The Statistical Analysis of Data.

Abstract

2 PrefaceThe main purpose of this book is to document the mathematics of propagation of error and regression analysis-two of the most common statistical analysis procedures in the physical sciences. Both procedures transform input data and their uncertainties to output parameters and their uncertainties.There are several important assumptions that make the mathematics rigorous. For example, each input datum is assumed to be a random sample from a probability distribution such as might occur from measurement. The difference between the sample value and the mean of the distribution is taken as the random error and the difference between the true value associated with that datum and the mean is taken as the systematic error. Another assumption involves the transformation between the input and output. Namely, each input error must be small enough that a change to its value (over a range that would include nearly all of its likely values) should cause only proportional changes in the output parameters.Sticking to the mathematics, several important experimental issues are left unaddressed. For example, while real physical variables are often nuanced, we take the simplistic view that the quantity underlying a datum is perfectly well defined and has an exact true value-the sample value one would get if the random and systematic error could be brought to zero. While impractical, this model should be kept in mind when constructing one for a real physical apparatus.The assumptions above are not overly restrictive for experiments in the physical sciences where small measurement errors are common. Together with the principle of maximum likelihood, they lead to a robust set of statistical formulas and procedures. Covariance matrices describe the behavior of the input and output variations, Jacobian matrices describe the linear relationships among those variations, and linear algebra makes the math readable and concise.The target audience for this book is anyone who wants to learn the mathematics of error propagation and regression analysis. I particularly enjoyed learning how applying the principle of maximum likelihood to regression analysis with independent variables governed by the Poisson, binomial, and exponential distributions leads to a chi-square minimization. In each case, the mathematics is quite specific about what to use for the variances appearing in the chi-square calculation so as to achieve the maximum likelihood 4 condition. Figuring out how to propagate an instrument's calibration uncertainty to the uncertainty in any results obtained using that instrument is one of the original problems I wanted to resolve. The mathematics turned out quite simple and straightforward.I hope the reader will find clarity and rigor in the treatment presented and beauty and simplicity in the final results.

Reverse Engineering, Some Board Games as a Key to the Development of Artificial Intelligence and Construction of Economic Models

“…The secretary problem (Moore, 1983). A group of N candidates for vacant secretary position differ in qualifications: from 1-best candidate to N-worst candidate, in random order, so that the best one has equal chances of applying as first, second, k (k = 3, 4, ..., N-1), or last.…”