DISSIMILARITIES Function(PV-WAVE Advantage)
Computes a matrix of dissimilarities (or similarities) between the rows (or columns) of a matrix.
Usage
result = dissimilarities (x)
Input Parameters
x—Array of size nrow by ncol containing the matrix.
Returned Value
result—Float or double array of size m by m containing the computed dissimilarities or similarities, where m = nrow (the default) or m = ncol, if the Columns keyword is specified.
Input Keywords
Double—If present and nonzero, double precision is used.
Columns—Determines whether distances are computed between rows or columns of x. Default: Distances are computed between rows.
Index—An input array containing the indices of the rows (or columns if the Columns keyword is set) to be used in computing the distance measure. Default: All rows(columns) are used.
Method—Method for computing the dissimilarities or similarities. See the
Discussion section for more information. Default:
Method = 0.
0
—Euclidean distance (
L2 norm)
1
—Sum of the absolute differences (
L1 norm)
2
—Maximum difference (L
∞ norm)
3
—Mahalanobis distance
4
—Absolute value of the cosine of the angle between the vectors
5
—Angle in radians (0,
π) between the lines through the origin defined by the vectors
6
—Correlation coefficient
7
—Absolute value of the correlation coefficient
8
—Number of exact matches
Scale—Scaling option. Scale is not used for methods 3 through 8. Default: Scale = 0.
0
—No scaling is performed.
1
—Scale each row (or column) by the standard deviation of the row (column).
2
—Scale each row (or column) by the range of the row (column).
Discussion
DISSIMILARITIES computes an upper triangular matrix (excluding the diagonal) of dissimilarities (or similarities) between the rows or columns of a matrix. Nine different distance measures can be computed. For the first three measures, three different scaling options can be employed. Output from DISSIMILARITIES is generally used as input to clustering or multidimensional scaling functions.
The following discussion assumes that the distance measure is being computed between the columns of the matrix, i.e., that the Columns keyword is set. If distances between the rows of the matrix are desired, leave the Columns keyword set to the default.
For Method = 0 to 2, each row of x is first scaled according to the value of Scale. The scaling parameters are obtained from the values in the row scaled as either the standard deviation of the row or the row range; the standard deviation is computed from the unbiased estimate of the variance. If Scale is 0, no scaling is performed, and the parameters in the following discussion are all 1.0. Once the scaling value (if any) has been computed, the distance between column i and column j is computed via the difference vector zk = (xk - yk)/sk, i = 1, ..., Ndstm, where:
Ndstm is the number of columns or, if the
Index keyword is specified, the number of elements in
Index xk denotes the
kth element in the
ith column
yk denotes the corresponding element in the
jth column.
For given zi, the metrics 0 to 2 are defined as:
0—
Euclidean distance
1—
L1 norm
2—
L∞ norm
Distance measures corresponding to Method = 3 through 8 do not allow for scaling. These measures are defined via the column vectors X = (xi), Y = (yi), and Z = (xi - yi) as follows:
3—
= Mahalanobis distance, where
is the usual unbiased sample estimate of the covariance matrix of the rows.
4—
= the dot product of X and Y divided by the length of X times the length of Y.
5—
θ, where
θ is defined in 4.
6—
ρ = the usual (centered) estimate of the correlation between
X and
Y.
7—The absolute value of
ρ (where
ρ is defined in 6).
8—The number of times
xi =
yi, where
xi and
yi are elements of
X and
Y.
For the Mahalanobis distance, any variable used in computing the distance measure that is (numerically) linearly dependent upon the previous variables in the Index vector is omitted from the distance measure.
Example
The following example illustrates the use of DISSIMILARITIES for computing the Euclidean distance between the rows of a matrix.
ncol = 2
nrow = 4
; Create an NROW x NCOL data set.
x = [[1., 1., 1., 1.], [1., 0., -1., 2.]]
; Call the routine using the default, Row orientation,
; no scaling and the
; 'Sum of the Absolute Differences' method
dist = DISSIMILARITIES(x, ind=[0,1], scale=0, method=1)
; Show the output
PRINT,""
PRINT," OUTPUT"
PRINT," -------------------"
PRINT,""
PRINT," dist"
PM, dist, Format="(4I)"
Output
OUTPUT
-------------------
dist
0 1 2 1
0 0 1 2
0 0 0 3
0 0 0 0
Version 2017.0
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