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## 25.2 Basic Statistical Functions

Octave also supports various helpful statistical functions.

Function File: mahalanobis (x, y)

Return the Mahalanobis' D-square distance between the multivariate samples x and y, which must have the same number of components (columns), but may have a different number of observations (rows).

Function File: center (x)
Function File: center (x, dim)

If x is a vector, subtract its mean. If x is a matrix, do the above for each column. If the optional argument dim is given, perform the above operation along this dimension

Function File: studentize (x, dim)

If x is a vector, subtract its mean and divide by its standard deviation.

If x is a matrix, do the above along the first non-singleton dimension. If the optional argument dim is given then operate along this dimension.

Function File: c = nchoosek (n, k)

Compute the binomial coefficient or all combinations of n. If n is a scalar then, calculate the binomial coefficient of n and k, defined as

 ``` / \ | n | n (n-1) (n-2) … (n-k+1) n! | | = ------------------------- = --------- | k | k! k! (n-k)! \ / ```

If n is a vector generate all combinations of the elements of n, taken k at a time, one row per combination. The resulting c has size ```[nchoosek (length (n), k), k]```.

`nchoosek` works only for non-negative integer arguments; use `bincoeff` for non-integer scalar arguments and for using vector arguments to compute many coefficients at once.

Function File: n = histc (y, edges)
Function File: n = histc (y, edges, dim)
Function File: [n, idx] = histc (…)

Produce histogram counts.

When y is a vector, the function counts the number of elements of y that fall in the histogram bins defined by edges. This must be a vector of monotonically non-decreasing values that define the edges of the histogram bins. So, `n (k)` contains the number of elements in y for which `edges (k) <= y < edges (k+1)`. The final element of n contains the number of elements of y that was equal to the last element of edges.

When y is a N-dimensional array, the same operation as above is repeated along dimension dim. If this argument is given, the operation is performed along the first non-singleton dimension.

If a second output argument is requested an index matrix is also returned. The idx matrix has same size as y. Each element of idx contains the index of the histogram bin in which the corresponding element of y was counted.

Function File: perms (v)

Generate all permutations of v, one row per permutation. The result has size `factorial (n) * n`, where n is the length of v.

As an example, `perms([1, 2, 3])` returns the matrix

 ``` 1 2 3 2 1 3 1 3 2 2 3 1 3 1 2 3 2 1 ```

Function File: values (x)

Return the different values in a column vector, arranged in ascending order.

As an example, `values([1, 2, 3, 1])` returns the vector `[1, 2, 3]`.

Function File: [t, l_x] = table (x)
Function File: [t, l_x, l_y] = table (x, y)

Create a contingency table t from data vectors. The l vectors are the corresponding levels.

Currently, only 1- and 2-dimensional tables are supported.

Function File: spearman (x, y)

Compute Spearman's rank correlation coefficient rho for each of the variables specified by the input arguments.

For matrices, each row is an observation and each column a variable; vectors are always observations and may be row or column vectors.

`spearman (x)` is equivalent to ```spearman (x, x)```.

For two data vectors x and y, Spearman's rho is the correlation of the ranks of x and y.

If x and y are drawn from independent distributions, rho has zero mean and variance `1 / (n - 1)`, and is asymptotically normally distributed.

Function File: run_count (x, n)

Count the upward runs along the first non-singleton dimension of x of length 1, 2, …, n-1 and greater than or equal to n. If the optional argument dim is given operate along this dimension

Function File: ranks (x, dim)

Return the ranks of x along the first non-singleton dimension adjust for ties. If the optional argument dim is given, operate along this dimension.

Function File: range (x)
Function File: range (x, dim)

If x is a vector, return the range, i.e., the difference between the maximum and the minimum, of the input data.

If x is a matrix, do the above for each column of x.

If the optional argument dim is supplied, work along dimension dim.

Function File: probit (p)

For each component of p, return the probit (the quantile of the standard normal distribution) of p.

Function File: logit (p)

For each component of p, return the logit of p defined as

 ```logit(p) = log (p / (1-p)) ```

Function File: cloglog (x)

Return the complementary log-log function of x, defined as

 ```cloglog(x) = - log (- log (x)) ```

Function File: kendall (x, y)

Compute Kendall's tau for each of the variables specified by the input arguments.

For matrices, each row is an observation and each column a variable; vectors are always observations and may be row or column vectors.

`kendall (x)` is equivalent to ```kendall (x, x)```.

For two data vectors x, y of common length n, Kendall's tau is the correlation of the signs of all rank differences of x and y; i.e., if both x and y have distinct entries, then

 ``` 1 tau = ------- SUM sign (q(i) - q(j)) * sign (r(i) - r(j)) n (n-1) i,j ```

in which the q(i) and r(i) are the ranks of x and y, respectively.

If x and y are drawn from independent distributions, Kendall's tau is asymptotically normal with mean 0 and variance `(2 * (2n+5)) / (9 * n * (n-1))`.

Function File: iqr (x, dim)

If x is a vector, return the interquartile range, i.e., the difference between the upper and lower quartile, of the input data.

If x is a matrix, do the above for first non-singleton dimension of x. If the option dim argument is given, then operate along this dimension.

Function File: cut (x, breaks)

Create categorical data out of numerical or continuous data by cutting into intervals.

If breaks is a scalar, the data is cut into that many equal-width intervals. If breaks is a vector of break points, the category has `length (breaks) - 1` groups.

The returned value is a vector of the same size as x telling which group each point in x belongs to. Groups are labelled from 1 to the number of groups; points outside the range of breaks are labelled by `NaN`.

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