52

4.5. Rayleigh distribution

The Rayleigh distribution is defined by the following functions:

PDF :

CDF :

The mean and variance of the distribution is :

;

The Rayleigh distribution is frequently used to model wave heights in oceanography, and in

communication theory to describe hourly median and instantaneous peak power of received radio

signals. It has been used to model the frequency of different wind speeds over a year at wind

turbine sites.

4.6. Parameter Estimation

We now in this section estimate the parameters of the Rayleigh distribution from which the

sample comes. Here we present the method of Maximum Likelihood Estimation as this method

gives simpler estimate as compared to the Method of moments and the Local frequency ratio

method of estimation.

53

4.6.1 Maximum Likelihood Estimation

Let be a random sample of n observations from the Rayleigh population with pdf

The likelihood function of this sample is

L = .

Taking logarithms on both sides

=

=

The likelihood equation is = 0

On simplification, we get

54

4.6.2 Frequency ratio method of Estimation

As explained in the Chapter 3, we now estimate the parameters by considering local frequency

ratio method by putting x = x1, x2 in the pdf of Rayleigh distribution, we get

The ratio of above frequencies is

Taking logarithms on both sides

–

We now in the next section demonstrate the two methods with an illustration

55

Illustration:

Generate first a random sample of 1000 observations from Rayleigh distribution using MATLAB

function and Construct a frequency distribution.

For example, using x= raylrnd (0.1, 1000 ) the following distribution is obtained.

x

(midvalue)

0.0178 0.0535 0.0891 0.1248 0.1604 0.1961 0.2317 0.2674 0.3030 0.3387

f 60 161 217 193 153 107 61 30 12 6

Taking the two maximum frequencies and corresponding mid-values, substituting in the β, we

get

The above procedure is repeated for 50 samples. The mean, Standard deviation , of

these 50 estimates were computed. The estimated bias was calculated as the mean minus the true

value of the parameter. The Mean Squared Error (MSE) was calculated as the bias squared plus

the variance. .

56

Table 4.6 : Simulation Statistics for Rayleigh ( 0.1)

From the above table, we notice that the actual value of and the mean estimated value of by

Method of MLE and Local Frequency Ratio method are almost same. Therefore, it can be taken

as a good estimator. Similar procedure is followed for different sample sizes and for different

values of i.e., 0.5,1,2,5,10.. Also, we have listed the histograms Figures 4.7 to 4.14 for the data

generated for the cases considered herein. The MATLAB programs were developed to obtain

these parameters by two methods are listed in the Appendix. The results are tabulated in Tables

4.7 to 4.11.

ns=50 ns=100 ns=200

MLE

method

Frequency

Ratio

method

MLE

method

Frequency

Ratio

method

MLE

method

Frequenc

y Ratio

method

Mean

0.1004 0.0996 0.1000 0.1045 0.0999 0.1024

Sd 0.0015 0.0089 0.0013 0.0153 0.0017 0.0129

1 0.0769 0.3918 0.1052 1.7882 -0.4270 1.2235

2

2.1236 3.1088 2.4792 8.4906 3.2829 5.2900

Bias 0.35e-003 -0.362e-003 -0.20e-003 0.0045 -0.14e-003 0.0024

MSE 0.002e-003 0.078e-003 0.01e-003 0.0003 0.003e-003 0.0002

57

4.7 Comparison of Method of MLE and Frequency Ratio Method

Table 4.7: Simulation Statistics for Rayleigh ( 0.5)

Table 4.8 : Simulation Statistics for Rayleigh ( 1)

ns=50 ns=100 ns=200

MLE

method

Frequenc

y Ratio

method

MLE

method

Frequency

Ratio

method

MLE

method

Frequency

Ratio

method

Mean

0.4988 0.5183 0.5006 0.5043 0.4988 0.5128

Sd 0.0101 0.0696 0.0076 0.0790 0.0076 0.0687

1 -0.6367 1.1831 -0.1080 1.7908 -0.1385 1.0257

2

3.2876 4.1211 2.6068 8.1241 2.8631 4.1796

Bias -0.0012 0.0183 6.14e-004 0.0043 -0.0012

0.0128

MSE 0.0001 0.0051 5.78e-005 0.0063 0.0001 0.0049

ns=50 ns=100 ns=200

MLE

method

Frequenc

y Ratio

method

MLE

method

Frequency

Ratio

method

MLE

method

Frequenc

y Ratio

method

Mean

0.9990 1.0228 0.9974 1.0221 0.9996 1.0221

Sd 0.0179 0.1276 0.0166 0.1310 0.0162 0.1287

1 0.0865 1.2572 0.1880 1.1413 -0.2462 0.7959

2

2.9688 7.5724 2.8529 6.8395 2.4603 3.8279

Bias -0.0010 0.0228 -0.0026 0.0221 -4.1e-005 0.0221

MSE 0.0003 0.0168 0.0003 0.0177 2.63e-005 0.0163

58

Table 4.9 : Simulation Statistics for Rayleigh ( 2 )

Table 4.10: Simulation Statistics for Rayleigh ( 5 )

ns=50 ns=100 ns=200

MLE

method

Frequenc

y Ratio

method

MLE

method

Frequenc

y Ratio

method

MLE

method

Frequenc

y Ratio

method

Mean

1.9957 2.0921 1.9963 2.0400 2.0019 2.0677

Sd 0.0344 0.3824 0.0333 0.2533 0.0325 0.2771

1 0.0993 1.0923 0.4259 0.8684 -0.3110 0.8010

2

2.2716 3.8626 3.5114 3.5532 3.4797 3.5994

Bias -0.0043 0.0921 -0.0037 0.0400 0.0019 0.0677

MSE 0.0012 0.1547 0.0011 0.0653 0.0011 0.0813

ns=50 ns=100 ns=200

MLE

method

Frequency

Ratio

method

MLE

method

Frequency

Ratio

method

MLE

method

Frequency

Ratio

method

Mean

5.0116 5.0856 5.0088 5.0128 4.9943 5.1833

Sd 0.0818 0.7520 0.0786 0.6429 0.0872 0.7289

1 0.0719 1.2018 0.0471 0.5044 0.0874 1.5580

2

2.1907 5.0922 2.3781 3.2951 3.0103 8.1958

Bias 0.0116 0.0856 0.0088 0.0128 -0.0057

0.1833

MSE 0.0068 0.5728 0.0063 0.4135 0.0076 0.5648

59

Table 4.11 Simulation Statistics for Rayleigh ( 10 )

ns=50 ns=100 ns=200

MLE

method

Frequency

Ratio

method

MLE

method

Frequenc

y Ratio

method

MLE

method

Frequency

Ratio

method

Mean

9.9824 10.0787 9.9823 10.2623 9.9764 10.2556

Sd 0.1614 0.9978 0.1691 1.3841 0.1528 1.3740

1 -0.2399 0.7448 -0.1151 1.208 -0.1385 1.0257

2

2.2674 3.5231 2.9061 6.2105 2.8634 4.1796

Bias -0.0176 0.0787 -0.0177 0.2623 -0.0236 0.2556

MSE 0.0264 1.0018 0.0265 1.9846 0.0239 1.9532

60

4.8 GRAPHS FOR DIFFERENT SAMPLE SIZES AND PARAMETERS

Figure 4.7 : Histogram for Rayleigh(0.5,50)

Figure 4.8 : Histogram for Rayleigh(1,50)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

50

100

150

200

250

Fre

quency

Bins

0 0.5 1 1.5 2 2.5 3 3.5 40

50

100

150

200

250

Fre

quency

Bins

61

Figure 4.9: Histogram for Rayleigh(1,100)

Figure 4.10: Histogram for Rayleigh (1,200)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

50

100

150

200

250

Fre

quency

Bins

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

50

100

150

200

250

300

Fre

quency

bins

62

Figure 4. 11 : Histogram for Rayleigh(2,50)

Figure 4. 12 : Histogram for Rayleigh(2,100)

0 1 2 3 4 5 6 7 80

50

100

150

200

250

Fre

quency

Bins

0 1 2 3 4 5 6 7 80

50

100

150

200

250

Fre

quency

Bins

63

Figure 4. 13 : Histogram for Rayleigh(2,200)

Figure 4. 14 : Histogram for Rayleigh(5,50)

0 1 2 3 4 5 6 7 80

50

100

150

200

250

Fre

quency

Bins

0 5 10 15 20 250

50

100

150

200

250

300

Bins

Fre

quency

64

MATLAB PROGRAMS

GAMMA DISTRIBUTION %PROGRAM TO ESTIMATE THE PARAMETERS BY METHOD OF MOMENTS AND

LOCAL FREQUENCY RATIO METHOD.

n=1000;k=8;a=4;b=4;ns=100;

for j=1:ns,

ex=gamrnd(a,b,n,1);

meanx = mean(ex);varx = var(ex);

stdx = std(ex);

bb(j)= varx/meanx;

aa(j) = (meanx^2)/varx;

maxx=max(ex);h=maxx/k;

[fr,r]=hist(ex,h/2:h:maxx);

hist(ex,h/2:h:maxx)

[u,v]=sort(fr);

u=fliplr(u);v=fliplr(v);

y=v(1:3);x=r(y);f=u(1:3);

x12=x(1)-x(2);x23=x(2)-x(3);

lx12=log(x(1)/x(2));lx23=log(x(2)/x(3));

lf12=log(f(1)/f(2));lf23=log(f(2)/f(3));

em=[x12,lx12;x23,lx23];rhs=[lf12;lf23];

est=em\rhs;

aest=-est(1);best=est(2)+1;

a1(j)=best;b1(j)=1./aest;

65

clear ex fr r u v ;

end

%BY METHOD OF MOMENTS

a2=aa;b2=bb;

ma2=mean(a2);sda2=std(a2);

mb2=mean(b2);sdb2=std(b2);

a2z=(a2-ma2)/sda2;b2z=(b2-mb2)./sdb2;

rb1a2=mean(a2z.^3);b2a2=mean(a2z.^4);

rb1b2=mean(b2z.^3);b2b2=mean(b2z.^4);

%bias

ba2=a2-a;mba2=mean(ba2);bb2=b2-b;mbb2=mean(bb2);

%mean square error

msea2=sda2^2+ mba2^2;mseb2=sdb2^2+ mbb2^2;

mom = [ma2, sda2,rb1a2,b2a2,mb2, sdb2,rb1b2,b2b2]

mombias = [mba2,msea2,mbb2,mseb2]

%BY Frequency Ratio METHOD

a1=a1;b1=b1;

ma1=mean(a1);sda1=std(a1); mb1=mean(b1);sdb1=std(b1);

a1z=(a1-ma1)/sda1;b1z=(b1-mb1)/sdb1;

rb1a1=mean(a1z.^3);b2a1=mean(a1z.^4);

rb1b1=mean(b1z.^3);b2b1=mean(b1z.^4);

%bias

ba1=a1-a;mba1=mean(ba1);bb1=b1-b;mbb1=mean(bb1);

%mean square error

66

msea1=sda1^2+ mba1^2;mseb1=sdb1^2+ mbb1^2;

locmom = [ma1, sda1,rb1a1,b2a1,mb1, sdb1,rb1b1,b2b1]

locbias=[mba1,msea1,mbb1,mseb1]

RAYLEIGH DISTRIBUTION

% PROGRAM TO FIND THE ESTIMATES OF PARAMETERS BY MLE AND

FREQUENCY RATIO METHOD.

n=1000;k=10;b=10;ns=100;

for j=1:ns

x=raylrnd(b,n,1);

z=sum(x.^2);

best=sqrt(z/(2*n));

b1(j)=best;

maxx=max(x);

h=maxx/k;

[fr,r]=hist(x,h/2:h:maxx);

hist(x,h/2:h:maxx)

[u,v]=sort(fr);

u=fliplr(u);v=fliplr(v);

y=v(1:3);x=r(y);f=u(1:3);

lx12=log(x(1)/x(2));lf12=log(f(1)/f(2));

num=((x(2)^2)-(x(1)^2));

den=2*(lf12-lx12);

estb=sqrt(num/den);

67

b2(j)=estb;

end

%LOCAL METHOD

b2=b2;

mb2=mean(b2);sdb2=std(b2);b2z=(b2-mb2)/sdb2;

rbb2=mean(b2z.^3);b2b2=mean(b2z.^4);

%bias

bb2=b2-b; mbb2=mean(bb2);

%mean square error

mseb2=sdb2^2+ mbb2^2;

localmoments = [mb2, sdb2,rbb2,b2b2]

locbias=[mbb2,mseb2]

%MLE METHOD

b1=b1;

mb1=mean(b1);sdb1=std(b1); b1z=(b1-mb1)/sdb1;

rbb1=mean(b1z.^3);b2b1=mean(b1z.^4);

%bias

bb1=b1-b; mbb1=mean(bb1);

%mean square error

mseb1=sdb1^2+ mbb1^2;

MLEMETHOD = [mb1,sdb1,rbb1,b2b1]

mlebias = [mbb1,mseb1]