9.2.1  Covariance and correlation: covariance correlation covariance_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 the 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.

Examples.

• Input:
  covariance([1,2,3,4],[1,4,9,16])
  or:
  covariance([[1,1],[2,4],[3,9],[4,16]])
  Output:

  25 4

• Input:
  covariance([1,2,3,4],[1,4,9,16],[3,1,5,2])
  or:
  covariance([1,2,3,4],[1,4,9,16],[[3,0,0,0],[0,1,0,0],[0,0,5,0],[0,0,0,2]])
  or:
  covariance([["XY",1,4,9,16],[1,3,0,5,0],[2,0,1,0,0],[3,0,0,5,0],[4,0,0,0,2]],-1)
  Output:

  662 121

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 the 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.

Example.
Input:

Output:

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 the 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.
Input:

Output: