September 28, 2000 Improved Simultaneous Data Reconciliation, Bias Detection and Identification Using Mixed Integer Optimization Methods Presented by: Tyler A. Soderstrom

Presentation Overview Background MILP Method Extension to Nonlinear Problems Inclusion of Statistical Tests as Constraints Multiple System Models as Constraints Correlated Data Conclusions

Background Data Reconciliation – –Optimal estimates for noisy measurements Bias Detection / Identification – –Determine presence and location of bias Problems are closely related –Presence of bias skews reconciliation results –Common techniques require reconciliation residuals

MILP Method Type of systems considered – Process data matrix –

MILP Method Problem Formulation –

MILP Method Realizable form –

MILP Method Bias constraint region l  ll U  l U  ll U  l U

Tuning Issues Horizon Length Binary variable weighting Bias bounding and thresholding

Extension to Nonlinear Problems Straightforward constraint replacement – with Modified problem is a MINLP –Tougher class of problem –Solution technology is not as mature –Global solution is not guaranteed

Solution Methods Outer Approximation / Equality Relaxation –DICOPT (J. Viswanathan and I. E. Grossman) Random Search –Genetic Algorithm (J. Holland) Meta-Heuristics –Tabu Search (F. Glover)

Basic Method Extensions Make use of Past Information Formulate Extensions as Additional Problem Constraints –Incorporate Common Statistical Tests –Include Empirical Data Model –Compensate for Non-Ideal Process Data May Require Modifications to Objective

Using Previous Estimates Moving Horizon Estimation Problem Previous Estimates of Process Variables and Biases Can be Made Available Including Past Information In Current Problem Execution Improves Stability and Convergence of Estimates Objective is Pathway to Past Information

Objective Modifications Add to Objective Φ Where and are Estimates from Previous Execution Convert to Realizable Form

Realizable Form Objective Φ Additional Constraints

Bias Penalty Term Inclusion Depends on Objectives –Most Will be Zero –May Delay Identification of Bias –If Bias is Persistent, Improve Estimate Convergence Not Important for Optimization Engine –Previous Solution Warm Starts Current Run

Statistical Tests as Constraints Tests are based on Hypothesis Testing –Test Statistic is Proposed –Statistic is Calculated with Current Data –If Statistic Exceeds a Threshold Value (Related to Level of Confidence) Bias is Present Statistic is defined as a problem variable Definition added as a problem constraint Constraint bounding Statistic below threshold Forces the no bias conclusion at solution

Mathematical Description Hypothesis Testing –H 0 there is no bias in the process data –H 1 there is at least 1 bias in the process data –Choice Depends on value of test statistic at a given level of significance Test StatisticZ Add the Following Constraints to Problem Definition Constraint Z   h(y) Null Enforcement Constraint |Z  < Z 

Objective Modifications Previous Description may be Infeasible Define new Constraint Violation Variable and Change Constraint Null Enforcement Constraint: |Z|  < Z   Penalize Variable in Objective Objective:  =  old + w i 

MILP Example

Embedded PC Test Principal Component Test –Form Matrix s.t. it contains eigenvectors of –y e contains principle component scores –scores are normal zero mean, unit variance

Embedded PC Test Principal Component Test – Threshold Value at confidence level  –Perform Test on Averaged Measurements Enhance Power of Test Single set of additional constraints is used in the formation of

Embedded PC Test Additional Constraints

Performance Measures Average number of Type I Errors Overall Power

Simulation Results

Discussion of Results No Benefit to OP –Estimates from basic method usually pass tests without specific enforcement Test Enforcement Increases AVTI –Other biases forced to become active to lower statistics Usually Global Statistical Tests Used First –require nonlinear equations

Non-Ideal Data Compensation Serially Correlated Data Requires New Measurement Noise Model Error Sequence,, Forms a Stationary Process Assume no Cross Correlation

Statistical Tests When Data are Serially Correlated Tests on Individual Vectors are Unaffected Statistical Tests are Often Used on Vectors of Averaged Measurement –Increase Power of the Test –Autocorrelation Invalidates Test Assumptions –Procedure Must be Modified

Statistical Tests When Data are Serially Correlated Most Test Statistics Require Covariance of N Averaged Measurements: –Time Independent Data: –Autocorrelated Data:

Methods of Dealing With Serially Correlated Data Correcting the Variance –Requires all Autocorrelation Coefficients –Coefficients can be calculated analytically if noise model is known (e.g. time series) –Otherwise coefficients can be estimated Prewhitening –Filtering Approach –Requires expression of noise model

Methods of Dealing With Serially Correlated Data Prewhitening (cont.) –Calculate a “approximately independent” sequence –Apply tests designed for independent data

Implementing Compensation Within MIP Framework Correcting the Variance –Unknown Correlation Model Estimate with bias free data Include, calculated using as parameters in MIP program –Known Correlation Model calculated analytically are used as parameters in MIP program Include Modified Test as Set of Constraints

Implementing Compensation Within MIP Framework Prewhitening –Uncorrelated residuals written as functions of noise model parameters and measurements –Equations included as constraints –Tests on uncorrelated residuals included as constraints

Conclusions MIP bias detection / identification performs better than several other methods High Power / Low Occurrence of False Identification Straightforward implementation Method Enhancements –Consider Past Information –Include Statistical Tests in Constraints Univariate Tests Do Not Improve Performance Global Tests May Help, but require nonlinear equations –Handling Autocorrelated Data

Future Work Investigate Additional Constraints with Nonlinear Models –Nonlinear Statistical Tests –Improve sensitivity to small bias Compare solution methods on larger nonlinear models Extend Method to Dynamic Models –Discrete vs. Continuous –Linear and Nonlinear –Computational Issues