Introductory Forecasting Concepts

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Introductory Forecasting Concepts
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Outline

• Information vs. intuition and the value of data
• Past vs. future performance
• Accuracy and its limits
• Prediction intervals
• Forecast accuracy and decisions
• Aggregate forecasting
• Averaging and smoothing
• Monitoring forecasting systems
• Dealing with alerts and outliers

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Forecasting

• The art of using information to predict future
  behavior of some time series
• There is no magic math that allows clairvoyance

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Forecasting

• This course focuses on using historical data for
  forecasting
• This should not diminish the importance of other
  sources of information and common sense
• Information consists of
• Historical data on our time series
• Insight/knowledge and common sense
• Dont confuse information with intuition/guessing/
  human pattern recognition

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Information and Intuition

• Example from a forecasting paper in the electric
  power industry
• Total kilowatt-hour usage in Georgia,
  Mississippi, and the gulf region appears to
  follow a 12 month seasonal pattern while Alabama
  follows a 13 month season.
• What do you think?

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On the other hand..

• People see Elvis visage in a cloud
• Humans see patterns in purely random systems
• Dont confuse information with intuition
• Lets try a case study. Forecast a real time
  series from scratch using intuition (this is real
  data!)

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Well guess same as last month
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Well guess same as last month plus a little more
for a possible trend
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This is easy, who needs forecasting
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Continue with our successful method guess the
same as last month plus a little more for a
possible trend
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Definitely looks like a trend
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Trend might be a tad steeper than I thought
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Opps
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Momentary deviation, trend will continue
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See, I told you this was easy!
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Trend will continue
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Opps, another momentary fluctuation
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Trend should continue
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Oh oh!
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Sales has leveled off Lets average last few
points
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Oh oh, maybe things are going down hill
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Lets be conservative and Assume a negative trend
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Thank goodness, we are still basically level
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Well guess same as last month
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This stuff is easy
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We have for sure leveled off
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Big trouble!!! Chief forecaster Smith and CEO
Smothers fired!
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New chief forecaster points out the obvious trend
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Remarkable turnaround in sales. New CEO Smithers
given credit
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Still looks like a trend to me
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Maybe not!
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Level except for anomaly
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Have things turned around?
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Ill hedge my bets
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Things have turned around. Perhaps Smithers truly
is a genius
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Trend up!
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Not bad!
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Revise trend a tad
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Smithers makes cover of Fortune
Smithers
Smothers
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This is easy!!
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No big deal, trend continues
(in an unrelated matter Smithers cashes out
stock options)
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Heads will surely roll soon
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Lets be cautiously optimistic
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Smithers called before board
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Perhaps we over reacted
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We will guess level
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Back to normal!
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Smithers fired!
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What have we learned?

• Our Actual sales appears to be a great leading
  indicator of our forecast
• It is supposed to work the other way around!!!!
• If we add up the (absolute value of) our forecast
  errors, we get 226.2
• If we had simply guessed same as last month we
  get 175.1

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Information vs. Intuition

• Our intuition (ability to recognize a pattern)
  was poor given almost no information or data.
  Never-the-less we saw patterns.
• For monthly data we can be tempted to over
  think forecasting.
• Now some additional information
• Source of data is monthly sales of Australian Red
  Wine
• We also have a few years of data

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We observe

• Series behavior includes
• Clear seasonal behavior
• A clear upward trend
• An increase in amplitude as the level increases

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Value of Data

• Given data, we can forecast this series quite
  accurately.
• This assumes stable behavior
• Recommend at least 4 - 5 seasons of data.
• Monthly demand thus prefers 4 to 5 years of data
• With 2 years of data, we are essentially
  forecasting on the basis of two points if there
  is seasonal behavior

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1st Law of Forecasting Past vs. Future
Performance

• In forecasting, we assume the future will behave
  like the past
• If behavior changes, our forecasts can be really
  really terrible

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Here is an easy series to forecast!
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No method based on historical behavior could see
this coming
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Corollary to the 1st Law of Forecasting

• In the real world, the future often does not
  behave like the past.
• The behavior of many time series changes fairly
  frequently.

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2nd Law of ForecastingForecast Accuracy
Limitations

• Even given that the future behaves like the past,
  there is a limit to how accurate forecasts can be
  (or nothing can be predicted with complete
  accuracy)
• Typical times series have a signal component
  and a noise component
• The achievable accuracy depends on the magnitude
  of the noise component

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Corollary to the 2nd Law of Forecasting

• All forecasts will be wrong
• The key issue is How close will the forecast be
  to the actual value?
• It is crucial to attempt to quantify the expected
  accuracy of a forecast

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Prediction Intervals

• We can create prediction intervals of the basic
  form
• There is a xx probability that the actual
  future value will fall in the range
• Forecast ? K
• Example

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Now
Future forecast
Past Data
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Prediction Intervals

• Details become complex (and will come later)
• Prediction intervals based on assumptions
• The future behaves like the past
• The model used to calculate the prediction
  interval is an accurate model of the data
• Prediction intervals tend to be optimistic
  (because of the assumptions)
• Never-the-less, look how wide they are!

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3rd Law of Forecasting

• The further into the future you attempt to
  forecast, the greater will be the forecast error.
  
• This is typically true even if the future behaves
  like the past.
• (as we can see from the prediction intervals)

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Corollary to 3rd Law of Forecasting

• Major decisions are often based on long term
  forecasts.
• e.g. building a new plant
• Considering risk is even more important in these
  cases
• Example Fiber optic production capacity

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Decisions and the 2nd Law of Forecasting

• Decisions will be based on the forecast
• (Otherwise there is no need to forecast!)
• One should consider the ramifications of a
  decision over the range of possible outcomes
• That is forecasts have inherent error, thus the
  decisions based on forecasts have inherent risk
• Rational decision making includes analysis of
  risk
• You would never buy stock solely on the basis of
  expected return

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Sales forecast
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Forecast prediction interval
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Forecast prediction interval
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Forecasting and Decisions

• We forecast as part of a decision making process
• It is important to keep in mind the decisions
  that are based on the forecast
• Keeping this link in mind can help guide us
• When and how often to forecast
• At what level of aggregation to forecast
• Etc.

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Aggregate Forecasting

• Suppose we forecast demand for product X at the
  individual customer level.
• Suppose we also need a forecast of total demand
  for product X for production planning
• Should we add the individual forecasts together
  to get the forecast for overall demand for
  product X?
• Or should we forecast total demand for product X
  directly using total demand data?

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Research is Inconclusive

• Define Disaggregated forecasting
• Develop individual forecasting methods for each
  customer
• Add the forecasts together to get forecast of
  total demand
• Define Direct forecasting
• Use the total monthly demand time series data to
  develop a forecasting method

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Argument for disaggregated forecasting

• By aggregating into total demand, we have, by
  definition, lost information
• Patterns existing in customer specific data that
  we could exploit for improved forecasting may be
  lost/obscured through aggregation
• Under idealized conditions, it can be shown that
  disaggregated forecasting will be at least as
  good as direct forecasting
• Assumptions for this are arguable however

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Argument for Direct Forecasting

• Idealized circumstances never true in practice
• Each individual forecast requires specification
  of models and model parameter estimates both of
  which are subject to error
• These errors are compounded in disaggregated
  forecasting
• Direct forecasting does not suffer from this. We
  develop one simple model.

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Empirical Results Support Both Viewpoints

• Direct method tends to be better when
• Individual series are positively correlated (tend
  to move together)
• e.g. all customers effected by the economy
• Lower level forecasts are ad hoc and do not
  exploit all the information available
• e.g. simple automated methods are applied

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Empirical Results Support Both Viewpoints

• Disaggregated method tends to be better if
• Individual series are negatively correlated (tend
  to move oppositely)
• e.g. sales of Product X Product Y ? constant
• Information at the lower levels is carefully
  exploited
• e.g. expertise is applied to forecast individual
  customer demands
• Warning be wary of demand forecasts from Sales
  Dept

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Disaggregated vs. Direct

• One can use composite forecasts (combine
  disaggregated and direct forecasts
• One can try both and test which works better in
  practice
• What if the sum of disaggregated forecasts is not
  equal to forecast of the sum?
• Performance difference tends to be greatest for
  near term forecasts (1 or 2 periods ahead)

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Forecasting Fundamentals

• Averaging
• Smoothing
• Modeling
• Most forecasting techniques combine modeling and
  smoothing

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Averaging

• Is a basic concept in statistics
• Is important in understanding simple smoothing
  based forecasts
• Understand the trade off between averaging and
  data currency

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Standard deviation of an average

• The standard deviation of an average is inversely
  proportional to the square root of the number of
  data points in the average
• The larger the sample, the better the average
  approximates the true mean
• Ball players may hit 450 in the first two weeks,
  but not by the end of the season

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Standard deviation of an average

• As sample size increases, the accuracy of an
  average increases
• The increase in accuracy decreases in N
• i.e. the marginal value of more samples decreases
• Aside How big should N be? Answer 5
• (an obvious gross over generalization)

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Knee of the curve
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Forecasting with Moving Averages

• Consider the time series generated by
• 10 random white noise
• (the best forecast would be 10 if we knew how
  the series behaved beforehand)
• We will average the last 3 points (a 3 period
  moving average)
• Then we average the last 10 points (a 10 point
  moving average)

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Averaging

• Averaging can be seen to smooth or filter out
  the noise.
• Why not use the average of all the data?
• Because things rarely remain the same

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Averaging

• Long averages are good since they reduce the
  variance and thus the forecast error
• Short averages are good because the respond more
  quickly to changes
• How much data should we include in our average?
• There is a trade off

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Moving Averages

• Moving averages are common forecasting techniques
• Their best feature is their simplicity everyone
  can understand them
• Consider the weights attached to past data points
  in a moving average

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Moving Averages

• We want less weight on past data points so the
  forecast can adapt to changes
• Why then do we put equal weight (0.2) on points 4
  and 5 periods ago?
• Why do we give a weight of 0.2 to 5 periods ago
  but zero weight to six periods ago?
• Why not discount weight based on age?

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Moving Average
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Moving Average
Exponential Smoothing
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Average age of data is the same in this example
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Average age of data is the same in this example
ES gives decreasingly less weight to past data
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Exponential Smoothing

• With equivalent average age, the forecast error
  variance will be the same (under a constant
  model)
• Since more weight is on recent points, ES should
  adapt to changes more quickly
• The weighting scheme is more intuitively
  appealing
• In general, ES is preferred to MA
• The only disadvantage is its complexity

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Time Series Modeling

• The science of extracting a signal from noise
• Requires a fair amount of data
• Can be very effective as long as the behavior of
  the time series does not change much.

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Simple Models

• Constant Mean Random Noise
• Trend Random Noise
• Trend Seasonal Factor Random Noise
• These are the basis for the models in SAP and
  other common planning systems

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Monitoring and Alerts

• Especially useful when automatically forecasting
  many time series
• Basic idea
• Compare forecast to actual value
• If error is too big, issue an alert
• Alert dealt with via human intervention
• Automatic adaptation methods also exist
• These are not without dangers

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Monitoring

• We can monitor for
• Magnitude of forecast errors
• Forecast bias

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Looks like something happened around here
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Could devise Control Limits
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Story clearer using AD
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Clearer still using Smoothed AD
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Monitoring for Bias

• Example
• Can you spot anything in the following plot of
  forecast error Actual Forecast?

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• From time 0 to 25, White noise with a mean of
  zero
• From time 26 to 50, White noise with a mean of
  0.2
• After time 26, the forecast is too low by 0.2 on
  average
• Now consider Error Total
• Sum of all errors up to now
• Should stay close to zero

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Tracking Signal

• Standardize Error Total
• Tracking Signal Error total / Smoothed MAD

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Alerts and Outliers

• It is crucial to investigate alerts and outliers
  and find their cause
• It may be a fluke or an error that should be
  ignored
• It may be the most important point in the data
  set indicating entirely new behavior
• i.e. it contains much information

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Example

• Lets look at my monthly power bill and that of
  three colleagues
• This example is real!

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Actual
Alert!
Forecast
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Outliers and Alerts

• Forecast is way off.
• What happened?
• What should be our next forecast?
• Close to the prior pattern?
• Close to the most recent observation?

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Example

• Lets investigate my power bill outlier
• Facts of the case
• From Jan to July 31, we were on sabbatical and
  rented out our house
• We returned Aug. 1
• My family still wonders what those switchy
  thingys on the wall are
• They like fresh air and AC together

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Outliers

• After investigation, we know what to do
• We expect power usage to stay at its new, higher
  level
• We can adjust our forecast accordingly
• We might also adjust our business practice in
  important new ways on the basis of what we have
  learned
• Throwing out outliers simply because they are
  outliers is very bad practice

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Data with many zeros

• 100 APCI Demand data sets

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How do we deal with this?
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Assume a Demand Distribution
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Poisson fit to data

• We could fit a Poisson to the data
• We could use the Poisson mean as the forecast
• This assumes each demand outcome iid
• There exist methods that try to track the
  parameter(s) of the demand distribution over time
• see parameter driven state space models

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Thank you