CourseKata Chapter 5

Simple Model: Prediction + Residual

Mansour Abdoli, PhD

Today: Representing Distributions with 1 Number

Session Goals

By the end of today, you can:

• Explain error in mathematical & statistical models
• Choose a statistic to represent a distribution
• Compute and interpret residuals
• Use lm() to build a simple model based on mean.
• Use predict() and resid() to evaluate the performance of a model.

Mathematical vs. Statistical Estimation

Mathematical Modeling

Assume we need to estimate the area of a circle:

Using squares

Using octagons

• Area = Estimate + Error

• We can control the error

Statistical Modeling of Distributions

Students’ Height:

• Two Important questions:
  • How to estimate the distribution?
  • How to predict future samples?
• Answer:
  1. Model the DGP process
  2. Estimate the model parameters.
  3. Control the error as much as possible.

A Simple DGP Model

Student’s Height:

 min   Q1 median   Q3  max
  59 63.5   65.5 68.1 76.5

     mean       sd   n missing
 65.94682 3.547939 157       0

• DGP: Data (Height) = Model + Error
• Estimating Model (Center):
  • Mode (Most likely): \((62.5, 67.5]\)
  • Median (Middle number): \(65.5\)
  • Mean (Center of mass): \(65.95\)

Evaluating A DGP Model

• DGP: Data (Height) = Model + Error

• Estimating Model (Center):
  • Mode (Most likely): \((62.5, 67.5]\)
  • Median (Middle number): \(65.5\)
  • Mean (Center of mass): \(65.95\)

• Estimating Error (Residual)
  • Mode:
    • Residual\(=1_{[Height==Mode]} \in \{0,1\}\)
  • Median & Mean:
    • Residual\(=Height - Center\)

Long-run Behavior

Hands on Simulation of Simple Models

A Toy Problem:

Predicting the next random selection from: \(2, 2, 2, 4, 6, 12\)

• Form a Team (of 3 or 4)
• Create a table with 15-20 rows:

#	Guess	Actural	Match(0/1)	Actual-Guess
1	-	-	-	-
2	-	-	-	-

• Decide on a predicting procedure:

\[\begin{array}{ll} - \text{Mode} & - \text{Median}\\ - \text{Mean} & - \text{Random Guess} \end{array}\]

A Toy Problem (continue):

Predicting the next random selection from: \(2, 2, 2, 4, 6, 12\)

Repeate 15-20 times:

• Guess the next value
• Record it under Guess

• Roll a die (real or digital)
• Use it to pick the next observation.
• Record the observation under Actual.

• Compute Match and Diff columns.
• How good was your guessing procedure?

Simulatoin of the Toy Problem

 min Q1 median  Q3 max
   2  2      3 5.5  11

 mean       sd n missing
  4.5 3.563706 6       0

Simulatoin Results

Simulation Results

Mean + Residual as Simple Model

• Mean:
  • Balances Residuals: \(\sum e_i=\sum (y_i-\bar{y}) = 0\)
  • Minimizes \(\sum e_i^2=\sum (y_i-\bar{y})^2\)
• R functions:

Models and Explaning Variability

• Mean-Model is the Null Hypothesis
  • All variations is due to chance
• The Alternative Hypothesis:
  • Added variables can explain some variability

Statistical Modeling Notation

Modeling Observed Data by Mean

• Data = Observed Mean + Residual \[y_i = b_0 + e_i\]
  • \(b_0 = \bar y\) and
  • \(e_i = y_i - \bar y\)

Modeling (Population) GDP by Mean

• Data = Population Mean + Error \[y_i = \beta_0 + \varepsilon_i\]
  • \(\beta_0 = \mu\) and
  • \(\varepsilon_i = y_i - \mu\)