Sample mean and sample covariance

Sample mean and sample covariance

Sample mean and sample covariance are statistics computed from a collection of data, thought of as being random.

ample mean and covariance

Given a random sample extstyle mathbf{x}_{1},ldots,mathbf{x}_{N} from an extstyle n-dimensional random variable extstyle mathbf{X} (i.e., realizations of extstyle N independent random variables with the same distribution as extstyle mathbf{X}), the sample mean is

: mathbf{ar{x=frac{1}{N}sum_{k=1}^{N}mathbf{x}_{k}.

In coordinates, writing the vectors as columns,

: mathbf{x}_{k}=left [ egin{array} [c] {c}x_{1k}\ vdots\ x_{nk}end{array} ight] ,quadmathbf{ar{x=left [ egin{array} [c] {c}ar{x}_{1}\ vdots\ ar{x}_{n}end{array} ight] ,

the entries of the sample mean are

: ar{x}_{i}=frac{1}{N}sum_{k=1}^{N}x_{ik},quad i=1,ldots,n.

The sample covariance of extstyle mathbf{x}_{1},ldots,mathbf{x}_{N} is the extstyle n by extstyle n matrix extstyle mathbf{Q}=left [ q_{ij} ight] with the entries given by

: q_{ij}=frac{1}{N-1}sum_{k=1}^{N}left( x_{ik}-ar{x}_{i} ight) left( x_{jk}-ar{x}_{j} ight)

The sample mean and the sample covariance matrix are unbiased estimates of the mean and the covariance matrix of the random variable extstyle mathbf{X}. The reason why the sample covariance matrix has extstyle N-1 in the denominator rather than extstyle N is essentially that the population mean E(X) is not known and is replaced by the sample mean extstylear{x}. If the population mean E(X) is known, the analogous unbiased estimate

: q_{ij}=frac{1}{N}sum_{k=1}^{N}left( x_{ik}-E(X_i) ight) left( x_{jk}-E(X_j) ight)

with the population mean indeed does have extstyle N. This is an example why in probability and statistics it is essential to distinguish between upper case letters (random variables) and lower case letters (realizations of the random variables).

The maximum likelihood estimate of the covariance

: q_{ij}=frac{1}{N}sum_{k=1}^{N}left( x_{ik}-ar{x}_{i} ight) left( x_{jk}-ar{x}_{j} ight)

for the Gaussian distribution case has extstyle N as well. The difference of course diminishes for large extstyle N.

Weighted samples

In a weighted sample, each vector extstyle extbf{x}_{k} is assigned a weight extstyle w_{k}geq0. Without loss of generality, assume that the weights are normalized:

: sum_{k=1}^{N}w_{k}=1.

(If they are not, divide the weights by their sum.)Then the weighted mean extstyle mathbf{ar{x and the weighted covariance matrix extstyle mathbf{Q}=left [ q_{ij} ight] are given by

: mathbf{ar{x=sum_{k=1}^{N}w_{k}mathbf{x}_{k}

and Mark Galassi, Jim Davies, James Theiler, Brian Gough, Gerard Jungman, Michael Booth, and Fabrice Rossi. [http://www.gnu.org/software/gsl/manual GNU Scientific Library - Reference manual, Version 1.9] , 2007. [http://www.gnu.org/software/gsl/manual/html_node/Weighted-Samples.html Sec. 20.6 Weighted Samples] ]

: q_{ij}=frac{sum_{k=1}^{N}w_{k}left( x_{ik}-ar{x}_{i} ight) left( x_{jk}-ar{x}_{j} ight) }{1-sum_{k=1}^{N}w_{k}^{2.

If all weights are the same, extstyle w_{k}=1/N, the weighted mean and covariance reduce to the sample mean and covariance above.

References

ee also

*Unbiased estimation of standard deviation
*Estimation of covariance matrices
*Scatter matrix
*Arithmetic mean
*Estimation theory
*Linear regression
*Weighted least squares
*Weighted mean
*Standard error (statistics)


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