Tom, list,
hmmm .... I worry that you do not fully understand probability theory
(not meaning to be rude ...) - or maybe I have not understood your post!
The situation that p(x) = 0 is true for all continuous random
variables, but this is not a problem since we typically (for continuous
random variables) consider, as you suggest, the probability of a small
interval or range, which is just the integral of the probability
distribution function over this range (= area under the curve). So far
so good - this is basic probability theory.
Your dwarf analogy is very bad however!
Edzer said it correctly when he pointed out that kriging, but in general
all statistics, is fundamentally based on some model, and assumptions.
This is true, in my perception, of all rigorous approaches to problem
solving in general (I am not meaning to start a debate here) - the key
thing is that you state and test your assumptions, and clearly define
your model. The debate is then, is your model appropriate and are your
assumptions valid - in real applications this MUST always be a debate!
If your variable really follows a Gaussian distribution (or multivariate
Gaussian in the standard geostatistical case) then this will be a good
model and confidence intervals, expressed as +/- standard deviation are
good (but note these are confidence intervals about the mean - your
prediction!). If you want to look at the probability of another value
(not the mean) then you have to do something else - this is not what
kriging variance is! But it will give you the numbers to compute the
probability of being in any range, or exceeding a value or whatever else
(from the Gaussian distribution function) BUT ONLY IF your model is
correct and your assumptions are also good.
Unless I am missing something very subtle in your arguments then there
is no issue here!
By the way, thanks to all for a good discussion of the merit and
demerits of simulation as opposed to prediction!
cheers
Dan
tom andrews wrote:
Dear Mario
I agree with You.
Suppose that ordinary kriging predicts that I am a dwarf (or giant)
with the height H.
Since for e.g. gaussian distribution holds
P(H)=0
we have to introduce
P(H - s < h < H + s) = p
where s is a square root of kriging variance and p is an area under
the gaussian curve for the interval (H-s,H+s) in its tail.
Knowing a total area under the curve we can compute probability.
No matter, am I a dwarf or not, the probability of being a dwarf
(with some height tolerance) is not high so kriging variance will
be small (it means that predicted value "comes from" the tail of
distribution not that estimation is "good").
Prediction intervals have no meaning.
Suppose that ordinary kriging predicts that I have a mean height.
Now, kriging variance (minimized error variance) is the estimator of
variance of random variable (sigma^2) and reaches maximum.
Probability that I have a mean height +/- sigma is high and known
but it is only property of distribution of random variable.
Any prediction intervals have no meaning too.
Best Regards
tom
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