Introduction to Machine Learning
NPFL 054
http://ufal.mff.cuni.cz/course/npfl054
Barbora Hladká hladka@ufal.mff.cuni.cz
Martin Holub holub@ufal.mff.cuni.cz
Charles University, Faculty of Mathematics and Physics, Institute of Formal and Applied Linguistics
NPFL054, 2022 Hladká & Holub Lecture 6, page 1/42
Lecture #6
Outline
• Logistic regression
• Evaluation of binary classifiers
NPFL054, 2022 Hladká & Holub Lecture 6, page 2/42
Decision boundary
A task of binary classification: Y = {0, 1}
Decision boundary takes a form of function f and partitions a feature space into two sets, one for each class.
NPFL054, 2022 Hladká & Holub Lecture 6, page 3/42
Binary classification Hyperplane
Hyperplane is a linear decision boundary of the form Θ > x = 0
where direction of hθ 1 , θ 2 , . . . , θ m i is perpendicular to the hyperplane and θ 0 determines position of the hyperplane with respect to the origin
NPFL054, 2022 Hladká & Holub Lecture 6, page 4/42
Hyperplane
• point if m = 1, line if m = 2, plane if m = 3, . . .
• we can use hyperplane for classification so that
f (x) =
1 if θ 0 + θ 1 x 1 + · · · + θ m x m ≥ 0 0 if θ 0 + θ 1 x 1 + · · · + θ m x m < 0
• linear classifiers classify examples using hyperplanes
NPFL054, 2022 Hladká & Holub Lecture 6, page 5/42
Binary classification
Can we use linear regression?
NPFL054, 2022 Hladká & Holub Lecture 6, page 6/42
Can we use linear regression?
Fit the data with a linear function f
NPFL054, 2022 Hladká & Holub Lecture 6, page 7/42
Binary classification
Can we use linear regression?
Classify
• if f (x) ≥ 0.5, predict 1
• if f (x) < 0.5, predict 0
NPFL054, 2022 Hladká & Holub Lecture 6, page 8/42
Can we use linear regression?
Add one more training instance
NPFL054, 2022 Hladká & Holub Lecture 6, page 9/42
Binary classification
Can we use linear regression?
We are heading for the logistic regresession algorithm.
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0.0 0.2 0.4 0.6 0.8 1.0
A
1Y
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NPFL054, 2022 Hladká & Holub Lecture 6, page 10/42
NPFL054, 2022 Hladká & Holub Lecture 6, page 11/42
Logistic regression
Logistic regression is a classification algorithm.
Its target hypothesis f for a binary classification has a form of sigmoid function
f (x; Θ) = 1
1 + e −Θ > x = e Θ > x 1 + e Θ > x
−6 −4 −2 0 2 4 6
0.00.20.40.60.81.0
Sigmoid function
z
g(z)
• g (z ) = 1+e 1 −z
• lim z→+∞ g (z ) = 1
• lim z→−∞ g (z ) = 0
NPFL054, 2022 Hladká & Holub Lecture 6, page 12/42
f (x; Θ) = 1 1 + e −θ 0 −θ 1 x 1
−4 −2 0 2 4
0.00.20.40.60.81.0
A1
Y
θ0=0 θ0=3 θ0= −3 θ1=2
NPFL054, 2022 Hladká & Holub Lecture 6, page 13/42
Logistic regression
f (x; Θ) = 1 1 + e −θ 0 −θ 1 x 1
−4 −2 0 2 4
0.00.20.40.60.81.0
A1
Y
θ1=2 θ1= −2 θ1=6 θ0=0
NPFL054, 2022 Hladká & Holub Lecture 6, page 14/42
Classification rule
Predict a target value using ˆ f (x; ˆ Θ) so that
• if ˆ f (x; ˆ Θ) ≥ 0.5, i.e. ˆ Θ > x ≥ 0, predict 1
• if ˆ f (x; ˆ Θ) < 0.5, i.e. ˆ Θ > x < 0, predict 0
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0.00.20.40.60.81.0
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Y
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NPFL054, 2022 Hladká & Holub Lecture 6, page 15/42
Logistic regression Derivation
Interpretation of f (x; Θ): it models the conditional probability Pr(y = 1|x; Θ) f (x; Θ) = Pr(y = 1|x; Θ)
1. categorical attribute Y = {0, 1}
2.
(( (( (( (( (( (
y = θ 0 + θ 1 x 1 · · · + θ m x m , see above → model Pr(Y = y |x), e.g.
Pr(Y = 1|x)
3.
(( (( (( (( (( (( (( ( (
Pr(Y = 1|x) = θ 0 + θ 1 x 1 · · · + θ m x m , see above
4. Model odds(Pr(Y = 1|x)) = Pr(Y Pr(Y =1|x) =0|x) = 1−Pr(Y Pr(Y =1|x) =1|x) ∈ h0, +∞)
NPFL054, 2022 Hladká & Holub Lecture 6, page 16/42
odds = Pr(success)/ Pr(failure) Example: Titanic data set
> d <- read.csv("train.csv")
> attach(d)
> table(Sex, Survived) Survived
Sex 0 1
female 81 233 male 468 109
> detach()
• the odds of surviving for male:
Pr(Survived = 1|Sex = male)/ Pr(Survived = 0|Sex = male) = 109 486 = 0.23
• the odds of surving for female:
Pr(Survived = 1|Sex = female)/ Pr(Survived = 0|Sex = female) = 233 81 = 2.88
• the ratio of the odds for female to the odds for male 2.88/0.23 = 12.52
NPFL054, 2022 Hladká & Holub Lecture 6, page 17/42
Logit
5. Transform h0, +∞) to (−∞, +∞): model
logit(Pr(Y = 1|x)) = ln(odds(Pr(Y = 1|x))) = ln( Pr(Y = 1|x) 1 − Pr(Y = 1|x) ) 6. Use linear regression
ln( Pr(Y = 1|x)
1 − Pr(Y = 1|x) ) = θ 0 + θ 1 x 1 + · · · + θ m x m i.e.,
Pr(Y = 1|x) = 1
1 + e −θ 0 −θ 1 x 1 −···−θ m x m
f (x i ; Θ) = Pr(Y i = 1|x i ; Θ) = 1
1+e −Θ > xi
NPFL054, 2022 Hladká & Holub Lecture 6, page 18/42
Binary features
• Use female = {1, 0} instead of Sex = {female, male}
• in Linear regression y = θ 0 + θ 1 ∗ female
• θ 0 is the average y for male
• θ 0 + θ 1 is the average y for female
• θ 1 is the average difference in y between female and male
• in Logistic regresion p = Pr(Survive = 1|x, Θ), ln 1−p p = θ 0 + θ 1 ∗ female
• If female == 0
• p = p 1 → ln( 1−p p 1
1 ) = θ 0 → 1−p p 1
1 = e θ 0
• the intercept θ 0 is the log odds for men
• If female == 1
• p = p 2 → 1−p p 2
2 = e θ 0 +θ 1
• odds ratio = 1−p p 2
2 / 1−p p 1
1 = e θ 1
• the parameter θ 1 is the log odds ratio between female and male
NPFL054, 2022 Hladká & Holub Lecture 6, page 19/42
Parameter interpretation Numerical features
• θ i gives an average change in logit(f (x)) with one-unit change in A i holding all other features fixed
NPFL054, 2022 Hladká & Holub Lecture 6, page 20/42
• Loss function
L(Θ) = −
n
X
i=1
y i log P(y i |x i ; Θ) + (1 − y i ) log(1 − P(y i |x i ; Θ))
See Maximum Likelihood Principle for derivation of this loss function.
• Optimization problem
Θ ? = argmin Θ L(Θ)
NPFL054, 2022 Hladká & Holub Lecture 6, page 21/42
Parameter estimates
L(Θ) = − P n
i=1 y i log P(y i |x i ; Θ) + (1 − y i ) log(1 − P(y i |x i ; Θ))
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