). The general ARMA model was described in the 1951 thesis of Peter Whittle, *Hypothesis testing in time series analysis*, and it was popularized in the 1970 book by George E. P. Box and Gwilym Jenkins.

In statistics, econometrics and signal processing, an **autoregressive** (**AR**) **model** is a representation of a type of random process; as such, it is used to describe certain time-varying processes in nature, economics, etc. The autoregressive model specifies that the output variable depends linearly on its own previous values and on a stochastic term ; thus the model is in the form of a stochastic difference equation. Together with the moving-average (MA) model, it is a special case and key component of the more general autoregressive–moving-average (ARMA) and autoregressive integrated moving average (ARIMA) models of time series, which have a more complicated stochastic structure; it is also a special case of the vector autoregressive model (VAR), which consists of a system of more than one interlocking stochastic difference equation in more than one evolving random variable.

In time series analysis, the **lag operator** (L) or **backshift operator** (B) operates on an element of a time series to produce the previous element. For example, given some time series

In statistics and econometrics, and in particular in time series analysis, an **autoregressive integrated moving average** (**ARIMA**) model is a generalization of an autoregressive moving average (ARMA) model. Both of these models are fitted to time series data either to better understand the data or to predict future points in the series (forecasting). ARIMA models are applied in some cases where data show evidence of non-stationarity in the sense of mean, where an initial differencing step can be applied one or more times to eliminate the non-stationarity of the mean function. When the seasonality shows in a time series, the seasonal-differencing could be applied to eliminate the seasonal component. Since the ARMA model, according to the Wold's decomposition theorem, is theoretically sufficient to describe a **regular** wide-sense stationary time series, we are motivated to make stationary a non-stationary time series, e.g., by using differencing, before we can use the ARMA model. Note that if the time series contains a **predictable** sub-process, the predictable component is treated as a non-zero-mean but periodic component in the ARIMA framework so that it is eliminated by the seasonal differencing.

In time series analysis, the **Box–Jenkins method,** named after the statisticians George Box and Gwilym Jenkins, applies autoregressive moving average (ARMA) or autoregressive integrated moving average (ARIMA) models to find the best fit of a time-series model to past values of a time series.

A **cyclostationary process** is a signal having statistical properties that vary cyclically with time. A cyclostationary process can be viewed as multiple interleaved stationary processes. For example, the maximum daily temperature in New York City can be modeled as a cyclostationary process: the maximum temperature on July 21 is statistically different from the temperature on December 20; however, it is a reasonable approximation that the temperature on December 20 of different years has identical statistics. Thus, we can view the random process composed of daily maximum temperatures as 365 interleaved stationary processes, each of which takes on a new value once per year.

In statistics and econometrics, an **augmented Dickey–Fuller test** (**ADF**) tests the null hypothesis that a unit root is present in a time series sample. The alternative hypothesis is different depending on which version of the test is used, but is usually stationarity or trend-stationarity. It is an augmented version of the Dickey–Fuller test for a larger and more complicated set of time series models.

In statistics, the **Durbin–Watson statistic** is a test statistic used to detect the presence of autocorrelation at lag 1 in the residuals from a regression analysis. It is named after James Durbin and Geoffrey Watson. The small sample distribution of this ratio was derived by John von Neumann. Durbin and Watson applied this statistic to the residuals from least squares regressions, and developed bounds tests for the null hypothesis that the errors are serially uncorrelated against the alternative that they follow a first order autoregressive process. Note that the distribution of this test statistic does not depend on the estimated regression coefficients and the variance of the errors.

In probability theory, the **inverse Gaussian distribution** is a two-parameter family of continuous probability distributions with support on (0,∞).

In the analysis of data, a **correlogram** is a chart of correlation statistics. For example, in time series analysis, a plot of the sample autocorrelations versus is an **autocorrelogram**. If cross-correlation is plotted, the result is called a **cross-correlogram**.

The **Ljung–Box test** is a type of statistical test of whether any of a group of autocorrelations of a time series are different from zero. Instead of testing randomness at each distinct lag, it tests the "overall" randomness based on a number of lags, and is therefore a portmanteau test.

**Maximum entropy spectral estimation** is a method of spectral density estimation. The goal is to improve the spectral quality based on the principle of maximum entropy. The method is based on choosing the spectrum which corresponds to the most random or the most unpredictable time series whose autocorrelation function agrees with the known values. This assumption, which corresponds to the concept of maximum entropy as used in both statistical mechanics and information theory, is maximally non-committal with regard to the unknown values of the autocorrelation function of the time series. It is simply the application of maximum entropy modeling to any type of spectrum and is used in all fields where data is presented in spectral form. The usefulness of the technique varies based on the source of the spectral data since it is dependent on the amount of assumed knowledge about the spectrum that can be applied to the model.

In probability and statistics, the **Tweedie distributions** are a family of probability distributions which include the purely continuous normal, gamma and Inverse Gaussian distributions, the purely discrete scaled Poisson distribution, and the class of compound Poisson–gamma distributions which have positive mass at zero, but are otherwise continuous. Tweedie distributions are a special case of exponential dispersion models and are often used as distributions for generalized linear models.

In statistics, the **Breusch–Godfrey test** is used to assess the validity of some of the modelling assumptions inherent in applying regression-like models to observed data series. In particular, it tests for the presence of serial correlation that has not been included in a proposed model structure and which, if present, would mean that incorrect conclusions would be drawn from other tests or that sub-optimal estimates of model parameters would be obtained.

In time series analysis, the **moving-average model**, also known as **moving-average process**, is a common approach for modeling univariate time series. The moving-average model specifies that the output variable depends linearly on the current and various past values of a stochastic term.

In probability theory and statistics, the **Poisson distribution**, named after French mathematician Siméon Denis Poisson, is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.

- Box, G. E. P.; Jenkins, G. M.; Reinsel, G. C. (2008).
*Time Series Analysis, Forecasting and Control*(4th ed.). Hoboken, NJ: Wiley. ISBN 9780470272848. - Brockwell, Peter; Davis, Richard (2009).
*Time Series: Theory and Methods*(2nd ed.). New York: Springer. ISBN 9781441903198. - Enders, Walter (2004).
*Applied Econometric Time Series*(Second ed.). New York: John Wiley. pp. 65–67. ISBN 0-471-23065-0.

This article incorporates public domain material from the National Institute of Standards and Technology document: " http://www.itl.nist.gov/div898/handbook/pmc/section4/pmc4463.htm ".

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.