variance (and mean) of a vector or matrix of frequency-weighted real or complex numbers
[s [,mc]] = variancef(x, fre [,orien [,m]]) [s, mc] = variancef(x) [s, mc] = variancef(x, fre, "r"|1 ) [s, mc] = variancef(x, fre, "c"|2 ) [s, mc] = variancef(x, fre, "*" , %nan) [s, mc] = variancef(x, fre, "r"|1, %nan) [s, mc] = variancef(x, fre, "c"|2, %nan) s = variancef(x, fre, "*", m) s = variancef(x, fre, "r", m) s = variancef(x, fre, "c", m)
vector or matrix of real or complex numbers
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.
the orientation of the computation. Valid values are:
m
is used.The known mean of the underlying statistical distribution law (assuming that it is known).
m
must be scalarm
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
.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.
x
elements. It is a scalar or a column vector or a row vector according to orien
.orien
-wise mean of weighted x
elements (= mean(x, fre,..)
), as computed before and used as reference in the variance.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).
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 | ![]() | ![]() |
Wonacott, T.H. & Wonacott, R.J.; Introductory Statistics, fifth edition, J.Wiley & Sons, 1990.
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