sample variance. My question is, why estimate the overall variance by the

sill value when data are actually correlated?

Meng-ying

On Tue, 7 Dec 2004, Isobel Clark wrote:

> Meng-Ying

>

> We are talking about estimating the variance of a set

> of samples where spatial dependence exists.

>

> The classical statistical unbiassed estimator of the

> population variance is s-squared which is the sum of

> the squared deviations from the mean divided by the

> relevant degrees of freedom. If the samples are not

> inter-correlated, the relevant degrees of freedom are

> (n-1). This gives the formula you find in any

> introductory statistics book or course.

>

> If samples are not independent of one another, the

> degrees of freedom issue becomes a problem and the

> classical estimator will be biassed (generally too

> small on average).

>

> In theory, pairs of samples beyond the range of

> influence on a semi-variogram graph are independent of

> one another. In theory, the variance of the difference

> betwen two values which are uncorrelated is twice the

> variance of one sample around the population mean.

> This is thought to be why Matheron defined the

> semi-variogram (one-half the squared difference) so

> that the final sill would be (theoretically) equal to

> the population variance.

>

> There are computer software packages which will draw a

> line on your experimental semi-variogram at the height

> equivalent to the classically calculated sample

> variance. Some people try to force their

> semi-variogram models to go through this line. This is

> dumb as the experimental sill is a better estimate

> because it does have the degrees of freedom it is

> supposed to have.

>

> I am not sure whether this is clear enough. If you

> email me off the list, I can recommend publications

> which might help you out.

>

> Isobel

> http://geoecosse.bizland.com/books.htm

>

> --- Meng-Ying Li <mengyl@...> wrote:

> > Hi Isobel,

> >

> > Could you explain why it would be a better estimate

> > of the variance when

> > independance is considered? I'd rather think that we

> > consider the

> > dependance when the overall variance are to be

> > estimated-- if there

> > actually is dependance between values.

> >

> > Or are you talking about modeling sill value by the

> > stablizing tail on

> > the experimental variogram, instead of modeling by

> > the calculated overall

> > variance?

> >

> > Or, are we talking about variance of different

> > definitions? I'd be

> > concerned if I missed some point of the original

> > definition for variances,

> > like, the variance should be defined with no

> > dependance beween values or

> > something like that. Frankly, I don't think I took

> > the definition of

> > variance too serious when I was learning stats.

> >

> >

> > Meng-ying

> >

> > > Digby

> > >

> > > I see where you are coming from on this, but in

> > fact

> > > the sill is composed of those pairs of samples

> > which

> > > are independent of one another - or, at least,

> > have

> > > reached some background correlation. This is why

> > the

> > > sill makes a better estimate of the variance than

> > the

> > > conventional statistical measures, since it is

> > based

> > > on independent sampling.

> > >

> > > Isobel

> >

>