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Algorithms - Similarity
Written by Jan Schulz   
Thursday, 15 May 2008 19:31

Minkowski distance

 

Objective

The Minkowski distance (e.g. Kruskal 1964) is a generalised metric that includes others as special cases of the generalised form. Although theoretically infinite measures exist by varying the order of the equation just three have gained importance.

Equation

Equation for the Minkowski distance

In the equation dMKD is the Minkowski distance between the data record i and j, k the index of a variable, n the total number of variables y and λ the order of the Minkowski metric. Although it is defined for any λ > 0, it is rarely used for values other than 1, 2 and ∞. As infinity can not be displayed in computer arithmetics the Minkowski metric is transformed for λ = ∞ and it becomes:

Equation for the Chebyshev distance
Or in easier words the Minkowski metric of the order ∞ returns the distance along that axis on which the two objects show the greatest absolute difference.

Overview over different orders of the Minkowski metric
The way distances are measured by the Minkowski metric of different orders between two objects with three variables (here displayed in a coordinate system with x-, y- and z-axes). The unfolded cube shows the way the different orders of the Minkowski metric measure the distance between the two points.

Synonyms

Different names for the Minkowski distance or Minkowski metric arise form the order:

  • λ = 1 is the Manhattan distance. Synonyms are L1-Norm, Taxicab or City-Block distance. For two vectors of ranked ordinal variables the Mahattan distance is sometimes called Footruler distance.
  • λ = 2 is the Euclidean distance. Synonyms are L2-Norm or Ruler distance. For two vectors of ranked ordinal variables the Euclidean distance is sometimes called Spearman distance.
  • λ = ∞ is the Chebyshev distance. Synonym are Lmax-Norm or Chessboard distance.

Usage

The Minkowski distance is often used when variables are measured on ratio scales with an absolute zero value. Variables with a wider range can overpower the result. Even a few outliers with high values bias the result and disregard the alikeness given by a couple of variables with a lower upper bound.

Algorithm

 

The algorithm controls whether the data input matrix is rectangular or not. If not the function returns FALSE and a defined, but empty output matrix. When the matrix is rectangular the Minkowski distance of the respective order is calculated. Therefore the dimensions of the respective arrays of the output matrix and the titles for the rows and columns set. As the result is a square matrix, which is mirrored along the diagonal only values for one triangular half and the diagonal are computed. When errors occur during computation the function returns FALSE.

 

Source

Function dist_Minkowski (InputMatrix : t2dVariantArrayDouble; MinkowskiOrder: Double; Var OutputMatrix : t2dVariantArrayDouble) : Boolean;
// The function dist_MinkowskiDistance calculates the Minkowski distance of
// the order given in MinkowskiOrder. The cases are expected in the rows.
// The variables are expected in the columns. Function returns FALSE if
// at least one cell can not be calculated. The result matrix is returned
// in OutputMatrix.
// (c) Dr. Jan Schulz, 04.May 2008; www.code10.info
Var OutputMatrixSize : Integer;
InputCols : Integer;
InputRows : Integer;
RunnerY : Integer;
RunnerX : Integer;
i : Integer;
Summed : Double;
FirstVal : Double;
SecondVal : Double;
Begin
// if one dimension is zero or matrix not rectangular or Minkowski order is zero
If Not mtx_IsRectangular (InputMatrix, InputRows, InputCols) Or (MinkowskiOrder = 0) THen
Begin
//create an empty matrix, return FALSE and exit
mtx_Create (OutputMatrix, 1, 1, 0, 'Erroneous Minkowski distance matrix');
dist_Minkowski := False;
Exit;
end;

// Minkowski order needs to be positive
MinkowskiOrder := Abs (MinkowskiOrder);

// let's expect the best case ...
dist_Minkowski := True;

// define and set the row dimension of the result matrix
OutputMatrixSize := High (InputMatrix.Cells) + 1;
SetLength (OutputMatrix.Cells, OutputMatrixSize);

// create the column dimension of the array
For RunnerY := Low (OutputMatrix.Cells) to High (Outputmatrix.Cells) do
Begin
SetLength (OutputMatrix.Cells [RunnerY], OutputMatrixSize);
end;

// define title of matrix
OutputMatrix.MatrixName := 'Minkowski distance matrix of order ' + FloatToStr (MinkowskiOrder);

// Set Row/Col-Title for the new matrix
SetLength (OutputMatrix.RowTitle, OutputMatrixSize);
SetLength (OutPutMatrix.ColTitle, OutPutMatrixSize);
For RunnerY := Low (InputMatrix.RowTitle) to High (InputMatrix.RowTitle) do
Begin
// names for rows and columns are the same in this triangualary matrix
OutputMatrix.RowTitle [RunnerY] := InputMatrix.RowTitle [RunnerY];
OutputMatrix.ColTitle [RunnerY] := InputMatrix.RowTitle [RunnerY];
end;

// compare every object
For RunnerY := Low (OutputMatrix.Cells) to High (OutputMatrix.Cells) do
Begin
//with every other object
For RunnerX := Low (OutputMatrix.Cells) to RunnerY do
Begin
Summed := 0;

//include all variables in analysis
For i := 0 to High (InputMatrix.Cells [0]) do
Begin
FirstVal := InputMatrix.Cells [RunnerX, i];
SecondVal := InputMatrix.Cells [RunnerY, i];

// add the power of the absolute difference
Summed := Summed + Power (Abs (FirstVal - SecondVal), MinkowskiOrder);
end;

Summed := Power (Summed, 1/MinkowskiOrder);
// set the calculated value on both sides of diagonal and diagonal itself
OutputMatrix.Cells [RunnerX, RunnerY] := Summed;
OutputMatrix.Cells [RunnerY, RunnerX] := Summed;
end;
end;
end;
 

Example

For a data matrix aInputMatrix of the type t2dVariantArrayDouble, populated with:

Data

Var1

Var2

Var3

Case1

1

1

1

Case2

1

1

0

Case3

2

2

2

Case4

10

10

10

Case5

11

11

11

Case6

10

5

0

the call of:

aBooleanVar := dist_Minkowski (aInputMatrix, 1, aOutputMatrix);

returns the respective Minkowski matrix of the first order in aOutputMatrix:

Manhattan

distance

Case1

Case2

Case3

Case4

Case5

Case6

Case1

0

1

3

27

30

14

Case2

1

0

4

28

31

13

Case3

3

3

0

24

27

13

Case4

27

28

24

0

3

15

Case5

30

31

27

3

0

18

Case6

14

13

13

15

18

0

and the call of:

aBooleanVar := dist_Minkowski (aInputMatrix, 2, aOutputMatrix);

returns the respective Minkowski matrix of the second order in aOutputMatrix:

Euclidean

distance

Case1

Case2

Case3

Case4

Case5

Case6

Case1

0

1.000

1.732

15.588

17.321

9.899

Case2

1.000

0

2.449

16.186

17.916

9.849

Case3

1.732

2.449

0

13.856

15.588

8.775

Case4

15.588

16.186

13.856

0

1.732

11.180

Case5

17.321

17.916

15.588

1.732

0

12.570

Case6

9.899

9.849

8.775

11.180

12.570

0

Characteristic for the Minkowski distance is to represent the absolute distance between objects independently from their distance to the origin. This is contrary to several other distance or similarity/dissimilarity measurements. Thus, the distance between the objects Case1 and Case3 is the same as between Case4 and Case5 for the above data matrix, when investigated by the Minkowski metric.

Literature

Kruskal J.B. (1964): Multidimensional scaling by optimizing goodness of fit to a non metric hypothesis. Psychometrika 29(1):1-27.

 

Last Updated on Friday, 18 March 2011 18:19
 
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