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 Hamming distance   Algorithms - Similarity
Written by Jan Schulz
Thursday, 15 May 2008 19:22 Hamming distance

Objective

The Hamming distance (Hamming 1950) is a metric expressing the distance between two objects by the number of mismatches among their pairs of variables. It is mainly used for string and bitwise analyses, but can also be useful for numerical variables. Although the basic Hamming distance is a metric, the here presented version allows to define a threshold. Variables having an absolute difference below the threshold are considered as equal. Using values larger than 0 for this threshold the triangle inequality could be violated for some calculated distances. Using 0 as threshold it is the original and metric Hamming distance, thresholds below 0 are not defined.

Equation In the equation dHAD is the Hamming distance between the objects i and j, k is the index of the respective variable reading y out of the total number of variables n. The Hamming distance itself gives the number of mismatches between the variables paired by k.

Synonyms

There are no common synonyms.

Usage

The Hamming distance is an important measurement for the detection of errors in information transmission (Hamming 1950). Beyond this application its usage is valuable for the investigation of e.g. ranked variables of entities coded by numerical tokens.

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 Hamming distance 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_Hamming (InputMatrix : t2dVariantArrayDouble; aMaxDiff: Double; Var OutputMatrix : t2dVariantArrayDouble) : Boolean;
// The function CalcHammingDistanceMatrix calculates the Hamming distance between
// several cases, which 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;
UnequalItems : Integer;
FirstVal : Double;
SecondVal : Double;
Begin
// if one dimension is zero or matrix not rectangular
If Not mtx_IsRectangular (InputMatrix, InputRows, InputCols) THen
Begin
//create an empty matrix, return FALSE and exit
mtx_Create (OutputMatrix, 1, 1, 0, 'Erroneous Hamming distance matrix');
dist_Hamming := False;
Exit;
end;

// maximum allowed difference between two variables needs to be positive
aMaxDiff := Abs (aMaxDiff);

// let's expect the best case ...
dist_Hamming := 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 := 'Hamming distance matrix';

// 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
UnequalItems := 0;

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

// are variables equal within a certain range?
If Abs (FirstVal - SecondVal) > aMaxDiff THen
Begin
// calculation impossible as invalid numbers were found
UnequalItems := UnequalItems + 1;
end;
end;

// set the calculated value on both sides of diagonal and diagonal itself
OutputMatrix.Cells [RunnerX, RunnerY] := UnequalItems;
OutputMatrix.Cells [RunnerY, RunnerX] := UnequalItems;
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_Hamming (aInputMatrix, 0, aOutputMatrix);

returns the respective matrix of the original Hamming distance in aOutputMatrix:

 Hamming distance Case1 Case2 Case3 Case4 Case5 Case6 Case1 0 1 3 3 3 3 Case2 1 0 3 3 3 2 Case3 3 3 0 3 3 3 Case4 3 3 3 0 3 2 Case5 3 3 3 3 0 3 Case6 3 2 3 2 3 0

The Hamming distance simply counts the number of differences between the paired variables. Thus the distances are unaffected by the distance of the object from the origin.

Literature

Hamming R.W. (1950): Error detecting and error correcting codes. Bell System Technical Journal 26(2):147-160.

Last Updated on Friday, 18 March 2011 18:17 