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Quantitative Variable Distances

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6 Jan 2009CPOL1 min read 36.9K   7   3
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

C#
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

C#
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

C#
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

C#
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)


Written By
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Philippines Philippines
My name : Aresh Saharkhiz.
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Education : BS Computer Science
MS Computer Science
Interests : Genetic Programming
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Programming: (language == statementEnd(semicolon)


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Comments and Discussions

 
GeneralMy vote of 2 Pin
ARon_6-Jan-09 9:47
ARon_6-Jan-09 9:47 
GeneralRe: My vote of 2 Pin
saharkiz6-Jan-09 19:12
saharkiz6-Jan-09 19:12 
Generaldouble max = double.MinValue; Pin
Axel Rietschin6-Jan-09 6:58
professionalAxel Rietschin6-Jan-09 6:58 

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