Partial autocorrelation function
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time series analysis In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
, the partial autocorrelation function (PACF) gives the
partial correlation In probability theory and statistics, partial correlation measures the degree of association between two random variables, with the effect of a set of controlling random variables removed. When determining the numerical relationship between two ...
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 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 of a random variabl ...
, 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 (AR) 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 A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure. Models c ...
or in an extended
ARIMA Arima, officially The Royal Chartered Borough of Arima is the easternmost and second largest in area of the three boroughs of Trinidad and Tobago. It is geographically adjacent to Sangre Grande and Arouca at the south central foothills of th ...
(p,d,q) model.


Definition

Given a time series z_t, the partial autocorrelation of lag k, denoted \phi_, is the
autocorrelation 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 of a random variable ...
between z_t and z_ with the linear dependence of z_t on z_ through z_ removed. Equivalently, it is the autocorrelation between z_t and z_ that is not accounted for by lags 1 through k-1, inclusive.\phi_ = \operatorname(z_, z_),\textk= 1,\phi_ = \operatorname(z_ - \hat_,\, z_ - \hat_),\textk\geq 2,where \hat_ and \hat_t are linear combinations of \ that minimize the
mean squared error In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference between ...
of z_ and z_t respectively. For
stationary process In mathematics and statistics, a stationary process (or a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose unconditional joint probability distribution does not change when shifted in time. Con ...
es, the coefficients in \hat_ and \hat_t are the same, but reversed: \hat_=\beta_1z_+\cdots+\beta_z_\qquad\text\qquad\hat_t=\beta_1z_+\cdots+\beta_z_.


Calculation

The theoretical partial autocorrelation function of a stationary time series can be calculated by using the Durbin–Levinson Algorithm:\phi_ = \fracwhere \phi_ = \phi_ - \phi_ \phi_ for 1 \leq k \leq n - 1 and \rho(n) is the autocorrelation function. The formula above can be used with sample autocorrelations to find the sample partial autocorrelation function of any given time series.


Examples

The following table summarizes the partial autocorrelation function of different models: The behavior of the partial autocorrelation function mirrors that of the autocorrelation function for autoregressive and moving-average models. For example, the partial autocorrelation function of an AR(''p'') series cuts off after lag ''p'' similar to the autocorrelation function of an MA(''q'') series with lag ''q''. In addition, the autocorrelation function of an AR(''p'') process tails off just like the partial autocorrelation function of an MA(''q'') process.


Autoregressive model identification

Partial autocorrelation is a commonly used tool for identifying the order of an autoregressive model. As previously mentioned, the partial autocorrelation of an AR(''p'') process is zero at lags greater than ''p''. If an AR model is determined to be appropriate, then the sample partial autocorrelation plot is examined to help identify the order. The partial autocorrelation of lags greater than ''p'' for an AR(''p'') time series are approximately independent and
normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...
with a
mean There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set. For a data set, the ''arithme ...
of 0. Therefore, a
confidence interval In frequentist statistics, a confidence interval (CI) is a range of estimates for an unknown parameter. A confidence interval is computed at a designated ''confidence level''; the 95% confidence level is most common, but other levels, such as 9 ...
can be constructed by dividing a selected
z-score In statistics, the standard score is the number of standard deviations by which the value of a raw score (i.e., an observed value or data point) is above or below the mean value of what is being observed or measured. Raw scores above the mean ...
by \sqrt. Lags with partial autocorrelations outside of the confidence interval indicate that the AR model's order is likely greater than or equal to the lag. Plotting the partial autocorrelation function and drawing the lines of the confidence interval is a common way to analyze the order of an AR model. To evaluate the order, one examines the plot to find the lag after which the partial autocorrelations are all within the confidence interval. This lag is determined to likely be the AR model's order.


References

{{Statistics, analysis Time domain analysis Covariance and correlation Time series