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Dataplot Vol 2 Vol 1

COSINE DISTANCE
COSINE SIMILARITY
ANGULAR COSINE DISTANCE
ANGULAR COSINE SIMILARITY

Name:
    COSINE DISTANCE (LET)
    COSINE SIMILARITY (LET)
    ANGULAR COSINE DISTANCE (LET)
    ANGULAR COSINE SIMILARITY (LET)
Type:
    Let Subcommand
Purpose:
    Compute the cosine distance (or cosine similarity, angular cosine distance, angular cosine similarity) between two variables.
Description:
    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.

Syntax 1:
    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.
Syntax 2:
    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.
Syntax 3:
    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.
Syntax 4:
    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.
Examples:
    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

Note:
    Dataplot statistics can be used in a number of commands. For details, enter

Default:
    None
Synonyms:
    None
Related Commands: Reference:
    John Foreman (2014), "Data Smart", Wiley.
Applications:
    Robust Clustering
Implementation Date:
    2017/06
Program:
     
    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)

    plot generated by sample program

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

     
        plot generated by sample program
        

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Date created: 07/03/2017
Last updated: 07/03/2017

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