Online Probabilistic Model Identification Using Adaptive Recursive MCMC Pedram Agand

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Online Probabilistic Model Identiﬁcation Using

Adaptive Recursive MCMC

Pedram Agand

Department of Computer Science,

Simon Fraser University

Burnaby, Canada.

pagand@sfu.ca

Mo Chen

Department of Computer Science,

Simon Fraser University

Burnaby, Canada.

0000-0001-8506-3665

Hamid D. Taghirad

Department of Electrical Engineering,

K. N. Toosi University of Technology

Tehran, Iran.

0000-0002-0615-6730

Abstract—Although the Bayesian paradigm offers a formal

framework for estimating the entire probability distribution

over uncertain parameters, its online implementation can be

challenging due to high computational costs. We suggest the

Adaptive Recursive Markov Chain Monte Carlo (ARMCMC)

method, which eliminates the shortcomings of conventional online

techniques while computing the entire probability density func-

tion of model parameters. The limitations to Gaussian noise, the

application to only linear in the parameters (LIP) systems, and

the persistent excitation (PE) needs are some of these drawbacks.

In ARMCMC, a temporal forgetting factor (TFF)-based variable

jump distribution is proposed. The forgetting factor can be

presented adaptively using the TFF in many dynamical systems

as an alternative to a constant hyperparameter. By offering

a trade-off between exploitation and exploration, the speciﬁc

jump distribution has been optimised towards hybrid/multi-

modal systems that permit inferences among modes. These trade-

off are adjusted based on parameter evolution rate. We demon-

strate that ARMCMC requires fewer samples than conventional

MCMC methods to achieve the same precision and reliability.

We demonstrate our approach using parameter estimation in

a soft bending actuator and the Hunt-Crossley dynamic model,

two challenging hybrid/multi-modal benchmarks. Additionally,

we compare our method with recursive least squares and the

particle ﬁlter, and show that our technique has signiﬁcantly more

accurate point estimates as well as a decrease in tracking error

of the value of interest.

Index Terms—MCMC, Bayesian optimization, hybrid/multi-

modal systems, temporal forgetting factor.

I. INTRODUCTION

Bayesian methods are powerful tools to not only obtain a

numerical estimate of a parameter but also to give a measure of

conﬁdence [1]. This is obtained by calculating the probability

distribution of parameters rather than a point estimate, which

is prevalent in frequentist paradigms [2], [3]. One of the main

advantages of probabilistic frameworks is that they enable

decision making under uncertainty. In addition, knowledge

fusion is signiﬁcantly facilitated in probabilistic frameworks;

different sources of data or observations can be combined

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according to their level of certainty in a principled manner [4].

Using credible intervals instead of conﬁdence intervals [5], an

absence of over parameterized phenomena [1], and evaluation

in the presence of limited number of observed data [6] are

other distinct features of this framework.

Nonetheless, Bayesian inference requires high computa-

tional effort for obtaining the whole probability distribution

and requires prior general knowledge about the noise dis-

tribution. Consequently, strong simplifying assumptions were

often made (e.g. calculating speciﬁc features of the model

parameters distribution rather than the whole distribution).

One of the most effective methods for Bayesian inferences is

Markov Chain Monte Carlo (MCMC). In the ﬁeld of system

identiﬁcation, MCMC variants such as the one proposed by

[7] are mostly focused on ofﬂine system identiﬁcation. This is

partly due to computational challenges which prevent its real-

time use. There are extensive research in the literature which

investigate how to increase sample efﬁciency (e.g. [8], [9]).

The authors in [10] ﬁrst introduced reversible jump Markov

chain Monte Carlo (RJMCMC) as a method to address the

model selection problem. In this method, an extra pseudo-

random variable is deﬁned to address dimension mismatch.

There are further extensions of MCMC in the literature;

however, there is a lack of variants suitable for online esti-

mation. One example which claims to have often much faster

convergence is No-U-Turn Sampler (NUTS), which can adapt

proposals “on the ﬂy” [11].

If the general form of the relation between inputs and

outputs is known either by physical relation or knowing the

basis function, parametric identiﬁcation techniques are an

effective way to model a system. To identify the parameters in

a known model, researchers propose frequency ( [12]) or time

domain ( [13]) approaches. Noisy measurements, inaccuracy,

inaccessibility, and costs are typical challenges that limit direct

measurement of unknown parameters in a physical/practical

system [14]. Parameter identiﬁcation techniques have been

extensively used in different subject areas including but not

limited to chemistry, robotics, fractional models and health

sectors [15]–[19]. For instance, motion ﬁltering and force

prediction in robotics applications are important ﬁelds of study

with interesting challenges which makes them suitable test

cases for Bayesian inferences [20].

arXiv:2210.12595v2 [cs.LG] 19 Oct 2023

Different parametric identiﬁcation methods have been pro-

posed in the literature for linear and Gaussian noise [21];

however, in cases of nonlinear hybrid/multi-modal systems

(e.g. Hunt-Crossley or ﬂuid soft bending actuator) or systems

with non-Gaussian noise (e.g. impulsive disturbance), there is

no optimal solution for the identiﬁcation problem. In addition,

in cases that the assumed model for the system is not com-

pletely valid, MCMC implementation of Bayesian approach

can provide reasonable inferences. Authors in [22] utilize the

least square methods for nonlinear physical modeling of air

dynamics in pneumatic soft actuator. A method to determine

the damping term in the Hunt-Crossley model was proposed

in [23]. A single-stage method for the estimation of the Hunt-

Crossley model is proposed by [24] which requires some

restrictive conditions to calculate the parameters. Moreover,

the method does not offer any solution for discontinuity in

the dynamic model, which is common in the transition phase

from contact to free motion and vice versa [25].

This paper proposes a new technique, Adaptive Recursive

Markov Chain Monte Carlo (ARMCMC), to address several

weaknesses of traditional online identiﬁcation methods, such

as solely being applicable to systems Linear in Parameters

(LIP), having Persistent Excitation (PE) requirements, and

assuming Gaussian noise. ARMCMC is an online method that

takes advantage of the previous posterior distribution whenever

there is no abrupt change in the parameter distribution. To

achieve this, we deﬁne a new variable jump distribution that

accounts for the degree of model mismatch using a temporal

forgetting factor (TFF). The TFF is computed from a model

mismatch index and determines whether ARMCMC employs

modiﬁcation or reinforcement to either restart or reﬁne the

estimated parameter distribution. As this factor is a function of

the observed data rather than a simple user-deﬁned constant, it

can effectively adapt to the underlying dynamics of the system.

We demonstrate our method using two different examples:

a soft bending actuator and the Hunt-Crossley model. We

show favorable performance compared to the state-of-the-art

baselines.

II. PRELIMINARIES

A. Problem statement

In the Bayesian paradigm, parameter estimations are given

in the form of the posterior probability density function (pdf);

this pdf can be continuously updated as new data points are

received. Consider the following general model:

Y=Fj(X, θj) + νj,∀j∈ {1,2, . . . , m},(1)

where Y,X,θ, and νare concurrent output, input, model

parameters set and noise vectors, respectively. For a hybrid

system, there can be mdifferent nonlinear functions (Fj)

with different parameters set (θj) for each mode. For multi-

modal systems, there can be mdifferent noise distribution νj

for each mode. To calculate the posterior pdf, the observed

data (input/output pairs) along with a prior distribution are

time

Data

Interval

Algorithm

Interval

𝑡!𝑡"𝑡"+ 𝑁#+ 1

𝑡 − 1 𝑡

……

Phase: A

𝑡 + 1

…

Fig. 1: Data timeline and different phases of ARMCMC

algorithm. For algorithm at time t: Phase (A) Data collection

[Nsdata points are packed for the next algorithm time

step], Phase (B) Adjustment [the method is applying to the

most recent pack], (C) Execution [after the min evaluation of

algorithm, the results will be updated on posterior distribution

and any byproduct value of interest].

combined via Bayes’ rule [26]. The Bayesian rule is deﬁned

as follows [1].

P(θ|D) = P(D|θ)P(θ)

P(D),(2)

where, P(θ)is the prior probability of parameters, P(D) =

RP(D|θ)P(θ)dθ, is called as evidence, P(D|θ)is the like-

lihood function, and P(θ|D)is posterior probability of pa-

rameters given the data. We will be applying updates to the

posterior pdf using batches of data points; hence, it will be

convenient to partition the data as follows:

Dt={(X, Y )tm+1,(X, Y )tm+2,··· ,(X, Y )tm+Ns+1},(3)

where Ns=Ts/T is the number of data points in each

data pack with T, Tsbeing the data and algorithm sampling

times, respectively. This partitioning is convenient for online

applications, as Dt−1should have been previously collected

so that the algorithm can be executed from tmto tm+Ns+1,

an interval which we will deﬁne as algorithm time step t.

Ultimately, inferences are completed at tm+Ns+ 2. Fig.

1 illustrates the timeline for the data and the algorithm.

It is worth mentioning that the computation can be done

simultaneously with the task in the adjacent algorithm step

(e.g. phase A of algorithm t, phase B of algorithm t−1and

phase C of algorithm t−2can all be done simultaneously).

According to Bayes’ rule (2) and assuming data points are

independent and identically distributed ( i.i.d) in Eq. (1), we

have

P(θt|[Dt−1, Dt]) = PDt|θt, Dt−1P(θt|Dt−1)

RPD1|τt, Dt−1P(τt|Dt−1)dτt,

(4)

where θtdenotes the parameters at current algorithm time

steps. P(θt|Dt−1)is the prior distribution over parameters,

which is also the posterior distribution at the previous algo-

rithm time step. Probability PDt|θt, Dt−1is the likelihood

function which is obtained by sampling from the one-step-

ahead prediction:

Yt|t−1=F(Dt−1, θt),(5)

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OnlineProbabilisticModelIdentificationUsingAdaptiveRecursiveMCMCPedramAgandDepartmentofComputerScience,SimonFraserUniversityBurnaby,Canada.pagand@sfu.caMoChenDepartmentofComputerScience,SimonFraserUniversityBurnaby,Canada.0000-0001-8506-3665HamidD.TaghiradDepartmentofElectricalEngineering,K.N.ToosiU...

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Online Probabilistic Model Identification Using Adaptive Recursive MCMC Pedram Agand

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