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Bivariate Random Variables: Definitions, Generation, and Graphical Analysis
Published in P. A. W. Lewis, E. J. Orav, Simulation Methodology for Statisticians, Operations Analysts, and Engineers, 2017
The most basic statistical tool for looking at bivariate data (or successive pairs of variables from higher dimensional data) is the scatter plot: every pair (Xi, Yi) in a sample of n pairs is used as a pair of coordinates in a two-dimensional plot. We have already used this idea in Chapter 4 to demonstrate what we mean by “really” random pseudo-random pairs: if the joint distribution is uniform in the plane, we should see approximately equal numbers of points in similar-sized little squares and no “pattern” in the scatter plot. Moreover, the idea of no “pattern” can be quantified by chi-square tests of fit (Conover, 1980, Chap. 4).
Graphical displays of data and descriptive statistics
Published in Andrew Metcalfe, David Green, Tony Greenfield, Mahayaudin Mansor, Andrew Smith, Jonathan Tuke, Statistics in Engineering, 2019
Andrew Metcalfe, David Green, Tony Greenfield, Mahayaudin Mansor, Andrew Smith, Jonathan Tuke
You may have noticed that for several of the data sets there was more than one variable measured for each item. For example, the Ballymun noise items are 5-minute intervals. For each interval, there are two measurements Leq and L10. The robot arm data had 14 measurements on each arm. If there are two variables measured for each item the data are referred to as bivariate. In general, if there are several variables measured for each item the data are referred to as multivariate. Bivariate data can be displayed with scatter plots and bivariate histograms. Multivariate data can be displayed with parallel coordinate plots.
Introducing coordination in hand position analysis during a steering wheel-based tracking task using fuzzy sets
Published in Theoretical Issues in Ergonomics Science, 2022
Pierre Loslever, Jessica Schiro, François Gabrielli, Philippe Pudlo
Other angle measurement examples could include the trunk and head inclination in the body vibration study of occupational drivers (Raffler et al. 2016). The second type of application is used in eye movement studies, e.g. when X and Y directions are combined to give areas of interest (Loslever, Popieul, and Simon 2003; Lehtonen et al. 2013). The remaining type of application concerns all cases when two variables should be combined, e.g. position and speed in a 2D tracking task (Miyake, Loslever, and Hancock 2001) or medial/lateral and anterior/posterior displacements in studies of the sitting posture (Grooten et al. 2013). As shown in Figure 2, for all these examples, any summarising procedure (across time, individuals, etc.) may yield a lower information loss with membership values than with usual indicators; see (Anscombe 1973; Loslever et al. 2012) for examples where different bivariate data sets yield identical summaries (including the usual correlation coefficient).
Assessing correlation to evaluate performance of a two component system and its impact on specifying separate production requirements
Published in Quality Engineering, 2019
We considered an example where knowing the correlation is important in determining separate production requirements for components in a system. Separate production requirements for the components may be too restrictive, however. Even if the component production requirements are not met, the system requirement may still be met. In the example in the previous section, we collected bivariate data (tA, tB). Often, the tA and tB will be collected separately. One has to assume that the correlation ρ has not changed or is bounded by some value so that whether the system meets its requirement can be evaluated. In the example, suppose that we had analyzed the tA and tB data separately and had obtained draws from the posterior distribution of and Then, in the example of evaluating P, we could replace the draws from the ρ posterior distribution with its upper bound, say Then the posterior median of P is 0.9982 and 95% credible interval is We would conclude that the system requirement 0.95 had been met.
A one-sided procedure for monitoring variables defined on contingency tables
Published in Journal of Quality Technology, 2019
Sotiris Bersimis, Athanasios Sachlas
From the early seventies, methods for monitoring multivariate Binomial and Poisson processes were proposed (Patel 1973). Marcucci (1985) extended the p-chart to three or more attribute levels in order to monitor multinomial processes. Lu et al. (1998) presented the mnp-chart, a Shewhart-type chart for monitoring multivariate attribute processes that uses the weighted sum of the number of nonconforming units of each quality characteristic. Recently, Skinner et al. (2003, 2004) proposed generalized linear model–based control charts for monitoring multiple discrete counts. Chiu and Kuo (2008) proposed the mp-chart for monitoring multivariate Poisson count data. Balakrishnan et al. (2009) proposed a control chart for bivariate data using run statistics. Chen et al. (2011) presented two statistical models in order to examine the influence of inspection error on the multinomial control charts. Weiß (2012) proposed several control charts designed to continuously monitor purely categorical processes (data are not collected in samples, but a full inspection is performed).