R_SimuSTAT_2

Prof. Ke-Sheng Cheng

Dept. of Bioenvironmental Systems Eng.

National Taiwan University

• Outline– Density and CDF plots– Plot the empirical cumulative distribution function

(ECDF) of a set of random numbers. – Simulation of discrete and continuous random

variables– Correlation

04/10/23Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

2

• A random variable is associated with its probability density function (PDF) and cumulative distribution function (CDF).

• Each probability density function has one or more parameters which characterize the location and shape of the PDF.

• We can observe how changes in parameters can affect the shape of PDF and CDF by plotting the PDF and CDF of different parameter settings.

• Plotting of PDF and CDF can be done using the plot and lines functions in R.

04/10/23Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

3

04/10/23Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

4

04/10/23Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

5

• Suppose that a random variable X is defined on the sample space of a random experiment. The random experiment is conducted n times and yields a set of n random numbers, say {x1, x2, …, xn}. This set of random numbers is called a random sample of size n of the random variable.

• Although the random variable X is associated with a (theoretical) CDF, an empirical CDF (ECDF) of the random sample can be considered as a sample of the theoretical CDF and can constructed by

04/10/23Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

6

– Sort the random sample {x1, x2, …, xn} in ascending order such that { y1=min(x1, x2, …, xn), …, yn=max(x1, x2, …, xn ) }

– Let

– Construct the plot of y vs Fn(y).

• The plot.ecdf function yields an ECDF plot.• Alternatively, the ecdf function can also be used

to establish an ECDF plot.04/10/23

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

7

. toequalor an smaller th are which nsobservatio ofnumber )( yym n

ymyFn

)()(

yyyF

niyyyn

iyF

yyyF

nn

iin

n

,1)(

1,,2,1;,)(

0)(

1

1

04/10/23Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

8

plot.ecdf(x)

To be continued

04/10/23Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

9

04/10/23Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

10

04/10/23Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

11

• Correlation between detrended cumulative-sum series.

04/10/23Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

12

04/10/23Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

13

04/10/23Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

14

04/10/23Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

15