Chi-square function

Moreover, because the Mahalanobis distance is a chi-square function, as is the SIMCA distance used to define the class space in the SIMCA method (Sect. 4.3), it is possible to use Coomans diagrams (Sect. 4.3) both to visualize the results of modelling and classification (distance from two category centroids) and to compare two different methods (Mahalanobis distance from the centroids versus SIMCA distance). 

The Chi-Squared function was used with the 70% confidence interval and 40(2 X (19 + 1)) degrees of freedom. The +1 is added to account for the possibility that another failure may occur in the next moment of operation. 

The functions chi2pdf(x,v) and chi2cdf(x,v) are available in the MATLAB Statistics Toolbox for calculating the probability density function and cumulative distribution function for chi-square function. The following examples demonstrate the application of chi-square distribution as a tool to analyze studies, tests, or surveys. 

It has been shown, in Sect. 4.2.1, how to estimate from a sample mean the limits within which the true mean of a population will fall with a desired level of confidence C. It may be desirable to estimate, now, the limits within which the population standard deviation will fall with a desired level of confidence C from the sample standard deviation s. The target is always the same to estimate true values when the entire population of data is not available. To this purpose, the method of the so called (Chi-squared) function may be used. Again, as in aU preceding sections, it must be assumed that the distribution is normal. The relation between 5 and A is given by... 

Note Impedance spectra were modeled using complex nonlinear least-squares (CNLS) fitting program EQUIVCRT developed by B. A. Boukamp. The Chi-square function gives a good indication of the quality of the fit a value of... 

Limits to the mean and standard deviation have been discussed in the previous sections based upon Student s t function and the Chi-squared function. While the theory of confidence intervals for these two quantities is well developed, such is not the case for the general nonlinear fitting of parameters to a data set. This will be discussed in the next chapter on general parameter estimation. For such cases, about the only approach to confidence intervals is through the Monte Carlo simulation of a number of test data sets. This approach is also applicable to limits on the mean and standard deviation and will be discussed here partly as background for the next chapter and as another approach to obtaining confidence intervals on statistical quantities. 

It may be decided that the gamma prior cannot be greater than a certain value xf. This has the effect of true Ling the normalizing denominator in equation 2.6-10," and leads to equation 2.6-17, where P(x v) is the cumulative integral from 0 to over the chi-squared density function with V degrees of freedom, a is the prescribed confidence fraction, and = 2 A" (t+Tr). Thus, the effect of the truncated gamma prior is to modify the confidence interval to become an effective confidence interval of a ... 

In order to determine the quality (or the validity) of fit of a particular function to the data points given, a comparison of the deviation of the curve from the data to the size of the experimental error can be made. The deviations (i.e., the scatter off the curve) should be of the same order of magnitude as the experimental error, so that the quantity chi-squared is defined as... 

The chi-square distribution gives the probability for a continuous random variable bounded on the left tail. The probability function has a shape parameter... 

CHIDIST is the Excel function for the one-tailed probability of the chi-squared distribution. 

The goodness-of-fit between the experimental and theoretically calculated CBED rocking curves is described by a merit function, and in the present study we use the chi-square merit function defined as... 

The F distribution, similar to the chi square, is sensitive to the basic assumption that sample values were selected randomly from a normal distribution. The Microsoft Excel function FDIST(X, dfh df2) gives the upper percent points of Table 3-8, where X is the tabular value. The function FINV(Percent, dfh df2) gives the table value. 

As was indicated in Section 7.2, the vector of measurement adjustments, e, has a multivariate normal distribution with zero mean and covariance matrix V. Thus, the objective function value of the least square estimation problem (7.21), ofv = eT l> 1 e, has a central chi-square distribution with a number of degrees of freedom equal to the rank of A. 

The least-squares method is also widely applied to curve fitting in phase-modulation fluorometry the main difference with data analysis in pulse fluorometry is that no deconvolution is required curve fitting is indeed performed in the frequency domain, i.e. directly using the variations of the phase shift and the modulation ratio M as functions of the modulation frequency. Phase data and modulation data can be analyzed separately or simultaneously. In the latter case the reduced chi squared is given by... 

Figure 4.11. Reduced chi-square for fitting a single Gaussian distribution function of decays with either a discrete single or double exponential model as a function of dis tribud on width (/f)...

There are several mathematically different ways to condnct the minimization of S [see Refs. 70-75]. Many programs yield errors of internal consistency (i.e., the standard deviations in the calculated parameters are due to the deviations of the measured points from the calculated function), and do not consider external errors (i.e., the uncertainty of the measured points). The latter can be accommodated by weighting the points by this uncertainty. The overall rehabU-ity of the operation can be checked by the (chi square) test [71], i.e., S (L + N - ) should be in the range 0.5-1.5 for a reasonable consistency between the measured points and the calculated parameters. 

The last term of Equation a-7 is the chi-square (x ) probability function and... 

For common statistics, such as the Student s t value, chi-square, and Fisher F, Excel has functions that return the critical value at a given probability and degrees of freedom (e.g., =TINV (0.05,10) for the two-tailed Lvalue at a probability of 95% and 10 degrees of freedom), or which accept a calculated statistic and give the associated probability (e.g., =TDIST( t, 10, 2 ) for 10 degrees of freedom and two tails). Table 2.3 gives common statistics calculated in the course of laboratory quality control. 

Suppose a polydisperse system is investigated experimentally by measuring the number of particles in a set of different classes of diameter or molecular weight. Suppose further that these data are believed to follow a normal distribution function. To test this hypothesis rigorously, the chi-squared test from statistics should be applied. A simple graphical examination of the hypothesis can be conducted by plotting the cumulative distribution data on probability paper as a rapid, preliminary way to evaluate whether the data conform to the requirements of the normal distribution. 

The log-likelihood function at the maximum likelihood estimates is -28.993171. For the model with only a constant term, the value is -31.19884. The t statistic for testing the hypothesis that (3 equals zero is 5.16577/2.51307 = 2.056. This is a bit larger than the critical value of 1.96, though our use of the asymptotic distribution for a sample of 10 observations might be a bit optimistic. The chi squared value for the likelihood ratio test is 4.411, which is larger than the 95% critical value of 3.84, so the hypothesis that 3 equals zero is rejected on the basis of these two tests. 

The decay parameters [a (X) and rj are recovered from the experimentally measured phase shift and demodulation factor by the method of non-linear least squares (24,25). The goodness-of-fit between the assumed model (c subscript) and the experimentally measured (m subscript) data is determined by the chi-squared (x2) function ... 

The chi-square distribution was discussed briefly in the earlier section on probability distributions. Suppose we have (k+1) independent standard normal variables. We then define % as the sum of the squares of these (k+1) variables. It can be shown that the probability density function of % is ... 

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