Intro to prediction and Ordinary Least Squares

STAT 20: Introduction to Probability and Statistics

Warmup

10:00

For each of the following scenarios identify:

• the target variable \(Y\) and its type
• (time permitting, the features \(X_j\), their types, and how many they are (\(q\)))
• whether it’s a regression or a classification task

Secenario A

• Amazon sets the price for a specific book, dependent on the book’s weight and whether it is hardback or paperback.

Careful!! This is a Causality question. “What would happen if I were to set the price of the book at some value \(x\)?”

Secenario B

• Abebooks guesses the time it’ll take to sell a specific book, based on the price it can freely set, the book’s weight and whether it is hardback or paperback.

Careful again! This is a Causal Prediction question. As a prediction problem, it’s a regression because time should be modeled as a continuous variable. But it’s also causal because again your’re wondering “What would happen if I were to set the price of the book at some value \(x\)?”

Secenario C

• Ebay guesses the final sale price for a specific book, based on the initial bid price the seller has set, the book’s weight and whether it is hardback or paperback.

Finally! This is a straight-up Prediction task. It’s a Regression because sale price should be modeled as a continuous variable.

Secenario D

• A Waymo engineer is working on a model to determine whether the self-driving car is seeing a real car or a reflection of a car.

This is a Prediction task. It’s a Classification task, because whether the car is real should be modeled as a discrete (Bernoulli) variable.

Secenario E

• A marine biologist is trying to guess the nose-to-tail length of the typical manatee in the waters off the coast of Florida, broken down by sex and age groups.

You fell for it again! This is not a Prediction task. It’s a Generalization task, because they’re estimating various population parameters (conditional means). These are hidden, non observable quantities.

Secenario F

• The Weather Channel predicts tomorrow’s temperature based on historical data, tomorrow’s expected weather conditions and yesterday’s temperature.

This is a Prediction task. It’s a Regression task, because temperature is modeled as a continuous variable. Careful, the historical data isn’t a feature, it’s either the sample used to fit the model or some other miscellaneous info that will help with writinf down the model.

Secenario G

• The UCB Admissions office has to predict whether a student will accept its admission offer or not based on the student’s high school and their intended major.

This is a Prediction task. It’s a Classification task, because whether the car is real should be modeled as a binary (Bernoulli) variable.

Secenario H

• A philologist examines a 2nd century BCE papyrus from Egypt, full of holes, and written in Coptic. They make a guess about the missing letters in the holes, based on the surrounding words, the dating, the geographic origin of the papyrus.

Careful! This is a Prediction task. Althout this thing was written 2200 years ago, we are still “predicting” the value of an unseen random variable (the missing letter). It’s a Classification task, because a letter from the Coptic alphabet is a nominal categorical variable.

Which is better?

• Large MSE or small MSE?

Small. MSE measures error!

Concept Question 1. Sample Mean Squared Error (MSE)

01:00

• Imagine the following sample (\(n=3\)) with true values of \(Y\) and the predictions from some model \(\hat{Y}_{\mbox{model}}\). The units of \(Y\) are kilograms.

	Y	Ŷ
1	2.3	2.8
2	−1.6	−1.2
3	4.3	3.6

• Calculate the sample MSE of the model. \(\widehat{\mbox{MSE}} =?\)

\(\widehat{\mbox{MSE}} = \frac1{3} \left((-0.5 \, kg)^2 +(-0.4 \, kg)^2 +(0.7 \, kg)^2 \right) = 0.3 \, kg^2\)

Concept Question 2.

• Which of these two features \(X_j\) will yield the better prediction \(\hat{Y}\), based on the sample we collected and plotted below?

\(X_1\) on the left seems more strongly associated with \(Y\) than \(X_2\) is.

Concept Question 3

• If your model \(\hat{Y}\) is awesome, what will be the typical size of the residual in either case?

Roughly 3-ish kg using \(X_1\) and roughly 7-ish kg using \(X_2\) depending on where you look on the \(x\)-axis.

Concept Question 4

• Here is a function f.

f <- function(x, y) {
  y*(x + 3) 
}

What will the following line of code return?

f(3,5)

This question reminds students that if function arguments are unnamed, they are assumed to have been input in the order they were originally specified as in the function definition. x will be assigned 3, and y will be assigned 5, so the correct answer is 30.

Break

05:00

Worksheet: Method of Least Squares

40:00