Sum of squared residuals

SXY(,) sum of squared residuals per column on diagonal, and sums of products of residuals in cells above diagonal... 

The number of factors r, s and t, assigned to each mode, is generally different. They are chosen such as to be less than the dimensions of the original three-way table in order to achieve a considerable amount of data reduction. The elements of Z represent the magnitude of the factors and the extent of their interaction [57]. Computationally, the core matrix Z and the loadings matrices A, B and C are derived such as to minimize the sum of squared residuals. 

In contrast to the Tucker3 model, described above, the number of factors in each mode is identical. It is chosen to be much smaller than the original dimensions of the data table in order to achieve a considerable reduction of the data. The elements of the loading matrices A, B, C are computed such as to minimize the sum of squared residuals. 

Goodness-of-fit tests may be a simple calculation of the sum of squared residuals for each organ in the model [26] or calculation of a log likelihood function [60], In the former case,... 

SSR = sum of squared residuals N = number of observations C° = mean of the observed drug concentration C = predicted drug concentration rii = number of experimental repetitions S2 = variance of the observed concentrations at each data point LL = log likelihood function... 

Although we cannot clearly determine the reaction order from Figure 3.9, we can gain some insight from a residual plot, which depicts the difference between the predicted and experimental values of cA using the rate constants calculated from the regression analysis. Figure 3.10 shows a random distribution of residuals for a second-order reaction, but a nonrandom distribution of residuals for a first-order reaction (consistent overprediction of concentration for the first five datapoints). Consequently, based upon this analysis, it is apparent that the reaction is second-order rather than first-order, and the reaction rate constant is 0.050. Furthermore, the sum of squared residuals is much smaller for second-order kinetics than for first-order kinetics (1.28 X 10-4 versus 5.39 xl0 4). 

In E-Z Solve, a separate equation entry is required for each data pair. The software then determines the values of m and b which minimize the sum of squared residuals (SSR). The E-Z Solve syntax is shown below. 

Although equation (A) can be integrated analytically (resulting in equations 3.4-9 and 3.4-10), for the sake of illustration we presume that we must integrate equation (A) numerically. For a given kA and n, numerical integration of Equation (A) provides a predicted cA(f) profile which may be compared against the experimental data. Values of kA and n are adjusted until the sum of squared residuals between the experimental and predicted concentrations is minimized. 

We will now investigate the sampling properties of the statistic representing the weighted sum of squared residuals i1 given by equation (5.4.13). We first observe that the slightly different expression (y — l)rSy i(y —is zero since... 

The model itself can be tested against the sum of squared residuals c2=4.01. If, as a first approximation, we admit that intensities are normally distributed (which may not be too incorrect since all the values seem to be distant from zero by many standard deviations), c2 is distributed as a chi-squared variable with 5 — 3 = 2 degrees of freedom. Consulting statistical tables, we find that there is a probability of 0.05 that a chi-squared variable with two degrees of freedom exceeds 5.99, a value much larger than the observed c2. We therefore accept to the 95 percent confidence level the hypothesis that the linear signal addition described by the mass balance equations is correct, o... 

Replacing z in the preceding equations with the sum of squared residuals, ssq, the first derivatives... 

There exists even a further simplification which makes it possible to directly use the sum of squared residuals RSS. In this so-called generalized CV (GCV), the values hu in Equation 4.47 are replaced by trace (H)/n = 1/hu/n, leading to a good approximation of the MSE ... 

In Table II the sums of squared residuals (RSS) of Set I are found calculated by the TTFA type model solved by PLS. All 13 potential profiles were predicted from the 40 air samples, while in reality there were only 9 active. The first row contains the RSS s from PLS models predicting one source profile at a time, the second row from the PLS model predicting all the source profiles simultaneously. From the difference of the RSS s between the first nine and the last four profiles it is clear that in this data set there were only nine sources active. These results are Intended only to Illustrate what kind of information is provided by the PLS solution. 

Infrared data in the 1575-400 cm region (1218 points/spec-trum) from LTAs from 50 coals (large data set) were used as input data to both PLS and PCR routines. This is the same spe- tral region used in the classical least-squares analysis of the small data set. Calibrations were developed for the eight ASTM ash fusion temperatures and the four major ash elements as oxides (determined by ICP-AES). The program uses PLSl models, in which only one variable at a time is modeled. Cross-validation was used to select the optimum number of factors in the model. In this technique, a subset of the data (in this case five spectra) is omitted from the calibration, but predictions are made for it. The sum-of-squares residuals are computed from those samples left out. A new subset is then omitted, the first set is included in the new calibration, and additional residual errors are tallied. This process is repeated until predictions have been made and the errors summed for all 50 samples (in this case, 10 calibrations are made). This entire set of... 

The dimensionality of the model, a, is estimated so as to give the model as good predictive properties as possible. Geometrically, this corresponds to the fitting of an a-dimensional hyperplane to the object points in the measurement space. The fitting is made using the least squares criterion, i.e. the sum of squared residuals is minimized for the class data set. 

Bias corrections are sometimes applied to MLEs (which often have some bias) or other estimates (as explained in the following section, [mean] bias occurs when the mean of the sampling distribution does not equal the parameter to be estimated). A simple bootstrap approach can be used to correct the bias of any estimate (Efron and Tibshirani 1993). A particularly important situation where it is not conventional to use the true MLE is in estimating the variance of a normal distribution. The conventional formula for the sample variance can be written as = SSR/(n - 1) where SSR denotes the sum of squared residuals (observed values, minus mean value) is an unbiased estimator of the variance, whether the data are from a normal distribution... 

The best regression was determined using a minimization routine incorporated into the program which minimized the sum of squared residuals between the calculated and observed EXAFS oscillation, x and % The result was visually checked comparing the k and k weighted Fourier transforms of the regression and contributions of each regressed shell to the acquired data. 

Suppose b is the least squares coefficient vector in the regression of y on X and c is any other Kx vector. Prove that the difference in the two sums of squared residuals is... 

The result follows immediately from the result which precedes (6-19). Since the sum of squared residuals must be at least as large, the coefficient of determination, COD = 1 - sum of squares / X, (y, - y )2, must be no larger. 

To estimate the variance components for the random effects model, we also computed the group means regression. The sum of squared residuals from the LSDV estimator is 444,288. The sum of squares from the group means regression is 22382.1. The estimate of a,.2 is 444,288/93 = 4777.29. The estimate of a 2 is 22,382.1/2 - (1/20)4777.29 = 10,952.2. The model is then reestimated by FGLS using these estimates ... 

For testing the hypotheses that the sets of dummy variable coefficients are zero, we will require the sums of squared residuals from the restrictions. These are... 

Prove that Newton s method for minimizing the sum of squared residuals in the linear regression model will converge to the minimum in one iteration. 

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