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 (the time lags) is an **autocorrelogram**. If cross-correlation is plotted, the result is called a **cross-correlogram**.

- Applications
- Importance
- Estimation of autocorrelations
- Alternate estimate
- Statistical inference with correlograms
- Software
- Related techniques
- References
- Further reading
- External links

The correlogram is a commonly used tool for checking randomness in a data set. If random, autocorrelations should be near zero for any and all time-lag separations. If non-random, then one or more of the autocorrelations will be significantly non-zero.

In addition, correlograms are used in the model identification stage for Box–Jenkins autoregressive moving average time series models. Autocorrelations should be near-zero for randomness; if the analyst does not check for randomness, then the validity of many of the statistical conclusions becomes suspect. The correlogram is an excellent way of checking for such randomness.

Sometimes, **corrgrams**, color-mapped matrices of correlation strengths in multivariate analysis,^{ [1] } are also called correlograms.^{ [2] }^{ [3] }

The correlogram can help provide answers to the following questions:

- Are the data random?
- Is an observation related to an adjacent observation?
- Is an observation related to an observation twice-removed? (etc.)
- Is the observed time series white noise?
- Is the observed time series sinusoidal?
- Is the observed time series autoregressive?
- What is an appropriate model for the observed time series?
- Is the model

- valid and sufficient?

- Is the formula valid?

^{ [4] }

Randomness (along with fixed model, fixed variation, and fixed distribution) is one of the four assumptions that typically underlie all measurement processes. The randomness assumption is critically important for the following three reasons:

- Most standard statistical tests depend on randomness. The validity of the test conclusions is directly linked to the validity of the randomness assumption.
- Many commonly used statistical formulae depend on the randomness assumption, the most common formula being the formula for determining the standard error of the sample mean:

where *s* is the standard deviation of the data. Although heavily used, the results from using this formula are of no value unless the randomness assumption holds.

- For univariate data, the default model is

If the data are not random, this model is incorrect and invalid, and the estimates for the parameters (such as the constant) become nonsensical and invalid.

The autocorrelation coefficient at lag *h* is given by

where *c _{h}* is the autocovariance function

and *c*_{0} is the variance function

The resulting value of *r _{h}* will range between −1 and +1.

Some sources may use the following formula for the autocovariance function:

Although this definition has less bias, the (1/*N*) formulation has some desirable statistical properties and is the form most commonly used in the statistics literature. See pages 20 and 49–50 in Chatfield for details.

In the same graph one can draw upper and lower bounds for autocorrelation with significance level :

- with as the estimated autocorrelation at lag .

If the autocorrelation is higher (lower) than this upper (lower) bound, the null hypothesis that there is no autocorrelation at and beyond a given lag is rejected at a significance level of . This test is an approximate one and assumes that the time-series is Gaussian.

In the above, *z*_{1−α/2} is the quantile of the normal distribution; SE is the standard error, which can be computed by Bartlett's formula for MA(*ℓ*) processes:

- for

In the picture above we can reject the null hypothesis that there is no autocorrelation between time-points which are adjacent (lag = 1). For the other periods one cannot reject the null hypothesis of no autocorrelation.

Note that there are two distinct formulas for generating the confidence bands:

1. If the correlogram is being used to test for randomness (i.e., there is no time dependence in the data), the following formula is recommended:

where *N* is the sample size, *z* is the quantile function of the standard normal distribution and α is the significance level. In this case, the confidence bands have fixed width that depends on the sample size.

2. Correlograms are also used in the model identification stage for fitting ARIMA models. In this case, a moving average model is assumed for the data and the following confidence bands should be generated:

where *k* is the lag. In this case, the confidence bands increase as the lag increases.

Correlograms are available in most general purpose statistical libraries.

Correlograms:

Corrgrams:

**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 **binomial distribution** with parameters *n* and *p* is the discrete probability distribution of the number of successes in a sequence of *n* independent experiments, each asking a yes–no question, and each with its own Boolean-valued outcome: *success* or *failure*. A single success/failure experiment is also called a Bernoulli trial or Bernoulli experiment, and a sequence of outcomes is called a Bernoulli process; for a single trial, i.e., *n* = 1, the binomial distribution is a Bernoulli distribution. The binomial distribution is the basis for the popular binomial test of statistical significance.

In probability theory, a **normal****distribution** is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is

In statistics, **correlation ** or **dependence ** is any statistical relationship, whether causal or not, between two random variables or bivariate data. In the broadest sense **correlation** is any statistical association, though it commonly refers to the degree to which a pair of variables are linearly related. Familiar examples of dependent phenomena include the correlation between the height of parents and their offspring, and the correlation between the price of a good and the quantity the consumers are willing to purchase, as it is depicted in the so-called demand curve.

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

The statistical **power** of a binary hypothesis test is the probability that the test correctly rejects the null hypothesis when a specific alternative hypothesis is true. It is commonly denoted by , and represents the chances of a "true positive" detection conditional on the actual existence of an effect to detect. Statistical power ranges from 0 to 1, and as the power of a test increases, the probability of making a type II error by wrongly failing to reject the null hypothesis decreases.

The ** t-test** is any statistical hypothesis test in which the test statistic follows a Student's

In statistics, **ordinary least squares** (**OLS**) is a type of linear least squares method for estimating the unknown parameters in a linear regression model. OLS chooses the parameters of a linear function of a set of explanatory variables by the principle of least squares: minimizing the sum of the squares of the differences between the observed dependent variable in the given dataset and those predicted by the linear function of the independent variable.

In statistics, **simple linear regression** is a linear regression model with a single explanatory variable. That is, it concerns two-dimensional sample points with one independent variable and one dependent variable and finds a linear function that, as accurately as possible, predicts the dependent variable values as a function of the independent variable. The adjective *simple* refers to the fact that the outcome variable is related to a single predictor.

In statistics, a **fixed effects model** is a statistical model in which the model parameters are fixed or non-random quantities. This is in contrast to random effects models and mixed models in which all or some of the model parameters are random variables. In many applications including econometrics and biostatistics a fixed effects model refers to a regression model in which the group means are fixed (non-random) as opposed to a random effects model in which the group means are a random sample from a population. Generally, data can be grouped according to several observed factors. The group means could be modeled as fixed or random effects for each grouping. In a fixed effects model each group mean is a group-specific fixed quantity.

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.

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.

**Bootstrapping** is any test or metric that uses random sampling with replacement, and falls under the broader class of resampling methods. Bootstrapping assigns measures of accuracy to sample estimates. This technique allows estimation of the sampling distribution of almost any statistic using random sampling methods.

The **Hurst exponent** is used as a measure of long-term memory of time series. It relates to the autocorrelations of the time series, and the rate at which these decrease as the lag between pairs of values increases. Studies involving the Hurst exponent were originally developed in hydrology for the practical matter of determining optimum dam sizing for the Nile river's volatile rain and drought conditions that had been observed over a long period of time. The name "Hurst exponent", or "Hurst coefficient", derives from Harold Edwin Hurst (1880–1978), who was the lead researcher in these studies; the use of the standard notation *H* for the coefficient also relates to his name.

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. If we are interested in finding to what extent there is a numerical relationship between two variables of interest, using their correlation coefficient will give misleading results if there is another, confounding, variable that is numerically related to both variables of interest. This misleading information can be avoided by controlling for the confounding variable, which is done by computing the partial correlation coefficient. This is precisely the motivation for including other right-side variables in a multiple regression; but while multiple regression gives unbiased results for the effect size, it does not give a numerical value of a measure of the strength of the relationship between the two variables of interest.

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 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.

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 survey methodology, the **design effect** is the ratio between the variances of two estimators to some parameter of interest. Specifically the ratio of an actual variance of an estimator that is based on a sample from some sampling design, to the variance of an alternative estimator that would be calculated (hypothetically) using a sample from a simple random sample (SRS) of the same number of elements. It measures the expected effect of the design structure on the variance of some estimator of interest. The design effect is a positive real number that can indicate an inflation, or deflation in the variance of an estimator for some parameter, that is due to the study not using SRS.

In econometrics, **Prais–Winsten estimation** is a procedure meant to take care of the serial correlation of type AR(1) in a linear model. Conceived by Sigbert Prais and Christopher Winsten in 1954, it is a modification of Cochrane–Orcutt estimation in the sense that it does not lose the first observation, which leads to more efficiency as a result and makes it a special case of feasible generalized least squares.

- ↑ Friendly, Michael (19 August 2002). "Corrgrams: Exploratory displays for correlation matrices" (PDF).
*The American Statistician*. Taylor & Francis.**56**(4): 316–324. doi:10.1198/000313002533 . Retrieved 19 January 2014. - 1 2 "CRAN – Package corrgram".
*cran.r-project.org*. 29 August 2013. Retrieved 19 January 2014. - 1 2 "Quick-R: Correlograms".
*statmethods.net*. Retrieved 19 January 2014. - ↑ "1.3.3.1. Autocorrelation Plot".
*www.itl.nist.gov*. Retrieved 2018-08-20. - ↑ "Visualization § Autocorrelation plot".

- Hanke, John E.; Reitsch, Arthur G.; Wichern, Dean W.
*Business forecasting*(7th ed.). Upper Saddle River, NJ: Prentice Hall. - Box, G. E. P.; Jenkins, G. (1976).
*Time Series Analysis: Forecasting and Control*. Holden-Day. - Chatfield, C. (1989).
*The Analysis of Time Series: An Introduction*(Fourth ed.). New York, NY: Chapman & Hall.

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