Chapter 16 Sections 3'2 3'7'3, 4'0,

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Materials for Lecture 10

• Chapter 6
• Chapter 16 Sections 3.2 - 3.7.3, 4.0,
• Lecture 10 Demo Distributions.xls
• Lecture 10 Empirical Distributions.xls

2
Empirical Probability Distribution

• Non-Parametric Distributions
• Discrete Uniform
• Empirical
• GRK and GRKS
• Triangle
• Parametric Distributions
• Fixed form, such as Uniform, Normal, Beta, Gamma,
  etc. and are estimated by UPES

3
Discrete Uniform Empirical

• Discrete Uniform Empirical distribution used
  where only fixed values can occur
• Each value has an equal probability of being
  drawn
• No interpolation between observed values
• Function might be used for things such as,
• Discrete number of labors
• Number of steers on a truck
• Simulating a fair die

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Discrete Uniform Empirical Distribution
 
- Parameters are DE(x1, x2, x3, , xn) based on
history
- Discrete Empirical means that each observed
value of Xi, has an equal probability of being
observed
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Discrete Uniform Empirical

• Simulate this type of random variable two ways in
  Simetar
• Discrete empirical with equal probabilities
• DEMPIRICAL(A1A5)
• RANDSORT(A1A5)

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Discrete Empirical -- Alphanumeric

• RANDSORT(I1I5)
• Random shuffle of names highlight 5 cells and
• RANDSORT(I1I5, Option) then hit Ctrl Shift
  Enter
• Option can be set to
• 0 causes it to draw a sample every time press F9
• 1 causes Simetar to make only one draw, so get
  one sample

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Empirical Distribution

• An empirical distribution is defined totally by
  the observations for the data, no distribution
  form is assumed
• Estimate parameters and simulate an empirical
  distribution
• Estimate the forecasted values means (?) or
  forecasts (Y)
• Calculate the deviation from the mean or forecast
  
• Sort the deviations from the mean or forecast
  from low to high
• Assign a cumulative probability to each data
  point (usually equal probability).
• The cumulative probabilities go from zero to one
• Assume the distribution is continuous, so
  interpolate between the observed points
• Use the Inverse Transform formula to simulate the
  distribution
• This requires simulation of a USD and
  interpolation
• Use Emp icon to estimate parameters

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Using the Empirical Distribution

• Empirical distribution should be used if
• The random variable is continuous over its range,
  
• You have limited observations for the variable,
  and/or
• You cannot easily estimate parameters for the
  true PDF
• Simulate crop yields as an Empirical distribution
  when you have only 10 historical values
• In this situation we know
• Yield can be any positive value
• We dont have enough observations to test for
  normality
• We know the 10 random values were observed with a
  probability of 1/10, or one observation each year

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PDF and CDF for an Empirical Dist.
Probability Density Function
Cumulative Distribution Function
F(x)
f(x)
1.0
X
0.0
max
min
min
max
X
We interpolate the Dark Black line in the CDF
based on the discrete CDF and use it as the
approximation for a continuous distribution
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Inverse Transform for the Empirical Distribution
1.0
F(x)
Start with a random USD, say
U(0,1) 0.45
Interpolate the ? axis using the USD value
0.0
Y1
Y2
Y6
Y3
Y4
Y5
Y7
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Simulate Empirical Distribution

• Empirical distribution is usually simulated as ?
  or Y as percent deviations from mean or trend
• percent deviates from mean (Yt ?t )/?t
• Parameters are
• Mean of the data is either ?t or Yt
• Sorted deviations from mean or forecasted Y are
• St Sort (Yt ?t )/?t or
• St Sort (Yt Yt)/ Yt
• Probabilities for Sts, are called F(St) or F(x)
  values and MUST range from 0.0 to 1.0
• Use the parameters to simulate random variable ?
  
• ? ?t (1 EMP(St, F(St), USD) )

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Empirical Distribution -- No Trend

• Given a random variable, ?, with 11 observations
• Develop the parameters if simulating the variable
  using the mean to forecast the deterministic
  component

• Parameter for deterministic component is the
  mean or the second column
• Calculate the stochastic component or ê as
  êi Yi ?
• Convert the residual to a fraction of forecast
  mean value Devi êi / ?
• Sort the Devi values from low to high (Si) and
  calculate the probabilities of Si or F(Si)
• Simulate ? in two steps
  Stoch Devi EMP(Si, F(Si), USD)
• Stoch ?Ti ?Ti (1 Stoch Devi)
• Derived from Devi (Yi- ?) / ?
  or (? Devi) Yi ?
  so Yi ? (? Devi)

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Empirical Dist. -- With Trend

• Parameters for EMP() if deterministic component
  is the trend forecast

• Calculate the stochastic component
  or ê as êi Yi Yi
• Convert the residual to a fraction of forecast
  value Devi êi / Yi
• Sort the Devi values from low to high (Si) and
  calculate the probabilities of Si or F(Si)
• Simulate ? as follows
• Stoch Devi EMP(Si, F(Si), USD )
• ?Ti YTi (1 Stoch Devi)
• Derived from Stoch Devi (Yi - Yi) / Yi
  or Yi Yi (Yi Stoch Devi)
  or Y Stochi Yi (Yi
  Stoch Devi)

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Simulate Emp Distribution with Simetar

• Let Si be in B1B10 and F(Si) in A1A10
• If Si expressed as actual values
• EMP(Si ) or EMP(B1B10)
• If Si expressed as residuals from mean
• ? EMP(B1B10, A1A10)
• If Si expressed as fractional deviates from
  trend or Si (? / Y)
• Y (1 EMP(B1B10, A1A10))

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Simulating an Emp Distribution

• Advantages of Emp Distribution
• It lets the data define the shape of the
  distribution
• Does not force an assumed distribution shape on
  the variable
• The larger the number of observations in the
  sample the closer Emp will approximate the true
  distribution
• Disadvantages of Emp Distribution
• It has finite min and max values, this is an
  advantage
• It does not adhere to known probabilities and
  parameters
• Parameters can be difficult to estimate w/o
  Simetar

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Simulating an Emp Distribution

• Advantages of specifying the Sis as a fraction
  of forecasted values
• Guarantees the relative risk for a random
  variable is the same as the historical period
• Coefficient of variation for the sample data is
  constant over time CV (s / ?) 100
• Allows you to use any mean (Y or ?) for the
  simulated planning horizon and it will have the
  same CV as the historical period
• Historical ? can be 100 and the mean for the
  forecast period Y can be 150 and the ? values
  will have the same CV as the historical data.