Tutorial of Time series analysis in Excel | Examples of Econometric data modeling and forecast | Time Series | Tutorial | Excel | NumXL. PASSaGE: Pattern Analysis, Spatial Statistics, and Geographic Exegesis. Moran's I. The white and black squares are perfectly dispersed so Moran's I would be −1. If the white squares were stacked to one half of the board and the black squares to the other, Moran's I would be close to +1. A random arrangement of square colors would give Moran's I a value that is close to 0. In statistics, Moran's I is a measure of spatial autocorrelation developed by Patrick Alfred Pierce Moran.[1][2] Spatial autocorrelation is characterized by a correlation in a signal among nearby locations in space.
Spatial autocorrelation is more complex than one-dimensional autocorrelation because spatial correlation is multi-dimensional (i.e. 2 or 3 dimensions of space) and multi-directional. Moran's I is defined as where is the number of spatial units indexed by and is the variable of interest; is the mean of ; and is an element of a matrix of spatial weights. The expected value of Moran's I under the null hypothesis of no spatial autocorrelation is Its variance equals Sources[edit] See also[edit]
Durbin–Watson statistic. In statistics, the Durbin–Watson statistic is a test statistic used to detect the presence of autocorrelation (a relationship between values separated from each other by a given time lag) in the residuals (prediction errors) 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 (von Neumann, 1941). Durbin and Watson (1950, 1951) 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.
Computing and interpreting the Durbin–Watson statistic[edit] If et is the residual associated with the observation at time t, then the test statistic is where T is the number of observations. If the design matrix of the regression is known, exact critical values for the distribution of is distributed as where statistic, i.e. Ljung–Box test. The Ljung–Box test (named for Greta M. Ljung and George E. P. Box) 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.
This test is sometimes known as the Ljung–Box Q test, and it is closely connected to the Box–Pierce test (which is named after George E. The Ljung–Box test is widely applied in econometrics and other applications of time series analysis. Formal definition[edit] The Ljung–Box test can be defined as follows. H0: The data are independently distributed (i.e. the correlations in the population from which the sample is taken are 0, so that any observed correlations in the data result from randomness of the sampling process). Ha: The data are not independently distributed.
The test statistic is:[2] where n is the sample size, where Box-Pierce test[edit] See also[edit] Chi-squared distribution. The chi-squared distribution is used in the common chi-squared tests for goodness of fit of an observed distribution to a theoretical one, the independence of two criteria of classification of qualitative data, and in confidence interval estimation for a population standard deviation of a normal distribution from a sample standard deviation.
Many other statistical tests also use this distribution, like Friedman's analysis of variance by ranks. History and name[edit] This distribution was first described by the German statistician Friedrich Robert Helmert in papers of 1875/1876,[7] where he computed the sampling distribution of the sample variance of a normal population. Thus in German this was traditionally known as the Helmertsche ("Helmertian") or "Helmert distribution". Definition[edit] is distributed according to the chi-squared distribution with k degrees of freedom. This is usually denoted as Characteristics[edit] Probability density function[edit] Cumulative distribution function[edit] Credit Research Centre ~ Specialists in credit research. Basel Analytics Inc.