Processing Random Data: Statistics For Engineers And Scientists

by Robert V Edwards

Overview

Two features of Processing Random Data differentiate it from other similar books: the focus on computing the reproducibility error for statistical measurements, and its comprehensive coverage of Maximum Likelihood parameter estimation techniques. The book is useful for dealing with situations where there is a model relating to the input and output of a process, but with a random component, which could be noise in the system or the process itself could be random, like turbulence. Parameter estimation techniques are shown for many different types of statistical models, including joint Gaussian. The Cramer-Rao bounds are described as useful estimates of reproducibility errors.Finally, using an example with a random sampling of turbulent flows that can occur when using laser anemometry the book also explains the use of conditional probabilities.

Product Details

ISBN-13:	9789812568342
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	07/05/2006
Pages:	152
Product dimensions:	6.20(w) x 9.00(h) x 0.60(d)

Table of Contents

Preface     vii
Random Variables     1
Basic Concepts     1
Expected Values     4
Probability Distribution Functions     4
Properties of probability density functions     10
The Expected Value Process     11
Variance and Standard Deviation     12
Moments and Moment Generating Functions     14
Common Types of Distributions     15
Uniform distribution     15
Binomial distribution     16
Using the binomial distribution     19
Poisson distribution     20
Gaussian (or normal) distribution     23
Student-t distribution     24
Sum of two Gaussian variables     26
The Chi-squared distribution     26
The error function     27
Functions of More Than One Random Variable     28
(Bayes theorem)     28
Joint Gaussian distributions     31
Change of Variable     32
Measurement of Samples of Random Numbers     37
Variance of the Measured Mean     37
Estimate of the Variance     39
Variance of the Measured Variance     41
Non-Independent Random Variables     45
Histograms     47
Statistical experiments involving the uniform and binomial distributions     48
Confidence Limits     51
Tests of hypotheses     53
Time Series-Random Variables that are Functions of Time     61
Averages     61
The Autocovariance and Autocorrelation Function     62
Variance of the average of a signal with correlation     63
The Power Spectrum of a Random Signal     66
Short review of Fourier transforms     67
Processing Time Series by Computer     69
Estimation of the Autocorrelation     70
Error covariance for autocorrelation estimates     70
Random, correlated signal generator     75
Estimation of the Power Spectrum of Time Series Data     76
Generic properties of the digital Fourier transform     77
The variance of spectrum estimates     79
Batch Mode Autocorrelation     82
Parameter Estimation     87
Motivation     87
Maximum Likelihood Estimation     91
Gaussian processes     92
Poisson processes     94
Non-independent Gaussian statistics     95
Chi-squared distributions      97
Residuals     98
Parameter Error Estimates     100
Expected errors for Gaussian statistics     103
Error estimates for joint Gaussian statistics     105
Error estimates for Poisson processes     106
Error estimates for Chi-square distributions     106
A Priori Error Estimation     106
Maximum A Posteriori Estimation     108
Random Sampling     115
Basic Concepts of Random Sampling     115
Independent Sampling     116
Mean and variance of the mean with random sampling     117
Spectrum and autocorrelation estimate using randomly sampled data     119
Effect of finite sample intervals [Delta]t     124
Sample and Hold Autocorrelation and Spectrum Estimation     124
Non-Independent Sampling     127
Sample and hold (rate depends on f)     129
Controlled sampling     132
Autocorrelation estimation     133
Photon Detection     134
Moments     134
Autocorrelation estimation     135
Index     139