STATISTICS Joint and Conditional Distributions Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University

Joint cumulative distribution function Let be k random variables all defined on the same probability space ( ,A, P[]). The joint cumulative distribution function of , denoted by , is defined as for all . 6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Discrete joint density 6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Marginal discrete density If X and Y are bivariate joint discrete random variables, then and are called marginal discrete density functions. 6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Continuous Joint Density Function The k-dimensional random variable ( ) is defined to be a k-dimensional continuous random variable if and only if there exists a function such that for all . is defined to be the joint probability density function. 6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Marginal continuous probability density function If X and Y are bivariate joint continuous random variables, then and are called marginal probability density functions. 6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Conditional distribution functions for discrete random variables If X and Y are bivariate joint discrete random variables with joint discrete density function , then the conditional discrete density function of Y given X=x, denoted by or , is defined to be 6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Conditional distribution functions for continuous random variables If X and Y are bivariate joint continuous random variables with joint continuous density function , then the conditional probability density function of Y given X=x, denoted by or , is defined to be 6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Stochastic independence of random variables 6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Expectation of function of a k-dimensional discrete random variable 6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Conditional Expectation E[Y|X] : Conditional expectation as a random variable. 6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Expectation of Random Sum of Random Variables Let N be a random variable which can assume positive integer values 1, 2, 3.... Let Xi be a sequence of independent random variables which are also independent of N and have a common mean E[X] independent of i. Then the expectation of the sum of N Xi’s can be expressed as Homework problem 6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Covariance 6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

If two random variables X and Y are independent, then Therefore, 6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

However, does not imply that two random variables X and Y are independent. 6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

A measure of linear correlation: Pearson coefficient of correlation 6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Covariance and Correlation Coefficient Suppose we have observed the following data. We wish to measure both the direction and the strength of the relationship between Y and X. 6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Examples of joint distributions Duration and total depth of storm events. (bivariate gamma, non-causal relation) Hours spent for study and test score. (causal relation) 6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Bivariate Normal Distribution Bivariate normal density function 6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Conditional normal density 6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Bivariate normal simulation I. Using the conditional density 6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

(x,y) scatter plot 6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Histogram of X 6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Histogram of Y 6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Bivariate normal simulation II. Using the PC Transformation 6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

(x,y) scatter plot 6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Histogram of X 6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Histogram of Y 6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Multivariate normal simulation using R The mvtnorm package in R dmvnorm rmvnorm pmvnorm qmvnorm 6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Conceptual illustration of bivariate gamma simulation 6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Stochastic simulation of bivariate gamma distribution Importance of the bivariate gamma distribution Many environmental variables are non-negative and asymmetric. The gamma distribution is a special case of the more general Pearson type III distribution. Total depth and storm duration have been found to be jointly distributed with gamma marginal densities. 6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Simulation of bivariate gamma distribution based on the frequency factor which is well-known to scientists and engineers in water resources field. The proposed approach aims to yield random vectors which have not only the desired marginal distributions but also a pre-specified correlation coefficient between component variates. 6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Rationale of BVG simulation using frequency factor From the view point of random number generation, the frequency factor can be considered as a random variable K, and KT is a value of K with exceedence probability 1/T. Frequency factor of the Pearson type III distribution can be approximated by Standard normal deviate [A] 6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

General equation for hydrological frequency analysis For gamma distribution 6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Gamma density 6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Assume two gamma random variables X and Y are jointly distributed. The two random variables are respectively associated with their frequency factors KX and KY . Equation (A) indicates that the frequency factor KX of a random variable X with gamma density is approximated by a function of the standard normal deviate and the coefficient of skewness of the gamma density. 6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Thus, random number generation of the second frequency factor KY must take into consideration the correlation between KX and KY which stems from the correlation between U and V. 6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Conditional normal density Given a random number of U, say u, the conditional density of V is expressed by the following conditional normal density with mean and variance . 6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Flowchart of BVG simulation (1/2) 6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

Flowchart of BVG simulation (2/2) 6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

[B] 6/17/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.