Bivariate normal distribution

FIGURE 2.8 Covariance matrices X for different distributions of two-dimensional data. In the main diagonal ofX are the variances ol x, and x2, respectively. For each covariance matrix 200 bivariate normally distributed points have been simulated. 

In SIMCA the distribution of the object in the inner model space is not considered, so the probability density in the inner space is constant and the overall PD appears as shown in Figs. 29, 30 for the enlarged and reduced SIMCA models. In CLASSY, Kernel estimation is used to compute the PD in the inner model space, whereas the errors in the outer space are considered, as in SIMCA, uncorrelated and with normal multivariate distribution, so that the overall distribution, in the inner and outer space of a one-dimensional model, looks like that reported in Fig. 31. Figures 32, 33 show the PD of the bivariate normal distribution and Kernel distribution (ALLOC) for the same data matrix as used for Fig. 31. Although in the data set of French wines no really important differences have been detected between SIMCA (enlarged model), ALLOC and CLASSY, it seems that CLASSY should be chosen when the number of objects is large and the distribution on the components of the inner model space is very different from a rectangular distribution. 

Suppose x and x2 have the bivariate normal distribution described in Section 3.8. Consider an extension of Example 3.4, where the bivariate normal distribution is obtained by transforming two independent standard normal variables. Obtain the distribution of z exp(yi)exp(y2) where y and y2 have a bivariate normal distribution and are correlated. Solve this problem in two ways. First, use the transformation approach described in Section 3.6.4. Second, note that z exp(yl+y2) = exp(vv), so you can first find the distribution of w, then use the results of Section 3.5 (and, in fact, Section 3.4.4 as well). 

Graphical Illustration of Selected Bivariate Normal Distributions... 

FIGURE 3.5 Scatterplots (a) of a bivariate normal distribution (100 points) with a correlation of 0.75, S = 0.44. Ellipses are drawn at 80 and 95% confidence intervals. Contour plots (b) and mesh plots (c) of the corresponding bivariate normal distribution functions are also shown. 

Mean and standard deviation values (xi, i, X2, 2) for every variable can be calculated, and the scatterplot with the n points can be used to see the form of the association between the two variables. In the case of random samples and assuming a bivariate normal distribution, the 95% confidence ellipse (xi —... 

Figure 4.3 Contours of constant density 17) for the bivariate normal distribution. Each contour encloses a probability content of 0.95.

Si, 2, mi, m2 are approximations for oi, 02, )ii, H2 respectively). The bivariate normal density function has a bell shape form, and it is centered at the point (pi, p.2) that represents the centroid of the distribution. A multivariate normal distribution can be defined similarly to a bivariate normal distribution. 

The representation of this equation for anything greater than two variates is difficult to visualize, but the bivariate form (m = 2) serves to illustrate the general case. The exponential term in Equation (26) is of the form x Ax and is known as a quadratic form of a matrix product (Appendix A). Although the mathematical details associated with the quadratic form are not important for us here, one important property is that they have a well known geometric interpretation. All quadratic forms that occur in chemometrics and statistical data analysis expand to produce a quadratic smface that is a closed ellipse. Just as the univariate normal distribution appears bell-shaped, so the bivariate normal distribution is elliptical. 

Figure 8 Bivariate normal distributions as probability contour plots for data having different covariance relationships...

From Figure 17.1 consider the situation where S and T have a bivariate normal distribution and the data are obtained from a single study. One can model the relationship between S and T and Z as three distinct linear regressions ... 

If data are collected from a random population (X, Y) from a bivariate normal distribution and predictions about Y given X are desired, then from the previous paragraphs it may be apparent that the linear model assuming fixed x is applicable because the observations are independent, normally distributed, and have constant variance with mean 0o + 0iX. Similar arguments can be made if inferences are to be made on X given Y. Thus, if X and Y are random, all calculations and inferential methods remain the same as if X were fixed. 

To illustrate these concepts a modification of the simulation suggested by Allison (2000) will be analyzed. In this simulation 10,000 observations of three variables were simulated Y, xi, and x2. Such a large sample size was used to ensure that sampling variability was small, xi and x2 were bivariate normally distributed random variables with mean 0 and variance 1 having a correlation of 0.5. Y was then generated using... 

Figure A.4 Joint pdf of the bivariate normal distribution and the marginal distributions for X and Y. In this example, both X and Y have a standard normal distribution with a correlation between variables of 0.6.

Notice that the conditional pdf is normally distributed wherever the joint pdf is sliced parallel to X. In contrast, the marginal distribution of Y or X evaluates the pdf across all of X or Y, respectively. Hence, the mean and variance of the marginal and condition distribution from a bivariate normal distribution can be expressed as ... 

An extension to the bivariate normal distribution is when there are more than two random variables under consideration and their joint distribution follows a multivariate normal (MVN) distribution. The pdf for the MVN distribution with p random variables can be written as... 

In the two-dimensional nonsingular case, the multivariate normal distribution reduces to the bivariate normal distribution. This bivariate normal distribution is a generalization of the familiar univariate normal distribution for a single r.v. X. Let jXx and cr be the mean and standard deviation of X, /Ay and ay be the mean and... 

The theoretical concept of correlation arises in conjunction with the bivariate normal distribution function. That function has five parameters. If the two variables are X tmd Y, the peuameters are the means (/x, /Xy) and the variances (correlation coefficient, p (rho). This chapter does not deal with the theoretical bivariate (or multivariate) normal distribution. However, in practice, the sample correlation coefficient, r, is a useful measure of linear association. It is a dimensionless ratio ranging from —1.0 (perfect inverse linear agreement) through zero (orthogonal or Unearly unrelated) to +1.0 (perfect direct linear agreement). The value can be obtained from Eq. (10) and used as an index without any assertion whatever being made about distribution form. 

Next, Ui and are transformed into standard normal random variables Z, and W and then the joint distribution of Z and W introduces the correlation through a bivariate normal distribution. 

Following the methodology presented in section 2, next application concerns to construction of tolerance regions through the implementation of the parametric version. Before applying the parametric approach, we must ensure that our output vector (C,R) follows a bivariate normal distribution and obtain its parameters. Theorem 1 guarantees its asymptotic bivariate normality, that is, (C,R) N2 (M, S), where the mean vector M = (C(m), R(m)) and the covariance matrix isS = DYD/n are obtained from m and V their equivalent parameters in the input vector and D is the matrix of partial derivates of C and R calculated in m. 

The riverbed levels upstream and downstream, Z and Zv respectively, are imcertain and their uncertainty will be quantified by a bivariate normal distribution N(/m., E). Indeed, as upstream and downstream section are quite close it seems more reasonable to model them as possibly dependent variables. 29 couples of data (zm z ) are available to perform Bayesian inference and setting the posterior distribution of fi and E. The prior distributions for both components of vector /i are normal with means equal to 56 and 50 m for ix and fj,2 respectively and standard deviations equal to 1 m. This prior translates a vague knowledge around two reference values. Concerning the prior of E, a classical choice is the inverse-Wishart distribution ... 

As bo and bj are random variables that foUow a bivariate normal distribution, yo becomes also a random variable that follows a normal distribution and, so, its variance can be estimated using error-propagation rules and eqns (2.9) and (2.11) as ... 

We now proeeed by extending the methodology to general joint distributions. As mentioned in the introduetion it is always possible to transform any pair of random variables into a pair of independent standard normal variables using the Rosenblatt transformation. As before we denote the environmental variables by T and H. The Rosenblatt transformation is a transformation T sueh that if (T, IT) = (T,H), then V and H are independent standard normal variables. As explained above for these transformed variables we can easily find the environmental contour, which we denote dB and a suitable circular set within the contour, which we denote by d- Unless the Rosenblatt transformation is linear, which happensonly when (T,H) has a bivariate normal distribution, we cannot simply obtain the environmental contour for T and H by using the inverse Rosenblatt transformation. That is, except for the linear case, we typically have that P ) dB- See (Huseby et al., 2013). However, provided that d is not too close to dB, we may still have that B ... 

State-of-the-art calculations for correlations analysis are based on Bravais-Pearson (precondition bivariate normal distributed characteristics), Spearman and Kendall (rank based analysis independent of characteristic distribution models), explanation in (Sachs 2002). The results of the application of these correlation analyses show the interdependences of the product characteristics (step 3, Fig. 1). For this case study Spearman correlation, shown in equation 4, was used. 

Table A.7 Minitab commands for drawing Markov chain Monte Carlo sample from a correlated bivariate normal distribution using either a normal random-walk candidate density or an independent candidate density...

A correlated bivariate normal distribution. The function blvnormMH can use the Metropolis-Hastings algorithm to draw a sample from a correlated bivariate normal target density using either an independent candidate density or a random-walk candidate density when we are drawing both parameters in a single draw. Also,... 

To estimate the parameters of distribution X and R, where U = X + R, we assume that (X, R) follows a non-degenerate bivariate normal distribution. For this distribution, it is well known (Bickel and Doksum, 1977) that the marginal distribution of X is an univariate normal distribution with mean px and standard deviation Ox, while the marginal distribution of R is also normally distributed with mean pr and standard deviation ctr. Let kt represent the proportion of total sales until reorder point t, 5t the correlation between X and R and pt the correlation between X and U. We estimate kt, 6t and Pt from historical data and use the formulas developed in Fisher... 

For the bivariate normal distribution (X,R), note that the updated distribution R/X = x is also normally distributed with mean pjyx = Pr + 5t... 

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