Multivariate Linear Regression with Self-Estimation of Errors
A common approach to the analysis of a sample of data is to search for linear correlations between the variables that describe the sample. Most of the fitting techniques used in that context derive from a statistically rigorous method of estimation, the maximum likelihood method. In its simplest and most widely used form, the method is applied to the case of two variables, only one of which is affected by error; while in the more general case all variables under consideration are beset by errors. Moreover, what we call generically “errors” can actually have various origins: they can either be real measurement errors, or rather originate from a scatter intrinsic in the sample, appearing when the variables included in the analysis are not sufficient to completely describe the physical attributes of the objects. The former case is easier to analyze because measurement errors can, in principle, be estimated and standard multivariate regression techniques can be adopted; the latter, instead, is more difficult to deal with as one must extract simultaneously from the data both the direction of the main correlation and the properties of the scatter. Principal components analysis is usually adopted for this purpose but by this technique arbitrary structure is introduced in the residuals.
KeywordsMultivariate Linear Regression Error Matrix Dimensional Parameter Space Main Correlation Standard Maximum Likelihood
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