Tolerance Factors

The use of curve-fitting or regression can have the additional result of the tolerance factor that defines confidence limits for the best-fit function.  Often this is completed as a second effort following determination of the best-fit function.  The traditional theory involves posing random errors about the data and minimizing deviations either by forming a linear system and using matrix solution or by a trial and error process that varies parameters and gauges progress toward minimizing the deviation of fit.  Usually the word ‘random error’ is synonymous with ‘unknown error’ or assuming a normally distributed error.  Thus the standard deviation, used for the tolerance factor, follows a chi-squared distribution.

This picture is extended to multiple dimensional fitting functions and thus, the random errors are from a multivariate normal distribution.  Therefore the tolerance factors for the fitting function are determined by a multivariate chi-squared distribution.  Considering the data could be correlated, the Wishart distribution provides a basis for determining the multivariate tolerance factors.  This essentially is a student-t like result, giving a tolerance factor for multiplying the standard deviation to get a confidence interval corresponding to a desired probability or coverage.  To generate these tolerance factors, a multivariate theory (to any dimension), using  a core of the Wishart distribution, is constructed and then Monte-Carlo is applied to randomly sample the distributions and generate a finite population (N instances).  The higher N gives more stable answers that converge to some precision.

The tolerance factor project was initiated as a statistics application.  This is a different point of view as error and uncertainty are often described using traditional statistics.  Instead, the duals arithmetic has every number formatted without use of statistics and Monte-Carlo is unnecessary to perform an uncertainty analysis.  The project is intended to generate the tolerance factors two possible ways.  The first implementation is to convert the existing traditional numbers and calculation steps to duals arithmetic and repeat the Monte-Carlo process while propagating error.  The challenge is to have a rational way to source errors from the random number generator.  The second implementation is to re-derive the multivariate tolerance factor theory to avoid Monte-Carlo sampling and distribution assumptions.  A goal is to obtain tolerance factors without distributions named normal, chi-squared, multi-variate normal and Wishart.  Duals arithmetic, with multi-dimensional error vectors, is a path toward this goal.

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