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In statistics, the Matérn covariance, also called the Matérn kernel, [1] is a covariance function used in spatial statistics, geostatistics, machine learning, image analysis, and other applications of multivariate statistical analysis on metric spaces. It is named after the Swedish forestry statistician Bertil Matérn. [2]
In probability theory and statistics, the covariance function describes how much two random variables change together (their covariance) with varying spatial or temporal separation. For a random field or stochastic process Z ( x ) on a domain D , a covariance function C ( x , y ) gives the covariance of the values of the random field at the two ...
The sample covariance matrix (SCM) is an unbiased and efficient estimator of the covariance matrix if the space of covariance matrices is viewed as an extrinsic convex cone in R p×p; however, measured using the intrinsic geometry of positive-definite matrices, the SCM is a biased and inefficient estimator. [1]
We employ the Matlab routine for 2-dimensional data. The routine is an automatic bandwidth selection method specifically designed for a second order Gaussian kernel. [14] The figure shows the joint density estimate that results from using the automatically selected bandwidth. Matlab script for the example
i.e., the inverse-gamma distribution, where () is the ordinary Gamma function. The Inverse Wishart distribution is a special case of the inverse matrix gamma distribution when the shape parameter α = ν 2 {\displaystyle \alpha ={\frac {\nu }{2}}} and the scale parameter β = 2 {\displaystyle \beta =2} .
The covariance function is a crucial design choice, since it stipulates the properties of the Gaussian process and thereby the behaviour of the model. The covariance function encodes information about, for instance, smoothness and periodicity, which is reflected in the estimate produced.
In statistics, a covariate represents a source of variation that has not been controlled in the experiment and is believed to affect the dependent variable. [8] The aim of such techniques as ANCOVA is to remove the effects of such uncontrolled variation, in order to increase statistical power and to ensure an accurate measurement of the true relationship between independent and dependent ...
Throughout this article, boldfaced unsubscripted and are used to refer to random vectors, and Roman subscripted and are used to refer to scalar random variables.. If the entries in the column vector = (,, …,) are random variables, each with finite variance and expected value, then the covariance matrix is the matrix whose (,) entry is the covariance [1]: 177 ...