Meng-Ying

No, I do not think we are communicating.

The variance of data values is not affected by

correlation between the sample values.

The estimated variance for the population IS affected

by correlation between the sample values. Statistical

inference about the population is based on the

assumption that samples were taken randomly and

independently from that population.

It is the process of estimation of unknown parameters

by classical statistical theory which requires these

assumptions.

Geostatistical inference does not require absence of

correlation, quite the contrary. The semi-variogram

graph is constructed on the assumption that there is a

correlation between samples and that this depends on

distance and direction between the pair of samples.

If we have a stationary situation, where the mean and

variance are constant over the study area, the

semi-variogram generally reaches a sill value. The

distance at which this happens is interpreted as that

distance beyond which the correlation is zero. Sample

pairs at this distance or greater can be used to

estimate the variance, since the statistical

assumptions are now satisifed.

Isobel

http://geoecosse.bizland.com/whatsnew.htm
--- Meng-Ying Li <

mengyl@...> wrote:

> Hi Isobel,

>

> I understand all points you pointed out, but I'm not

> sure why the variance

> should be defined as data NOT SPATIALLY CORRELATED

> when they may or may

> not be correlated.

>

> Thanks for the clarification, though, I don't think

> I'd be able to

> clarify the things you clarifies. You're good.

>

>

> Meng-ying

>

> On Wed, 8 Dec 2004, Isobel Clark wrote:

>

> > Meng-Ying

> >

> > I don't know how to say this any other way. At

> > distances larger than the range of influence,

> samples

> > are NOT SPATIALLY CORRELATED.

> >

> > The variance of the difference between two

> > uncorrelated samples is twice the variance of one

> > sample around the mean.

> >

> > The semi-variogram is one-half of the variance of

> the

> > difference.

> >

> > Hence the sill is (theoretically) equal to the

> > variance. The sill is based on all pairs of

> samples

> > found at a distance greater thn the range of

> > influence.

> >

> > The classical statistical estimator of the

> variance is

> > only unbiassed if the correct degrees of freedom

> are

> > used. If the samples are correlated, n-1 is NOT

> the

> > correct degrees of freedom.

> >

> > All explained in immense detail in Practical

> > Geostatistics 2000, Clark and Harper,

> > http://geoecosse.hypermart.net

> >

> > Did I get it clear this time?

> > Isobel

> >

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

> > > I understand why it is not appropriate to force

> the

> > > sill so it matches the

> > > 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

>

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