20.2.1  Covariance and correlation

The covariance of two random variables measures their connectedness; i.e., whether they tend to change with each other. If X and Y are two random variables, then the covariance is the expected value of (X−X)(Y−Ȳ), where X and Ȳ are the means of X and Y, respectively. The covariance command calculates covariances.

• covariance takes two mandatory and one optional argument:
  • X and Y, two lists.
  • Optionally, W, a list of weights or a matrix (w_jk) where w_jk is the weight of the pair (x_j,y_k).
  If the arguments are all lists, then can be entered as the columns of a single matrix.

  If the arguments consist of two lists and a matrix, to make it simpler to enter data in a spreadsheet the lists X and Y and the matrix W can be combined into a single matrix, by augmenting W with the list Y on the top and the transpose of the list X on the left, with a filler in the upper left hand corner:

  ⎡ ⎢ ⎣       “XY” Y       XT W ⎤ ⎥ ⎦

  For this, you have to give covariance a second argument of -1.

• covariance(X,Y ⟨,W⟩) returns the covariance of X and Y.

Examples

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The linear correlation coefficient of two random variables is another way to measure their connectedness. Given random variables X and Y, their correlation is defined as cov(X,Y)/(σ(X)σ(Y)), cov(X,Y) is the covariance of X and Y, and σ(X) and σ(Y) are the standard deviations of X and Y, respectively.

The correlation command finds the correlation of two lists and take the same types of arguments as the covariance command.

• correlation takes two mandatory and one optional argument:
  • X and Y, two lists.
  • Optionally, W, a list of weights or a matrix (w_jk) where w_jk is the weight of the pair (x_j,y_k).
  If the arguments are all lists, then can be entered as the columns of a single matrix.

  If the arguments consist of two lists and a matrix, to make it simpler to enter data in a spreadsheet the lists X and Y and the matrix W can be combined into a single matrix, by augmenting W with the list Y on the top and the transpose of the list X on the left, with a filler in the upper left hand corner:

  ⎡ ⎢ ⎣       "XY" Y       XT W ⎤ ⎥ ⎦

  For this, you have to give correlation a second argument of -1.

• correlation(X,Y ⟨,W⟩) returns the correlation of X and Y.

Example

The covariance_correlation command will compute both the covariance and correlation simultaneously, and return a list with both values. This command takes the same type of arguments as the covariance and correlation commands.

• covariance_correlation takes two mandatory and one optional argument:
  • X and Y, two lists.
  • Optionally, W, a list of weights or a matrix (w_jk) where w_jk is the weight of the pair (x_j,y_k).
  If the arguments are all lists, then can be entered as the columns of a single matrix.

  If the arguments consist of two lists and a matrix, to make it simpler to enter data in a spreadsheet the lists X and Y and the matrix W can be combined into a single matrix, by augmenting W with the list Y on the top and the transpose of the list X on the left, with a filler in the upper left hand corner:

  ⎡ ⎢ ⎣       "XY" Y       XT W ⎤ ⎥ ⎦

  For this, you have to give covariance_correlation a second argument of -1.

• covariance_correlation(X,Y ⟨,W⟩) returns a list consisting of the covariance and the correlation of X and Y.

Example