Cumulative Normal Distribution Function

The cumulative normal distribution function is actually an integral formula. There is no explicit closed form solution. A good approximation can be found at Milton Abramowiz and Irene A Stegun. Handbook of Mathematical Functions. National Bureau of Standards, 1964.

The following code is a java function for calculating the cumulative normal distribution.

public double CNDF(double z)
{
double mean = calculateMean();
double sd = calculateSD();
z = z - mean;
if (z> 6.0) return 1.0;
if (z<-6.0) return 0;
double b1 = 0.31938153;
double b2 = -0.356563782;
double b3 = 1.781477937;
double b4 = -1.821255978;
double b5 = 1.330274429;
double p = 0.2316419;
double c2 = 0.3989423;
double a = Math.abs(z);
double t = 1.0/(1.0 + a*p);
double b = c2 * Math.exp((-z)*(z/2.0));
double n = ((((b5 * t + b4)*t+b3)*t+b2)*t+b1)*t;
n = 1.0 - b*n;
if (z<0.0) n = 1.0 - n;
return n;
}

Normality Testing

There are many algorithms for testing the normality of a data set.
Some of the tests include:

1. Kolmogorov-Smirnov test
2. Lilliefors test
3. Shapiro-Wilks’ W test

Shapiro-Wilks test seems to the best because of its good power properties. But, Kolmogorov-Smirnov test is easier to implement.

NSF grant proposal: Organization

I found Expert Opinon's link useful and interesting.

CLASSPATH

I have worked only on C# for the past one year. Today, I got a chance to work on my favorite programming language, Java. I always forget to set the CLASSPATH after I install Java SDK. This solved a lot of issues.

Types of output

Binary classification problem - A learning problem with binary outputs
Multi-class classification problem - A learning problem with finite number of categories
Regression - A learning problem with real-valued outputs

Report

The report that I would recommend in a typical NN experiment can be divided into the following sections:

1. Objective of the experiment
2. Data description - Preprocessing of the data (if any)
3. Architecture of the Neural Network used
4. Pseudo code
5. Results
6. Discussion
7. Conclusion
8. Matlab code as an appendix

Matlab script

I used this MATLAB script using Neural Network toolbox. The important features were selected based on a threshold.

---

clear all;
P = load('train1.txt');
T = load('trainout1.txt');
Ps = load('test1.txt');
Ts = load('testout1.txt');
Net = newff(minmax(P), [300, 200, 150, 1], {'logsig', 'logsig', 'logsig', 'logsig'}, 'trainscg' );
Net.trainParam.epochs = 1000;
Net.trainParam.goal = 0.0001;
[Net_t, TR] = train(Net,P,T);
y = sim(Net_t,Ps);
% figure
x=[1:37]; % to draw the x axis values
efficiency = 1 - (sum(abs(round(y)-Ts)))/37;
efficiency
% plot(x, y, 'bo--', x, Ts, 'r*-');
% legend('network','actual');

% Selection of Features
Threshold = 0.90;
for i=1:300
disp 'Current Iteration'
disp(i)
FP = P;
FPs = Ps;
FP(i,:) = [];
FPs(i,:) = [];
Net = newff(minmax(FP), [299, 200, 150, 1], {'logsig', 'logsig', 'logsig', 'logsig'}, 'trainscg' );
Net.trainParam.epochs = 1000;
Net.trainParam.goal = 0.0001;
[Net_t, TR] = train(Net,FP,T);
y = sim(Net_t,FPs);
efficiency = 1 - (sum(abs(round(y)-Ts)))/37;
disp(efficiency);
if (efficiency < Threshold)
disp(i)
disp(' is an Important Feature');
end
end

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