View source for Correlation

[[Category:Statistical Functions]]
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==Correlation(X, Y'', I, w{{Release|6.2||, resultIfNoVariation}}'')==
Computes an estimate of the correlation or weighted correlation between two uncertain quantities «X» and «Y».  The correlation is a measure of the degree of linear dependence and ranges between -1 and 1.  A correlation of 0 indicates that the two quantities appear to be independent (although if there is a non-linear relationship, this is not necessarily the case), a positive value indicates that they tend to increase together, while a negative correlation indicates that an increase in one quantity tends to be accompanied by a decrease in the other.

When «X» and «Y» are both uncertain quantities, the correlation is computed by [[Correlation]](X, Y).

If there is no variation in «X» or «Y», the result is [[NaN]]{{Release|6.2||, unless you specify the optional parameter «resultIfNoVariation» which is returned in this case}}.

== Correlation of data ==
If you have a data set containing two variables, <code>A</code> and <code>B</code>, where data points are indexed by <code>J</code>, the correlation of <code>A</code> and <code>B</code> is computed using 
:<code>Correlation(A, B, J)</code>

Here <code>J</code> is referred to as the ''running index''.

If you have an array in which you want to find the correlation of two columns, then you will apply the subscript operator to extract each column.  For example, the following computes the correlation between historical revenue in 2002 and 2003 (where data points are indexed by <code>J</code>).
:<code>Correlation(HistoricalRevenue[Year = 2002], HistoricalRevenue[Year = 2003], J)</code>

== Weighted Correlation ==
Unweighted correlation treats all data or sample points as equally weighted.  Weighted correlation computes the correlation when each data point may have a different weight.  The optional «w» parameter may be used to specify a weight, which should be indexed by the running index (or by [[Run]] if no running index is specified).  For example, the following specifies an importance weight:
:<code>Correlation(X, Y, w: sampleImportance)</code>

The global sample weighting, specified by the system variable [[SampleWeighting]], is used by default.

== Computing a sample correlation matrix ==
Suppose each sample point is a vector along index <code>I</code>.  A correlation matrix is a 2-D square symmetric matrix where each element <code>(m, n)</code> indicates the correlation of column <code>I = m</code> and column <code>I = n</code>.  To compute a sample correlation matrix for a given set of data, create a second index, <code>I2</code> as a copy of <code>I</code>:

:<code>Index I2 := CopyIndex(I)</code>

With this index, the correlation matrix, index by <code>I</code> and <code>I2</code>, is computed from data <code>X</code> using:
:<code>Correlation(X, X[I = I2])</code>

Or if data points are listed along an index other than [[Run]], say <code>J</code>, this would be:
:<code>Correlation(X, X[I = I2], J)</code>

== Mathematical details ==
Weighted correlation is given by
:<math>
{\sum_i { w_i \hat{x}_i \hat{y}_i } \over {\sqrt{\sum_i w_i \hat{x}_i^2 \sum_j w_j \hat{y}_j^2} } }
</math>

== Null values ==
When either parameter contains a [[Null]] value, that data points is ignored.  The computed [[Correlation]] is then based only on those points with non-[[Null]] values.  Note that for certain statistical significance tests, you may need to adjust the degrees of freedom used to match the number of non-[[Null]] values.

You cannot compute the [[Correlation]] when there is only a single data point.  When only one data point is supplied, or when there is only one data point that is non-[[Null]] in both «x» and «y» parameters, the result of [[Correlation]] is [[NaN]].

When there are zero data points, or when every data point is [[Null]] in one or the other of «x» or «y», then the result of [[Correlation]] is [[Null]].  (''Note'': Prior to Analytica 4.4, the result in this case was [[NaN]].  The change to [[Null]] in 4.4 makes the handling of [[Null]] by [[Correlation]]() consistent with other Analytica functions.)

== See Also ==
* [[SampleCorrelation]]
* [[RankCorrel]] -- the Rank Correlation function
* [[Variance]]
* [[Covariance]]
* [[SampleCovariance]]  -- multi-dimensional case, covariance matrix
* [[Statistical Functions and Importance Weighting]]