Probability and Statistics in Environmental Modeling

Concepts of Probability An experiment is a well-defined process with observable outcomes. The set or collection of all outcomes of an experiment is called the sample space, S An event E is any collection or subset of outcomes from the sample sample. Classical definition of probability P(E) => 0  P(E)  1 Consider the experiment of drawing a single card from a well-shuffled, standard deck of cards. Look at the following events: A: The card drawn is an ace. The number of outcomes in S is 52. P(A)=4/52 Complement of E = the event consisting of all outcomes in S which are not in E; P(E ')=1-P(E)

Probability Rules Mutually exclusive events events that can not occur at the same time –P(E or F)=P(E)+P(F) Not mutually exclusive events –P(E or F)=P(E)+P(F)-P(E and F) Conditional probability –5 balls w 1,w 2,w 3, r 1, r 2 P(wr)=P(r|w)P(w)=2/4*3/5=6/20 E F S

Probability Rules Bayes Theorem –S is partitioned in a finite number of arbitrary events S=A 1 +A 2 +…+A n Then for any event B –P(B)=P(B|A 1 )P(A 1 )+ P(B|A 2 )P(A 2 )+ … + P(B|A n )P(A n ) Example –An environmental system is in state A i with probability P(A i ) –The system probabilities to undergo transition from state A i to state B are P(B|A i ) –The probability that after a transition the system is in state B is P(B)=P(B|A 1 )P(A 1 )+ P(B|A 2 )P(A 2 )+ … + P(B|A n )P(A n )

Probability Density Function When the number of possible outcomes is infinite P(E)=n(E)/n(S) is not meaningful If we can associate E with a numerical value (called random variable), X, in between -  and  then we can define a probability density function Properties of pdfs F(x) is called cumulative distribution function

Normal Distribution Used for modeling several populations –Heights, weights –Measurement errors –Measurement of Intelligence f(x:, ,  ) = –f (x:, ,  ) is the probability distribution function –  is the mean –  is the standard deviation We can also say X~N( ,  2 )

Random Variables Expected Value, E(X) –Found by integrating the product of a dummy variable and the density function over the entire span of possibilities –E(X) = –Represents the long-run average result from a series of repeated random experiments.

Random Variables Variance –The deviation of the individual values from the expected value –Uses the probability distribution function –Var(X)=

Random Variables Examples of random variables –uniformly distributed –normally distributed –log-normally distributed (their logarithms are normally distributed rvs) –exponentially distributed

Estimation of Probability Density Function Step 1. Determined using the observed data the pdf. Step 2. Identify and/or parameterize the pdf Example. Characterization of the rainfall intensity pdf

Use of Probability Theory and Statistical Modeling in Real Life Applications Mitigation of the effect of observational and modeling uncertainty on the design and operation of systems Optimal estimation of state variables of observed systems Assimilation of observed data in models

Mitigation of the Effect of Uncertainty on Operation and Design Assume a prediction (assessment) model –y=f(x,a)+  1. Generate random variables X  f X and  f  2. Predict Y=f(X,a)+  3. Calculate E(Y), Var(Y) 4. Calculate E(C(Y,a)) where C(Y,a) is a cost associated with Y and a. 5 Choose a that minimizes E(C(Y,a))

xspan=0:100; %parameter definition Lr=zeros(100,101); Dr=zeros(100,101); global ka kd U U=16.4; y0=[10 0]'; %initial concentrations are given in mg/L for i=1:100, ka= *randn; kd= *randn; while ka < 0 | kr < 0, ka= *randn; kd= *randn; end [x,y] = ODE45('dydx_sp',xspan,y0) ; Lr(i,:)=y(:,1)'; Dr(i,:)=y(:,2)'; end subplot 211 plot(x, mean(Lr,1),'linewidth',1.25) hold on plot(x, mean(Lr,1)+std(Lr,0,1),'--', … 'linewidth',1.25) plot(x, mean(Lr,1)-std(Lr,0,1),'--', … 'linewidth',1.25) ylabel('mg L^{-1}') title('BOD vs. distance') subplot 212 plot(x, mean(Dr,1),'r','linewidth',1.25); hold on plot(x, mean(Dr,1)+std(Dr,0,1),'r--', … 'linewidth',1.25); plot(x, mean(Dr,1)-std(Dr,0,1),'r--', … 'linewidth',1.25); xlabel('Distance (mi)') title('DO Deficit vs. distance') print -djpeg bod_mc.jpeg

Optimal Estimation of State Variables or Parameter If –X is state variable or parameter set to be estimated –Z is the set of observations Z=z m (X) then the optimal estimate maximizes –P(X|Z) From Bayes Theorem –P(X|Z)=P(X&Z)/P(Z)=P(Z|X)P(X)/P(Z)(1) –If Z|X and X are considered normally distributed then (1)=> –F=(Z-Z m (X)) T W Z (Z-Z m (X))+(X a -X) T W X (X a -X)(2)

Assimilation of Data in Models Same as for estimation but there is a time component a a dynamic model –X i is state variable or parameter set to be estimated at time i –Z is the set of observations Z i =z m (X i ) –X i+1 =f(X i ) From Bayes Theorem –F=(Z i -Z m (f(X i-1 ))) T W Z (Z-Z m (f(X i-1 )))+(X a -X i-1 ) T W X (X a -X i-1 )

Assimilation of Data in Models Example –Data assimilation in a tropospheric ozone model –Physical model –Observations are provided by air quality monitoring stations and meteorological stations –Special numerical technique are used to minimize F