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Statistical Preliminaries
Published in Jaakko Astola, Pauli Kuosmanen, Fundamentals of Nonlinear Digital Filtering, 2020
Jaakko Astola, Pauli Kuosmanen
Example 2.18. The mean square error is a natural error criterion and it is especially suitable when the observations obey a Gaussian distribution. However, it is not good in the case of impulsive noise where large deviations occur often. This is because in this case the minimization of the quadratic term makes the estimator too heavily influenced by the worst observations; the ones corrupted by impulses. The effect of outliers can be alleviated by using a less rapidly increasing function. A natural choice is the mean absolute error. Thus, we seek the estimator T(X1, X2, …, Xn) for S minimizing E{|S−T(X1,X2,…,XN)|},
Chapter 13 Mathematical and statistical techniques
Published in B H Brown, R H Smallwood, D C Barber, P V Lawford, D R Hose, Medical Physics and Biomedical Engineering, 2017
All experimental data show a degree of ‘spread’ about the measure of central tendency. Some measure that describes this dispersion or variation quantitatively is required. The range of the data, from the maximum to the minimum value, is one possible candidate but it only uses two pieces of data. An average deviation obtained from averaging the difference between each data point and the location of central tendency is another possibility. Unfortunately, in symmetrical distributions there will be as many positive as negative deviations and so they will cancel one another out ‘on average’. The dispersion would be zero. Taking absolute values by forming the modulus of the deviation is a third alternative, but this is mathematically clumsy. These deliberations suggest that the square of the deviation should be used to remove the negative signs that troubled us with the symmetric distribution. The sum of the squares of the deviations divided by the number of data points is the mean square deviation or variance. To return to the original units the square root is taken. So this is the root mean square (rms) deviation or the standard deviation.
Two-Level Factorial Design
Published in Mark J. Anderson, Patrick J. Whitcomb, DOE Simplified, 2017
Mark J. Anderson, Patrick J. Whitcomb
The sum of squares for model and residual are shown in the first column of data in the ANOVA, shown in Table 3.9. The next column lists the degrees of freedom (df) associated with the sum of squares (derived from the effects). Each effect is based on two averages, high versus low, so it contributes 1 degree of freedom (df) for the sum of squares. Thus, you will see 3 df for the three effects in the model pool and 4 df for the four effects in the residual pool. This is another simplification made possible by restricting the factorial to two levels. The next column in the ANOVA is the mean square: the sum of squares divided by the degrees of freedom (SS/df). The ratio of mean squares (MSModel/MSResidual) forms the F value of 31.5 (= 781.0/24.8).
Efficient removal of hexavalent chromium from water by Bacillus sp. Y2-7 with production of extracellular polymeric substances
Published in Environmental Technology, 2023
Xuehan Wang, Ying Zhang, Xiaojie Sun, Xianchao Jia, Yin Liu, Xinfeng Xiao, Hongge Gao, Lin Li
Furtherly, in order to obtain the optimal conditions for chromium bioremediation, response surface optimization experiments were performed. The Kaiser-Meyer-Olkin (KMO) index value was chosen to evaluate adequacy of sample selection. In KMO test, KMO value was 0.52 greater than 0.5. It suggested that data of influence of four variables (glucose concentration, pH, inoculation dose and temperature) on Cr(VI) removal were suitable for subsequent factors analysis. The contribution rates of above factors were shown in Table 1(a). The eigenvalues of the glucose concentration and pH were 2.81 and 1.03, respectively. And the sum of the eigenvalues of the first two factors accounted for 95.81% of the total eigenvalues. Therefore, the first two factors were extracted as the main factors. The scores of Factor1 and Factor2 were obtained by using SPSS (Table 1(b)). In analysis of variance (ANOVA; Table 3), the F value represents the ratio between the mean square of factors in different groups and the mean square of factors in the same group. The F and P values of the model were 141.22 and < 0.0001, respectively, indicating that the regression model was highly significant (P < 0.01). According to ANOVA and KMO, the influence of tested factors on Cr(VI) removal was arranged as follows: glucose concentration > pH > inoculation dose > temperature.
A statistical fatigue life prediction model applicable for fibre-reinforced roller-compacted concrete pavement
Published in Road Materials and Pavement Design, 2023
H. Rooholamini, A. Karimi, J. Safari
Table 7 lists the within and between group mean square values found through dividing the sum of the squared deviations from the mean by the degree of freedom (df) which the number of groups or different design mixes minus one for within groups; for between groups, it is the total number of tests at each stress level minus the number of groups. The ANOVA statistics (F) is the mean square of between groups divided by that of within groups. If the p-value related to F-value (last column in Table 7) is more than 0.05, the difference between the average fatigue life of RCCP mixes is statistically insignificant at 95% confidence level.
Analysis of factors influencing absorbed energy in CFRP and Kevlar hybrid laminate subjected to low-velocity impact
Published in Australian Journal of Multi-Disciplinary Engineering, 2023
Ritesh K Dundi, Pratheek Nagesh Amin, N. Rajesh Mathivanan
The ANOVA technique is used to evaluate the model’s suitability. According to this technique, the model is deemed appropriate within the confidence limit if the calculated value of the F ratio of the produced model does not exceed the standard tabulated value of F ratio for a desired degree of confidence. The difference between the mean square due to a factor and the error mean square is known as the variance ratio, or F in ANOVA tables. F ratio can be utilised in resilient design to understand the relative factor impacts qualitatively. If F has a high value, the effect of that component is significant relative to the error variance.