Analysis of Nonlinear Time Series Model for Independent Component Analysis

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Analysis of Nonlinear Time Series Model for
Independent Component Analysis
Kaoru Fueda Okayama University, Japan

• Introduction
• Estimation of autoregressive function
• Estimation of derivatives
• Links with the other methods
• Conclusion and problems

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1. INTRODUCTION

• Motivation
• (curse of dimensionality)

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Example 1. Signals separated by Independent
Component Analysis 2. Chaotic time series data
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Model
Aims
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Existing methods

• a. Embedding dimension
• Cheng and Tong (1995)

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b. Dimension reduction

• 1. Projection pursuit regression
• Friedman and Stuetzle (1981, JASA)
• 2. Average derivative estimation
• Hardle and Stoker (1989, JASA)
• 3. Sliced inverse regression
• Li (1991, JASA)
• 4. Adaptive estimation of dimension reduction
  space
• Xia, Tong, Li and Zhu (2002, JRSS)

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2. ESTIMATION OF AUTOREGRESSIVE FUNCTION F

• 2.1 Basic idea local linear regression

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An illustration of the basic idea
Regression surface
Tangent plain at this point
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Let
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2.2 Goodness of the model
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2.3 Bandwidth selection

• Note that

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2.4 Parameter selection
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3. Estimation of derivatives

• 3.1 Basic Idea
• Usually the local linear regression estimates the
  derivatives of regression function. (Fan and
  Gijbels, 1996)
• However, sometime it fails to estimate.
• example

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Problem 1Fail to estimate derivativesProblem
2Couldnt check if we fail to estimate
derivativesNote that such ill condition occurs
at some point (Not all)
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3.2 Reparameterization
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3.2.A Diagonalize
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3.2.B Adaptive variable selection
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Thus least weighted sum of square problem is
transformed to
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4. Links with the other methods
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5. Conclusion and problems