This document discusses linear regression with multiple variables. It introduces notation for multiple features and describes using gradient descent to minimize the cost function and update the parameters simultaneously for each feature. It also covers best practices for gradient descent, including feature scaling, choosing a learning rate, and using polynomial regression. Finally, it describes using the normal equation to analytically solve for the parameters and issues that can arise with non-invertibility.

1. Linear Regression with
multiple variables
Multiple features
Machine Learning

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Size (feet2) Price ($1000)
2104 460
1416 232
1534 315
852 178
… …
Multiple features (variables).

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Size (feet2) Number of
bedrooms
Number of
floors
Age of home
(years)
Price ($1000)
2104 5 1 45 460
1416 3 2 40 232
1534 3 2 30 315
852 2 1 36 178
… … … … …
Multiple features (variables).
Notation:
= number of features
= input (features) of training example.
= value of feature in training example.

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Hypothesis:
Previously:

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For convenience of notation, define .
Multivariate linear regression.

7. Linear Regression with
multiple variables
Gradient descent for
multiple variables
Machine Learning

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Hypothesis:
Cost function:
Parameters:
(simultaneously update for every )
Repeat
Gradient descent:

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(simultaneously update )
Gradient Descent
Repeat
Previously (n=1):
New algorithm :
Repeat
(simultaneously update for
)

11. Linear Regression with
multiple variables
Gradient descent in
practice I: Feature Scaling
Machine Learning

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E.g. = size (0-2000 feet2)
= number of bedrooms (1-5)
Feature Scaling
Idea: Make sure features are on a similar scale.
size (feet2)
number of bedrooms

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Feature Scaling
Get every feature into approximately a range.

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Replace with to make features have approximately zero mean
(Do not apply to ).
Mean normalization
E.g.

16. Linear Regression with
multiple variables
Gradient descent in
practice II: Learning rate
Machine Learning

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Gradient descent
- “Debugging”: How to make sure gradient
descent is working correctly.
- How to choose learning rate .

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Example automatic
convergence test:
Declare convergence if
decreases by less than
in one iteration.
0 100 200 300 400
No. of iterations
Making sure gradient descent is working correctly.

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Making sure gradient descent is working correctly.
Gradient descent not working.
Use smaller .
No. of iterations
No. of iterations No. of iterations
- For sufficiently small , should decrease on every iteration.
- But if is too small, gradient descent can be slow to converge.

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Summary:
- If is too small: slow convergence.
- If is too large: may not decrease on
every iteration; may not converge.
To choose , try

22. Linear Regression with
multiple variables
Features and
polynomial regression
Machine Learning

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Housing prices prediction

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Polynomial regression
Price
(y)
Size (x)

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Choice of features
Price
(y)
Size (x)

27. Linear Regression with
multiple variables
Normal equation
Machine Learning

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Gradient Descent
Normal equation: Method to solve for
analytically.

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Intuition: If 1D
Solve for
(for every )

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Size (feet2) Number of
bedrooms
Number of
floors
Age of home
(years)
Price ($1000)
1 2104 5 1 45 460
1 1416 3 2 40 232
1 1534 3 2 30 315
1 852 2 1 36 178
Size (feet2) Number of
bedrooms
Number of
floors
Age of home
(years)
Price ($1000)
2104 5 1 45 460
1416 3 2 40 232
1534 3 2 30 315
852 2 1 36 178
Examples:

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examples ; features.
E.g. If

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is inverse of matrix .
Octave: pinv(X’*X)*X’*y

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training examples, features.
Gradient Descent Normal Equation
• No need to choose .
• Don’t need to iterate.
• Need to choose .
• Needs many iterations.
• Works well even
when is large.
• Need to compute
• Slow if is very large.

35. Linear Regression with
multiple variables
Normal equation
and non-invertibility
(optional)
Machine Learning

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Normal equation
- What if is non-invertible? (singular/
degenerate)
- Octave: pinv(X’*X)*X’*y

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What if is non-invertible?
• Redundant features (linearly dependent).
E.g. size in feet2
size in m2
• Too many features (e.g. ).
- Delete some features, or use regularization.