cmu statistics - nonparametric modal regression · 2015. 9. 18. · nonparametric modal regression...

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

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


Introduction


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.


Conditional Local Modes


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).


Example for Modal Regression

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0.0 0.2 0.4 0.6 0.8 1.0

−0.

4−

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0.0

0.2

0.4

0.6

0.8

X

Y


Asymptotic Theory


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)).


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.


Asymptotic Theory

To conduct statistical inferences, we ignore the bias and focus on thestochastic variation.

Theorem


√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).


Asymptotic Theory


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 .



Asymptotic Theory


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 .



Confidence Sets

We use the bootstrap to approximate ∆n. Define another uniform metric∆̂n = supx Haus(M̂∗n(x), M̂n(x)).

Theorem


√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.


Confidence Sets


Theorem

Under regularity conditions,√nhd+3∆̂n ≈ supf ∈F |B(f )| for certain function space F .

√nhd+3∆̂n ≈

√nhd+3∆n.




Confidence Sets


Theorem

Under regularity conditions,√nhd+3∆̂n ≈ supf ∈F |B(f )| for certain function space F .√nhd+3∆̂n ≈

√nhd+3∆n.




Example for Confidence Sets

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Prediction Sets

We can use modal regression to construct a compact prediction set.

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Prediction Sets

We can use modal regression to construct a compact prediction set.

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Bandwidth Selection

We can choose smoothing parameter h via minimizing the size ofprediction set.Namely, we choose

h∗ = argminh>0

Vol(P̂1−α

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where P̂1−α is the prediction set.


Example: Bandwidth Selection

0.00 0.05 0.10 0.15 0.20

1.0

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Size of 95% Prediction interval

h

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Optimal h


Regression Clustering

We can use modal regression to do ‘clustering’–exploring the hiddenstructures.

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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.


Thank you!More information and R source code can be found in

http://www.stat.cmu.edu/~yenchic


http://www.stat.cmu.edu/~yenchic

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.


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.


Modal Regression VS Density Ridges


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 .


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.


3D examples

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cmu statistics - nonparametric modal regression · 2015. 9. 18. · nonparametric modal regression...

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