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Statistical

Distribution Fitting

Simulation with Arena — Statistical Distribution Fitting (^) C5/

Some Issues in Fitting Input

Distributions

• Not an exact science — no “right” answer
• Consider physical or logical process that

generates the data

• Consider range of distribution
  • Infinite both ways (e.g., normal)
  • Positive (e.g., exponential, gamma)
  • Bounded (e.g., beta, uniform)
• Consider ease of parameter manipulation to

affect means, variances - decision variables

• Outliers, multimodal data
  • Maybe split data set (see textbook for details)
  • Consider theoretical vs. empirical

Simulation with Arena — Statistical Distribution Fitting (^) C5/

Chi-Squared Test

• Formalizes this notion of

distribution fit

• Oi represents the number of

observed data values in the i -

th interval.

• pi is the probability of a data

value falling in the i -th interval

under the hypothesized

distribution.

• So we would expect to

observe Ei = npi , if we have n

observations

frequency

data values

O i

pdf

data values

p i

Simulation with Arena — Statistical Distribution Fitting (^) C5/

Chi-Squared Test

• So the chi-squared statistic is
• By assuming that the O i - E i

terms are normally

distributed,

• it can be shown that the distribution of the statistic is

approximately chi-squared with k-s-1 degrees of freedom

• s is the number of parameters of the distribution
• Hint: consider

∑















 (^) −

k

i i

i i

E

O E

∑ = (^) 

















 − =

k

i i

i

i

p

p n

O

χ 0

Simulation with Arena — Statistical Distribution Fitting (^) C5/

Chi-Squared Test

• If the expected frequencies E i

are too small, then

the test statistic will not reflect the departure of

the observed from the expected frequencies.

• The test can reject because of noise
• In practice a minimum of E (^) i ≥ 5 is used
• If E (^) i is too small for a given interval, then adjacent intervals

can be combined

• For discrete distributions
  • each possible discrete value can be a class interval
  • combine adjacent values if the E (^) i ’s are too small

Simulation with Arena — Statistical Distribution Fitting (^) C5/

Chi-Squared Test

• For continuous data
  • intervals that give equal probabilities should be used, not

equal length intervals

• this gives a better power for the test
  • the power of test is the probability of rejecting a false hypothesis
• it is not known what probability gives the highest power, but

we want

npi ≥ 5

≥ 5 k

n

k

pi

1

5

n k ≤

Ei ≥ 5

Simulation with Arena — Statistical Distribution Fitting (^) C5/

Eyeballing

• Another method of seeing if a distribution fits sample data

is the q-q plot

• x is the q-quantile of a random variable X with cdf F if F(x)=q or x=F -1(q)
• Take a data sample {x 1 ,…x (^) n} and order them to get y 1 ≤ y 2 ≤ ... ≤ y (^) n
• y (^) j is an estimate of the (j - 0.5)/n quantile
• Plot y (^) j versus F-1( (j - 0.5)/n )
• This should give a straight line

0

10

20

30

40

50

60

70

80

0 5 10 15 20 Order Statistics

Exponential Quantile

Simulation with Arena — Statistical Distribution Fitting (^) C5/

Eyeballing

• Note:
  • Will never actually be a straight line
  • Order statistics are not independent
  • One point above line will likely be followed by another
  • The variance at the extremes is larger
  • So for exponential, you will likely see more discrepancy at larger values

Order Statistics

Exponential Quantile

Simulation with Arena — Statistical Distribution Fitting (^) C5/

Comparing the Two Tests

• The Chi-Squared Test
  • Not just a maximum deviation, but a sum of squared

deviations

• Uses more of the information in the data
• So it needs more data to be accurate
• Is more accurate if it has enough data
• The Kolmogorov-Smirnov Test
• Just a maximum deviation
• Needs less data to be accurate
• Is less accurate with more data

Simulation with Arena — Statistical Distribution Fitting (^) C5/

Empirical Distribution

• “Fit” Empirical distribution (continuous or

discrete): Fit/Empirical

• Can interpret results as a Discrete or Continuous

distribution

• Discrete: get pairs ( Cumulative Probability, Value)
• Continuous: Arena will linearly interpolate within the data range

according to these pairs (so you can never generate values outside

the range, which might be good or bad)

• Empirical distribution can be used when “theoretical”

distributions fit poorly, or intentionally

• When sampling from the empirical distribution, you are just

re-sampling from the data

Simulation with Arena — Statistical Distribution Fitting (^) C5/

Multivariate and Correlated Input

Data

• Usually we assume that all generated random

observations across a simulation are

independent (though from possibly different

distributions)

• Sometimes this isn’t true:
  • If a clerk starts to get long jobs, they may get tired and slow

down

• A “difficult” part requires long processing in both the Prep

and Sealer operations

• Ignoring such relations can invalidate model

Simulation with Arena — Statistical Distribution Fitting (^) C5/

Checking for Auto-Correlation

• Suppose we have a series of inter-arrival times
  • What is the relationship between the j-th observation and

the (j-1)st?

• What is the relationship between the j-th observation and

the (j-2)nd?

• We are talking about auto-correlation as the

series is correlated with itself

• How many steps back we are looking is called the

lag

Simulation with Arena — Statistical Distribution Fitting (^) C5/

Time Series Models

• If the auto-correlation calculations show a

correlation, then you may have to use a time-

series model

• Such models are auto-regression models and

moving average models

• Using the auto-correlation and another concept

called the partial auto-correlation, you can fit

these models

• The details are too much for this course

Simulation with Arena — Statistical Distribution Fitting (^) C5/

Multivariate Input Data

• A “difficult” part requires long processing in both

the Prep and Sealer operations

• The service times at the Prep and Sealer areas would be

correlated

• Some multivariate models are quite easy, for instance the

multivariate normal model

• You can also use the multiplication rule, to specify the

marginal distribution of one time and then specify the other

time conditional on the first time

f (^) X , Y ( x , y ) = fX | Y ( x | y ) fY ( y )