COSINE DISTANCE
COSINE SIMILARITY
ANGULAR COSINE DISTANCE
ANGULAR COSINE SIMILARITY

COSINE DISTANCE (LET)
COSINE SIMILARITY (LET)
ANGULAR COSINE DISTANCE (LET)
ANGULAR COSINE SIMILARITY (LET)

Let Subcommand

Compute the cosine distance (or cosine similarity, angular cosine distance, angular cosine similarity) between two variables.

The cosine similarity is defined as
\( \mbox{Cosine Similarity} = \frac{\sum_{i=1}^{n}{x_{i} y_{i}}} {\sqrt{\sum_{i=1}^{n}{x_{i}^{2}}} \sqrt{\sum_{i=1}^{n}{y_{i}^{2}}}} \)

The cosine distance is then defined as

\( \mbox{Cosine Distance} = 1 - \mbox{Cosine Similarity} \)

The cosine distance above is defined for positive values only. It is also not a proper distance in that the Schwartz inequality does not hold. However, the following angular definitions are proper distances:

\( \mbox{angular cosine distance} = \frac{1/\mbox{cosine similarity}} {\pi} \)

\( \mbox{angular cosine similarty} = 1 - \mbox{angular cosine distance} \)

If negative values are encountered in the input, the cosine distances will not be computed. However, the cosine similarities will be computed.

NOTE: The 2018/08 version of Dataplot updated the definition for the angular cosine distance to

\( \mbox{angular cosine distance} = \frac{\mbox{c} \arccos(\mbox{cosine similarity})} {\pi} \)

with \( \arccos \) designating the arccosine function and where c = 2 if there are no negative values and c = 1 if there are negative values.

LET <par> = COSINE DISTANCE <y1> <y2>
                        <SUBSET/EXCEPT/FOR qualification>
where <y1> is the first response variable;
            <y2> is the second response variable;
            <par> is a parameter where the computed cosine distance is stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

LET <par> = COSINE SIMILARITY <y1> <y2>
                        <SUBSET/EXCEPT/FOR qualification>
where <y1> is the first response variable;
            <y2> is the second response variable;
            <par> is a parameter where the computed cosine similarity is stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

LET <par> = ANGULAR COSINE DISTANCE <y1> <y2>
                        <SUBSET/EXCEPT/FOR qualification>
where <y1> is the first response variable;
            <y2> is the second response variable;
            <par> is a parameter where the computed angular cosine distance is stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

LET <par> = ANGULAR COSINE SIMILARITY <y1> <y2>
                        <SUBSET/EXCEPT/FOR qualification>
where <y1> is the first response variable;
            <y2> is the second response variable;
            <par> is a parameter where the computed angular cosine similarity is stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

LET A = COSINE DISTANCE Y1 Y2
LET A = COSINE SIMILARITY Y1 Y2
LET A = ANGULAR COSINE DISTANCE Y1 Y2
LET A = ANGULAR COSINE SIMILARITY Y1 Y2
LET A = SHORTEST HALF MIDMEAN Y1 SUBSET TAG > 2


LET A = COSINE DISTANCE Y1 Y2 SUBSET Y1 > 0 SUBSET Y2 > 0

Dataplot statistics can be used in a number of commands. For details, enter
HELP STATISTICS

None

None

EUCLIDEAN DISTANCE	=	Compute the Euclidean distance.
MANHATTAN DISTANCE	=	Compute the Manhattan distance.
MATRIX DISTANCE	=	Compute various distance metrics for a matrix.
CORRELATION	=	Compute the correlation between two variables.

John Foreman (2014), "Data Smart", Wiley.

Robust Clustering

2017/06
2018/08: Modified formula for angular cosine distance

 
SKIP 25
READ IRIS.DAT Y1 TO Y4 X
.
LET COSDIST  = COSINE DISTANCE Y1 Y2
LET COSADIST = ANGULAR COSINE DISTANCE Y1 Y2
LET COSSIMI  = COSINE SIMILARITY Y1 Y2
LET COSASIMI = ANGULAR COSINE SIMILARITY Y1 Y2
SET WRITE DECIMALS 4
TABULATE COSINE DISTANCE Y1 Y2 X

            Cross Tabulate COSINE DISTANCE
 
(Response Variables: Y1       Y2      )
---------------------------------------------
       X          |   COSINE DISTANCE
---------------------------------------------
         1.0000   |            0.0027
         2.0000   |            0.0049
         3.0000   |            0.0056


.
XTIC OFFSET 0.2 0.2
X1LABEL GROUP ID
LET NDIST = UNIQUE X
XLIMITS 1 NDIST
MAJOR X1TIC MARK NUMBER NDIST
MINOR X1TIC MARK NUMBER 0
CHAR X
LINE BLANK
LABEL CASE ASIS
CASE ASIS
TITLE CASE ASIS
TITLE OFFSET 2
.
MULTIPLOT CORNER COORDIANTES 5 5 95 95
MULTIPLOT SCALE FACTOR 2
MULTIPLOT 2 2
.
Y1LABEL Cosine Distance
TITLE Cosine Distance (Sepal Length and Sepal Width)
COSINE DISTANCE PLOT Y1 Y2 X
.
Y1LABEL Cosine Similarity
TITLE Cosine Similarity (Sepal Length and Sepal Width)
COSINE SIMILARITY PLOT Y1 Y2 X
.
Y1LABEL Angular Cosine Distance
TITLE Angular Cosine Distance (Sepal Length and Sepal Width)
COSINE ANGULAR DISTANCE PLOT Y1 Y2 X
.
Y1LABEL Angular Cosine Similarity
TITLE Angular Cosine Similarity (Sepal Length and Sepal Width)
ANGULAR COSINE SIMILARITY PLOT Y1 Y2 X
.
END OF MULTIPLOT
JUSTIFICATION CENTER
MOVE 50 98
TEXT Distance/Similarity Measures (IRIS.DAT)



.
BOOTSTRAP SAMPLES 1000
CHAR X ALL
LINE BLANK ALL
BOOTSTRAP COSINE DISTANCES PLOT Y1 Y2 X