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Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional.

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Presentation on theme: "Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional."— Presentation transcript:

1 Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Joint and Conditional Distributions Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University

2 Lab for Remote Sensing Hydrology and Spatial Modeling Dept 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.

3 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Discrete joint density

4 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

5 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

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

7 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 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.

8 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

9 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Marginal continuous probability density function If X and Y are bivariate joint continuous random variables, then and are called marginal probability density functions.

10 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 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

11 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

12 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 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

13 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

14 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

15 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

16 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

17 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

18 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Stochastic independence of random variables

19 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Expectation of function of a k- dimensional discrete random variable

20 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

21 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

22 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Covariance

23 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

24 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University If two random variables X and Y are independent, then Therefore,

25 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University However, does not imply that two random variables X and Y are independent.

26 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University A measure of linear correlation: Pearson coefficient of correlation

27 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 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.

28 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

29 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

30 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

32 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

33 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

34 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

35 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 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)

36 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Bivariate Normal Distribution Bivariate normal density function

37 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Conditional normal density

38 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

39 Bivariate normal simulation I. Using the conditional density

40

41 (x,y) scatter plot

42 Histogram of X

43 Histogram of Y

44 Bivariate normal simulation II. Using the PC Transformation

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46

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48 (x,y) scatter plot

49 Histogram of X

50 Histogram of Y

51 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Conceptual illustration of Bivariate gamma simulation


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