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All authors have read and agreed to the published version of the manuscript. Reliability for the temperature at region A, Reliability for the temperature at region B. Non-probabilistic reliability under different threshold values (Example 4.2). Non-probabilistic reliability https://wizardsdev.com/ under different threshold values (Example 4.1). Mohammad Noori, a recognized researcher in AI techniques for struc¬tural health monitoring, and random vibrations, is an ASME Fellow and professor of mechanical engineering, at California Polytechnic State University.

Since summated scales are an assembly of interrelated items designed to measure underlying constructs, it is very important to know whether the same set of items would elicit the same responses if the same questions are recast and re-administered to the same respondents. Variables derived from test instruments are declared to be reliable only when they provide stable and reliable responses over a repeated administration of the test. This includes defining each construct and identifying their constituent domains and/or dimensions. Next, we select items or indicators for each construct based on our conceptualization of these construct, as described in the scaling procedure in Chapter 5.

## The Why and What of Factor Analysis

This is a data reduction technique which aggregates a given set of items to a smaller set of factors based on the bivariate correlation structure discussed above using a statistical technique called principal components analysis. These factors should ideally correspond to the underling theoretical constructs that we are trying to measure. The general norm for factor extraction is that each extracted factor should have an eigenvalue greater than 1.0. A more sophisticated technique for evaluating convergent and discriminant validity is the multi-trait multi-method approach. This technique requires measuring each construct using two or more different methods (e.g., survey and personal observation, or perhaps survey of two different respondent groups such as teachers and parents for evaluating academic quality). This is an onerous and relatively less popular approach, and is therefore not discussed here.

This paper aims at developing methodologies for such analyses, which are well suited for application to complex systems. A measure can be reliable but not valid, if it is measuring something very consistently but is consistently measuring the wrong construct. Likewise, a measure can be valid but not reliable if it is measuring the right construct, but not doing so in a consistent manner. Using the analogy of a shooting target, as shown in Figure 7.1, a multiple-item measure of a construct that is both reliable and valid consists of shots that clustered within a narrow range near the center of the target. A measure that is valid but not reliable will consist of shots centered on the target but not clustered within a narrow range, but rather scattered around the target.

## A nested hierarchy of second order upper bounds on system failure probability

Meanwhile, the results obtained through the proposed method is very close to that by using MCS, indicating that the proposed method based on the multi-CEM is suitable for analyzing the NPR of AR glasses. According to the measured samples, the uncertainty domain of the mentioned four variables is modeled as the multi-CEM with two clusters as follows. In this example, the three-dimensional bounded uncertain variables x1, x2, and x3 are considered, which are all assumed to follow the normal distribution in theory with the median values 3.358, 3.064, and 3.800, respectively. However, the samples of the uncertain variables are limited, which are listed in Table 3. It is the objective of this book to provide a more in-depth understanding of the vehicle-bridge interaction from the random vibration perspective.

Yet, the probabilistic models of the uncertainties are often unavailable in the problems due to the lack of samples, and the precision of the conventional non-probabilistic models are not satisfactory when the samples are of multi-cluster distribution. Firstly, a Gaussian mixture model is built for the multi-cluster samples with performing expectation maximization algorithm, based on which the multi-CEM can be constructed. In the structural reliability analysis, two cases, respectively, considering whether the components of the multi-CEM are intersected or not are researched in detail. In the end, two numerical examples and a practical application are conducted and analyzed to testify the effectiveness of the method. An effective non-probabilistic reliability analysis method based on the multi-cluster ellipsoidal model is presented. A multi-CEM is constructed for the samples according to the ellipsoid critical contour feature of the GMM, which can deal with the multi-cluster distribution characteristics of the uncertain variables.

## Data Availability Statement

In addition, a triangle algorithm that can reduce the amount of computation in solving connectivity matrix is proposed to establish a seismic connectivity reliability calculation model with low-discrepancy sequence sampling. The ground motion attenuation model and the magnification effect of site soil were used to perform a seismic hazard analysis of the demonstration area, and the peak ground acceleration and the spatial distribution characteristics of the power system were obtained. Based on the results of the fragility analysis of a 110 kV substation and a 330 kV substation in Xi’an, Shaanxi Province, the standard Monte Carlo simulation and the Sobol sequence quasi-Monte Carlo simulation are carried out.

Hao P., Wang Y., Liu C., Wang B., Wu H. A novel non-probabilistic reliability-based design optimization algorithm using enhanced chaos control method. Where x1 and x2 are the two uncertain-but-bounded variables, and a means the threshold value. The available samples of the uncertain variables are shown in Figure 7 and listed in Table 1. It can be intuitively observed that the samples are distributed in two separate clusters.

## The Variability and Reliability of Standardized UX Scales

Where Rk denotes the k-th critical contour ellipsoid, which can be determined according to the critical elliptical contour feature of the GMM to ensure that the samples of each cluster are entirely encircled by the ellipsoid. Finally, the multi-CEM of the uncertainties can be constructed multi-scale analysis as the following Equation . The first statement invoked the procedure PROC CORR that implements the option ALPHA to do Cronbach’s alpha analysis on all observations with no missing values . The VAR statement lists down all the variables to be processed for the analysis.

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However, these kinds of methods require a large number of samples to compute the probability density functions, which is often unfeasible in practical engineering. In order to make up for the shortcomings of traditional probability methods, non-probabilistic methods including interval methods, polygon methods, and ellipsoid clustering methods, have been extensively studied. Yet, it is not difficult to see that they generally failed to describe the uncertainty domain in a compact and accurate way when processing variables with correlation and multi-cluster distribution characteristics. This leads to that though the designed structure can meet the basic functional requirements, it is probably not the most lightweight and efficient one. As shown in Figure 1, although the samples of the uncertain variables are of limited sizes, they still have complex distributions and multiple clustering characteristics.

## Theory of Measurement

When the ellipsoidal model tends to be stable, the number of scattered points can be determined. Through Gaussian clustering analysis and EM algorithm, the optimal Gaussian function for these additional samples is constructed, whose parameters are listed in Table 5. Via determining the critical contour ellipsoid, the AEM is then constructed. According to the sample distribution of the uncertain variables, two typical situations can be considered for the constructed multi-CEM.

- For example, a typical five-point item had a standard deviation of 1 (25% of max range).
- However, the interval model cannot consider the correlation of samples, resulting in the conservative results of uncertainty measurement.
- Peng Y.B., Zhou T., Li J. Surrogate modeling immersed probability density evolution method for structural reliability analysis in high dimensions.
- Once you have created a scale, you should test to see if it is reliable; that is, to see if the scale items are internally consistent.
- Please note that these are basic tests to see if your scale is internally reliable.