Download 7_Stochastic

Why Do Stochastic
Simulations?
• Empirical evaluation of
statistical tests or methods
– Want to know how well t-test
performs when have unequal
variances between groups
• “Better” represent system
dynamics
– Real world appears stochastic
• Develop management strategies
that are robust to variability
• Fun
Sources of Stochasticity
or Uncertainty
• Measurement error
– Variance
– “True” error or bias
• Process variability
– Dynamic rates are variable
• Model Uncertainty
• Walk through spreadsheet
example
Where Does Process
Error Enter the Model?
dPrey
(K – Prey)
-------- = Prey * r * ------------dt
K
dPrey
(K – Prey)
-------- = Prey * r * ------------t
dt
K
r ----> Implies that r varies over time
t
Where Does Measurement
Error Enter the Model?
dPrey
(K – Prey)
-------- = Prey * r * ------------dt
K
Observed Prey = Actual Prey + error
Key is that the observed prey abundance does
not enter into differential equation – at least in
concept. In practice, it may have to if that is
the only measure we have.
Where Does Modeling
Error Occur?
dPrey
(1000 – Prey)
-------- = Prey * r * ----------------dt
1000
dPrey
(1500 – Prey)
-------- = Prey * r * ----------------dt
1500
dPrey
(K – Prey)
-------- = Prey * r * ------------- - a*Predator
dt
K
Basic Concepts
• Discrete random variables
– Probability mass function
f(x)
x
Basic Concepts
• Continuous random variables
– Probability density function
f(x)
x
Basic Concepts
• Cumulative Distribution Function
f(x)
x
1.00
F(x)
x
Basic Concepts
• Expected values
If c is a constant and X and Y are random variables
E(cX) = c E(X)
E(c+X) = c + E(X)
E(X + Y) = E(X) + E(Y)
E(XY) = E(X) E(Y) if X and Y are independent
E(XY) ≈
Note that in general, E(f(X)) ≠ f(E(X))
eg., E(ln(X)) ≠ ln(E(X))
Basic Concepts
• Variance
If c is a constant and X and Y are random variables
Var(cX) = c2 Var(X)
Var(c+X) = Var(X)
Var(X + Y) = Var(X) + Var(Y) if X and Y are independent
Var(X - Y) = Var(X) + Var(Y) if X and Y are independent
Var(X + Y) = Var(X) + Var(Y) + 2Cov(X,Y)
Var(X - Y) = Var(X) + Var(Y) - 2Cov(X,Y)
if X and Y are independent
Var(XY) ≈ (E(X))2Var(Y) + (E(Y))2Var(X) – Var(X)Var(Y)
Basic Concepts
Using these principles – can shift mean and variance
To get “target” values
Example – Generate N(0,1), but want N(5,9)
1st multiply each value by 3 – this gives you a variance of 9
2nd add 5 to each value – this shifts mean from 0 to 5
Generally – want to multiply first to get target variance and
Secondly add value to shift mean. If try to do in reverse
order, you may shift the mean when you multiply if the
Mean is not 0