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Nonparametric Modal Regression
Yen-Chi Chen
Christopher R. Genovese Ryan J. Tibshirani Larry Wasserman
Department of StatisticsCarnegie Mellon University
August 4, 2015
Yen-Chi Chen (CMU-Stats) Modal Regression August 4, 2015 1 / 20
![Page 2: CMU Statistics - Nonparametric Modal Regression · 2015. 9. 18. · Nonparametric Modal Regression Yen-Chi Chen Christopher R. Genovese Ryan J. Tibshirani Larry Wasserman Department](https://reader036.vdocuments.us/reader036/viewer/2022071104/5fde0f91ef084b40404b043e/html5/thumbnails/2.jpg)
Motivating Examples for Modal Regression
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0.0 0.2 0.4 0.6 0.8 1.0
−0.
4−
0.2
0.0
0.2
0.4
0.6
0.8
X
Y
Local RegressionModal Regression
Yen-Chi Chen (CMU-Stats) Modal Regression August 4, 2015 2 / 20
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Introduction
Yen-Chi Chen (CMU-Stats) Modal Regression August 4, 2015 3 / 20
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Definition for Modal Regression
We assume x ∈ K, a compact support.
Regression function–the conditional mean:
m(x) = E(Y |X = x) =
∫yp(y |x)dy .
Modal function–the conditional (local) modes:
M(x) = Mode(Y |X = x) =
{y :
d
dyp(y |x) = 0,
d2
dy2p(y |x) < 0
}.
Equivalently,
M(x) =
{y :
∂
∂yp(x , y) = 0,
∂2
∂y2p(x , y) < 0
}.
M(x) is a multi-value function.
Yen-Chi Chen (CMU-Stats) Modal Regression August 4, 2015 4 / 20
![Page 5: CMU Statistics - Nonparametric Modal Regression · 2015. 9. 18. · Nonparametric Modal Regression Yen-Chi Chen Christopher R. Genovese Ryan J. Tibshirani Larry Wasserman Department](https://reader036.vdocuments.us/reader036/viewer/2022071104/5fde0f91ef084b40404b043e/html5/thumbnails/5.jpg)
Definition for Modal Regression
We assume x ∈ K, a compact support.
Regression function–the conditional mean:
m(x) = E(Y |X = x) =
∫yp(y |x)dy .
Modal function–the conditional (local) modes:
M(x) = Mode(Y |X = x) =
{y :
d
dyp(y |x) = 0,
d2
dy2p(y |x) < 0
}.
Equivalently,
M(x) =
{y :
∂
∂yp(x , y) = 0,
∂2
∂y2p(x , y) < 0
}.
M(x) is a multi-value function.
Yen-Chi Chen (CMU-Stats) Modal Regression August 4, 2015 4 / 20
![Page 6: CMU Statistics - Nonparametric Modal Regression · 2015. 9. 18. · Nonparametric Modal Regression Yen-Chi Chen Christopher R. Genovese Ryan J. Tibshirani Larry Wasserman Department](https://reader036.vdocuments.us/reader036/viewer/2022071104/5fde0f91ef084b40404b043e/html5/thumbnails/6.jpg)
Definition for Modal Regression
We assume x ∈ K, a compact support.
Regression function–the conditional mean:
m(x) = E(Y |X = x) =
∫yp(y |x)dy .
Modal function–the conditional (local) modes:
M(x) = Mode(Y |X = x) =
{y :
d
dyp(y |x) = 0,
d2
dy2p(y |x) < 0
}.
Equivalently,
M(x) =
{y :
∂
∂yp(x , y) = 0,
∂2
∂y2p(x , y) < 0
}.
M(x) is a multi-value function.
Yen-Chi Chen (CMU-Stats) Modal Regression August 4, 2015 4 / 20
![Page 7: CMU Statistics - Nonparametric Modal Regression · 2015. 9. 18. · Nonparametric Modal Regression Yen-Chi Chen Christopher R. Genovese Ryan J. Tibshirani Larry Wasserman Department](https://reader036.vdocuments.us/reader036/viewer/2022071104/5fde0f91ef084b40404b043e/html5/thumbnails/7.jpg)
Definition for Modal Regression
We assume x ∈ K, a compact support.
Regression function–the conditional mean:
m(x) = E(Y |X = x) =
∫yp(y |x)dy .
Modal function–the conditional (local) modes:
M(x) = Mode(Y |X = x) =
{y :
d
dyp(y |x) = 0,
d2
dy2p(y |x) < 0
}.
Equivalently,
M(x) =
{y :
∂
∂yp(x , y) = 0,
∂2
∂y2p(x , y) < 0
}.
M(x) is a multi-value function.
Yen-Chi Chen (CMU-Stats) Modal Regression August 4, 2015 4 / 20
![Page 8: CMU Statistics - Nonparametric Modal Regression · 2015. 9. 18. · Nonparametric Modal Regression Yen-Chi Chen Christopher R. Genovese Ryan J. Tibshirani Larry Wasserman Department](https://reader036.vdocuments.us/reader036/viewer/2022071104/5fde0f91ef084b40404b043e/html5/thumbnails/8.jpg)
Conditional Local Modes
Yen-Chi Chen (CMU-Stats) Modal Regression August 4, 2015 5 / 20
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Conditional Local Modes
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![Page 10: CMU Statistics - Nonparametric Modal Regression · 2015. 9. 18. · Nonparametric Modal Regression Yen-Chi Chen Christopher R. Genovese Ryan J. Tibshirani Larry Wasserman Department](https://reader036.vdocuments.us/reader036/viewer/2022071104/5fde0f91ef084b40404b043e/html5/thumbnails/10.jpg)
Conditional Local Modes
Yen-Chi Chen (CMU-Stats) Modal Regression August 4, 2015 5 / 20
![Page 11: CMU Statistics - Nonparametric Modal Regression · 2015. 9. 18. · Nonparametric Modal Regression Yen-Chi Chen Christopher R. Genovese Ryan J. Tibshirani Larry Wasserman Department](https://reader036.vdocuments.us/reader036/viewer/2022071104/5fde0f91ef084b40404b043e/html5/thumbnails/11.jpg)
Conditional Local Modes
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Estimator for Modal Regression
Our estimator is the plug-in from the KDE:
M̂n(x) =
{y :
∂
∂yp̂n(x , y) = 0,
∂2
∂y2p̂n(x , y) < 0
}.
Finding conditional local modes is hard in general.
Partial mean shift: a simple algorithm for computing M̂n(x), theplug-in estimator of the KDE, from the data (Einbeck et. al. 2006).
Yen-Chi Chen (CMU-Stats) Modal Regression August 4, 2015 6 / 20
![Page 13: CMU Statistics - Nonparametric Modal Regression · 2015. 9. 18. · Nonparametric Modal Regression Yen-Chi Chen Christopher R. Genovese Ryan J. Tibshirani Larry Wasserman Department](https://reader036.vdocuments.us/reader036/viewer/2022071104/5fde0f91ef084b40404b043e/html5/thumbnails/13.jpg)
Estimator for Modal Regression
Our estimator is the plug-in from the KDE:
M̂n(x) =
{y :
∂
∂yp̂n(x , y) = 0,
∂2
∂y2p̂n(x , y) < 0
}.
Finding conditional local modes is hard in general.
Partial mean shift: a simple algorithm for computing M̂n(x), theplug-in estimator of the KDE, from the data (Einbeck et. al. 2006).
Yen-Chi Chen (CMU-Stats) Modal Regression August 4, 2015 6 / 20
![Page 14: CMU Statistics - Nonparametric Modal Regression · 2015. 9. 18. · Nonparametric Modal Regression Yen-Chi Chen Christopher R. Genovese Ryan J. Tibshirani Larry Wasserman Department](https://reader036.vdocuments.us/reader036/viewer/2022071104/5fde0f91ef084b40404b043e/html5/thumbnails/14.jpg)
Estimator for Modal Regression
Our estimator is the plug-in from the KDE:
M̂n(x) =
{y :
∂
∂yp̂n(x , y) = 0,
∂2
∂y2p̂n(x , y) < 0
}.
Finding conditional local modes is hard in general.
Partial mean shift: a simple algorithm for computing M̂n(x), theplug-in estimator of the KDE, from the data (Einbeck et. al. 2006).
Yen-Chi Chen (CMU-Stats) Modal Regression August 4, 2015 6 / 20
![Page 15: CMU Statistics - Nonparametric Modal Regression · 2015. 9. 18. · Nonparametric Modal Regression Yen-Chi Chen Christopher R. Genovese Ryan J. Tibshirani Larry Wasserman Department](https://reader036.vdocuments.us/reader036/viewer/2022071104/5fde0f91ef084b40404b043e/html5/thumbnails/15.jpg)
Example for Modal Regression
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Example for Modal Regression
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Yen-Chi Chen (CMU-Stats) Modal Regression August 4, 2015 7 / 20
![Page 17: CMU Statistics - Nonparametric Modal Regression · 2015. 9. 18. · Nonparametric Modal Regression Yen-Chi Chen Christopher R. Genovese Ryan J. Tibshirani Larry Wasserman Department](https://reader036.vdocuments.us/reader036/viewer/2022071104/5fde0f91ef084b40404b043e/html5/thumbnails/17.jpg)
Asymptotic Theory
Yen-Chi Chen (CMU-Stats) Modal Regression August 4, 2015 8 / 20
![Page 18: CMU Statistics - Nonparametric Modal Regression · 2015. 9. 18. · Nonparametric Modal Regression Yen-Chi Chen Christopher R. Genovese Ryan J. Tibshirani Larry Wasserman Department](https://reader036.vdocuments.us/reader036/viewer/2022071104/5fde0f91ef084b40404b043e/html5/thumbnails/18.jpg)
Error Measurement
To measure the errors, we consider the following two losses:
the pointwise loss
∆n(x) = Haus(M̂n(x),M(x)),
where Haus(A,B) is the Hausdorff distance.
the uniform loss
∆n = supx
∆n(x) = supx
Haus(M̂n(x),M(x)).
Yen-Chi Chen (CMU-Stats) Modal Regression August 4, 2015 9 / 20
![Page 19: CMU Statistics - Nonparametric Modal Regression · 2015. 9. 18. · Nonparametric Modal Regression Yen-Chi Chen Christopher R. Genovese Ryan J. Tibshirani Larry Wasserman Department](https://reader036.vdocuments.us/reader036/viewer/2022071104/5fde0f91ef084b40404b043e/html5/thumbnails/19.jpg)
Error Measurement
To measure the errors, we consider the following two losses:
the pointwise loss
∆n(x) = Haus(M̂n(x),M(x)),
where Haus(A,B) is the Hausdorff distance.
the uniform loss
∆n = supx
∆n(x) = supx
Haus(M̂n(x),M(x)).
Yen-Chi Chen (CMU-Stats) Modal Regression August 4, 2015 9 / 20
![Page 20: CMU Statistics - Nonparametric Modal Regression · 2015. 9. 18. · Nonparametric Modal Regression Yen-Chi Chen Christopher R. Genovese Ryan J. Tibshirani Larry Wasserman Department](https://reader036.vdocuments.us/reader036/viewer/2022071104/5fde0f91ef084b40404b043e/html5/thumbnails/20.jpg)
Error Measurement
To measure the errors, we consider the following two losses:
the pointwise loss
∆n(x) = Haus(M̂n(x),M(x)),
where Haus(A,B) is the Hausdorff distance.
the uniform loss
∆n = supx
∆n(x) = supx
Haus(M̂n(x),M(x)).
Yen-Chi Chen (CMU-Stats) Modal Regression August 4, 2015 9 / 20
![Page 21: CMU Statistics - Nonparametric Modal Regression · 2015. 9. 18. · Nonparametric Modal Regression Yen-Chi Chen Christopher R. Genovese Ryan J. Tibshirani Larry Wasserman Department](https://reader036.vdocuments.us/reader036/viewer/2022071104/5fde0f91ef084b40404b043e/html5/thumbnails/21.jpg)
Rate of Convergence
Both the pointwise and the uniform losses obey the commonnonparametric rate:
Theorem
Under regularity conditions,
∆n(x) = O(h2) + OP
(√1
nhd+3
)
∆n = O(h2) + OP
(√log n
nhd+3
).
Rate = Bias +√
Variance.
Yen-Chi Chen (CMU-Stats) Modal Regression August 4, 2015 10 / 20
![Page 22: CMU Statistics - Nonparametric Modal Regression · 2015. 9. 18. · Nonparametric Modal Regression Yen-Chi Chen Christopher R. Genovese Ryan J. Tibshirani Larry Wasserman Department](https://reader036.vdocuments.us/reader036/viewer/2022071104/5fde0f91ef084b40404b043e/html5/thumbnails/22.jpg)
Rate of Convergence
Both the pointwise and the uniform losses obey the commonnonparametric rate:
Theorem
Under regularity conditions,
∆n(x) = O(h2) + OP
(√1
nhd+3
)
∆n = O(h2) + OP
(√log n
nhd+3
).
Rate = Bias +√
Variance.
Yen-Chi Chen (CMU-Stats) Modal Regression August 4, 2015 10 / 20
![Page 23: CMU Statistics - Nonparametric Modal Regression · 2015. 9. 18. · Nonparametric Modal Regression Yen-Chi Chen Christopher R. Genovese Ryan J. Tibshirani Larry Wasserman Department](https://reader036.vdocuments.us/reader036/viewer/2022071104/5fde0f91ef084b40404b043e/html5/thumbnails/23.jpg)
Rate of Convergence
Both the pointwise and the uniform losses obey the commonnonparametric rate:
Theorem
Under regularity conditions,
∆n(x) = O(h2) + OP
(√1
nhd+3
)
∆n = O(h2) + OP
(√log n
nhd+3
).
Rate = Bias +√
Variance.
Yen-Chi Chen (CMU-Stats) Modal Regression August 4, 2015 10 / 20
![Page 24: CMU Statistics - Nonparametric Modal Regression · 2015. 9. 18. · Nonparametric Modal Regression Yen-Chi Chen Christopher R. Genovese Ryan J. Tibshirani Larry Wasserman Department](https://reader036.vdocuments.us/reader036/viewer/2022071104/5fde0f91ef084b40404b043e/html5/thumbnails/24.jpg)
Asymptotic Theory
To conduct statistical inferences, we ignore the bias and focus on thestochastic variation.
Theorem
Under regularity conditions,
√nhd+3∆n ≈ sup{Empirical Process} ≈ sup{Gaussian process}.√nhd+3∆n ≈ supf ∈F |B(f )| for certain function space F .
However, this is not enough for statistical inference (unknown quantities inthe Gaussian Process).
Yen-Chi Chen (CMU-Stats) Modal Regression August 4, 2015 11 / 20
![Page 25: CMU Statistics - Nonparametric Modal Regression · 2015. 9. 18. · Nonparametric Modal Regression Yen-Chi Chen Christopher R. Genovese Ryan J. Tibshirani Larry Wasserman Department](https://reader036.vdocuments.us/reader036/viewer/2022071104/5fde0f91ef084b40404b043e/html5/thumbnails/25.jpg)
Asymptotic Theory
To conduct statistical inferences, we ignore the bias and focus on thestochastic variation.
Theorem
Under regularity conditions,√nhd+3∆n ≈ sup{Empirical Process} ≈ sup{Gaussian process}.
√nhd+3∆n ≈ supf ∈F |B(f )| for certain function space F .
However, this is not enough for statistical inference (unknown quantities inthe Gaussian Process).
Yen-Chi Chen (CMU-Stats) Modal Regression August 4, 2015 11 / 20
![Page 26: CMU Statistics - Nonparametric Modal Regression · 2015. 9. 18. · Nonparametric Modal Regression Yen-Chi Chen Christopher R. Genovese Ryan J. Tibshirani Larry Wasserman Department](https://reader036.vdocuments.us/reader036/viewer/2022071104/5fde0f91ef084b40404b043e/html5/thumbnails/26.jpg)
Asymptotic Theory
To conduct statistical inferences, we ignore the bias and focus on thestochastic variation.
Theorem
Under regularity conditions,√nhd+3∆n ≈ sup{Empirical Process} ≈ sup{Gaussian process}.√nhd+3∆n ≈ supf ∈F |B(f )| for certain function space F .
However, this is not enough for statistical inference (unknown quantities inthe Gaussian Process).
Yen-Chi Chen (CMU-Stats) Modal Regression August 4, 2015 11 / 20
![Page 27: CMU Statistics - Nonparametric Modal Regression · 2015. 9. 18. · Nonparametric Modal Regression Yen-Chi Chen Christopher R. Genovese Ryan J. Tibshirani Larry Wasserman Department](https://reader036.vdocuments.us/reader036/viewer/2022071104/5fde0f91ef084b40404b043e/html5/thumbnails/27.jpg)
Asymptotic Theory
To conduct statistical inferences, we ignore the bias and focus on thestochastic variation.
Theorem
Under regularity conditions,√nhd+3∆n ≈ sup{Empirical Process} ≈ sup{Gaussian process}.√nhd+3∆n ≈ supf ∈F |B(f )| for certain function space F .
However, this is not enough for statistical inference (unknown quantities inthe Gaussian Process).
Yen-Chi Chen (CMU-Stats) Modal Regression August 4, 2015 11 / 20
![Page 28: CMU Statistics - Nonparametric Modal Regression · 2015. 9. 18. · Nonparametric Modal Regression Yen-Chi Chen Christopher R. Genovese Ryan J. Tibshirani Larry Wasserman Department](https://reader036.vdocuments.us/reader036/viewer/2022071104/5fde0f91ef084b40404b043e/html5/thumbnails/28.jpg)
Confidence Sets
We use the bootstrap to approximate ∆n. Define another uniform metric∆̂n = supx Haus(M̂∗n(x), M̂n(x)).
Theorem
Under regularity conditions,
√nhd+3∆̂n ≈ supf ∈F |B(f )| for certain function space F .√nhd+3∆̂n ≈
√nhd+3∆n.
The set {(x , y) : y ∈ M̂n(x)⊕ t̂1−α, x ∈ K
}is an asymptotic valid confidence set for M; t̂1−α is the upper 1− αquantile of ∆̂n.
Yen-Chi Chen (CMU-Stats) Modal Regression August 4, 2015 12 / 20
![Page 29: CMU Statistics - Nonparametric Modal Regression · 2015. 9. 18. · Nonparametric Modal Regression Yen-Chi Chen Christopher R. Genovese Ryan J. Tibshirani Larry Wasserman Department](https://reader036.vdocuments.us/reader036/viewer/2022071104/5fde0f91ef084b40404b043e/html5/thumbnails/29.jpg)
Confidence Sets
We use the bootstrap to approximate ∆n. Define another uniform metric∆̂n = supx Haus(M̂∗n(x), M̂n(x)).
Theorem
Under regularity conditions,√nhd+3∆̂n ≈ supf ∈F |B(f )| for certain function space F .
√nhd+3∆̂n ≈
√nhd+3∆n.
The set {(x , y) : y ∈ M̂n(x)⊕ t̂1−α, x ∈ K
}is an asymptotic valid confidence set for M; t̂1−α is the upper 1− αquantile of ∆̂n.
Yen-Chi Chen (CMU-Stats) Modal Regression August 4, 2015 12 / 20
![Page 30: CMU Statistics - Nonparametric Modal Regression · 2015. 9. 18. · Nonparametric Modal Regression Yen-Chi Chen Christopher R. Genovese Ryan J. Tibshirani Larry Wasserman Department](https://reader036.vdocuments.us/reader036/viewer/2022071104/5fde0f91ef084b40404b043e/html5/thumbnails/30.jpg)
Confidence Sets
We use the bootstrap to approximate ∆n. Define another uniform metric∆̂n = supx Haus(M̂∗n(x), M̂n(x)).
Theorem
Under regularity conditions,√nhd+3∆̂n ≈ supf ∈F |B(f )| for certain function space F .√nhd+3∆̂n ≈
√nhd+3∆n.
The set {(x , y) : y ∈ M̂n(x)⊕ t̂1−α, x ∈ K
}is an asymptotic valid confidence set for M; t̂1−α is the upper 1− αquantile of ∆̂n.
Yen-Chi Chen (CMU-Stats) Modal Regression August 4, 2015 12 / 20
![Page 31: CMU Statistics - Nonparametric Modal Regression · 2015. 9. 18. · Nonparametric Modal Regression Yen-Chi Chen Christopher R. Genovese Ryan J. Tibshirani Larry Wasserman Department](https://reader036.vdocuments.us/reader036/viewer/2022071104/5fde0f91ef084b40404b043e/html5/thumbnails/31.jpg)
Confidence Sets
We use the bootstrap to approximate ∆n. Define another uniform metric∆̂n = supx Haus(M̂∗n(x), M̂n(x)).
Theorem
Under regularity conditions,√nhd+3∆̂n ≈ supf ∈F |B(f )| for certain function space F .√nhd+3∆̂n ≈
√nhd+3∆n.
The set {(x , y) : y ∈ M̂n(x)⊕ t̂1−α, x ∈ K
}is an asymptotic valid confidence set for M; t̂1−α is the upper 1− αquantile of ∆̂n.
Yen-Chi Chen (CMU-Stats) Modal Regression August 4, 2015 12 / 20
![Page 32: CMU Statistics - Nonparametric Modal Regression · 2015. 9. 18. · Nonparametric Modal Regression Yen-Chi Chen Christopher R. Genovese Ryan J. Tibshirani Larry Wasserman Department](https://reader036.vdocuments.us/reader036/viewer/2022071104/5fde0f91ef084b40404b043e/html5/thumbnails/32.jpg)
Example for Confidence Sets
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![Page 33: CMU Statistics - Nonparametric Modal Regression · 2015. 9. 18. · Nonparametric Modal Regression Yen-Chi Chen Christopher R. Genovese Ryan J. Tibshirani Larry Wasserman Department](https://reader036.vdocuments.us/reader036/viewer/2022071104/5fde0f91ef084b40404b043e/html5/thumbnails/33.jpg)
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Yen-Chi Chen (CMU-Stats) Modal Regression August 4, 2015 13 / 20
![Page 34: CMU Statistics - Nonparametric Modal Regression · 2015. 9. 18. · Nonparametric Modal Regression Yen-Chi Chen Christopher R. Genovese Ryan J. Tibshirani Larry Wasserman Department](https://reader036.vdocuments.us/reader036/viewer/2022071104/5fde0f91ef084b40404b043e/html5/thumbnails/34.jpg)
Extensions
Yen-Chi Chen (CMU-Stats) Modal Regression August 4, 2015 14 / 20
![Page 35: CMU Statistics - Nonparametric Modal Regression · 2015. 9. 18. · Nonparametric Modal Regression Yen-Chi Chen Christopher R. Genovese Ryan J. Tibshirani Larry Wasserman Department](https://reader036.vdocuments.us/reader036/viewer/2022071104/5fde0f91ef084b40404b043e/html5/thumbnails/35.jpg)
Prediction Sets
We can use modal regression to construct a compact prediction set.
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Modal Regression
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Yen-Chi Chen (CMU-Stats) Modal Regression August 4, 2015 15 / 20
![Page 36: CMU Statistics - Nonparametric Modal Regression · 2015. 9. 18. · Nonparametric Modal Regression Yen-Chi Chen Christopher R. Genovese Ryan J. Tibshirani Larry Wasserman Department](https://reader036.vdocuments.us/reader036/viewer/2022071104/5fde0f91ef084b40404b043e/html5/thumbnails/36.jpg)
Prediction Sets
We can use modal regression to construct a compact prediction set.
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Yen-Chi Chen (CMU-Stats) Modal Regression August 4, 2015 15 / 20
![Page 37: CMU Statistics - Nonparametric Modal Regression · 2015. 9. 18. · Nonparametric Modal Regression Yen-Chi Chen Christopher R. Genovese Ryan J. Tibshirani Larry Wasserman Department](https://reader036.vdocuments.us/reader036/viewer/2022071104/5fde0f91ef084b40404b043e/html5/thumbnails/37.jpg)
Bandwidth Selection
We can choose smoothing parameter h via minimizing the size ofprediction set.Namely, we choose
h∗ = argminh>0
Vol(P̂1−α
),
where P̂1−α is the prediction set.
Yen-Chi Chen (CMU-Stats) Modal Regression August 4, 2015 16 / 20
![Page 38: CMU Statistics - Nonparametric Modal Regression · 2015. 9. 18. · Nonparametric Modal Regression Yen-Chi Chen Christopher R. Genovese Ryan J. Tibshirani Larry Wasserman Department](https://reader036.vdocuments.us/reader036/viewer/2022071104/5fde0f91ef084b40404b043e/html5/thumbnails/38.jpg)
Example: Bandwidth Selection
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Optimal h
Yen-Chi Chen (CMU-Stats) Modal Regression August 4, 2015 17 / 20
![Page 39: CMU Statistics - Nonparametric Modal Regression · 2015. 9. 18. · Nonparametric Modal Regression Yen-Chi Chen Christopher R. Genovese Ryan J. Tibshirani Larry Wasserman Department](https://reader036.vdocuments.us/reader036/viewer/2022071104/5fde0f91ef084b40404b043e/html5/thumbnails/39.jpg)
Regression Clustering
We can use modal regression to do ‘clustering’–exploring the hiddenstructures.
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Yen-Chi Chen (CMU-Stats) Modal Regression August 4, 2015 18 / 20
![Page 40: CMU Statistics - Nonparametric Modal Regression · 2015. 9. 18. · Nonparametric Modal Regression Yen-Chi Chen Christopher R. Genovese Ryan J. Tibshirani Larry Wasserman Department](https://reader036.vdocuments.us/reader036/viewer/2022071104/5fde0f91ef084b40404b043e/html5/thumbnails/40.jpg)
Concluding Remarks
Modal regression is very similar to mixture regression.
However, our approach is purely nonparametric–no Gaussianassumption, free from number of mixture components.
Fast to compute–no need to use EM algorithm.
Yen-Chi Chen (CMU-Stats) Modal Regression August 4, 2015 19 / 20
![Page 41: CMU Statistics - Nonparametric Modal Regression · 2015. 9. 18. · Nonparametric Modal Regression Yen-Chi Chen Christopher R. Genovese Ryan J. Tibshirani Larry Wasserman Department](https://reader036.vdocuments.us/reader036/viewer/2022071104/5fde0f91ef084b40404b043e/html5/thumbnails/41.jpg)
Concluding Remarks
Modal regression is very similar to mixture regression.
However, our approach is purely nonparametric–no Gaussianassumption, free from number of mixture components.
Fast to compute–no need to use EM algorithm.
Yen-Chi Chen (CMU-Stats) Modal Regression August 4, 2015 19 / 20
![Page 42: CMU Statistics - Nonparametric Modal Regression · 2015. 9. 18. · Nonparametric Modal Regression Yen-Chi Chen Christopher R. Genovese Ryan J. Tibshirani Larry Wasserman Department](https://reader036.vdocuments.us/reader036/viewer/2022071104/5fde0f91ef084b40404b043e/html5/thumbnails/42.jpg)
Concluding Remarks
Modal regression is very similar to mixture regression.
However, our approach is purely nonparametric–no Gaussianassumption, free from number of mixture components.
Fast to compute–no need to use EM algorithm.
Yen-Chi Chen (CMU-Stats) Modal Regression August 4, 2015 19 / 20
![Page 43: CMU Statistics - Nonparametric Modal Regression · 2015. 9. 18. · Nonparametric Modal Regression Yen-Chi Chen Christopher R. Genovese Ryan J. Tibshirani Larry Wasserman Department](https://reader036.vdocuments.us/reader036/viewer/2022071104/5fde0f91ef084b40404b043e/html5/thumbnails/43.jpg)
Thank you!More information and R source code can be found in
http://www.stat.cmu.edu/~yenchic
Yen-Chi Chen (CMU-Stats) Modal Regression August 4, 2015 20 / 20
![Page 44: CMU Statistics - Nonparametric Modal Regression · 2015. 9. 18. · Nonparametric Modal Regression Yen-Chi Chen Christopher R. Genovese Ryan J. Tibshirani Larry Wasserman Department](https://reader036.vdocuments.us/reader036/viewer/2022071104/5fde0f91ef084b40404b043e/html5/thumbnails/44.jpg)
reference
1. Chen, Yen-Chi, Christopher R. Genovese, and Larry Wasserman. ”Density Level Sets: Asymptotics, Inference, andVisualization.” Submitted to the Journal of American Statistical Association. arXiv preprint arXiv:1504.05438 (2015).
2. Chen, Yen-Chi, Christopher R. Genovese, and Larry Wasserman. ”Asymptotic theory for density ridges.” To appear inthe Annals of Statistics. arXiv preprint arXiv:1406.5663 (2014).
3. Chen, Yen-Chi, Christopher R. Genovese, Ryan J. Tibshirani, and Larry Wasserman. ”Nonparametric ModalRegression.” Under review of the Annals of Statistics. arXiv preprint arXiv:1412.1716 (2014).
4. Chernozhukov, Victor, Denis Chetverikov, and Kengo Kato. ”Gaussian approximation of suprema of empiricalprocesses.” The Annals of Statistics 42, no. 4 (2014): 1564-1597.
5. Chernozhukov, Victor, Denis Chetverikov, and Kengo Kato. ”Anti-concentration and honest, adaptive confidencebands.” The Annals of Statistics 42, no. 5 (2014): 1787-1818.
6. Einbeck, Jochen, and Gerhard Tutz. ”Modelling beyond regression functions: an application of multimodal regression tospeedflow data.” Journal of the Royal Statistical Society: Series C (Applied Statistics) 55, no. 4 (2006): 461-475.
7. Genovese, Christopher R., et al. ”Nonparametric ridge estimation.” The Annals of Statistics 42.4 (2014): 1511-1545.
8. Ozertem, Umut, and Deniz Erdogmus. ”Locally defined principal curves and surfaces.” The Journal of Machine
Learning Research 12 (2011): 1249-1286.
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Regularity Conditions
(A1) The joint density p ∈ BC4(Cp) for some Cp > 0.
(A2) There exists λ2 > 0 such that for any (x , y) ∈ K×K withpy (x , y) = 0, |pyy (x , y)| > λ2.
(K1) The kernel function K ∈ BC2(CK ) and satifies∫R
(K (α))2(z) dz <∞,∫Rz2K (α)(z) dz <∞,
for α = 0, 1, 2.
(K2) The collection K is a VC-type class, i.e. there exists A, v > 0 suchthat for 0 < ε < 1,
supQ
N(K, L2(Q),CK ε
)≤(A
ε
)v
,
where N(T , d , ε) is the ε-covering number for a semi-metric space(T , d) and Q is any probability measure.
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Modal Regression VS Density Ridges
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Mixture Regression
A general mixture model:
p(y |x) =
K(x)∑j=1
πj(x)φj(y ;µj(x), σ2j (x)),
where each φj(y ;µj(x), σ2j (x)) is a density function, parametrized by a
mean µj(x) and variance σ2j (x).
Common assumptions:
(MR1) K (x) = K ,
(MR2) πj(x) = πj for each j ,
(MR3) µj(x) = βTj x for each j ,
(MR4) σ2j (x) = σ2
j for each j , and
(MR5) φj(x) is Gaussian for each j .
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Mixture Inference versus Modal Inference
Mixture-based Mode-basedDensity estimation Gaussian mixture Kernel density estimate
Clustering K -means Mean-shift clustering
Regression Mixture regression Modal regressionAlgorithm EM Mean-shift
Complexity parameter K (number of components) h (smoothing bandwidth)
Type Parametric model Nonparametric model
Table: Comparison for methods based on mixtures versus modes.
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3D examples
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Yen-Chi Chen (CMU-Stats) Modal Regression August 4, 2015 26 / 20