Test Random Numbers

7/28/2019 Test Random Numbers

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Test forRandomNumbers

Pablo Hinojosa NavaSelected topics of computational

science
http://www.ua.ac.be/main.aspx?c=.OODE2012&n=105102&ct=105102&e=290432&detail=2001WETCCWhttp://www.ua.ac.be/main.aspx?c=.OODE2012&n=105102&ct=105102&e=290432&detail=2001WETCCWhttp://www.ua.ac.be/main.aspx?c=.OODE2012&n=105102&ct=105102&e=290432&detail=2001WETCCWhttp://www.ua.ac.be/main.aspx?c=.OODE2012&n=105102&ct=105102&e=290432&detail=2001WETCCWhttp://www.ua.ac.be/main.aspx?c=.OODE2012&n=105102&ct=105102&e=290432&detail=2001WETCCW


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Index

1. Tests (Hypothesis testing)a. Kolmogorov-Smirnof testb. Chi-square test

c. Runs up and runs downd. Runs above and below mean

2. Simulating Coin Tossing Experiment


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Tests properties

Uniformity-Frequency test

a)Kolmogorov-Smirnov

b)Chi-Square testIndependence

-Runs

c)Runs up and runs downd)Runs above and below mean

-Autocorrelation-Gap-Poker


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Begining of the Tests

1. Formulate H0

(null hypothesis) in confrontation with H1

(alternative hypothesis)H

0-> the numbers are distributed uniformly or

independent on the interval [0,1]2. Level of significance must be stated (confidenceinterval)

is frequently set to 0,01 or 0,053. Measure the quantity of interest Q

4. Check Qt from the distribution table based on and:v= n-1 (n=number of intervals) (Chi-Square test)N (total number of numbers) (the rest of the tests)

5. If Q < Qtthen H

0is true else false


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a) Kolmogorov-Smirnov test

1. Arrange the numbers in ascending order2. Compute

Q+ = max1 i N

{i/N - ri

}

Q- = max1 i N

{ ri

- i/N }

3. Choose Q (maximum between Q+ and Q-)4. Determine the critical value Q

tfrom the

next table for the specified and N5. If Q < Q

tthen numbers are uniformly

distributed else not


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a) Kolmogorov-Smirnov testtable


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a) Kolmogorov-Smirnov testexample

Example 2.9 Suppose that the five numbers0.44, 0.81, 0.14, 0.05, 0.93 are generated.So, N = 5. Let = 0.05. The calculations can

be facilitated by use of Table 2.2.


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b)Chi-Square test

1. Fix the number of intervals n of equal length in [0, 1].2. Compute E = N/n

3. Compute the observed number Oiin ith interval for

i = 1, 2, . . . , n.4. Compute

5. Determine the critical value, Qt, from the next table for

the specified and n.

6. If Q < Qt then numbers are uniformly distributedelse not


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b)Chi-Square test

table


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Independence testsSome definitions

1. Run. A run is defined as a succession of similar eventspreceded and followed by a different event2. Length of run. The length of a run is the number of

events that occur in the run3. Up run. An up run is a sequence of numbers each ofwhich is succeeded by a larger number4. Down run. A down run is a sequence of numbers eachof which is succeeded by a smaller number

For the random number to be tested we give + or a -depending on whether theyare followed by a larger number or a smaller number.


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Example

Consider this numbers:0.87, 0.15, 0.23, 0.45, 0.69, 0.32, 0.30, 0.19,0.24, 0.18, 0.65, 0.82, 0.93, 0.22, 0.81.

The sequence of + and is as follows: + + + + + + + +

So, the number of runs is 8, the number ofup runs is 4 and the number of down runs is4.


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c)Runs up and runs down test I

1. Find X, the total number of runs in thegiven sequence of random numbers to betested.

2. Compute

3. Let N denote the total number of random

numbers in the sequence.4. If N 20, proceed further. Else, nothingcan be said about the independency of thenumbers.


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c)Runs up and runs down test II

5. Compute

X number of runs

6. Determine the critical value, Qt, from the

next tables for the specified

7. If Q < Qt, the numbers are independent,

else, they are not.


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c)Runs up and runs down tableI


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c)Runs up and runs down tableII


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c)Runs up and runs downexample


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d)Runs above and below mean

The following steps will be carried out for theruns above and below mean procedure:1. Find X, the total number of runs in the

given sequence of random numbers to betested.2. Find the number n

1of random numbers

above the mean (total number of + signs)3. Find the number n

2of random numbers

below the mean (total number signs).

The cross check is N = n1 + n2 .


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d)Runs above and below meanII

6. If n1

> 20 or n2

> 20, proceed further.

Else, nothing can be said about the indepen-

dency of the given random numbers


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d)Runs above and below meanIII

8. Determine the critical value, Qt, from the

last tables for the specified .9. If Q < Q

t, the numbers are independent,

else, they are not.


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d)Runs above and below meanexample


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Simulating Coin Tossing Experiment

http://nlvm.usu.edu/en/nav/frames_asid_305_g_4_t_5.html

http://pbskids.org/cyberchase/games/probability/
http://pbskids.org/cyberchase/games/probability/http://pbskids.org/cyberchase/games/probability/http://pbskids.org/cyberchase/games/probability/http://nlvm.usu.edu/en/nav/frames_asid_305_g_4_t_5.htmlhttp://nlvm.usu.edu/en/nav/frames_asid_305_g_4_t_5.html

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Test Random Numbers