Ch3. Power-spectrum estimation for sensing the environment (1/2)

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Ch3. Power-spectrum estimation for sensing the environment (1/2)

in Cognitive Dynamic Systems, S. Haykin

Course: Autonomous Machine Learning

Soojeong Kim

Cloud and Mobile Systems Laboratory School of Computer Science and Engineering

Seoul National University http://cmslab.snu.ac.kr

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CONTENTS

Power spectrum

Part B

Power spectrum estimation

Part C

Multitaper method

Part D

Multitaper method in space

Part E

C1. Parametric methods C2. Nonparametric methods

Review: Sensing the environment

Part A

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Review : Sensing the environment

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Ill-posed inverse problem

Ill-posed problem : have lack of well-posed conditions (existence, uniqueness, continuity)

Two views of perception : ill-posed inverse problem, Bayesian inference

Inverse problem : uncover underlying physical laws from the stimuli = find a mapping from stimuli to the state

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Sensing the environment

Solving an ill-posed inverse problem

Power spectrum estimation is an example of sensing the environment

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Power spectrum

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What is power spectrum?

The average power of the incoming signal expressed as a function of frequency

Signal is measured by sensors as a time series(time domain)

Power spectrum focus on signal power on frequency(frequency domain)

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Why power spectrum estimation is useful?

Impossible to predict future signal

Estimate the shape of the signal by only finite set of data

Ex : Radar, Radio, Speech Recognition

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Power spectrum in a theory

Incoming signals are picked up as a stochastic(random) process

Only finite set of data is used(applying theory is impossible)

Power spectrum is an ill-posed inverse problem

Fourier transform

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Power spectrum estimation

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Power spectrum Estimation

Parametric methods(model-based)

Non-parametric methods(model-free)

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Parametric method example

Build a model 𝐻 𝑒𝑗𝑗𝜋𝑓

- Apply prior knowledge of Fourier transform(assume 𝑒𝑗𝑗𝜋𝑓 relationship) - H is a function to convert time domain signals to frequency domain signals

- H contains some parameters/coefficients

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Parametric method example

Controlled input signal

- Model output is only depends on the model, H

- Controlled signal is given -> input signal power on frequency is known value

Controlled signal

Model output

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Parametric method example

Problem : Finding parameters/coefficient of 𝐻

- Finding parameters is a typical machine learning problem - Finding parameters of H to produce acceptable output

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Parametric method example

After finding good parameters of H

- Use the model as an estimator - Model is built! , problem is solved

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Power spectrum Estimation

Parametric methods(model-based)

Non-parametric methods(model-free)

If model is good : produce good quality estimates with less sample data If model is bad : mislead conclusion

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Nonparametric method(periodogram)

Only finite set of signal data is used in practice

Use estimates of 𝑆 𝑓 , 𝑆̂ 𝑓 = 𝑁1 |𝑋𝑁(𝑓)|𝑗

𝑥 𝑛 → 𝑋

𝑁

(𝑓) → 𝑆 𝑓 ( Fourier transform)

𝑆 𝑓 = lim 𝑛→∞ 1

𝑁 𝐸[|𝑋𝑁(𝑓)|] 2

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Nonparametric method(periodogram)

Same effect as

𝑥 𝑛 × 𝑎 𝑛 → 𝑥

𝑛 → 𝑋

𝑁

(𝑓) → 𝑆̂ 𝑓 ( Fourier transform)

𝐷𝑒𝑓𝐷𝑛𝑒. 𝑎 𝑛 = 1 𝐷𝑓 𝑛 𝐷𝑖 𝐷𝑛 𝑏𝑏𝑏𝑛𝑏 𝑏𝑓 𝑁, 0 𝑏𝑜𝑜𝑒𝑜𝑜𝐷𝑖𝑒

𝑎(𝑛) is taper

𝑆̂ 𝑓 = 1

𝑁 |𝑋

𝑁

(𝑓)|𝑗

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Nonparametric method(periodogram)

Not require any model to develop

Directly calculate the value using Fourier transform with taper

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Bias-variance dilemma with taper

2 Conditions for good estimator

1. Low bias

2. Low variance(not noisy)

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Bias-variance dilemma with taper

Bias from spectral leakage

Cause of spectral leakage

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Bias-variance dilemma with taper

Spectral leakage↓ by taper(not rectangle) -> bias↓

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Bias-variance dilemma with taper

bias↓ -> variance↑

Taper-> reduce effective sample size -> variance ↑

Bias-variance trade-off!

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Multitaper method

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Multitaper method

Address the trade-off using multiple tapers instead of one

K tapers are used on the same sample data K set of estimates are created

Average K set of estimates

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Tapers are Slepian sequences

Tapers following Slepian sequences

𝑘th taper : 𝑘 ↑→ 𝑏𝐷𝑎𝑖 ↑, 𝑣𝑎𝑜𝐷𝑎𝑛𝑣𝑒 ↓

𝑆� 𝑓 1

𝑆� 𝑓 7

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Multitaper method

All K estimates are averaged

Average of 𝑆� 𝑓 , 𝑆1 � 𝑓 , 𝑆𝑗 � 𝑓 , 3

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Multitaper method

All K estimates are averaged

Find appropriate K to compromise bias and variance

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Why multitaper method is good?

Reduce bias by tapers

Reduce variance by increasing effective sample size(N X K)

Easy to solve bias-variance trade-off by choosing K

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Multitaper method in space

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Multitaper method in space

Previous methods consider power of signal on frequency

Extended multitaper method to get power of signal on frequency and space

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Multitaper method in space

𝑋𝑚𝑘(𝑓) : Fourier transform of input signal by m-th sensor

𝑎𝑚𝑘: coefficients accounting for different localized area around the gridpoints

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A

Multitaper method in space

Apply singular value decomposition to matrix A

𝐴(𝑓) 𝑈 Σ 𝑉 ∗

Sensor related(M X M) Taper related(K X K) Power spectrum estimates(Diagonal)

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Thank You!

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