1. Choosing a Probability Distribution
Institute for Water Resources
2010
Choosing a Probability Distribution
Charles Yoe, Ph.D.

2. Probability x Consequence
Quantitative risk assessment requires you to use probability
Sometimes you will estimate the probability of an event
Sometimes you will use distributions to
Describe data
Model variability
Represent our uncertainty
What distribution do you use?

3. Probability—Language of Random Variables
Constant
Variables
Some things vary predictably
Some things vary unpredictably
Random variables
It can be something known but not known by us
A constant is a numerical characteristic that does not change. There are many important constants in life numerical and otherwise. Pi, Avogadro’s number, inches in a foot, pounds in a ton, and so on.
Ask: Tell me something that is constant. Your name, number of eyes, number of children.
Is there anything relative about a constant? Time? Place?
What varies predictably? Crowds at stadiums--if actual number is not important we sure know there will be many people at a game. Maybe who will be home for dinner at your house.
Our level of resolution is important in determining what is predictable. Big or little may be predictable, but the exact number is not.

4. Checklist for Choosing a Distributions From Some Data
Can you use your data?
Understand your variable
Source of data
Continuous/discrete
Bounded/unbounded
Meaningful parameters
Do you know them? (1st or 2nd order)
Univariate/multivariate
Look at your data—plot it
Use theory
Calculate statistics
Use previous experience
Distribution fitting
Expert opinion
Sensitivity analysis
How do we choose a distribution to represent our data?
What do your data look like?
Do you know from previous experience of your own or others?
Do we have theory that suggest what the distribution should be—sampling theory and statistics.
Use distribution fitting tests. We’ll do this later.
Expert opinion. Estimate the parameters of a distribution or estimate cdf.
Do analysis making different assumptions about the distribution.

5. First!
Do you have data?
If so, do you need a distribution or can you just use your data?
Answer depends on the question(s) you’re trying to answer as well as your data

6. Use Data
If your data are representative of the population germane to your problem use them
One problem could be bounding data
What are the true min & max?
Any dataset can be converted into a
Cumulative distribution function
General density function

7. Fitting Empirical Distribution to Data
If continuous & reasonably extensive
May have to estimate minimum & maximum
Rank data x(i) in ascending order
Calculate the percentile for each value
Use data and percentiles to create cumulative distribution function

8. When You Can’t Use Your Data
Given wide variety of distributions it is not always easy to select the most appropriate one
Results can be very sensitive to distribution choice
Using wrong assumption in a model can produce incorrect results=>poor decisions=> undesirable outcomes

9. Understand Your Data What is source of data? Experiments Observation
Surveys
Computer databases
Literature searches
Simulations
Test case
The source of the data may
affect your decision to use
it or not.
Understand your variable

10. Type of Variable? Is your variable discrete or continuous ?
Barges in a tow
Houses in floodplain
People at a meeting
Results of a diagnostic test
Casualties per year
Relocations and acquisitions
Type of Variable?
Average number of barges per tow
Weight of an adult striped bass
Sensitivity or specificity of a diagnostic test
Transit time
Expected annual damages
Duration of a storm
Shoreline eroded
Sediment loads
Is your variable discrete or continuous ?
Do not overlook this!
Discrete distributions- take one of a set of identifiable values, each of which has a calculable probability of occurrence
Continuous distributions- a variable that can take any value within a defined range  
Understand your variable

11. What Values Are Possible?
Is your variable bounded or unbounded?
Bounded-value confined to lie between two determined values
Unbounded-value theoretically extends from minus infinity to plus infinity
Partially bounded-constrained at one end (truncated distributions)
Use a distribution that matches
Understand your variable

12. Continuous Distribution Examples
Unbounded
Normal t
Logistic
Left Bounded
Chi-square
Exponential
Gamma
Lognormal
Weibull
Bounded
Beta
Cumulative
General/histogram
Pert
Uniform
Triangle
Understand your variable

13. Discrete Distribution Examples
Unbounded
None
Left Bounded
Poisson
Negative binomial
Geometric
Bounded
Binomial
Hypergeometric
Discrete
Discrete Uniform
Understand your variable

14. Are There Parameters
Does your variable have parameters that are meaningful?
Parametric--shape is determined by the mathematics describing a conceptual probability model
Require a greater knowledge of the underlying
Non-parametric—empirical distributions for which the mathematics is defined by the shape required
Intuitively easy to understand
Flexible and therefore useful
Understand your variable

15. Choose Parametric Distribution If
Theory supports choice
Distribution proven accurate for modelling your specific variable (without theory)
Distribution matches any observed data well
Need distribution with tail extending beyond the observed minimum or maximum
Understand your variable

16. Choose Non-Parametric Distribution If
Theory is lacking
There is no commonly used model
Data are severely limited
Knowledge is limited to general beliefs and some evidence
Understand your variable

17. Parametric and Non-Parametric
Normal
Lognormal
Exponential
Poisson
Binomial
Gamma
Uniform
Pert
Triangular
Cumulative
Understand your variable

18. Do You Know the Parameters?
Probability distribution with precisely known parameters (N(100,10)) is called a 1st order distribution
Probability distribution with some uncertainty about its parameters (N(m,s)) is called a 2nd order distribution
Risknormal(risktriang(90,100,103),riskuniform(8,11))
Understand your variable

19. Is It Dependent on Other Variables
Univariate and multivariate distributions
Univariate--describes a single parameter or variable that is not probabilistically linked to any other in the model
Multivariate--describe several parameters that are probabilistically linked in some way
Engineering relationships are often multivariate
Understand your variable

20. Continuing Checklist for Choosing a Distributions
Look at your data—plot it
Use theory
Calculate statistics
Use previous experience
Distribution fitting
Expert opinion
Sensitivity analysis
How do we choose a distribution to represent our data?
What do your data look like?
Do you know from previous experience of your own or others?
Do we have theory that suggest what the distribution should be—sampling theory and statistics.
Use distribution fitting tests. We’ll do this later.
Expert opinion. Estimate the parameters of a distribution or estimate cdf.
Do analysis making different assumptions about the distribution.

21. Plot--Old Faithful Eruptions
What do your data look like?
You could calculate Mean & SD and assume its normal
Beware, danger lurks
Always plot your data

22. Which Distribution? Examine your plot
Look for distinctive shapes of specific distributions
Single peaks
Symmetry
Positive skew
Negative values
Gamma, Weibull, beta are useful and flexible forms

23. Theory-Based Choice Most compelling reason for choice Formal theory
Central limit theorem
Theoretical knowledge of the variable
Behavior
Math—range
Informal theory
Sums normal, products lognormal
Study specific
Your best documented thoughts on subject

24. Calculate Statistics Summary statistics may provide clues Normal
Low coefficient of variation
Equal mean and median
Exponential has positive skew
Equal mean and standard deviation
Consider outliers

25. Outliers
Extreme observations can drastically influence a probability model
No prescriptive method for addressing them
If observation is an error remove it
If not what is data point telling you?
What about your world-view is inconsistent with this result? 
Should you reconsider your perspective? 
What possible explanations have you not yet considered?

26. Outliers (cont) Your explanation must be correct, not merely plausible
Consensus is poor measure of truth  
If you must keep it and can't explain it
Use conventional practices and live with skewed consequences
Choose methods less sensitive to such extreme observations (Gumbel, Weibull)

27. Previous Experience
Have you dealt with this issue successfully before? Have others?
What did other analyses or risk assessments use?
What does the literature reveal?

28. Goodness of Fit
Provides statistical evidence to test hypothesis that your data could have come from a specific distribution
H0 these data come from an “x” distribution
Small test statistic and large p mean accept H0
It is another piece of evidence not a determining factor

29. GOF Tests Chi-Square Test Most common—discrete & continuous
Data are divided into a number of cells, each cell with at least five
Usually 50 observations or more
Kolomogorov-Smirnov Test
More suitable for small samples than Chi-Square
Better fit for means than tails
Andersen-Darling Test
Weights differences between theoretical and empirical distributions at their tails greater than at their midranges
Desirable when better fit at extreme tails of distribution are desired

30. Kolmogorov-Smirnov Statistic
Blue = data
Red = true/hypothetical
Find biggest difference between the two
K-S statistic is largest difference consistent with your n α

31. Defining Distributions w/ Expert Opinion
Data never collected
Data too expensive or impossible
Past data irrelevant
Opinion needed to fill holes in sparse data
New area of inquiry, unique situation that never existed

32. What Experts Estimate The distribution itself
Judgment about distribution of value in population
E.g. population is normal
Parameters of the distribution
E.g. mean is x and standard deviation is y

33. Parametric or Non-parametric distributions
Modeling Techniques
Disaggregation (Reduction)
Subjective Probability Elicitation
PDF or CDF
Parametric or Non-parametric distributions

34. Elicitation Techniques Needed
Literature shows we do not assess subjective probabilities well
In part due to heuristics we use
Representativeness
Availability
Anchoring and adjustment
There are methods to counteract our heuristics and to elicit our expert knowledge

35. Sensitivity Analysis Unsure which is the best distribution?
Try several
If no difference you are free to use any one
Significant differences mean doing more work

36. Take Away Points
Choosing the best distribution is where most new risk assessors feel least comfortable.
Choice of distribution matters.
Distributions come from data and expert opinion.
Distribution fitting should never be the basis for distribution choice.

37. Questions?
Charles Yoe, Ph.D.