Forecasting Theory

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Forecasting Theory

• J. M. Akinpelu

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What is Forecasting?

• Forecast - to calculate or predict some future
  event or condition, usually as a result of
  rational study or analysis of pertinent data
• Websters Dictionary

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What is Forecasting?

• Forecasting methods
• Qualitative
• intuitive, educated guesses that may or may not
  depend on past data
• Quantitative
• based on mathematical or statistical models
• The goal of forecasting is to reduce forecast
  error.

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What is Forecasting?

• We will consider two types of forecasts based on
  mathematical models
• Regression forecasting
• Single-variable (time series) forecasting

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What is Forecasting?

• Regression forecasting
• We use the relationship between the variable of
  interest and the other variables that explain its
  variation to make predictions.
• The explanatory variables are non-stochastic.
• The explanatory variables are independent the
  variable of interest is dependent.

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What is Forecasting?

• Regression forecasting
• Height is the independent variable
• Weight is the dependent variable

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What is Forecasting?

• Single-variable (time series) forecasting
• We use past history of the variable of interest
  to predict the future.
• Predictions exploit correlations between past
  history and the future.
• Past history is stochastic.

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What is Forecasting?

• Single-variable (time series) forecasting

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Normal Distribution

• A continuous random variable X is normally
  distributed if its density function is given by
• In this case
• EX ?
• var(X) ? 2.

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Normal Density Function
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Maximum Likelihood Estimation

• Suppose that Y1, , Yn are continuous random
  variables with respective densities fi(y ?) that
  depend on some common parameter ? (which can be
  vector-valued). Assume that
• ? is unknown
• we observe y1, , yn.
• We want to estimate the value of ? associated
  with Y1, , Yn . Intuitively, we want to find the
  value of ? that is most likely to give rise to
  the data sample y1, , yn.

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Maximum Likelihood Estimation
Example Consider the data sample y1, , y20
below. Assume that all the densities are the
same, and that the unknown parameter is the mean.
Which of the two distributions most likely
produced the data sample below?
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Maximum Likelihood Estimation

• Assume that the observations are independent. We
  define the likelihood function
• In maximum likelihood estimation, we choose the
  value of ? that maximizes the likelihood
  function.

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Maximum Likelihood Estimation

• Furthermore, since logarithm is a monotone
  increasing function, then the value of ? that
  maximizes () also maximizes the log of the
  likelihood function

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Maximum Likelihood Estimation and Least Squares
Estimation

• Now assume that Yi is normally distributed with
  mean ?i(?), where ? is unknown. Assume also that
  all of the densities have a common known variance
  ? 2. Then the log likelihood function becomes

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Maximum Likelihood Estimation and Least Squares
Estimation

• Hence maximizing L(? y1, , yn) is equivalent
  to minimizing the sum of squared deviations
• The value of ? that minimizes S(?) is called the
  least squares estimate of ?.

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Regression Forecasting

• We suppose that Y is a variable of interest, and
  X1, , Xp are explanatory or predictor variables
  such that
• Y h(X1, , Xp ß).
• h is the mathematical model that determines the
  relationship between the variable of interest and
  the explanatory variables
• ? (ß0, , ßm)? are the model parameters.

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Regression Forecasting

• Further assume that
• we know h (i. e., the model is known), but we do
  not know ß
• we have noisy measurements of the variable of
  interest, Y
• yi h(xi1, , xip ß) ei.

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Regression Forecasting

• the random noise ei satisfy
• Eei 0 for all i.
• var(ei) ? 2, a constant that does not depend on
  i.
• The eis are uncorrelated.
• The eis are each normally distributed (which
  implies that they are independent), i.e.,
• ei N(0, ? 2).

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Regression Forecasting

• Note that since
• yi h(xi1, , xip ß) ei
• and
• ei N(0, ? 2)
• then
• yi N(h(xi1, , xip ß) , ? 2).

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Regression Forecasting

• For any values of the explanatory variables x1,
  , xp, if ? is known, we can predict y as
• y h(x1, , xp ß).
• Since ? is unknown, we use least squares
  estimation to estimate ?, which we denote by
  . In this case, we forecast y as

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Regression Forecasting
Example
What are the best values for ?0 and ?1?
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Regression Forecasting
Residuals are the differences between the
observed values and the predicted values. We
define the residual for the ith observation
as A good set of parameters is one for
which the residuals are small.
ei
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Regression Forecasting

• More specifically, if
• then we choose to minimize

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Regression Forecasting

• Examples of Regression Models

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Constant Mean Regression

• Suppose that the yis are a constant value plus
  noise
• yi ?0 ei,
• i.e., ? ?0. Hence
• yi N(?0, ? 2).
• We want to determine the value of ?0 that
  minimizes

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Constant Mean Regression

• Taking the derivative of S(?0) gives
• Finally setting this equal to zero leads to
• Hence the sample mean is the least squares
  estimator for ?0.

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Constant Mean Regression

• Example yi ?0 ei,

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Simple Linear Regression

• Consider the model
• yi ?0 ?1xi ei,
• i.e., ? (?0, ?1). Hence
• yi N(?0 ?1xi, ? 2).
• We want to determine the values of ?0 and ?1 that
  minimize

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Simple Linear Regression

• Setting the first partial derivatives equal to
  zero gives

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Simple Linear Regression

• Solving for ?0 and ?1 leads to the least squares
  estimates
• (This is left as a homework exercise.)

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Simple Linear Regression

• Define e (e1, , en)?, where
• The equations
• imply that

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Simple Linear Regression

• Example

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Simple Linear Regression

• Example continued

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Simple Linear Regression

• Example (continued)
• Regression equation

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General Linear Regression

• Consider the linear regression model
• or
• where xi (1, xi1, , xip)? and ? (?0, , ?p)?.

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General Linear Regression

• Suppose that we have n observations yi. We
  introduce matrix notation and define
• y (y1, , yn)?, e (e1, , en)?,
• Note that y is n ? 1, e is n ? 1, and X is n ? (p
  1).

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General Linear Regression

• Then we can write the regression model as
• Note that y has a mean vector and covariance
  matrix given by
• where I is the n ? n identity matrix.

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General Linear Regression

• Note that by var(y), we mean the matrix
• Note that this is a symmetric matrix.

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General Linear Regression

• We assume that the matrix X, which is called the
  design matrix, is of full rank. This means that
  the columns of the X matrix are not linearly
  related.
• A violation of this assumption would indicate
  that some of the independent variables are
  redundant, since at least one of the variables
  would contain the same information as a linear
  combination of the others.

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General Linear Regression

• In matrix notation, the least squares criterion
  can be expressed as minimizing
• The least squares estimator is given by
• (Proof is omitted.)

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General Linear Regression

• It follows that the prediction for Y is given by

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General Linear Regression

• Example Simple Linear Regression

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General Linear Regression

• Example Simple Linear Regression (p 1)

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General Linear Regression

• Example Simple Linear Regression (p 1)

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Properties of Least Squares Estimators

• The least squares estimator of ? is unbiased.
• The (p 1) ? (p 1) covariance matrix of the
  least squares estimator is given by
• If the errors are normally distributed then

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Properties of Least Squares Estimators

• It follow from the derivation of the least
  squares estimate that the residuals satisfy
• X?e 0.
• In particular,

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Testing the Regression Model

• How well does the regression line describe the
  relationship between the independent and
  dependent variables?

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Testing the Regression Model
Explained deviation
Unexplained deviation
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Testing the Regression Model

• Lets analyze these variations
• But

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Testing the Regression Model

• Hence
• Total sum of squares (SSTO)
• Sum of squares due to regression (SSR)
• Sum of squares due to error (SSE)

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Coefficient of Determination

• The coefficient of determination R2 is a measure
  of how well the model is doing in explaining the
  variation of the observations around their mean
• A large R2 (near 1) indicates that a large
  portion of the variation is explained by the
  model.
• A small value of R2 (near 0) indicates that only
  a small fraction of the variation is explained by
  the model.

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Correlation Coefficient

• The correlation coefficient R is the square root
  of the coefficient of determination. For simple
  linear regression, it can also be expressed as
• It varies between -1 and 1, and quantifies the
  strength of the association between the
  independent and dependent variables. A value of R
  close to 1 indicates a strong positive
  correlation a value close to -1 indicates a
  strong negative correlation. A value close to
  zero indicates weak or no correlation.

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Correlation
positive correlation
negative correlation
no correlation
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Testing the Regression Model

• Example Simple Linear Regression
• Regression equation

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Estimating the Variance

• So far we have assumed that we know the variance
  ? 2. But in general this value will be unknown.
  We can estimate ? 2 from the sample data by

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Confidence Interval for Regression Line

• Simple Linear Regression Suppose x0 is a
  specified value of the independent variable. A
  100?(1-?) confidence interval for the value of
  the mean of the dependent variable y0 at x0 is
  given by

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Prediction Interval for an Observation

• Simple Linear Regression A 100?(1-?) prediction
  interval for an observation y0 associated with x0
  is given by

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What is Forecasting (Revisited)?

• Statistical forecasting is not predicting
• a value
• Statistical forecasting is predicting
• the expected value
• variability about the expected value

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Homework

• Complete the proof for the result that for the
  simple linear regression model,
• Prove that if Y is a random variable with finite
  expected value, then the constant c that
  minimizes E(Y c)2 is c EY.

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Homework

• Suppose that the following data represent the
  total costs and the number of units produced by a
  company.
• Graph the relationship between X and Y.
• Determine the simple linear regression line
  relating Y to X.
• Predict the costs for producing 10 units. Give a
  95 confidence interval for the costs, and for
  the expected value (mean) of the costs associated
  with 10 units.
• Compute the SSTO, SSR, SSE, R and R2. Interpret
  the value of R2.

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Homework

• Consider the fuel consumption data on the next
  slide, and the following model which relates fuel
  consumption (Y) to the average hourly temperature
  (X1) and the chill index (X2)
• Plot Y versus X1 and Y versus X2.
• Determine the least squares estimates for the
  model parameters.
• Predict the fuel consumption when the temperature
  is 35 and the chill index is 10.
• Compute the SSTO, SSR, SSE and R2. Interpret the
  value of R2.

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Data for Problem 4
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References

• Bovas Abraham, Johannes Ledolter, Statistical
  Methods for Forecasting, Wiley Series in
  Probability and Mathematical Sciences, 1983.
• Stanton A. Glantz, Bryan K. Slinker, Primer of
  Applied Regression and Analysis of Variance,
  Second Edition, McGraw-Hill, Inc., 2001.
• Spyros Makridakis, Steven C. Wheelwright, Rob J.
  Hyndman, John Wiley Sons, Inc., 1998.