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Quantitative Variable Distances

, 6 Jan 2009 CPOL
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An overview of different distances under the Minkowski distance.

Introduction

In this article, I discuss and implement the following:

  • Minkowski Distance
  • Euclidean Distance
  • Manhattan Distance
  • Chebyshev Distance

The general metric for distance is the Minkowski distance. When lambda is equal to 1, it becomes the city block distance, and when lambda is equal to 2, it becomes the Euclidean distance. The special case is when lambda is equal to infinity (taking a limit), where it is considered as the Chebyshev distance.

MinkowskiDistance.gif

public static double MinkowskiDistance(double[] X, double[] Y, int order)
{
    int count = 0;
    double distance = 0.0;
    double sum = 0.0;
    if (X.GetUpperBound(0) != Y.GetUpperBound(0))
    {
        throw new System.ArgumentException("the number of elements" + 
                  " in X must match the number of elements in Y");
    }
    else
    {
        count = X.Length;
    }
    for (int i = 0; i < count; i++)
    {
        sum = sum + Math.Pow(Math.Abs(X[i] - Y[i]), order);
    }
    distance = Math.Pow(sum,(1/order));
    return distance;
}

Euclidean distance is the most common use of distance. When people talk about distance, this is what they are referring to. Euclidean distance, or simply 'distance', examines the root of square differences between the coordinates of a pair of objects. This is most generally known as the Pythagorean theorem.

EuclideanDistance.gif

public static double EuclideanDistance(double [] X, double []Y)
{
    int count = 0;
    double distance = 0.0;
    double sum = 0.0;
    if(X.GetUpperBound(0) != Y.GetUpperBound(0))
    {
        throw new System.ArgumentException("the number of elements" + 
                  " in X must match the number of elements in Y");
    }
    else
    {
        count = X.Length;
    }
    for (int i = 0; i < count; i++)
    {
        sum = sum + Math.Pow(Math.Abs(X[i] - Y[i]),2);
    }
    distance = Math.Sqrt(sum);
    return distance;
}

The taxicab metric is also known as rectilinear distance, L1 distance or L1 norm, city block distance, Manhattan distance, or Manhattan length, with the corresponding variations in the name of the geometry. It represents the distance between points in a city road grid. It examines the absolute differences between the coordinates of a pair of objects.

CityBlockDistance.gif

public static double ManhattanDistance(double [] X, double []Y)
{
    int count = 0;
    double distance = 0.0;
    double sum = 0.0;
    if(X.GetUpperBound(0) != Y.GetUpperBound(0))
    {
        throw new System.ArgumentException("the number of elements" + 
                  " in X must match the number of elements in Y");
    }
    else
    {
        count = X.Length;
    }
    for (int i = 0; i < count; i++)
    {
        sum = sum + Math.Abs(X[i] - Y[i]);
    }
    distance = sum;
    return distance;
}

Chebyshev distance is also called the Maximum value distance, defined on a vector space where the distance between two vectors is the greatest of their differences along any coordinate dimension. In other words, it examines the absolute magnitude of the differences between the coordinates of a pair of objects.

ChebyshevDistance.gif

public static double ChebyshevDistance(double[] X, double[] Y)
{
    int count = 0;
    if (X.GetUpperBound(0) != Y.GetUpperBound(0))
    {
        throw new System.ArgumentException("the number of elements" + 
                  " in X must match the number of elements in Y");
    }
    else
    {
        count = X.Length;
    }
    double[] newData = new double[count];
    for (int i = 0; i < count; i++)
    {
        newData[i] = Math.Abs(X[i] - Y[i]);
    }
    double max = double.MinValue;
    foreach (double num in newData)
    {
        if (num > max)
        {
            max = num;
        }
    }
    return max;
}

Sources

License

This article, along with any associated source code and files, is licensed under The Code Project Open License (CPOL)

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About the Author

saharkiz
Web Developer
Philippines Philippines
My name : Aresh Saharkhiz.
Origin: Unknown
 
Education : BS Computer Science
MS Computer Science
Interests : Genetic Programming
Neural Networks
Game AI
Programming: (language == statementEnd(semicolon)
 

http://sites.google.com/site/docaresh
 
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Comments and Discussions

 
GeneralMy vote of 2 PinmemberARon_6-Jan-09 10:47 
Three of the four functions could be simple call to the first function with different order variables. The subject is too simple; the author provided no insight in the theory or applications. Tell why you wrote this article. This is stuff I would expect a beginner to cruse through.
GeneralRe: My vote of 2 Pinmemberaresguya6-Jan-09 20:12 
Generaldouble max = double.MinValue; Pinmemberaxelriet6-Jan-09 7:58 

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