Download Document

Stochastic Optimization
Review of Probability Theory
Water Resources Planning and Management: M6L1
D Nagesh Kumar, IISc
Objectives
 To introduce the concept of probability
 To define random variables and its statistical properties
 To introduce commonly used probability distributions
2
Water Resources Planning and Management: M6L1
D Nagesh Kumar, IISc
Introduction
 Most water resources decision problems face the risk of uncertainty
 Uncertainty - Randomness of the variables
 Hydrologic random variables: rainfall in a command area, inflow to a
reservoir, evapo-transpiration of crops etc.
 Optimization models developed for water resources management - optimal
decisions with an indication of the associated hydrologic uncertainty
 Two classical approaches to deal with the hydrologic uncertainty in
optimization models are:
• Implicit Stochastic Optimization (ISO)
• Explicit Stochastic Optimization (ESO)
3
Water Resources Planning and Management: M6L1
D Nagesh Kumar, IISc
Introduction…
Implicit Stochastic Optimization (ISO)
 Hydrologic uncertainty is implicitly incorporated
 Optimization model is a deterministic model
 Hydrologic inputs are varied with a number of equi-probable sequences
 Deterministic optimization model is run once with each of the input sequences
 Output set is then statistically analyzed to generate a set of optimal decisions.
4
Water Resources Planning and Management: M6L1
D Nagesh Kumar, IISc
Introduction…
Explicit Stochastic Optimization (ESO)
 Stochastic nature of the inputs is explicitly included through their probability
distributions
 Optimization model is a stochastic model
 A single run of the model specifies the optimal decisions
 Two commonly used ESO techniques are:
 Chance Constrained Linear Programming (CCLP) and
 Stochastic Dynamic Programming (SDP) (will be discussed in the following lectures)
 A background of probability theory is essential for ESO
5
Water Resources Planning and Management: M6L1
D Nagesh Kumar, IISc
Concept of Probability
 Sample space S: Area containing all possible outcomes of an experiment
 An event is one subset of these outcomes
 Probability is a measure of the likelihood of occurrence of an event
 Probability can be assessed in two ways:
1.
Objective or posterior probability which is based on the observation of events and
2.
Subjective or prior probability which is based on experience or judgement.
 Three basic axioms of probability are:
(i)
Totality:
P(S) = 1 where S is the sample space
(ii)
Nonnegativity: P(A) ≥ 0 where A is an event
(iii) Mutually exclusive: If A and B are two mutually exclusive events, then
P  A  B = P(A) + P(B)
 For mutual exclusive events P  A  B = 0.
 The third axiom after relaxing mutual exclusiveness will be P = P(A) + P(B) - P  A  B
6
Water Resources Planning and Management: M6L1
D Nagesh Kumar, IISc
Random Variable
 Random variable (r.v): Variable whose value is not known or cannot be measured
with certainty (or is nondeterministic)
 Examples of random variables in water resources: Rainfall, streamflow, time
between hydrologic events (e.g. floods of a given magnitude), evaporation from a
reservoir, groundwater levels, re-aeration and de-oxygenation rates etc.
 Any function of a random variable is also a random variable
 Random variable is denoted using an upper case letter and the corresponding lower
case letter is used to denote the value that it takes
 For example, daily rainfall may be denoted as X. The value it takes on a particular
day is denoted as x.
 We then associate probabilities with events such as X ≥ x, 0  X  x
7
Water Resources Planning and Management: M6L1
D Nagesh Kumar, IISc
Random Variable…
 Random variable can be essentially classified into two categories:
 Discrete
 Continuous.
Discrete r.v.:
 X can take on only discrete values x1, x2, x3, ...,.
 Eg.: Number of rainy days in a year which may take values such as, 10, 20, etc.
 Can assume a finite number of values
Continuous r.v.:
 X can take on all real values in a range
 Most variables in hydrology are continuous random variables
 Number of values that a continuous random variable can assume is infinite.
8
Water Resources Planning and Management: M6L1
D Nagesh Kumar, IISc
Probability Distributions
 For
discrete
random
variables,
the
probability
distribution
is
called
a
probability mass function
 For
continuous random variables, the probability distribution is called a
probability density function (pdf)
 The cumulative distribution function (CDF), F(x), represents the probability that X is
less than or equal to x,
i.e.
F(x) = P(X  x).
 The probability mass function (PMF) of X is defined as
p(x) = P(X = x).
9
Water Resources Planning and Management: M6L1
D Nagesh Kumar, IISc
Probability Distributions…
(a) PMF and (b) CDF of a discrete random variable
F(x)
p(x)
1
. . .
F(x2)
. . .
x 1 x2 x3
. . . xN-1 x
N
x
(a)
F(x2) = p(x1) + p(x2)
x1 x2 x 3
. . . xN-1 x
N
x
(b)
 For a discrete random variable, there are spikes of probability associated with the
values that the random variable assumes.
 CDF appears as a staircase
10
Water Resources Planning and Management: M6L1
D Nagesh Kumar, IISc
Probability Distributions…
 For a continuous random variable, the probability density function (PDF) is:
f x  
dF x 
dx
where F(x) is the CDF of X.
(a) PDF and (b) CDF of a continuous random variable
F(x)
 
x2
F x   f x  dx
2

f(x)
1
F(x2)
0
x2
(a)
11
Water Resources Planning and Management: M6L1
x
x2
x
(b)
D Nagesh Kumar, IISc
Probability Distributions…
 Probability distributions of continuous random variables are smooth curves
 CDF of a continuous random variable denoted by F(x) is a non-decreasing function
with a maximum value of 1
 CDF represents the probability that X is less than or equal to x,
i.e. F(x) = P(X  x).
 Any function f(x) defined on the real line can be a valid probability density function
if and only if
i.
f(x)  0 for all x, and
ii.
f(x)  1 for all x.
 Given the PMF or PDF, the CDF can be obtained as
F x  
 px 
i i  n
F x  
for discrete random variables
x
 f x  dx

12
i
Water Resources Planning and Management: M6L1
for continuous random variables
D Nagesh Kumar, IISc
Probability Distributions…
 Area under the curve to the left of x = a is
f(x)
Area  Pa  X  b
P(X ≤ a)
 Area under the curve to the left of x = b is
P(X ≤ b)
 Area between x = a and x = b is
P[a ≤ X ≤ b].
0
a
b
x
Probability density function
 For a continuous random variable, probability of the random variable taking a value
exactly equal to a given value is zero because
d
P X  d   Pd  X  d    f x  dx  0
d
13
Water Resources Planning and Management: M6L1
D Nagesh Kumar, IISc
Probability Distributions…
F(x)
0.8
F-1(0.5) = 10
F-1(0.8) = 15
0.5
10
15
x
Cumulative density function
 For any given probability α, 0 ≤ α ≤ 1, the value x of the random variable can be
determined from the CDF as
x = F-1(α).
14
Water Resources Planning and Management: M6L1
D Nagesh Kumar, IISc
Statistical properties of random variables
 Population: Set of all the values taken by a random process
 Sample: Subset of the population
 Expected value of (X – x0)r is the rth moment of a random variable X about any
reference point X = x0.
 Mathematically,

  x  x  f xdx
E  X  x0  
r

r
0
for continuous case



N
E  X  x0     xi  x0  pxi 
r
r
i 1
for discrete case
where E[ ] is a statistical expectation operator.
 The first three moments describe the central tendency, variability and asymmetry
of the distribution of a random variable.
15
Water Resources Planning and Management: M6L1
D Nagesh Kumar, IISc
Statistical properties of random variables…
Expected Value or Mean
The central tendency is expressed as an expectation as
EX  

 x f x dx
for continuous case

N
E  X    xi p  xi 
for discrete case
i 1
 The mean of a r.v is denoted by μ is equal to the expected value,
i.e., μ = E[X].
16
Water Resources Planning and Management: M6L1
D Nagesh Kumar, IISc
Statistical properties of random variables…
Variance
 Second order central moment.
 Variance of a continuous r.v. is defined as

Var X    2  E  X   
2



 x    f x dx
2

 The positive square root of variance is called the standard deviation, σ.
 Coefficient of variation
C v


Skewness
 The asymmetry of PDF of a r.v. is measured by skew coefficient
 Defined as
17


  E  X   3  3
Water Resources Planning and Management: M6L1
D Nagesh Kumar, IISc
Example
Probability density function (pdf) of a random variable X is
f(x) = 6 x2
= 0
0 x1
else where
Determine (1) Cumulative distribution function (cdf); (2) Expected value, E(X); (3)
Variance, Var (X); (4) P[X  0.6]; and (5) P[0.4  X  0.7]
Solution:
1.
Cumulative distribution function
x
F x    f x  dx

x
  6 x 2 dx  2 x 3
0
18
Water Resources Planning and Management: M6L1
0  x 1
D Nagesh Kumar, IISc
Example…

2. Expected value, E(X):
3. Variance, Var (X)
1
E X    x f x dx   x 6 x 2 dx  3 / 2

0
Var X  

2


x


f  x dx


1
   x  3 / 2  6 x 2 dx  1.2
2
0
4. P[X  0.6]
PX  0.6  1  PX  0.6  1  F 0.6
 1  2  0.63  0.568
5. P[0.4  X  0.7]
P0.4  X  0.7  PX  0.7  PX  0.4
 F 0.7   F 0.4
19
Water Resources Planning and Management: M6L1
D Nagesh Kumar, IISc
Commonly used Probability Distributions
Three commonly used distributions in water resources are: Normal, Lognormal and
Exponential distributions.
Normal distribution
 Also called as Gaussian distribution
 Two parameters are involved in this distribution: mean and variance
 A normal random variable with mean μ and variance σ2 is denoted as
X ~ N(μ, σ2)
 PDF of the normal distribution given by f(x) is expressed as
 1  x   2 
1
f  x 
exp   
 
2
 2    
20
Water Resources Planning and Management: M6L1
for    x  
D Nagesh Kumar, IISc
Normal distribution
f(x)
 PDF: Bell-shaped and symmetric at x = μ
μ
 CDF of a normal distribution is
f  x 
x


x
Normal PDF
 1  x   2 
1
exp   
  dx
2

2
 
 
for    x  
 Normal random variables are usually transformed to standardized variate Z with zero
mean and unit variance
i.e.,
Z = (X - μ) / σ.
 Then PDF of Z can be expressed as
 z2 
 z  
exp 
2
 2
1
for    z  
 Values of (z) obtained by numerical integration are used in the computations
21
Water Resources Planning and Management: M6L1
D Nagesh Kumar, IISc
Example
The monthly streamflow at a reservoir site is represented by a random variable X which
follows normal distribution with a mean of 100 units and a standard deviation of 50
units. Find (1) P[X > 150]; (2) P[X ≤ 40] and (3) The flow value which will be
exceeded with a probability of 0.8.
Solution:
The monthly streamflow at a reservoir site is represented by a random variable X which
(1) P[X > 150]
PX  150  P X    /    150  100 / 50
 PZ  1  1  PZ  1
 1  0.8413  0.1587
22
Water Resources Planning and Management: M6L1
D Nagesh Kumar, IISc
Example…
(2) P[X ≤ 40]
PX  40  P X    /    40  100 / 50
 PZ  1.2
 0.1539
(3) To find P[X ≥ x] = 0.8
PX  x   0.8
PZ   x    /    0.8
1  PZ  z   0.8
PZ  z   0.2
z   x  100  / 50   0.84
x  58 units
23
Water Resources Planning and Management: M6L1
D Nagesh Kumar, IISc
Lognormal distribution
 Used when random variable cannot be negative
 A r.v. X is lognormally distributed if its logarithmic transform
Y=ln(X)
has a normal distribution with mean μlnX and variance σ2lnX
 The PDF of lognormal r.v. is
1
f  x 
2 X  ln X
24
 
1 ln X   ln X
exp   
 2 
 ln X

Water Resources Planning and Management: M6L1




2




for    X  
D Nagesh Kumar, IISc
Exponential distribution
 PDF of an exponential distribution with parameter λ is:
e  x
f x   
0
x0
x0
 λ > 0 is the parameter of the distribution
 Mean E[X] = 1 / λ
 Var (X) = E[X2] - E[X] 2 = 1 / λ2.
 CDF is given by:
1  e  x
F x    f x dx  
0

x
25
Water Resources Planning and Management: M6L1
x0
x0
D Nagesh Kumar, IISc
Thank You
Water Resources Planning and Management: M6L1
D Nagesh Kumar, IISc