Modeling and Simulation NETW 707

Transcription

1 Modelng and Smulaton NETW 707 Lecture 5 Tests for Random Numbers Course Instructor: Dr.-Ing. Magge Mashaly magge.ezzat@guc.edu.eg C

2 Propertes of Random Numbers Random Number Generators (RNGs) must satsfy propertes of Random Numbers: 1. Unformty 2. Independence But what f you are gven some numbers generated by any RNG How to guarantee that they are Unform & Independent random numbers??? Usng Tests for Random Numbers 2

3 Tests for Random Numbers There are two types of tests: 1. Frequency Tests - Compare the dstrbuton of the set of numbers to a unform dstrbuton - Examples for frequency tests: 1. Kolmogorov-Smrnov Test (KS) 2. Ch-Square Test 2. Autocorrelaton Test Tests the correlaton between numbers and compares the sample correlaton to the expected correlaton of zero 3

4 Kolmogorov Smrnov Test The KS test compares a contnuous CDF F(x) to an emprcal CDF S N (x) of the sample of N observatons. If the sample from the random number generator s R 1, R 2,, R N, then the emprcal CDF S N (x) s gven by: S N number of R 1, R2, ( x) N, R N whch are x 4

5 5 Kolmogorov Smrnov Test How to perform the test? 1. Rank the data from smallest to largest: 2. Compute: R N R R 2 1 N R N D 1 max N R D N 1 max 1

6 Kolmogorov Smrnov Test 3. Compute D=max(D +,D - ) 4. Locate the crtcal value D α n the Kolmogorov Smrnov Crtcal Values Table for the specfed sgnfcance level α and the gven sample sze N 5. Compare: If D D α Accept: No dfference between S N (x) and F (x ) If D > D α Reject: Dfference exsts between S N (x) and F (x) 6

7 Kolmogorov Smrnov Test Kolmogorov Smrnov Crtcal Values Table 7

8 Kolmogorov Smrnov Test: Example The fve numbers: 0.44, 0.81, 0.14, 0.05, 0.93 were generated and t s requred to test for unformty usng the Kolmogorov-Smrnov Test wth the level of sgnfcance α =

9 Kolmogorov Smrnov Test: Example Soluton: R N N R R 1 n D=max(D +,D - ) = max(0.26,0.21) = 0.26 In KS crtcal values table, For α = 0.05 and N = 5, the crtcal value D α = Snce D < D α, no dfference has been detected between the true dstrbuton of {R 1, R 2,, R N } and the unform dstrbuton 9

10 Kolmogorov Smrnov Test: Example Soluton: 10

11 Ch-square Test The Ch-square test uses the sample statstc: 2 0 Where: O = the number of observatons n the -th class E = the expected number n the -th class n = the number of classes. n 1 O E E 2 11

12 Ch-square Test 1. Rank the data from smallest to largest: R 1 R 2 R N 2. Dvde the Range R N -R 1 n n equdstant ntervals such that each nterval has at least 5 observatons. 3. Calculate: 2 0 n 1 O E E 2 12

13 Ch-square Test 4. For sgnfcance level α, utlze the table of (Percentage ponts of the ch square dstrbuton wth ν degrees of freedom) to determne χ α,n-1 5. Compare If χ 0 2 χ 2 α,n-1 Accept: No dfference between S N (x) and F(x) If χ 0 2 > χ 2 α,n-1 Reject: Dfference exsts between S N (x) and F(x) 13

14 Ch-square Test Percentage ponts of the CHI SQUARE dstrbuton wth ν degrees of freedom 14

15 Ch-square Test: Example Use the ch-square test wth α=0.05 to test whether the data shown next are unformly dstrbuted

16 Ch-square Test: Example Soluton: Interval O E O E O E O E 2 E 16

17 Ch-square Test: Example Soluton: The test uses n=10 ntervals of equal length, namely [0,0.1[, [0.1,0.2[,, [0.9,1] The value of χ 02 =3.4 From table (Percentage ponts of the CHI SQUARE dstrbuton wth ν degrees of freedom), the crtcal value of χ 0.05,9 =16.9 Snce χ 02 < χ 0.05,9, the hypothess of unform dstrbuton s not rejected. 17

18 Notes on Unformty Tests Both the Kolmogorov-Smrnov test and the ch-square test are acceptable for testng the unformty of sample data provded that the sample sze s large. The KS test can be appled to small sample szes, whereas the ch-square test s vald only for large samples, e.g.: N 50. The KS test s more powerful and s recommended 18

19 Test for Auto-correlaton The tests for auto-correlaton are concerned wth the dependence between numbers n a sequence. Example: Examnaton of the 5 th, 10 th, 15 th,,etc. ndcates a large number n that poston. 19

20 Test for Auto-correlaton The test requres the computaton of the autocorrelaton between every m numbers (m s known as the lag), startng wth the th number: The autocorrelaton ρ m of nterest shall be between numbers: R, R +m, R +2m, R +(M+1)m M s the largest nteger such that +(M+1)m N If the values are uncorrelated: For large values of M, the dstrbuton of the estmator of ρ m, denoted s approxmately normal. 20

21 Test for Auto-correlaton Test statstc s: Where: ˆ m ˆ m 1 M 1 13M 7 12 M 1 k 0 Z 0 s dstrbuted normally wth mean = 0 and varance = 1. Z M 0 R ˆ km m ˆ m R k 1 m

22 Test for Auto-correlaton Test Steps: 1. Compute Z 0 2. The hypothess of ndependence s not rejected f: z Z z / 2 0 / 2 Where α s the level of sgnfcance and z α/2 s obtaned from the standard normal dstrbuton table. / 2 -Z / 2 Z / 2 / 2 22

23 Test for Auto-correlaton The standard normal dstrbuton table 23

24 Test for Auto-correlaton Test Steps: 3. If numbers are correlated, determne the type of correlaton If ρ m > 0, the subsequence has postve autocorrelaton Hgh random numbers tend to be followed by hgh ones, and vce versa. If ρ m < 0, the subsequence has negatve autocorrelaton Low random numbers tend to be followed by hgh ones, and vce versa. 24

25 Test for Auto-correlaton: Example Test whether the 3 rd, 8 th, 13 th, random varables are correlated for the followng output usng α =

26 Test for Auto-correlaton: Example Soluton: = 3, m = 5, N = 30, 3+(M+1)5 30 M = 4 M 1 ˆ 1 m R kmr M k 0 ˆ ˆ ˆ M 12 k 1 m M

27 Test for Auto-correlaton: Example Soluton: The test statstc s gven by: ˆ m Z From the standard normal dstrbuton table, the crtcal value s: Snce 0 ˆ z m 0.05/ 2 z0.025 z Z0 z0.025 The hypothess of ndependence cannot be rejected 27

28 Test for Auto-correlaton: Example Soluton: The test statstc s gven by: ˆ m Z From the standard normal dstrbuton table, the crtcal value s: Snce 0 ˆ z m 0.05/ 2 z0.025 z Z0 z0.025 The hypothess of ndependence cannot be rejected 28

29 References NETW 707 Lectures sldes by A. Prof. Tallal El-Shabrawy, 2016 & 2017 NETW 707 Lectures sldes by Dr. Akram Al, 2014 Smulaton Modelng and Analyss by Averll M. Law, 5 th Edton, 2015 F. J. Massey, The Kolmogorov-Smrnov Test for Goodness of Ft, The Journal of the Amercan Statstcal Assocaton, Vol. 46, 1951, p.70 R. J. Wonnacolt, T.H. Wonnacolt, Statstcs: Dscoverng Its Power, New Yor,: John Wley and Sons, 1982, p