CHAPTER 3 RECURSIVE ESTIMATION FOR LINEAR MODELS

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CHAPTER 3 RECURSIVE ESTIMATION FOR LINEAR MODELS
Slides for Introduction to Stochastic Search and
Optimization (ISSO) by J. C. Spall

• Organization of chapter in ISSO
• Linear models
• Relationship between least-squares and
  mean-square
• LMS and RLS estimation
• Applications in adaptive control
• LMS, RLS, and Kalman filter for time-varying
  solution
• Case study Oboe reed data

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Basic Linear Model

• Consider estimation of vector ? in model that is
  linear in ?
• Model has classical linear form
• where zk is kth measurement, hk is corresponding
  design vector, and vk is unknown noise value
• Model used extensively in control, statistics,
  signal processing, etc.
• Many estimation/optimization criteria based on
  squared-error-type loss functions
• Leads to criteria that are quadratic in ?
• Unique (global) estimate ?

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Least-Squares Estimation

• Most common method for estimating ? in linear
  model is by method of least squares
• Criterion (loss function) has form
• where Zn z1, z2 ,, znT and Hn is n ? p
  concatenated matrix of hkT row vectors
• Classical batch least-squares estimate is
• Popular recursive estimates (LMS, RLS, Kalman
  filter) may be derived from batch estimate

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Geometric Interpretation of Least-Squares
Estimate when p 2 and n 3
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Recursive Estimation

• Batch form not convenient in many applications
• E.g., data arrive over time and want easy way
  to update estimate at time k to estimate at time
  k1
• Least-mean-squares (LMS) method is very popular
  recursive method
• Stochastic analogue of steepest descent algorithm
  
• LMS recursion
• Convergence theory based on stochastic
  approximation (e.g., Ljung, et al., 1992
  Gerencsér, 1995)
• Less rigorous theory based on connections to
  steepest descent (ignores noise) (Widrow and
  Stearns, 1985 Haykin, 1996)

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LMS in Closed-Loop Control

• Suppose process is modeled according to
  autoregressive (AR) form
• where xk represents state, ? and ?i are unknown
  parameters, uk is control, and wk is noise
• Let target (desired) value for xk be dk
• Optimal control law known (minimizes mean-square
  tracking error)
• Certainty equivalence principle justifies
  substitution of parameter estimates for unknown
  true parameters
• LMS used to estimate ? and ?i in closed-loop mode

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LMS in Closed-Loop Control for First-Order AR
Model
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Recursive Least Squares (RLS)

• Alternative to LMS is RLS
• Recall LMS is stochastic analogue of steepest
  descent (first order method)
• RLS is stochastic analogue of Newton-Raphson
  (second order method) ? faster convergence than
  LMS in practice
• RLS algorithm (2 recursions)
• Need P0 and to initialize RLS recursions

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Recursive Methods for Estimation of Time-Varying
Parameters

• It is common to have the underlying true ? evolve
  in time (e.g., target tracking, adaptive control,
  sequential experimental design, etc.)
• Time-varying parameters implies ? replaced with
  ?k
• Consider modified linear model
• Prototype recursive form for estimating ?k is
• where choice of Ak and ?k depends on specific
  algorithm

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Three Important Algorithms for Estimation of
Time-Varying Parameters

• LMS
• Goal is to minimize instantaneous squared-error
  criteria across iterations
• General form for evolution of true parameters ?k
• RLS
• Goal is to minimize weighted sum of squared
  errors
• Sum criterion creates inertia not present in
  LMS
• General form for evolution of ?k
• Kalman filter
• Minimizes instantaneous squared-error criteria
• Requires precise statistical description of
  evolution of ?k via state-space model
• Details for above algorithms in terms of
  prototype algorithm (previous slide) are in
  Section 3.3 of ISSO

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Case Study LMS and RLS with Oboe Reed Data

• an ill wind that nobody blows good.
• Comedian Danny Kaye in speaking of the oboe in
  the The Secret Life of Walter Mitty (1947)
• Section 3.4 of ISSO reports on linear and
  curvilinear models for predicting quality of oboe
  reeds
• Linear model has 7 parameters curvilinear has 4
  parameters
• This study compares LMS and RLS with batch
  least-squares estimates
• 160 data points for fitting models (reeddata-fit
  ) 80 (independent) data points for testing
  models (reeddata-test)
• reeddata-fit and reeddata-test data sets
  available from ISSO Web site

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Oboe with Attached Reed
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Comparison of Fitting Results for reeddata-fit
and reeddata-test

• To test similarity of fit and test data sets,
  performed model fitting using test data set
• This comparison is for checking consistency of
  the two data sets not for checking accuracy of
  LMS or RLS estimates
• Compared model fits for parameters in
• Basic linear model (eqn. (3.25) in ISSO) (p 7)
• Curvilinear model (eqn. (3.26) in ISSO) (p 4)
• Results on next slide for basic linear model

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Comparison of Batch Parameter Estimates for Basic
Linear Model. Approximate 95 Confidence
Intervals Shown in ,
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Comparison of Batch and RLS with Oboe Reed Data

• Compared batch and RLS using 160 data points in
  reeddata-fit and 80 data points for testing
  models in reeddata-test
• Two slides to follow present results
• First slide compares parameter estimates in pure
  linear model
• Second slide compares prediction errors for
  linear and curvilinear models

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Batch and RLS Parameter Estimates for Basic
Linear Model (Data from reeddata-fit )
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Mean and Median Absolute Prediction Errors for
the Linear and Curvilinear Models (Model fits
from reeddata-fit Prediction Errors from
reeddata-test)