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# A Normal / Exponential Random Generator and Histogram class

, 2 Dec 2002
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A fast random generator with normal or exponential distribution + a histogram class.

## Introduction

### Random Variable Generator

This article present a fast generator for Random Variable, namely normal and exponential distributions. The algorithm is due to George Marsaglia and Wai Wan Tsang in [1]. Here's the abstract of their paper:

We provide a new version of our ziggurat method for generating a random variable from a given decreasing density. It is faster and simpler than the original, and will produce, for example, normal or exponential variates at the rate of 15 million per second with a C version on a 400MHz PC. It uses two tables, integers ki and reals wi. Some 99% of the time, the required x is produced by: Generate a random 32-bit integer j and let i be the index formed from the rightmost 8 bits of j. If j < ki return x = j . wi.

They did 99,9% of the work, I just encapsulated their C code in a class wrapper.

### Histogram

In order to illustrate the generator, an Histogram template class, `THistogram` is provided.

## A little bit of mathematical background

The normal distribution holds an honored role in probability and statistics, mostly because of the central limit theorem, one of the fundamental theorems that forms a bridge between the two subjects.

The normal distribution is also called the Gaussian distribution, in honor of Carl Friedrich Gauss, who was among the first to use the distribution.

There are plenty tutorials and demo applets on normal distributions on the web so I won't go into mathematical details here. A very nice web site on this topic can be found at Virtual Laboratories in Probability and Statistics

The algorithm used to compute the mean, variance and other statistical properties are taken from the numerical recipies (see [2]).

## Using the generator

The `CRandomGenerator` provides two types of random variables: normal and exponential. These are computed by 2 static functions, RNOR and REXP:

• standard normal variable:
```float var = CRandomGenerator::RNOR();
```
You can also modify the mean and standard deviation of the generator:
```float fMean=...;
float fSdev=...;
float var = CRandomGenerator::RNOR(fMean, fSdev);
```
• exponential variable:
```float var = CRandomGenerator::REXP();
```

## Initializing the generator

Like any random generator, it has to be initialized with a "seed". Classically, one uses the current time to seed the generator. This is done implicitly in the default constructor of `CRandomGenerator`, so what you have to do is to build one and only one `CRandomGenerator` object in your application thread before using the generator. Doing that in your `CWinApp::InitInstance` function is a good idea.

```CWinApp::InitInstance
{
...
// build a CRandomGenerator to seed the generator
CRandomGenerator randn;
}
```

## Using the histogram

`THistogram` is a templated histogram class.

### Template parameters

• `T`, input data type: can be `float` or `double`,
• `TOut`, result data type: can be `float` or `double`, by default set to `double`

### Characteristics

The histogram is characterized by

• a region, defined by a minimum and maximum spectrum value (see `Get/SetMinSpectrum`,`Get/SetMaxSpectrum`)
• a step size (see `GetStep`)

### Computing the histogram

You can feed the histogram with data by different manners:

• Feeding a vector of data,
```vector< float > vData;
...  // computing the data

// Creating an histogram with 101 regions
THistogram< float > histo(101);
// Computing the histogram, min and max values are automatically computed...
histo.Compute( vData , true /* compute min, max */);
```
• Updating with a vector of data,
```vector< float > vData;
...
// updating the histogram
histo.Update( vData );
```
• Updating with a single data entry,
```float fData;
...
// updating the histogram
histo.Update( fData );
```
• Computing the statistical characteristics of a data set: you can compute the moments of a data sets (mean, standard deviation, variance, etc.) by using the `static` method `GetMoments`:
```float fMean, fAdev, fSdev,  fVar, fSkew,fKurt;
// Computing succesively the mean, absolute mean, standard deviation, variance, skewness and kurtosis
```
This method is used in the demo to compare the normal distribution generated by ```CRandomGenerator ``` and the theoretical probability density function.

### Retrieving results

• Get the histogram results by using `GetHistogram`.
• Get a normalized histogram (such that it's area is 1) by calling `GetNormalizedHistogram`,
• Get the coordinates of the regions can be accessed by using `GetLeftContainers` and `GetCenterContainers`.

The code to generate the plots looks like this:

```vector < double > vDistribution ( histo.GetNormalizedHistogram() );
vector < double > vLeftPositions ( histo.GetLeftContainers() );
CPGLLine2D* pLine;
...
pLine->SetDatas(vLeftPositions, vDistributions);
```

## Demo application

The demo application shows the normal and exponential distribution. The red curve represents the histogram of the random values and the blue curve represent the theoretical probability density function.

There are two projects in the demo application:

• `HistogramDemo` uses the Plot Graphic Library for visualization. There is another article on the PGL here. You will need the GDI+ binaries to make the demo application work ! (gdiplus.dll)
• `HistogramMatlab` uses the Matlab engine for visualization. You will need Matlab installed to make the demo application work !

Note that the source code is documented using Doxygen syntax.

## Update History

Nov,26,2002
 Added theoretical distribution in `CRandomGenerator`, Added histogram area computation, Fixed normalized histogram, added .
Sep,13,2002 Added a new demo project using Matlab.
Sep,11,2002 Fixed demo project and added binary in demo.

A list of licenses authors might use can be found here

## Share

 Engineer United States
Jonathan de Halleux is Civil Engineer in Applied Mathematics. He finished his PhD in 2004 in the rainy country of Belgium. After 2 years in the Common Language Runtime (i.e. .net), he is now working at Microsoft Research on Pex (http://research.microsoft.com/pex).

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 missing files zhshshen9-Dec-02 13:41 zhshshen 9-Dec-02 13:41
 GDI+ Jonathan de Halleux10-Dec-02 19:20 Jonathan de Halleux 10-Dec-02 19:20
 Re: GDI+ JoeSox23-Dec-02 7:35 JoeSox 23-Dec-02 7:35
 GDI+ Jonathan de Halleux24-Dec-02 2:32 Jonathan de Halleux 24-Dec-02 2:32
 Re: missing files ronc12-Feb-03 4:14 ronc 12-Feb-03 4:14
 Re: missing files Jonathan de Halleux4-Mar-03 8:02 Jonathan de Halleux 4-Mar-03 8:02
 Missing DLL? Jim A. Johnson2-Dec-02 10:46 Jim A. Johnson 2-Dec-02 10:46
 Fix... Jonathan de Halleux2-Dec-02 22:27 Jonathan de Halleux 2-Dec-02 22:27
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 doesn't compile deepsea200213-Sep-02 19:13 deepsea2002 13-Sep-02 19:13
 VC7 problem Jonathan de Halleux2-Oct-02 2:47 Jonathan de Halleux 2-Oct-02 2:47
 Very nice! Gary McHale11-Sep-02 18:40 Gary McHale 11-Sep-02 18:40
 Won't build on VC6 Jim A. Johnson9-Sep-02 12:21 Jim A. Johnson 9-Sep-02 12:21
 Updated... Jonathan de Halleux10-Sep-02 22:08 Jonathan de Halleux 10-Sep-02 22:08
 does not compile ! Mario M.9-Sep-02 6:23 Mario M. 9-Sep-02 6:23
 ??? Jonathan de Halleux9-Sep-02 6:26 Jonathan de Halleux 9-Sep-02 6:26
 Re: does not compile ! Chris Losinger9-Sep-02 6:44 Chris Losinger 9-Sep-02 6:44
 Re: does not compile ! Jonathan de Halleux10-Sep-02 22:09 Jonathan de Halleux 10-Sep-02 22:09
 Re: does not compile ! Mario M.11-Sep-02 1:35 Mario M. 11-Sep-02 1:35
 Re: does not compile ! Mario M.11-Sep-02 1:52 Mario M. 11-Sep-02 1:52
 VC7 problem Jonathan de Halleux2-Oct-02 2:48 Jonathan de Halleux 2-Oct-02 2:48
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