THE PRINCIPLES OF MONTE CARLO SIMULATION
Lecture Three:
Statistical Models and Stationarity
•Statistical Inference
–Parameter Estimation
–Confidence Limits on Parameters
–Hypothesis Testing
•Stationarity
•Representative Statistics
•Implication of Scale and Space/Time Coordinates
•Regression
Statistical Inference in MCS
Need of Statistical Inference
• Decision making requires a model of the populations
involved.
• Statistics allow us to infer those models.
• The formalism is:
– Interpret the values z(x) at location x (spatial, temporal or spatiotemporal) as an outcome of a random variable Z(x), which has
some probability distribution F.
– Also interpret the set of random variables {Z(x), for all x} as the
underlying random function.
• Need to assume stationarity of the data to be able to make
inference from the sample set.
• Sample set must be representative of the population.
The Decision of Stationarity
• Impossible to infer the random function Z(x) with only one
realization z(x) (the data).
What does F(z) mean if we only have one data per location
(space or time)?
• Assuming that the same random function applies to all
coordinates (locations or time) x, we can use the data z(x) to
infer the underlying random function Z(x).
Stationarity
• Stationarity works as an export license to use a set of data to
infer the population parameters: mean, covariance, …
The Decision of Stationarity
• Pooling data in a histogram assumes they come from the
same population (we assume stationarity, even if we don’t
know what that means!!)
• Evident example:
Would you estimate population
parameters using this histogram?
Clearly, there are two populations:
We must infer the population parameters
separately!
The Decision of Stationarity
• Stationarity can be under translations (homogeneity)
and/or rotations (isotropy).
• Stationarity is a property of the RF model. It is not a
characteristic of the phenomenon under study. Stationarity
is a decision made by the user to make reliable inference.
• Exploratory data analysis may indicate the existence of
several populations with significantly different statistics.
• Consider the possibility of subdividing the area into more
homogeneous subzones, conditioned by:
– The availability of enough data to infer the parameters of each
separate RF
– The ability to delineate the different populations both on the data
and at unsampled locations (may need qualitative or secondary
data).
Definition of Stationarity
• Stationarity of order 2 : A RF is said to be stationary of
order 2 when:
~ E{Z ( x)} = m
∀x
~
∀x
C (h) = E{Z ( x + h) ⋅ Z ( x )} − m 2
the stationarity of the covariance implies the stationarity of
the variance and the variogram
• Intrinsic stationarity: A RF is said to be intrinsic stationary
when:
∀x
~ E{Z ( x)} = m
~ Var{Z ( x + h) − Z ( x)} = E{[ Z ( x + h) − Z ( x)]2 } = 2γ (h) ∀x
That is, increments are stationary, but covariance is not.
Ergodic Fluctuations
• Given that the model statistics are inferred from sample
statistics that are uncertain because of limited sample size,
exact specification of the model statistics by limited data is
not possible.
• The stationary RF is said to be “ergodic” in the parameter µ, if
the corresponding realization statistics tends toward µ as the
size of field increases.
• Ergodic fluctuations allow one to account indirectly for the
uncertainty about sample statistics.
• Removal of ergodic fluctuations may lead to a false sense of
certainty.
Ergodic Fluctuations
φ1 = 12%
φ 2 = 9%
φ3 = 18%
Ergodic Average = 15%
Scale Effect