variancef

Arguments

x

vector or matrix of real or complex numbers

fre

vector or matrix of positive decimal integers = frequencies: fre(i,j) is the number of times that x(i,j) must be counted. fre and x have same sizes.

orien

the orientation of the computation. Valid values are:

• 1 or "r" : result is a row, after a column-wise computation.
• 2 or "c" : result is a column, after a row-wise computation.
• "*" : full undirectional computation (default); explicitly required when m is used.

m

The known mean of the underlying statistical distribution law (assuming that it is known).

• "*" mode (default): m must be scalar
• "r" or 1 mode: m is a row of length size(x,2). The variance along the column #j is computed using m(j) as the mean for the considered column. If m(j) is the same for all columns, it can be provided as a scalar m.
• "c" or 2 mode: m is a column of length size(x,1). The variance along the row #i is computed using m(i) as the mean for the considered row. If m(i) is the same for all rows, it can be provided as a scalar m.

When m is not provided, the variance is built dividing the quadratic distance of n values to mean(x,fre)(or mean(x,fre,"c") or mean(x,fre,"r")) by (n-1) (n being sum(fre) or sum(fre,"c") or sum(fre,"r")). If the elements of x are mutually independent, the result is then statistically unbiased.

Else, the variance is built dividing the quadratic distance of values to m by the number n of considered values.

If a true value m independent from x elements is used, x and m values are mutually independent, and the result is then unbiased.

When the special value m = %nan is provided, the variance is still normalized by n (not n-1) but is computed using m = mean(x, fre) instead (or m = mean(x,fre,"c") or m = mean(x,fre,"r")). This m does not bring independent information, and yields a statistically biased result.

s

The variance of weighted values of x elements. It is a scalar or a column vector or a row vector according to orien.

mc

Scalar or orien-wise mean of weighted x elements (= mean(x, fre,..)), as computed before and used as reference in the variance.

Description

This function computes the variance of the values of a vector or matrix x, each of them x(i,j) being counted fre(i,j) times. If x is complex, then variancef(x,fre,..) = variancef(real(x),fre,..) + variancef(imag(x),fre,..) is returned.

s = variancef(x,fre) (or s=variancef(x,fre,"*")) returns the scalar variance computed over all values of x.

s = variancef(x,fre,"r")(or equivalently s = variancef(x,fre,1)) returns a row s such that for each j, s(j) = variancef(x(:,j),fre(:,j),..).

s = variancef(x,fre,"c")(or equivalently s = variancef(x,fre,2)) returns a column s such that for each i, s(i) = variancef(x(i,:),fre(i,:),..).

When the mean m is provided, it is used as reference in the variance computation instead of being internally estimated from x (unless it is equal to the special value %nan: See m's description). This allows to compute the variance of a sample x with respect to a given statistical model (rather than extracting an empirical statistical dispersion in order to build the model).

Examples

x = [0.2113249 0.0002211 0.6653811; 0.7560439 0.9546254 0.6283918]
fre = [1 2 3; 3 4 3]
[s, m] = variancef(x, fre)
[s, m] = variancef(x, fre, "r")
[s, m] = variancef(x, fre, "c")

// Example #2:
x0 = grand(20, 7, "uin", -9, 10)+0.4
x = matrix((-9:10)+0.4, 5, 4)
fre = members(x, x0)        // Computes the frequencies of x's elements in x0
[s, m] = variancef(x, fre)  // Should be equal to variance(x0)
[s, m] = variance(x0)

// Example #2 (follow-up):
m = (-9+10)/2+0.4               // Known asymptotic mean (if x0 had an infinite number of elements)
s = variancef(x, fre, "*", m)   // Sample variance wrt the true mean
s0 = (10 - (-9))^2 /12            // Known asymptotic variance
s2 = variancef(x, fre, "*", %nan) // Takes m = meanf(x,fre) =>  always <= s

See Also

• variance — variance (and mean) of a vector or matrix (or hypermatrix) of real or complex numbers
• mtlb_var — Matlab var emulation function
• stdevf — standard deviation

Bibliography

Wonacott, T.H. & Wonacott, R.J.; Introductory Statistics, fifth edition, J.Wiley & Sons, 1990.

History