STAT131 Week7 L1a Exponential Distribution

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STAT131Week7 L1aExponential Distribution

• Anne Porter
• alp_at_uow.edu.au

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Lecture Outline

• Review Poisson
• Introduction to the Exponential
• Context/assumptions
• Probability
• Centre
• Spread
• Calibration of the model
• Goodness of fit

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Poisson

• Video Clip Blowhole

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Poisson The random variable of interest , X,
is the number of events occurring in a fixed
dimension of length t.

• The events occur in time or along any other
  dimensional
• continuum.
• In each infinitesimally small period of length ?
  the
• probability of an event is P(event)?? for some
  value ?.
• In any infinitesimally small period of length, ?,
  the
• probability of two or more events is zero ie two
  events
• do not occur simultaneously.
• The co-occurrence of events in any two
  non-overlapping
• periods is independent.
• (Griffiths et al, 1998)

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Exponential distribution

• How else can we think about the count data we
  examined as exemplifying the Poisson
  distribution?

• The time until the first event occurs and
  because
• the exponential process has no memory of the
  previous event
• The time until the next event or
• The time between events.

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Context or Problem - Exponential

• Given a Poisson process observed from time t0
  with a rate of events ?. Let Y be the time when
  the first event occurs. Then Y has an exponential
  distribution with parameter ?. Any two successive
  events also has an exponential distribution with
  parameter ?.

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ProbabilityExponential

The probability that the random event Y takes on
a value between time1 and time 2 is given by

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CentreExponential

• The mean of the Exponential(?) Random Variable Y
  is given as

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Spread Variance
Spread Variance
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Calibration of the Exponential Model

• To estimate ? based on a sample we set the
  
• mean of the distribution equal to the sample
  mean
• that is so

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Problem

• The random variable of interest is time to a the
  next car passing.The count of cars per 30 second
  interval is an homogenous Poisson (?).
• The data consist of a list of inter-car arrival
  times in seconds.
• Develop a model to describe the data

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Suggesting a model

• Context and assumptions
• Exploratory data analysis to reveal

If we have a homogeneous Poisson process with a
rate of events ? per unit of time then the time
from t0 to the first event is a random variable
with an exponential distribution.
Centre, shape, spread, outliers, allow
examination of theoretical assumptions
What plot might be useful?
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Stem- and leaf plot reveals an exponential
distribution(12 bins)

• Frequency Stem Leaf
• 7.00 0 2223333
• 34.00 0 . 55555566666667777788888999999999
  99
• 18.00 1 000111122223333444
• 9.00 1 . 555567889
• 12.00 2 001222223344
• 4.00 2 . 5678
• 6.00 3 011123
• 7.00 3 . 6777899
• 3.00 4 133
• 2.00 4 . 59
• 2.00 5 22
• 1.00 5 . 9
• 7.00 Extremes (64), (68), (72), (74),
  (86), (93)
• Stem width 10
• Each leaf 1 case(s)

Exponential shape
Centre Spread Outliers
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Histogram suggests and exponential shaped
distribution (20 class intervals or bins)
Little jagged rather than smooth so try fewer bins
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Histogram suggests and exponential shaped
distribution (10 class intervals or bins)
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How many bins should we use?

• let the data speak for themselves experiment
• balance between too smooth and too jagged to
  reveal shape

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Exploratory statistics

• TIME inter-arrival time (secs)
• Valid cases 112.0
• Missing cases .0 Percent missing .0
• Mean 21.0982 Std Err 1.8138
• Min 2.0000 Skewness 1.6881
• Median 13.5000 Variance 368.4858
• Max 93.0000 S E Skew .2284
• 5 Trim 18.9623 Std Dev 19.1960
• Range 91.0000 Kurtosis 2.6822
• IQR 21.5000 S E Kurt .4531

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Exploratory statistics Compare Mean and
Standard deviation
Theoretically and
That is
In our sample the mean 21.0982 and the standard
deviation19.1960
These are close about 9 difference as a
percentage of the mean
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Exploratory statistics Calibration

What should we do to calibrate the model?
This gives
Estimate ??
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Probabilities

• To find the probability of the time to be within
  a certain time interval we use
• 
  where
• P(0ltYlt20)
• P(20ltYlt40)
• P(40ltYlt60)
• P(60ltYlt80)
• P(Ygt80)

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Probabilities

• To find the probability of the time to be within
  a certain time interval we use
• 
  where
• P(0ltYlt20)

e -0.047x0-e -0.047x20
e0-e-0.94
1-0.3906 0.6094
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Probabilities

• P(0ltYlt20) 1- 0.3906 0.6094
• P(20ltYlt40)

e -0.047x20 -e-0.047x40
0.3906-0.1526 0.2380
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Probabilities

• P(0ltYlt20) 1- 0.3906 0.6094
• P(20ltYlt40) 0.3906 - 0.1526 0.2380
• P(40ltYlt60) 0.1526 - 0.0596 0.0930
• P(60ltYlt80) 0.0595 - 0.0233 0.0362
• P(Ygt80)

1-(0.60940.23800.09300.0362) 0.0234
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Expected counts

• To find the frequencies of inter-arrival times
  expected in each class interval

Multiply the probability of falling in an
interval by the total number of inter-arrival
times

• freq expected (0ltYlt20)

0.6094 x 112 68.25
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Finding expected counts for cells

• freq expected (0ltYlt20) 0.6094 x 112 68.25
• freq expected (20ltYlt40) 0.2380 x112 26.66
• freq expected (40ltYlt60) 0.0930 x 112 10.42
• freq expected (60ltYlt80) 0.0362 x 112 4.05
• freq expected (Ygt80)

112-(68.2526.6610.424.05)2.62
As the expected number in the last two intervals
amount to less than 5 amalgamate these cells to
have freq expected gt60 6.67
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Finding expected counts for cells

• freq expected (0ltYlt20) 0.6094 x 112 68.25
• freq expected (20ltYlt40) 0.2380 x112 26.66
• freq expected (40ltYlt60) 0.0930 x 112 10.42
• freq expected (Ygt60)
  6.67
• 
  112.00
• As the expected number in the last two intervals
  amount to less than 5 we will amalgamate these
  cells to have freq expected gt60 112-
  (68.2526.6610.42) 6.67

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• class interval freq freq
• Observed Expected
• (0ltYlt20) 68 68.25
• (20ltYlt40) 29 26.66
• (40ltYlt60) 8 10.42
• Ygt60 7 6.67
• total

0.0009 0.2054 0.5620 0.016
0.7846
112
112
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Decision

• 0.7486 ltg-p-1 ie lt 4-1-1 so the data
  can be considered to fit the model
• Informal Where we have one parameter l and
• g 4 cells and dg-p-12. If gt
• there is evidence of lack of fit
• BUT 0.7486lt 24 so there is little evidence
  that the data do not fit the model

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Decision

• Formal gt tabulated value
  (5.991)with a0.05 and dfg-p-1 then there is
  evidence the data do not fit the model
• As 0.7486lt5.991 there is little evidence of lack
  of fit between the exponential (.047) model and
  the data

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Assess Fit Observed compared to Expected
(Need to calculate probabilities and expected
counts first and this has been done with 5 not 4
bins as most appropriate Given in the last cell
the expected count is too small)
Good fit likely
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Simulation

• Simulate many samples of the same size (n112)
• Simulate them so as to have the same parameter(s)
• In this case ?0.047
• See if the data set is similar to those samples
  simulated and known to come from an exponential
  model (0.047)
• Same bins is sensible
• If the data set is typical of those simulated it
  is likely that the data follow the exponential
  (0.047) model

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Simulate Exponential (0.047) (n112 as per
original data set)Simulated samples can be
generated in SPSS
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Simulate Exponential (0.047) (n112 as per
original data set)
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Simulate Exponential (0.047) (n112 as per
original data set)
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Simulate Exponential (0.047) (n112 as per
original data set)
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Simulate Exponential (0.047) (n112 as per
original data set)
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Simulate Exponential (0.047) (n112 as per
original data set)
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Simulate Exponential (0.047) (n112 as per
original data set)
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Simulation

• If the data set is similar to those samples
  simulated and known to come from an exponential
  model (?) then it is likely that the data set
  also comes from that same population

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Quantile Plots
When the points fall on the straight line y(i)
qi where the qi are the quantiles expected with
the exponential (?).
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Quantile - quantile plots

• Sort (enter) the data in ascending order
• Determine q1, q2,..qn such that the data are
  divided into n1 areas
• F(q1)1/(n1) 1/113
• F(q2)2/(n1) 2/113 etc
• Find P(0ltYltqi)
• Solve for qi
• F(q1) 1/113
• F(qi) i/113 etc
• Find P(0ltYltqi) and solve for qi (See lecture
  notes)

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What do we do if the data does not fit the model?

• If the model does not fit, ask 'why not?' Could
  it be that one or more assumptions do not hold.
• Look at changes over time, constant rate or
  lack of independence in time periods
• Examine which cells have the largest lack of fit
• Look at theoretical relationships eg
  meanvariance or meanstandard deviation where
  they exist

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Poisson link to the ExponentialNext lecture

• No events between time zero and time t
• Is the same as
• The time of the first event is greater than t