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Justin Grimmer
Associate ProfessorDepartment of Political Science
Stanford University
November 10th, 2014
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 1 / 40
Supervised Learning Methods
1) Task
- Classify documents to pre existing categories- Measure the proportion of documents in each category
2) Objective function1) Penalized Regressions
- Ridge regression- LASSO regression
2) Classification Surface Support Vector Machines3) Measure Proportions Naive Bayes(ish) objective
3) Optimization
- Depends on method
4) Validation
- Obtain predicted fit for new data f (X i , θ)- Examine prediction performance compare classification to gold
standard
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 2 / 40
Regression modelsSuppose we have N documents, with each document i having labelyi ∈ {−1, 1} {liberal, conservative}
We represent each document i is x i = (xi1, xi2, . . . , xiJ).
f (β,X ,Y ) =N∑i=1
(yi − β
′x i
)2β = arg minβ
{N∑i=1
(yi − β
′x i
)2}
=(X
′X)−1
X′Y
Problem:
- J will likely be large (perhaps J > N)
- There many correlated variables
Predictions will be variable
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 3 / 40
Regression modelsSuppose we have N documents, with each document i having labelyi ∈ {−1, 1} {liberal, conservative}We represent each document i is x i = (xi1, xi2, . . . , xiJ).
f (β,X ,Y ) =N∑i=1
(yi − β
′x i
)2β = arg minβ
{N∑i=1
(yi − β
′x i
)2}
=(X
′X)−1
X′Y
Problem:
- J will likely be large (perhaps J > N)
- There many correlated variables
Predictions will be variable
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 3 / 40
Regression modelsSuppose we have N documents, with each document i having labelyi ∈ {−1, 1} {liberal, conservative}We represent each document i is x i = (xi1, xi2, . . . , xiJ).
f (β,X ,Y ) =N∑i=1
(yi − β
′x i
)2
β = arg minβ
{N∑i=1
(yi − β
′x i
)2}
=(X
′X)−1
X′Y
Problem:
- J will likely be large (perhaps J > N)
- There many correlated variables
Predictions will be variable
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 3 / 40
Regression modelsSuppose we have N documents, with each document i having labelyi ∈ {−1, 1} {liberal, conservative}We represent each document i is x i = (xi1, xi2, . . . , xiJ).
f (β,X ,Y ) =N∑i=1
(yi − β
′x i
)2β = arg minβ
{N∑i=1
(yi − β
′x i
)2}
=(X
′X)−1
X′Y
Problem:
- J will likely be large (perhaps J > N)
- There many correlated variables
Predictions will be variable
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 3 / 40
Regression modelsSuppose we have N documents, with each document i having labelyi ∈ {−1, 1} {liberal, conservative}We represent each document i is x i = (xi1, xi2, . . . , xiJ).
f (β,X ,Y ) =N∑i=1
(yi − β
′x i
)2β = arg minβ
{N∑i=1
(yi − β
′x i
)2}
=(X
′X)−1
X′Y
Problem:
- J will likely be large (perhaps J > N)
- There many correlated variables
Predictions will be variable
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 3 / 40
Regression modelsSuppose we have N documents, with each document i having labelyi ∈ {−1, 1} {liberal, conservative}We represent each document i is x i = (xi1, xi2, . . . , xiJ).
f (β,X ,Y ) =N∑i=1
(yi − β
′x i
)2β = arg minβ
{N∑i=1
(yi − β
′x i
)2}
=(X
′X)−1
X′Y
Problem:
- J will likely be large (perhaps J > N)
- There many correlated variables
Predictions will be variable
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 3 / 40
Regression modelsSuppose we have N documents, with each document i having labelyi ∈ {−1, 1} {liberal, conservative}We represent each document i is x i = (xi1, xi2, . . . , xiJ).
f (β,X ,Y ) =N∑i=1
(yi − β
′x i
)2β = arg minβ
{N∑i=1
(yi − β
′x i
)2}
=(X
′X)−1
X′Y
Problem:
- J will likely be large (perhaps J > N)
- There many correlated variables
Predictions will be variable
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 3 / 40
Regression modelsSuppose we have N documents, with each document i having labelyi ∈ {−1, 1} {liberal, conservative}We represent each document i is x i = (xi1, xi2, . . . , xiJ).
f (β,X ,Y ) =N∑i=1
(yi − β
′x i
)2β = arg minβ
{N∑i=1
(yi − β
′x i
)2}
=(X
′X)−1
X′Y
Problem:
- J will likely be large (perhaps J > N)
- There many correlated variables
Predictions will be variable
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 3 / 40
Regression modelsSuppose we have N documents, with each document i having labelyi ∈ {−1, 1} {liberal, conservative}We represent each document i is x i = (xi1, xi2, . . . , xiJ).
f (β,X ,Y ) =N∑i=1
(yi − β
′x i
)2β = arg minβ
{N∑i=1
(yi − β
′x i
)2}
=(X
′X)−1
X′Y
Problem:
- J will likely be large (perhaps J > N)
- There many correlated variables
Predictions will be variableJustin Grimmer (Stanford University) Text as Data November 10th, 2014 3 / 40
Mean Square Error
Suppose θ is some value of the true parameter
Bias:
Bias = E [θ − θ]
We may care about average distance from truth
E[(θ − θ)2] = E [θ2]− 2θE [θ] + θ2
= E [θ2]− E [θ]2 + E [θ]2 − 2θE [θ] + θ2
= E [θ2]− E [θ]2 + (E [θ − θ])2
= Var(θ) + Bias2
To reduce MSE, we are willing to induce bias to decrease variance methods that shrink coefficeints toward zero
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 4 / 40
Mean Square Error
Suppose θ is some value of the true parameterBias:
Bias = E [θ − θ]
We may care about average distance from truth
E[(θ − θ)2] = E [θ2]− 2θE [θ] + θ2
= E [θ2]− E [θ]2 + E [θ]2 − 2θE [θ] + θ2
= E [θ2]− E [θ]2 + (E [θ − θ])2
= Var(θ) + Bias2
To reduce MSE, we are willing to induce bias to decrease variance methods that shrink coefficeints toward zero
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 4 / 40
Mean Square Error
Suppose θ is some value of the true parameterBias:
Bias = E [θ − θ]
We may care about average distance from truth
E[(θ − θ)2] = E [θ2]− 2θE [θ] + θ2
= E [θ2]− E [θ]2 + E [θ]2 − 2θE [θ] + θ2
= E [θ2]− E [θ]2 + (E [θ − θ])2
= Var(θ) + Bias2
To reduce MSE, we are willing to induce bias to decrease variance methods that shrink coefficeints toward zero
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 4 / 40
Mean Square Error
Suppose θ is some value of the true parameterBias:
Bias = E [θ − θ]
We may care about average distance from truth
E[(θ − θ)2] = E [θ2]− 2θE [θ] + θ2
= E [θ2]− E [θ]2 + E [θ]2 − 2θE [θ] + θ2
= E [θ2]− E [θ]2 + (E [θ − θ])2
= Var(θ) + Bias2
To reduce MSE, we are willing to induce bias to decrease variance methods that shrink coefficeints toward zero
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 4 / 40
Mean Square Error
Suppose θ is some value of the true parameterBias:
Bias = E [θ − θ]
We may care about average distance from truth
E[(θ − θ)2]
= E [θ2]− 2θE [θ] + θ2
= E [θ2]− E [θ]2 + E [θ]2 − 2θE [θ] + θ2
= E [θ2]− E [θ]2 + (E [θ − θ])2
= Var(θ) + Bias2
To reduce MSE, we are willing to induce bias to decrease variance methods that shrink coefficeints toward zero
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 4 / 40
Mean Square Error
Suppose θ is some value of the true parameterBias:
Bias = E [θ − θ]
We may care about average distance from truth
E[(θ − θ)2] = E [θ2]− 2θE [θ] + θ2
= E [θ2]− E [θ]2 + E [θ]2 − 2θE [θ] + θ2
= E [θ2]− E [θ]2 + (E [θ − θ])2
= Var(θ) + Bias2
To reduce MSE, we are willing to induce bias to decrease variance methods that shrink coefficeints toward zero
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 4 / 40
Mean Square Error
Suppose θ is some value of the true parameterBias:
Bias = E [θ − θ]
We may care about average distance from truth
E[(θ − θ)2] = E [θ2]− 2θE [θ] + θ2
= E [θ2]− E [θ]2 + E [θ]2 − 2θE [θ] + θ2
= E [θ2]− E [θ]2 + (E [θ − θ])2
= Var(θ) + Bias2
To reduce MSE, we are willing to induce bias to decrease variance methods that shrink coefficeints toward zero
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 4 / 40
Mean Square Error
Suppose θ is some value of the true parameterBias:
Bias = E [θ − θ]
We may care about average distance from truth
E[(θ − θ)2] = E [θ2]− 2θE [θ] + θ2
= E [θ2]− E [θ]2 + E [θ]2 − 2θE [θ] + θ2
= E [θ2]− E [θ]2 + (E [θ − θ])2
= Var(θ) + Bias2
To reduce MSE, we are willing to induce bias to decrease variance methods that shrink coefficeints toward zero
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 4 / 40
Mean Square Error
Suppose θ is some value of the true parameterBias:
Bias = E [θ − θ]
We may care about average distance from truth
E[(θ − θ)2] = E [θ2]− 2θE [θ] + θ2
= E [θ2]− E [θ]2 + E [θ]2 − 2θE [θ] + θ2
= E [θ2]− E [θ]2 + (E [θ − θ])2
= Var(θ) + Bias2
To reduce MSE, we are willing to induce bias to decrease variance methods that shrink coefficeints toward zero
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 4 / 40
Mean Square Error
Suppose θ is some value of the true parameterBias:
Bias = E [θ − θ]
We may care about average distance from truth
E[(θ − θ)2] = E [θ2]− 2θE [θ] + θ2
= E [θ2]− E [θ]2 + E [θ]2 − 2θE [θ] + θ2
= E [θ2]− E [θ]2 + (E [θ − θ])2
= Var(θ) + Bias2
To reduce MSE, we are willing to induce bias to decrease variance methods that shrink coefficeints toward zero
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 4 / 40
Ridge Regression
Penalty for model complexity
f (β,X ,Y ) =N∑i=1
yi − β0 +J∑
j=1
βjxij
2
+ λJ∑
j=1
β2j︸ ︷︷ ︸Penalty
where:
- β0 intercept
- λ penalty parameter
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 5 / 40
Ridge Regression
Penalty for model complexity
f (β,X ,Y )
=N∑i=1
yi − β0 +J∑
j=1
βjxij
2
+ λJ∑
j=1
β2j︸ ︷︷ ︸Penalty
where:
- β0 intercept
- λ penalty parameter
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 5 / 40
Ridge Regression
Penalty for model complexity
f (β,X ,Y ) =N∑i=1
yi − β0 +J∑
j=1
βjxij
2
+ λJ∑
j=1
β2j︸ ︷︷ ︸Penalty
where:
- β0 intercept
- λ penalty parameter
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 5 / 40
Ridge Regression
Penalty for model complexity
f (β,X ,Y ) =N∑i=1
yi − β0 +J∑
j=1
βjxij
2
+ λ
J∑j=1
β2j︸ ︷︷ ︸Penalty
where:
- β0 intercept
- λ penalty parameter
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 5 / 40
Ridge Regression
Penalty for model complexity
f (β,X ,Y ) =N∑i=1
yi − β0 +J∑
j=1
βjxij
2
+ λ
J∑j=1
β2j︸ ︷︷ ︸Penalty
where:
- β0 intercept
- λ penalty parameter
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 5 / 40
Ridge Regression
Penalty for model complexity
f (β,X ,Y ) =N∑i=1
yi − β0 +J∑
j=1
βjxij
2
+ λ
J∑j=1
β2j︸ ︷︷ ︸Penalty
where:
- β0 intercept
- λ penalty parameter
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 5 / 40
Ridge Regression
Penalty for model complexity
f (β,X ,Y ) =N∑i=1
yi − β0 +J∑
j=1
βjxij
2
+ λ
J∑j=1
β2j︸ ︷︷ ︸Penalty
where:
- β0 intercept
- λ penalty parameter
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 5 / 40
Ridge Regression Optimization
βRidge = arg minβ {f (β,X ,Y )}
= arg minβ
N∑i=1
yi − β0 +J∑
j=1
βjxij
2
+ λJ∑
j=1
β2j
= arg minβ
{(Y − X
′β)
′(Y − X
′β) + λβ
′β}
=(X
′X + λI J
)−1X
′Y
Demean the data and set β0 = y =∑N
i=1yiN
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 6 / 40
Ridge Regression Optimization
βRidge = arg minβ {f (β,X ,Y )}
= arg minβ
N∑i=1
yi − β0 +J∑
j=1
βjxij
2
+ λ
J∑j=1
β2j
= arg minβ
{(Y − X
′β)
′(Y − X
′β) + λβ
′β}
=(X
′X + λI J
)−1X
′Y
Demean the data and set β0 = y =∑N
i=1yiN
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 6 / 40
Ridge Regression Optimization
βRidge = arg minβ {f (β,X ,Y )}
= arg minβ
N∑i=1
yi − β0 +J∑
j=1
βjxij
2
+ λ
J∑j=1
β2j
= arg minβ
{(Y − X
′β)
′(Y − X
′β) + λβ
′β}
=(X
′X + λI J
)−1X
′Y
Demean the data and set β0 = y =∑N
i=1yiN
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 6 / 40
Ridge Regression Optimization
βRidge = arg minβ {f (β,X ,Y )}
= arg minβ
N∑i=1
yi − β0 +J∑
j=1
βjxij
2
+ λ
J∑j=1
β2j
= arg minβ
{(Y − X
′β)
′(Y − X
′β) + λβ
′β}
=(X
′X + λI J
)−1X
′Y
Demean the data and set β0 = y =∑N
i=1yiN
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 6 / 40
Ridge Regression Optimization
βRidge = arg minβ {f (β,X ,Y )}
= arg minβ
N∑i=1
yi − β0 +J∑
j=1
βjxij
2
+ λ
J∑j=1
β2j
= arg minβ
{(Y − X
′β)
′(Y − X
′β) + λβ
′β}
=(X
′X + λI J
)−1X
′Y
Demean the data and set β0 = y =∑N
i=1yiN
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 6 / 40
Ridge Regression Intuition (1)
Suppose X′X = I J .
β =(X
′X)−1
X′Y
= X′Y
βridge =(X
′X + λI J
)−1X
′Y
= (I j + λI j) X′Y
= (I j + λI j) β
βRidgej =βj
1 + λ
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 7 / 40
Ridge Regression Intuition (1)
Suppose X′X = I J .
β =(X
′X)−1
X′Y
= X′Y
βridge =(X
′X + λI J
)−1X
′Y
= (I j + λI j) X′Y
= (I j + λI j) β
βRidgej =βj
1 + λ
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 7 / 40
Ridge Regression Intuition (1)
Suppose X′X = I J .
β =(X
′X)−1
X′Y
= X′Y
βridge =(X
′X + λI J
)−1X
′Y
= (I j + λI j) X′Y
= (I j + λI j) β
βRidgej =βj
1 + λ
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 7 / 40
Ridge Regression Intuition (1)
Suppose X′X = I J .
β =(X
′X)−1
X′Y
= X′Y
βridge =(X
′X + λI J
)−1X
′Y
= (I j + λI j) X′Y
= (I j + λI j) β
βRidgej =βj
1 + λ
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 7 / 40
Ridge Regression Intuition (1)
Suppose X′X = I J .
β =(X
′X)−1
X′Y
= X′Y
βridge =(X
′X + λI J
)−1X
′Y
= (I j + λI j) X′Y
= (I j + λI j) β
βRidgej =βj
1 + λ
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 7 / 40
Ridge Regression Intuition (1)
Suppose X′X = I J .
β =(X
′X)−1
X′Y
= X′Y
βridge =(X
′X + λI J
)−1X
′Y
= (I j + λI j) X′Y
= (I j + λI j) β
βRidgej =βj
1 + λ
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 7 / 40
Ridge Regression Intuition (1)
Suppose X′X = I J .
β =(X
′X)−1
X′Y
= X′Y
βridge =(X
′X + λI J
)−1X
′Y
= (I j + λI j) X′Y
= (I j + λI j) β
βRidgej =βj
1 + λ
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 7 / 40
Ridge Regression Intuition (2)
βj ∼ Normal(0, τ2)
yi ∼ Normal(β0 + x′iβ, σ
2)
p(β|X ,Y ) ∝J∏
j=1
p(βj)N∏i=1
p(yi |x i ,β)
∝J∏
j=1
1√2πτ
exp
(−β2j
2τ 2
)N∏i=1
1√2πσ
exp
(− (yi − β0 + x
′β)2
2σ2
)
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 8 / 40
Ridge Regression Intuition (2)
βj ∼ Normal(0, τ2)
yi ∼ Normal(β0 + x′iβ, σ
2)
p(β|X ,Y ) ∝J∏
j=1
p(βj)N∏i=1
p(yi |x i ,β)
∝J∏
j=1
1√2πτ
exp
(−β2j
2τ 2
)N∏i=1
1√2πσ
exp
(− (yi − β0 + x
′β)2
2σ2
)
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 8 / 40
Ridge Regression Intuition (2)
βj ∼ Normal(0, τ2)
yi ∼ Normal(β0 + x′iβ, σ
2)
p(β|X ,Y ) ∝J∏
j=1
p(βj)N∏i=1
p(yi |x i ,β)
∝J∏
j=1
1√2πτ
exp
(−β2j
2τ 2
)N∏i=1
1√2πσ
exp
(− (yi − β0 + x
′β)2
2σ2
)
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 8 / 40
Ridge Regression Intuition (2)
log p(β|X ,Y ) = −J∑
j=1
β2j2τ2−
N∑i=1
(yi − β0 + x ′β)2
2σ2
− 2σ2 log p(β|X ,Y ) =N∑i=1
(yi − β0 + x′β)2 +
J∑j=1
σ2
τ2β2j
where:
- λ = σ2
τ2β2j
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 9 / 40
Ridge Regression Intuition (2)
log p(β|X ,Y ) = −J∑
j=1
β2j2τ2−
N∑i=1
(yi − β0 + x ′β)2
2σ2
− 2σ2 log p(β|X ,Y ) =N∑i=1
(yi − β0 + x′β)2 +
J∑j=1
σ2
τ2β2j
where:
- λ = σ2
τ2β2j
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 9 / 40
Ridge Regression Intuition (2)
log p(β|X ,Y ) = −J∑
j=1
β2j2τ2−
N∑i=1
(yi − β0 + x ′β)2
2σ2
− 2σ2 log p(β|X ,Y ) =N∑i=1
(yi − β0 + x′β)2 +
J∑j=1
σ2
τ2β2j
where:
- λ = σ2
τ2β2j
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 9 / 40
Ridge Regression Intuition (2)
log p(β|X ,Y ) = −J∑
j=1
β2j2τ2−
N∑i=1
(yi − β0 + x ′β)2
2σ2
− 2σ2 log p(β|X ,Y ) =N∑i=1
(yi − β0 + x′β)2 +
J∑j=1
σ2
τ2β2j
where:
- λ = σ2
τ2β2j
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 9 / 40
Lasso Regression Objective Function/Optimization
Different Penalty for Model Complexity
f (β,X ,Y ) =N∑i=1
yi − β0 +J∑
j=1
βjxij
2
+ λ
J∑j=1
|βj |︸︷︷︸Penalty
- Optimization is non-linear (Absolute Value)
- Coordinate Descent- Start with Ridge- Sub-differential, update steps
- Induces sparsity sets some coefficients to zero
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 10 / 40
Lasso Regression Objective Function/Optimization
Different Penalty for Model Complexity
f (β,X ,Y ) =N∑i=1
yi − β0 +J∑
j=1
βjxij
2
+ λ
J∑j=1
|βj |︸︷︷︸Penalty
- Optimization is non-linear (Absolute Value)
- Coordinate Descent- Start with Ridge- Sub-differential, update steps
- Induces sparsity sets some coefficients to zero
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 10 / 40
Lasso Regression Objective Function/Optimization
Different Penalty for Model Complexity
f (β,X ,Y ) =N∑i=1
yi − β0 +J∑
j=1
βjxij
2
+ λ
J∑j=1
|βj |︸︷︷︸Penalty
- Optimization is non-linear (Absolute Value)
- Coordinate Descent
- Start with Ridge- Sub-differential, update steps
- Induces sparsity sets some coefficients to zero
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 10 / 40
Lasso Regression Objective Function/Optimization
Different Penalty for Model Complexity
f (β,X ,Y ) =N∑i=1
yi − β0 +J∑
j=1
βjxij
2
+ λ
J∑j=1
|βj |︸︷︷︸Penalty
- Optimization is non-linear (Absolute Value)
- Coordinate Descent- Start with Ridge
- Sub-differential, update steps
- Induces sparsity sets some coefficients to zero
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 10 / 40
Lasso Regression Objective Function/Optimization
Different Penalty for Model Complexity
f (β,X ,Y ) =N∑i=1
yi − β0 +J∑
j=1
βjxij
2
+ λ
J∑j=1
|βj |︸︷︷︸Penalty
- Optimization is non-linear (Absolute Value)
- Coordinate Descent- Start with Ridge- Sub-differential, update steps
- Induces sparsity sets some coefficients to zero
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 10 / 40
Lasso Regression Objective Function/Optimization
Different Penalty for Model Complexity
f (β,X ,Y ) =N∑i=1
yi − β0 +J∑
j=1
βjxij
2
+ λ
J∑j=1
|βj |︸︷︷︸Penalty
- Optimization is non-linear (Absolute Value)
- Coordinate Descent- Start with Ridge- Sub-differential, update steps
- Induces sparsity sets some coefficients to zero
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 10 / 40
Lasso Regression Intuition 1, Soft Thresholding
Suppose again X′X = I J
f (β,X ,Y ) = (Y − Xβ)′(Y − Xβ) + λ
J∑j=1
|βj |
= −2X′Yβ + β
′β + λ
J∑j=1
|βj |
The coefficient is
βLASSOj = sign(βj
)(|βj | − λ
)+
- sign(·) 1 or −1
-(|βj | − λ
)+
= max(|βj | − λ, 0)
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 11 / 40
Lasso Regression Intuition 1, Soft Thresholding
Suppose again X′X = I J
f (β,X ,Y ) = (Y − Xβ)′(Y − Xβ) + λ
J∑j=1
|βj |
= −2X′Yβ + β
′β + λ
J∑j=1
|βj |
The coefficient is
βLASSOj = sign(βj
)(|βj | − λ
)+
- sign(·) 1 or −1
-(|βj | − λ
)+
= max(|βj | − λ, 0)
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 11 / 40
Lasso Regression Intuition 1, Soft Thresholding
Suppose again X′X = I J
f (β,X ,Y ) = (Y − Xβ)′(Y − Xβ) + λ
J∑j=1
|βj |
= −2X′Yβ + β
′β + λ
J∑j=1
|βj |
The coefficient is
βLASSOj = sign(βj
)(|βj | − λ
)+
- sign(·) 1 or −1
-(|βj | − λ
)+
= max(|βj | − λ, 0)
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 11 / 40
Lasso Regression Intuition 1, Soft Thresholding
Suppose again X′X = I J
f (β,X ,Y ) = (Y − Xβ)′(Y − Xβ) + λ
J∑j=1
|βj |
= −2X′Yβ + β
′β + λ
J∑j=1
|βj |
The coefficient is
βLASSOj = sign(βj
)(|βj | − λ
)+
- sign(·) 1 or −1
-(|βj | − λ
)+
= max(|βj | − λ, 0)
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 11 / 40
Lasso Regression Intuition 1, Soft Thresholding
Suppose again X′X = I J
f (β,X ,Y ) = (Y − Xβ)′(Y − Xβ) + λ
J∑j=1
|βj |
= −2X′Yβ + β
′β + λ
J∑j=1
|βj |
The coefficient is
βLASSOj = sign(βj
)(|βj | − λ
)+
- sign(·) 1 or −1
-(|βj | − λ
)+
= max(|βj | − λ, 0)
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 11 / 40
Lasso Regression Intuition 1, Soft Thresholding
Suppose again X′X = I J
f (β,X ,Y ) = (Y − Xβ)′(Y − Xβ) + λ
J∑j=1
|βj |
= −2X′Yβ + β
′β + λ
J∑j=1
|βj |
The coefficient is
βLASSOj = sign(βj
)(|βj | − λ
)+
- sign(·) 1 or −1
-(|βj | − λ
)+
= max(|βj | − λ, 0)
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 11 / 40
Lasso Regression Intuition 1, Soft Thresholding
Suppose again X′X = I J
f (β,X ,Y ) = (Y − Xβ)′(Y − Xβ) + λ
J∑j=1
|βj |
= −2X′Yβ + β
′β + λ
J∑j=1
|βj |
The coefficient is
βLASSOj = sign(βj
)(|βj | − λ
)+
- sign(·) 1 or −1
-(|βj | − λ
)+
= max(|βj | − λ, 0)
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 11 / 40
Lasso Regression Intuition 1, Soft Thresholding
Compare soft assignment
βLASSOj = sign(βj
)(|βj | − λ
)+
With hard assignment, selecting M biggest components
βsubsetj = βj · I(|βj | ≥ |β(M)|
)Intuition 2: Prior on coefficients Double exponentialWhy does LASSO induce sparsity?
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 12 / 40
Lasso Regression Intuition 1, Soft Thresholding
Compare soft assignment
βLASSOj = sign(βj
)(|βj | − λ
)+
With hard assignment, selecting M biggest components
βsubsetj = βj · I(|βj | ≥ |β(M)|
)Intuition 2: Prior on coefficients Double exponentialWhy does LASSO induce sparsity?
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 12 / 40
Lasso Regression Intuition 1, Soft Thresholding
Compare soft assignment
βLASSOj = sign(βj
)(|βj | − λ
)+
With hard assignment, selecting M biggest components
βsubsetj = βj · I(|βj | ≥ |β(M)|
)Intuition 2: Prior on coefficients Double exponentialWhy does LASSO induce sparsity?
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 12 / 40
Lasso Regression Intuition 1, Soft Thresholding
Compare soft assignment
βLASSOj = sign(βj
)(|βj | − λ
)+
With hard assignment, selecting M biggest components
βsubsetj = βj · I(|βj | ≥ |β(M)|
)
Intuition 2: Prior on coefficients Double exponentialWhy does LASSO induce sparsity?
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 12 / 40
Lasso Regression Intuition 1, Soft Thresholding
Compare soft assignment
βLASSOj = sign(βj
)(|βj | − λ
)+
With hard assignment, selecting M biggest components
βsubsetj = βj · I(|βj | ≥ |β(M)|
)Intuition 2: Prior on coefficients Double exponential
Why does LASSO induce sparsity?
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 12 / 40
Lasso Regression Intuition 1, Soft Thresholding
Compare soft assignment
βLASSOj = sign(βj
)(|βj | − λ
)+
With hard assignment, selecting M biggest components
βsubsetj = βj · I(|βj | ≥ |β(M)|
)Intuition 2: Prior on coefficients Double exponentialWhy does LASSO induce sparsity?
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 12 / 40
Comparing Ridge and LASSO
−4 −2 0 2 4
−4
−2
02
4
●
Ridge Regression
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 13 / 40
Comparing Ridge and LASSO
−4 −2 0 2 4
−4
−2
02
4
●
LASSO Regression
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 13 / 40
Comparing Ridge and LASSOContrast β = ( 1√
2, 1√
2) and β = (1, 0)
Under ridge:
2∑j=1
β2j =1
2+
1
2= 1
2∑j=1
β2j = 1 + 0 = 1
Under LASSO
2∑j=1
|βj | =1√2
+1√2
=√
2
2∑j=1
|βj | = 1 + 0 = 1
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 14 / 40
Comparing Ridge and LASSOContrast β = ( 1√
2, 1√
2) and β = (1, 0)
Under ridge:
2∑j=1
β2j =1
2+
1
2= 1
2∑j=1
β2j = 1 + 0 = 1
Under LASSO
2∑j=1
|βj | =1√2
+1√2
=√
2
2∑j=1
|βj | = 1 + 0 = 1
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 14 / 40
Comparing Ridge and LASSOContrast β = ( 1√
2, 1√
2) and β = (1, 0)
Under ridge:
2∑j=1
β2j =1
2+
1
2= 1
2∑j=1
β2j = 1 + 0 = 1
Under LASSO
2∑j=1
|βj | =1√2
+1√2
=√
2
2∑j=1
|βj | = 1 + 0 = 1
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 14 / 40
Comparing Ridge and LASSOContrast β = ( 1√
2, 1√
2) and β = (1, 0)
Under ridge:
2∑j=1
β2j =1
2+
1
2= 1
2∑j=1
β2j = 1 + 0 = 1
Under LASSO
2∑j=1
|βj | =1√2
+1√2
=√
2
2∑j=1
|βj | = 1 + 0 = 1
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 14 / 40
Comparing Ridge and LASSOContrast β = ( 1√
2, 1√
2) and β = (1, 0)
Under ridge:
2∑j=1
β2j =1
2+
1
2= 1
2∑j=1
β2j = 1 + 0 = 1
Under LASSO
2∑j=1
|βj | =1√2
+1√2
=√
2
2∑j=1
|βj | = 1 + 0 = 1
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 14 / 40
Comparing Ridge and LASSOContrast β = ( 1√
2, 1√
2) and β = (1, 0)
Under ridge:
2∑j=1
β2j =1
2+
1
2= 1
2∑j=1
β2j = 1 + 0 = 1
Under LASSO
2∑j=1
|βj | =1√2
+1√2
=√
2
2∑j=1
|βj | = 1 + 0 = 1
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 14 / 40
Comparing Ridge and LASSOContrast β = ( 1√
2, 1√
2) and β = (1, 0)
Under ridge:
2∑j=1
β2j =1
2+
1
2= 1
2∑j=1
β2j = 1 + 0 = 1
Under LASSO
2∑j=1
|βj | =1√2
+1√2
=√
2
2∑j=1
|βj | = 1 + 0 = 1
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 14 / 40
Selecting λ
How do we determine λ? Cross validation (lecture on Thursday)
Applying models gives score (probability) of document belong to class threshold to classify
To the R code!
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 15 / 40
Selecting λ
How do we determine λ? Cross validation (lecture on Thursday)Applying models gives score (probability) of document belong to class threshold to classifyTo the R code!
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 15 / 40
Selecting λ
How do we determine λ? Cross validation (lecture on Thursday)Applying models gives score (probability) of document belong to class threshold to classifyTo the R code!
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 15 / 40
Assessing Models (Elements of Statistical Learning)
- Model Selection: tuning parameters to select final model (next week’sdiscussion)
- Model assessment : after selecting model, estimating error inclassification
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 16 / 40
Comparing Training and Validation Set
Text classification and model assessment
- Replicate classification exercise with validation set
- General principle of classification/prediction
- Compare supervised learning labels to hand labels
Confusion matrix
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 17 / 40
Comparing Training and Validation Set
Representation of Test Statistics from Dictionary week (along with somenew ones)
Actual Label
Classification (algorithm) Liberal Conservative
Liberal True Liberal False Liberal
Conservative False Conservative True Conservative
Accuracy =TrueLib + TrueCons
TrueLib + TrueCons + FalseLib + FalseCons
PrecisionLiberal =True Liberal
True Liberal + False Liberal
RecallLiberal =True Liberal
True Liberal + False Conservative
FLiberal =2PrecisionLiberalRecallLiberal
PrecisionLiberal + RecallLiberal
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 18 / 40
Comparing Training and Validation Set
Representation of Test Statistics from Dictionary week (along with somenew ones)
Actual Label
Classification (algorithm) Liberal Conservative
Liberal True Liberal False Liberal
Conservative False Conservative True Conservative
Accuracy =TrueLib + TrueCons
TrueLib + TrueCons + FalseLib + FalseCons
PrecisionLiberal =True Liberal
True Liberal + False Liberal
RecallLiberal =True Liberal
True Liberal + False Conservative
FLiberal =2PrecisionLiberalRecallLiberal
PrecisionLiberal + RecallLiberal
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 18 / 40
Comparing Training and Validation Set
Representation of Test Statistics from Dictionary week (along with somenew ones)
Actual Label
Classification (algorithm) Liberal Conservative
Liberal True Liberal False Liberal
Conservative False Conservative True Conservative
Accuracy =TrueLib + TrueCons
TrueLib + TrueCons + FalseLib + FalseCons
PrecisionLiberal =True Liberal
True Liberal + False Liberal
RecallLiberal =True Liberal
True Liberal + False Conservative
FLiberal =2PrecisionLiberalRecallLiberal
PrecisionLiberal + RecallLiberal
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 18 / 40
Comparing Training and Validation Set
Representation of Test Statistics from Dictionary week (along with somenew ones)
Actual Label
Classification (algorithm) Liberal Conservative
Liberal True Liberal False Liberal
Conservative False Conservative True Conservative
Accuracy =TrueLib + TrueCons
TrueLib + TrueCons + FalseLib + FalseCons
PrecisionLiberal =True Liberal
True Liberal + False Liberal
RecallLiberal =True Liberal
True Liberal + False Conservative
FLiberal =2PrecisionLiberalRecallLiberal
PrecisionLiberal + RecallLiberal
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 18 / 40
Comparing Training and Validation Set
Representation of Test Statistics from Dictionary week (along with somenew ones)
Actual Label
Classification (algorithm) Liberal Conservative
Liberal True Liberal False Liberal
Conservative False Conservative True Conservative
Accuracy =TrueLib + TrueCons
TrueLib + TrueCons + FalseLib + FalseCons
PrecisionLiberal =True Liberal
True Liberal + False Liberal
RecallLiberal =True Liberal
True Liberal + False Conservative
FLiberal =2PrecisionLiberalRecallLiberal
PrecisionLiberal + RecallLiberal
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 18 / 40
Comparing Training and Validation Set
Representation of Test Statistics from Dictionary week (along with somenew ones)
Actual Label
Classification (algorithm) Liberal Conservative
Liberal True Liberal False Liberal
Conservative False Conservative True Conservative
Accuracy =TrueLib + TrueCons
TrueLib + TrueCons + FalseLib + FalseCons
PrecisionLiberal =True Liberal
True Liberal + False Liberal
RecallLiberal =True Liberal
True Liberal + False Conservative
FLiberal =2PrecisionLiberalRecallLiberal
PrecisionLiberal + RecallLiberal
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 18 / 40
ROC Curve
ROC as a measure of model performance
RecallLiberal =True Liberal
True Liberal + False Conservative
RecallConservative =True Conservative
True Conservative + False Liberal
Tension:
- Everything liberal: RecallLiberal =1 ; RecallConservative = 0
- Everything conservative: RecallLiberal =0 ; RecallConservative = 1
Characterize Tradeoff:Plot True Positive Rate RecallLiberalFalse Positive Rate (1 - RecallConservative)
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 19 / 40
Precision/Recall Tradeoff
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
False Positive Rate
True
Pos
itive
Rat
e
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 20 / 40
Simple Classification Example
Analyzing house press releasesHand Code: 1,000 press releases
- Advertising
- Credit Claiming
- Position Taking
Divide 1,000 press releases into two sets
- 500: Training set
- 500: Test set
Initial exploration: provides baseline measurement at classifierperformancesImprove: through improving model fit
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 21 / 40
Example from First Model FitActual Label
Classification (Naive Bayes) Position Taking Advertising Credit Claim.Position Taking 10 0 0Advertising 2 40 2Credit Claiming 80 60 306
Accuracy =10 + 40 + 306
500= 0.71
PrecisionPT =10
10= 1
RecallPT =10
10 + 2 + 80= 0.11
PrecisionAD =40
40 + 2 + 2= 0.91
RecallAD =40
40 + 60= 0.4
PrecisionCredit =306
306 + 80 + 60= 0.67
RecallCredit =306
306 + 2= 0.99
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 22 / 40
Fit Statistics in R
RWeka library provides Amazing functionality.You can easily code them yourself
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 23 / 40
Support Vector Machines
Document i is an J × 1 vector of counts
x i = (x1i , x2i , . . . , xJi )
Suppose we have two classes, C1,C2.
Yi = 1 if i is in class 1
Yi = −1 if i is in class 2
Suppose they are separable:
- Draw a line between groups
- Goal: identify the line in the middle
- Maximum margin
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 24 / 40
Support Vector Machines: Maximum Margin Classifier(Bishop 2006)
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Justin Grimmer (Stanford University) Text as Data November 10th, 2014 25 / 40
Support Vector Machines: Maximum Margin Classifier(Bishop 2006)
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Justin Grimmer (Stanford University) Text as Data November 10th, 2014 25 / 40
Support Vector Machines: Maximum Margin Classifier(Bishop 2006)
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Justin Grimmer (Stanford University) Text as Data November 10th, 2014 25 / 40
Support Vector Machines: Maximum Margin Classifier(Bishop 2006)
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Justin Grimmer (Stanford University) Text as Data November 10th, 2014 25 / 40
Support Vector Machines: Maximum Margin Classifier(Bishop 2006)
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Justin Grimmer (Stanford University) Text as Data November 10th, 2014 25 / 40
Support Vector Machines: Algebra (Bishop 2006)
Goal create a score to classify:
s(x i ) = β′x i + b
- β Determines orientation of surface (slope)
- b determines location (moves surface up or down)
- If s(x i ) > 0→ class 1
- If s(x i ) < 0→ class 2
- |s(x i )|||β|| = Document distance from decision surface (margin)
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 26 / 40
Support Vector Machines: Algebra (Bishop 2006)
Objective function: maximum margin
mini [ |(s(x i )| ]: Point closest to decision surfaceWe want to identify β and b to maximize the margin:
arg maxβ,b
{1
||β||mini [ |(s(x i )| ]
}arg maxβ,b
{1
||β||mini [ |β
′x i + b| ]
}Constrained optimization problem Quadratic programming problem
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 27 / 40
Support Vector Machines: Algebra (Bishop 2006)
Objective function: maximum marginmini [ |(s(x i )| ]: Point closest to decision surface
We want to identify β and b to maximize the margin:
arg maxβ,b
{1
||β||mini [ |(s(x i )| ]
}arg maxβ,b
{1
||β||mini [ |β
′x i + b| ]
}Constrained optimization problem Quadratic programming problem
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 27 / 40
Support Vector Machines: Algebra (Bishop 2006)
Objective function: maximum marginmini [ |(s(x i )| ]: Point closest to decision surface
We want to identify β and b to maximize the margin:
arg maxβ,b
{1
||β||mini [ |(s(x i )| ]
}arg maxβ,b
{1
||β||mini [ |β
′x i + b| ]
}Constrained optimization problem Quadratic programming problem
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 27 / 40
Support Vector Machines: Algebra (Bishop 2006)
Objective function: maximum marginmini [ |(s(x i )| ]: Point closest to decision surface
We want to identify β and b to maximize the margin:
arg maxβ,b
{1
||β||mini [ |(s(x i )| ]
}
arg maxβ,b
{1
||β||mini [ |β
′x i + b| ]
}Constrained optimization problem Quadratic programming problem
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 27 / 40
Support Vector Machines: Algebra (Bishop 2006)
Objective function: maximum marginmini [ |(s(x i )| ]: Point closest to decision surface
We want to identify β and b to maximize the margin:
arg maxβ,b
{1
||β||mini [ |(s(x i )| ]
}arg maxβ,b
{1
||β||mini [ |β
′x i + b| ]
}
Constrained optimization problem Quadratic programming problem
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 27 / 40
Support Vector Machines: Algebra (Bishop 2006)
Objective function: maximum marginmini [ |(s(x i )| ]: Point closest to decision surface
We want to identify β and b to maximize the margin:
arg maxβ,b
{1
||β||mini [ |(s(x i )| ]
}arg maxβ,b
{1
||β||mini [ |β
′x i + b| ]
}Constrained optimization problem Quadratic programming problem
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 27 / 40
What About Overlap? (Bishop 2006)
- Rare that classes are separable.
- Define:
ξi = 0 if correctly classified
ξi = |s(x i )| if incorrectly classified
Tradeoff:
- Maximize margin between correctly classified groups
- Minimize error from misclassified documents
arg maxβ,b
{C
N∑i=1
ξi +1
||β||mini [ |β
′x i + b| ]
}
C captures tradeoff
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 28 / 40
What About Overlap? (Bishop 2006)
- Rare that classes are separable.
- Define:
ξi = 0 if correctly classified
ξi = |s(x i )| if incorrectly classified
Tradeoff:
- Maximize margin between correctly classified groups
- Minimize error from misclassified documents
arg maxβ,b
{C
N∑i=1
ξi +1
||β||mini [ |β
′x i + b| ]
}
C captures tradeoff
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 28 / 40
What About Overlap? (Bishop 2006)
- Rare that classes are separable.
- Define:
ξi = 0 if correctly classified
ξi = |s(x i )| if incorrectly classified
Tradeoff:
- Maximize margin between correctly classified groups
- Minimize error from misclassified documents
arg maxβ,b
{C
N∑i=1
ξi +1
||β||mini [ |β
′x i + b| ]
}
C captures tradeoff
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 28 / 40
What About Overlap? (Bishop 2006)
- Rare that classes are separable.
- Define:
ξi = 0 if correctly classified
ξi = |s(x i )| if incorrectly classified
Tradeoff:
- Maximize margin between correctly classified groups
- Minimize error from misclassified documents
arg maxβ,b
{C
N∑i=1
ξi +1
||β||mini [ |β
′x i + b| ]
}
C captures tradeoff
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 28 / 40
What About Overlap? (Bishop 2006)
- Rare that classes are separable.
- Define:
ξi = 0 if correctly classified
ξi = |s(x i )| if incorrectly classified
Tradeoff:
- Maximize margin between correctly classified groups
- Minimize error from misclassified documents
arg maxβ,b
{C
N∑i=1
ξi +1
||β||mini [ |β
′x i + b| ]
}
C captures tradeoff
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 28 / 40
What About Overlap? (Bishop 2006)
- Rare that classes are separable.
- Define:
ξi = 0 if correctly classified
ξi = |s(x i )| if incorrectly classified
Tradeoff:
- Maximize margin between correctly classified groups
- Minimize error from misclassified documents
arg maxβ,b
{C
N∑i=1
ξi +1
||β||mini [ |β
′x i + b| ]
}
C captures tradeoff
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 28 / 40
What About Overlap? (Bishop 2006)
- Rare that classes are separable.
- Define:
ξi = 0 if correctly classified
ξi = |s(x i )| if incorrectly classified
Tradeoff:
- Maximize margin between correctly classified groups
- Minimize error from misclassified documents
arg maxβ,b
{C
N∑i=1
ξi +1
||β||mini [ |β
′x i + b| ]
}
C captures tradeoff
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 28 / 40
What About Overlap? (Bishop 2006)
- Rare that classes are separable.
- Define:
ξi = 0 if correctly classified
ξi = |s(x i )| if incorrectly classified
Tradeoff:
- Maximize margin between correctly classified groups
- Minimize error from misclassified documents
arg maxβ,b
{C
N∑i=1
ξi +1
||β||mini [ |β
′x i + b| ]
}
C captures tradeoff
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 28 / 40
What About Overlap? (Bishop 2006)
- Rare that classes are separable.
- Define:
ξi = 0 if correctly classified
ξi = |s(x i )| if incorrectly classified
Tradeoff:
- Maximize margin between correctly classified groups
- Minimize error from misclassified documents
arg maxβ,b
{C
N∑i=1
ξi +1
||β||mini [ |β
′x i + b| ]
}
C captures tradeoff
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 28 / 40
What About Overlap? (Bishop 2006)
- Rare that classes are separable.
- Define:
ξi = 0 if correctly classified
ξi = |s(x i )| if incorrectly classified
Tradeoff:
- Maximize margin between correctly classified groups
- Minimize error from misclassified documents
arg maxβ,b
{C
N∑i=1
ξi +1
||β||mini [ |β
′x i + b| ]
}
C captures tradeoff
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 28 / 40
How to Handle Multiple Comparisons?
- Rare that we only want to classify two categories
- How to handle classification into K groups?1) Set up K classification problems:
- Compare each class to all other classes- Problem: can lead to inconsistent results- Solution(?): select category with largest “score”- Problem: scales are not comparable
2) Common solution: set up K (K − 1)/2 classifications- Perform vote to select class (still suboptimal)
3) Simultaneous estimation possible, much slower
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 29 / 40
How to Handle Multiple Comparisons?
- Rare that we only want to classify two categories
- How to handle classification into K groups?
1) Set up K classification problems:
- Compare each class to all other classes- Problem: can lead to inconsistent results- Solution(?): select category with largest “score”- Problem: scales are not comparable
2) Common solution: set up K (K − 1)/2 classifications- Perform vote to select class (still suboptimal)
3) Simultaneous estimation possible, much slower
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 29 / 40
How to Handle Multiple Comparisons?
- Rare that we only want to classify two categories
- How to handle classification into K groups?1) Set up K classification problems:
- Compare each class to all other classes- Problem: can lead to inconsistent results- Solution(?): select category with largest “score”- Problem: scales are not comparable
2) Common solution: set up K (K − 1)/2 classifications- Perform vote to select class (still suboptimal)
3) Simultaneous estimation possible, much slower
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 29 / 40
How to Handle Multiple Comparisons?
- Rare that we only want to classify two categories
- How to handle classification into K groups?1) Set up K classification problems:
- Compare each class to all other classes
- Problem: can lead to inconsistent results- Solution(?): select category with largest “score”- Problem: scales are not comparable
2) Common solution: set up K (K − 1)/2 classifications- Perform vote to select class (still suboptimal)
3) Simultaneous estimation possible, much slower
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 29 / 40
How to Handle Multiple Comparisons?
- Rare that we only want to classify two categories
- How to handle classification into K groups?1) Set up K classification problems:
- Compare each class to all other classes- Problem: can lead to inconsistent results
- Solution(?): select category with largest “score”- Problem: scales are not comparable
2) Common solution: set up K (K − 1)/2 classifications- Perform vote to select class (still suboptimal)
3) Simultaneous estimation possible, much slower
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 29 / 40
How to Handle Multiple Comparisons?
- Rare that we only want to classify two categories
- How to handle classification into K groups?1) Set up K classification problems:
- Compare each class to all other classes- Problem: can lead to inconsistent results- Solution(?): select category with largest “score”
- Problem: scales are not comparable
2) Common solution: set up K (K − 1)/2 classifications- Perform vote to select class (still suboptimal)
3) Simultaneous estimation possible, much slower
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 29 / 40
How to Handle Multiple Comparisons?
- Rare that we only want to classify two categories
- How to handle classification into K groups?1) Set up K classification problems:
- Compare each class to all other classes- Problem: can lead to inconsistent results- Solution(?): select category with largest “score”- Problem: scales are not comparable
2) Common solution: set up K (K − 1)/2 classifications- Perform vote to select class (still suboptimal)
3) Simultaneous estimation possible, much slower
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 29 / 40
How to Handle Multiple Comparisons?
- Rare that we only want to classify two categories
- How to handle classification into K groups?1) Set up K classification problems:
- Compare each class to all other classes- Problem: can lead to inconsistent results- Solution(?): select category with largest “score”- Problem: scales are not comparable
2) Common solution: set up K (K − 1)/2 classifications
- Perform vote to select class (still suboptimal)3) Simultaneous estimation possible, much slower
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 29 / 40
How to Handle Multiple Comparisons?
- Rare that we only want to classify two categories
- How to handle classification into K groups?1) Set up K classification problems:
- Compare each class to all other classes- Problem: can lead to inconsistent results- Solution(?): select category with largest “score”- Problem: scales are not comparable
2) Common solution: set up K (K − 1)/2 classifications- Perform vote to select class (still suboptimal)
3) Simultaneous estimation possible, much slower
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 29 / 40
How to Handle Multiple Comparisons?
- Rare that we only want to classify two categories
- How to handle classification into K groups?1) Set up K classification problems:
- Compare each class to all other classes- Problem: can lead to inconsistent results- Solution(?): select category with largest “score”- Problem: scales are not comparable
2) Common solution: set up K (K − 1)/2 classifications- Perform vote to select class (still suboptimal)
3) Simultaneous estimation possible, much slower
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 29 / 40
R Code to Run SVMs
library(e1071)
fit<- svm(T . , as.data.frame(tdm) , method =’C’,
kernel=’linear’)
where: method = ‘C’ → Classificationkernel=’linear’ → allows for distortion of feature space. Options:
- Linear
- Polynomial
- Radial
- sigmoid
preds<- predict(fit, data =
as.data.frame(tdm[-c(1:no.train),]))
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 30 / 40
Example of SVMs in Political Science Research
Hillard, Purpura, Wilkerson: SVMs to code topic/sub topics for policyagendas project
SVMs are under utilized in political science
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 31 / 40
ReadMe: Optimization for a Different Goal (Hopkins andKing 2010)
Naive Bayes (and next week, SVM): focused on individual documentclassification.
But what if we’re focused on proportions only?Hopkins and King (2010): method for characterizing distribution of classesCan be much more accurate than individual classifiers, requires fewerassumptions (do not need random sample of documents ) .
- King and Lu (2008): derive method for characterizing causes ofdeaths for verbal autopsies
- Hopkins and King (2010): extend the method to text documents
Basic intuition:
- Examine joint distribution of characteristics (without making NaiveBayes like assumption)
- Focus on distributions (only) makes this analysis possible
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 32 / 40
ReadMe: Optimization for a Different Goal (Hopkins andKing 2010)
Naive Bayes (and next week, SVM): focused on individual documentclassification.But what if we’re focused on proportions only?
Hopkins and King (2010): method for characterizing distribution of classesCan be much more accurate than individual classifiers, requires fewerassumptions (do not need random sample of documents ) .
- King and Lu (2008): derive method for characterizing causes ofdeaths for verbal autopsies
- Hopkins and King (2010): extend the method to text documents
Basic intuition:
- Examine joint distribution of characteristics (without making NaiveBayes like assumption)
- Focus on distributions (only) makes this analysis possible
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 32 / 40
ReadMe: Optimization for a Different Goal (Hopkins andKing 2010)
Naive Bayes (and next week, SVM): focused on individual documentclassification.But what if we’re focused on proportions only?Hopkins and King (2010): method for characterizing distribution of classes
Can be much more accurate than individual classifiers, requires fewerassumptions (do not need random sample of documents ) .
- King and Lu (2008): derive method for characterizing causes ofdeaths for verbal autopsies
- Hopkins and King (2010): extend the method to text documents
Basic intuition:
- Examine joint distribution of characteristics (without making NaiveBayes like assumption)
- Focus on distributions (only) makes this analysis possible
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 32 / 40
ReadMe: Optimization for a Different Goal (Hopkins andKing 2010)
Naive Bayes (and next week, SVM): focused on individual documentclassification.But what if we’re focused on proportions only?Hopkins and King (2010): method for characterizing distribution of classesCan be much more accurate than individual classifiers, requires fewerassumptions (do not need random sample of documents ) .
- King and Lu (2008): derive method for characterizing causes ofdeaths for verbal autopsies
- Hopkins and King (2010): extend the method to text documents
Basic intuition:
- Examine joint distribution of characteristics (without making NaiveBayes like assumption)
- Focus on distributions (only) makes this analysis possible
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 32 / 40
ReadMe: Optimization for a Different Goal (Hopkins andKing 2010)
Naive Bayes (and next week, SVM): focused on individual documentclassification.But what if we’re focused on proportions only?Hopkins and King (2010): method for characterizing distribution of classesCan be much more accurate than individual classifiers, requires fewerassumptions (do not need random sample of documents ) .
- King and Lu (2008): derive method for characterizing causes ofdeaths for verbal autopsies
- Hopkins and King (2010): extend the method to text documents
Basic intuition:
- Examine joint distribution of characteristics (without making NaiveBayes like assumption)
- Focus on distributions (only) makes this analysis possible
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 32 / 40
ReadMe: Optimization for a Different Goal (Hopkins andKing 2010)
Naive Bayes (and next week, SVM): focused on individual documentclassification.But what if we’re focused on proportions only?Hopkins and King (2010): method for characterizing distribution of classesCan be much more accurate than individual classifiers, requires fewerassumptions (do not need random sample of documents ) .
- King and Lu (2008): derive method for characterizing causes ofdeaths for verbal autopsies
- Hopkins and King (2010): extend the method to text documents
Basic intuition:
- Examine joint distribution of characteristics (without making NaiveBayes like assumption)
- Focus on distributions (only) makes this analysis possible
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 32 / 40
ReadMe: Optimization for a Different Goal (Hopkins andKing 2010)
Naive Bayes (and next week, SVM): focused on individual documentclassification.But what if we’re focused on proportions only?Hopkins and King (2010): method for characterizing distribution of classesCan be much more accurate than individual classifiers, requires fewerassumptions (do not need random sample of documents ) .
- King and Lu (2008): derive method for characterizing causes ofdeaths for verbal autopsies
- Hopkins and King (2010): extend the method to text documents
Basic intuition:
- Examine joint distribution of characteristics (without making NaiveBayes like assumption)
- Focus on distributions (only) makes this analysis possible
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 32 / 40
ReadMe: Optimization for a Different Goal (Hopkins andKing 2010)
Naive Bayes (and next week, SVM): focused on individual documentclassification.But what if we’re focused on proportions only?Hopkins and King (2010): method for characterizing distribution of classesCan be much more accurate than individual classifiers, requires fewerassumptions (do not need random sample of documents ) .
- King and Lu (2008): derive method for characterizing causes ofdeaths for verbal autopsies
- Hopkins and King (2010): extend the method to text documents
Basic intuition:
- Examine joint distribution of characteristics (without making NaiveBayes like assumption)
- Focus on distributions (only) makes this analysis possible
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 32 / 40
ReadMe: Optimization for a Different Goal (Hopkins andKing 2010)
Measure only presence/absence of each term [(Jx1) vector ]
x i = (1, 0, 0, 1, . . . , 0)
What are the possible realizations of x i?
- 2J possible vectors
Define:
P(x) = probability of observing xP(x |Cj) = Probability of observing x conditional on category Cj
P(X |C ) = Matrix collecting vectors
P(C ) = P(C1,C2, . . . ,CK ) target quantity of interest
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 33 / 40
ReadMe: Optimization for a Different Goal (Hopkins andKing 2010)
Measure only presence/absence of each term [(Jx1) vector ]
x i = (1, 0, 0, 1, . . . , 0)
What are the possible realizations of x i?
- 2J possible vectors
Define:
P(x) = probability of observing xP(x |Cj) = Probability of observing x conditional on category Cj
P(X |C ) = Matrix collecting vectors
P(C ) = P(C1,C2, . . . ,CK ) target quantity of interest
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 33 / 40
ReadMe: Optimization for a Different Goal (Hopkins andKing 2010)
Measure only presence/absence of each term [(Jx1) vector ]
x i = (1, 0, 0, 1, . . . , 0)
What are the possible realizations of x i?
- 2J possible vectors
Define:
P(x) = probability of observing xP(x |Cj) = Probability of observing x conditional on category Cj
P(X |C ) = Matrix collecting vectors
P(C ) = P(C1,C2, . . . ,CK ) target quantity of interest
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 33 / 40
ReadMe: Optimization for a Different Goal (Hopkins andKing 2010)
Measure only presence/absence of each term [(Jx1) vector ]
x i = (1, 0, 0, 1, . . . , 0)
What are the possible realizations of x i?
- 2J possible vectors
Define:
P(x) = probability of observing xP(x |Cj) = Probability of observing x conditional on category Cj
P(X |C ) = Matrix collecting vectors
P(C ) = P(C1,C2, . . . ,CK ) target quantity of interest
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 33 / 40
ReadMe: Optimization for a Different Goal (Hopkins andKing 2010)
Measure only presence/absence of each term [(Jx1) vector ]
x i = (1, 0, 0, 1, . . . , 0)
What are the possible realizations of x i?
- 2J possible vectors
Define:
P(x) = probability of observing xP(x |Cj) = Probability of observing x conditional on category Cj
P(X |C ) = Matrix collecting vectors
P(C ) = P(C1,C2, . . . ,CK ) target quantity of interest
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 33 / 40
ReadMe: Optimization for a Different Goal (Hopkins andKing 2010)
Measure only presence/absence of each term [(Jx1) vector ]
x i = (1, 0, 0, 1, . . . , 0)
What are the possible realizations of x i?
- 2J possible vectors
Define:
P(x) = probability of observing x
P(x |Cj) = Probability of observing x conditional on category Cj
P(X |C ) = Matrix collecting vectors
P(C ) = P(C1,C2, . . . ,CK ) target quantity of interest
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 33 / 40
ReadMe: Optimization for a Different Goal (Hopkins andKing 2010)
Measure only presence/absence of each term [(Jx1) vector ]
x i = (1, 0, 0, 1, . . . , 0)
What are the possible realizations of x i?
- 2J possible vectors
Define:
P(x) = probability of observing xP(x |Cj) = Probability of observing x conditional on category Cj
P(X |C ) = Matrix collecting vectors
P(C ) = P(C1,C2, . . . ,CK ) target quantity of interest
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 33 / 40
ReadMe: Optimization for a Different Goal (Hopkins andKing 2010)
Measure only presence/absence of each term [(Jx1) vector ]
x i = (1, 0, 0, 1, . . . , 0)
What are the possible realizations of x i?
- 2J possible vectors
Define:
P(x) = probability of observing xP(x |Cj) = Probability of observing x conditional on category Cj
P(X |C ) = Matrix collecting vectors
P(C ) = P(C1,C2, . . . ,CK ) target quantity of interest
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 33 / 40
ReadMe: Optimization for a Different Goal (Hopkins andKing 2010)
Measure only presence/absence of each term [(Jx1) vector ]
x i = (1, 0, 0, 1, . . . , 0)
What are the possible realizations of x i?
- 2J possible vectors
Define:
P(x) = probability of observing xP(x |Cj) = Probability of observing x conditional on category Cj
P(X |C ) = Matrix collecting vectors
P(C ) = P(C1,C2, . . . ,CK ) target quantity of interest
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 33 / 40
ReadMe: Optimization for a Different Goal (Hopkins andKing 2010)
Measure only presence/absence of each term [(Jx1) vector ]
x i = (1, 0, 0, 1, . . . , 0)
What are the possible realizations of x i?
- 2J possible vectors
Define:
P(x) = probability of observing xP(x |Cj) = Probability of observing x conditional on category Cj
P(X |C ) = Matrix collecting vectors
P(C ) = P(C1,C2, . . . ,CK ) target quantity of interest
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 33 / 40
ReadMe: Optimization for a Different Goal (Hopkins andKing 2010)
P(x)︸ ︷︷ ︸2Jx1
= P(x |C )︸ ︷︷ ︸2JxK
P(C )︸ ︷︷ ︸Kx1
Matrix algebra problem to solve, for P(C )Like Naive Bayes, requires two pieces to estimateComplication 2J >> no. documentsKernel Smoothing Methods (without a formal model)
- P(x) = estimate directly from test set
- P(x |C ) = estimate from training set
- Key assumption: P(x |C ) in training set is equivalent to P(x |C ) in testset
- If true, can perform biased sampling of documents, worry less aboutdrift...
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 34 / 40
Algorithm Summarized
- Estimate p(x) from test set
- Estimate p(x |C ) from training set
- Use p(x) and p(x |C ) to solve for p(C )
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 35 / 40
Assessing Model Performance
Not classifying individual documents → different standardsMean Square Error :
E[(θ − θ)2] = var(θ) + Bias(θ, θ)2
Suppose we have true proportions P(C )true. Then, we’ll estimate RootMean Square Error
RMSE =
√∑Jj=1(P(Cj)true − P(Cj))
J
Mean Abs. Prediction Error = |∑J
j=1(P(Cj)true − P(Cj))
J|
Visualize: plot true and estimated proportions
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 36 / 40
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 37 / 40
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 37 / 40
Using the House Press Release Data
Method RMSE APSE
ReadMe 0.036 0.056NaiveBayes 0.096 0.14SVM 0.052 0.084
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 38 / 40
Code to Run in R
Control file:filename truth trainingset
20July2009LEWIS53.txt 4 126July2006LEWIS249.txt 2 0
tdm<- undergrad(control=control, fullfreq=F)
process<- preprocess(tdm)
output<- undergrad(process)
output$est.CSMF ## proportion in each category
output$true.CSMF ## if labeled for validation set (but not
used in training set)
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 39 / 40
Model Selection!
Justin Grimmer (Stanford University) Text as Data November 10th, 2014 40 / 40
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