Chi-square method

A deconvolution problem in general does not have a unique solution. Instead there are an infinite number of possible solutions that can fit the same set of cascade impactor measurements. It is well recognized that, for most engineering applications, actual particle size distributions can be reasonably represented by a set of log-normal distributions. With this concern, in the following, a deconvolution method (chi-squared method) to extract particle size distributions from cascade impactor data is introduced, which is based on multimodal log-normal size distributions (Dzubay and Hasan, 1990). 

For each mode, three size distribution parameters (Ck, xk, and ok) need to be determined. These can be determined by a nonlinear least squared method (known as chi-squared method) which minimizes %2 defined by... 

For one failure, the following table applies. The times for larger numbers of failures can be calculated accordingly (i.e., from chi-square methods). 

There are several numerical methods to determine the parameters below we describe, as examples, the standard least-squares method and the chi-square method. 

The last word about the meaning of the intensity parameters has not been said yet, and further research is necessary. A problem for a rigorous comparison is the relatively large error in the Q), parameter (more than 5% or sometimes up to 10-20%). Parameter sets for the same system, but determined by different authors, may be substantially different. This is partially due to the fact that the values of the parameters depend on the transitions chosen for the fitting procedure (as the standard least-squares fitting procedure is chosen). Comparison between fits is only possible if the chi-square method is used to determine the parameters (Goldner and Auzel 1996). 

B.l Random Number Generation (See Sect. 4.2) B.1.1 Chi-Square Method... 

The Chi-Square method checks for uniformity by dividing a range F of uniformly distributed random numbers a into a series of k adjacent intervals as... 

The numerator is a random normally distributed variable whose precision may be estimated as V(N) the percent of its error is f (N)/N = f (N). For example, if a certain type of component has had 100 failures, there is a 10% error in the estimated failure rate if there is no uncertainty in the denominator. Estimating the error bounds by this method has two weaknesses 1) the approximate mathematics, and the case of no failures, for which the estimated probability is zero which is absurd. A better way is to use the chi-squared estimator (equation 2,5.3.1) for failure per time or the F-number estimator (equation 2.5.3.2) for failure per demand. (See Lambda Chapter 12 ),... 

Selection 2 is a similar calculation using the F-Number method (Section 2.5.3.2) 3 calculates the integral over the Chi-Squared distribution. When selected i nput the upper limit of integration... 

Frequency domain performance has been analyzed with goodness-of-fit tests such as the Chi-square, Kolmogorov-Smirnov, and Wilcoxon Rank Sum tests. The studies by Young and Alward (14) and Hartigan et. al. (J 3) demonstrate the use of these tests for pesticide runoff and large-scale river basin modeling efforts, respectively, in conjunction with the paired-data tests. James and Burges ( 1 6 ) discuss the use of the above statistics and some additional tests in both the calibration and verification phases of model validation. They also discuss methods of data analysis for detection of errors this last topic needs additional research in order to consider uncertainties in the data which provide both the model input and the output to which model predictions are compared. 

One critical limitation of MAXSLOPE is that the method locates the hitmax correctly only for certain types of latent distributions (e.g., normal distribution). For other kinds of distributions (e.g., chi-square distribution), the estimated location of the hitmax and the base rate may be substantially off the mark. In other words, when the underlying distributions are of the difficult kind, MAXSLOPE will detect taxonicity, but the estimated taxon base rate may be incorrect. Moreover, MAXSLOPE may fail to detect taxonicity under certain circumstances. Specifically, this will happen if ... 

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... 

Xi2( ) The 100(1 -ac)% point of the chi-squared distribution with 1 degree of freedom iji A parameter setting the relative contributions of the linearization and steepest descent methods in determining the correction vector b of Eq. (45)... 

Note that the approximation by the chi-square distribution is only possible for multivariate normally distributed data which somehow is in conflict if outliers are present that should be identified with this measure. We recommend that robust PCA is used whenever diagnostics is done because robust methods tolerate deviations from multivariate normal distribution. 

In general it can be shown that the approach following the signal-to-noise ratio method is mathematically very similar to the standard formulation of the chi-square test using observed and expected frequencies, and in practice they will invariably give very similar results. Altman (1991) (Section 10.7.4) provides more detail on this connection. 

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). 

Objects do not fall exactly into the inner model space, and a residual error on each variable can be computed. These residuals are uncorrelated variables, because each significant correlation is retained in the linear model. So, the variance of residuals is a chi-square variable, the SIMCA distance, and, multiplied by a suitable coefficient obtained from the F distribution, it fixes the boundary of the class space around the model, called the SIMCA box, that corresponds to the confidents hyperellipsoid of the bayesian method. Objects, both those used and those not used to obtain the... 

The XPS spectra were recorded on a Surface Science Laboratories small spot system using a monochromatized A1K X-ray radiation source. The take-off angle used for these measurements was 35°. Full details of the methods used in interpreting the XPS data have been described elsewhere [14], Data reduction was done using Surface Science Laboratories software version 8.0. This software utilizes a least squares curve fitting approach with only chi square statistics for goodness of the calculated fit to the experimental data. 

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