EM algorithm for Markov chains observed via Gaussian noise and point process information: Theory and case studies

Camilla Damian; Zehra Eksi; Rüdiger Frey

doi:10.1515/strm-2017-0021

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EM algorithm for Markov chains observed via Gaussian noise and point process information: Theory and case studies

Camilla Damian , Zehra Eksi und Rüdiger Frey

Veröffentlicht/Copyright: 21. November 2017

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Abstract

In this paper we study parameter estimation via the Expectation Maximization (EM) algorithm for a continuous-time hidden Markov model with diffusion and point process observation. Inference problems of this type arise for instance in credit risk modelling. A key step in the application of the EM algorithm is the derivation of finite-dimensional filters for the quantities that are needed in the E-Step of the algorithm. In this context we obtain exact, unnormalized and robust filters, and we discuss their numerical implementation. Moreover, we propose several goodness-of-fit tests for hidden Markov models with Gaussian noise and point process observation. We run an extensive simulation study to test speed and accuracy of our methodology. The paper closes with an application to credit risk: we estimate the parameters of a hidden Markov model for credit quality where the observations consist of rating transitions and credit spreads for US corporations.

Keywords: Expectation maximization (EM) algorithm; hidden Markov models; point processes; nonlinear filtering; goodness-of-fit tests; credit risk ratings

MSC 2010: 60G35; 62P05

Funding statement: Camilla Damian and Rüdiger Frey are grateful for the support by the Vienna Science and Technology Fund (WWTF) through project MA14-031.

A EM algorithm for Example 2.2

In what follows we are going to provide the steps of the EM algorithm corresponding to Example 2.2. To this, we need to define new processes due to the dependence of the variates λt+ and λt- on the rating observation Rti. Namely, we define the processes Cj⁢k and Bj⁢k, 1≤j,k≤K, with the following:

Bt+,j⁢k=∫0t1{Rs=ej}⁢〈Xs,ek〉⁢𝑑Ds+ and Ctj⁢k=∫0t1{Rs=ej}⁢〈Xs,ek〉⁢𝑑s.

We can define B-,j⁢k in a similar fashion. Now we have the following likelihood function:

L⁢(θ,θ′)=…+∫0tlog⁡(λ+⁢(Xs,Rs))⁢𝑑Ds+-∫0tλ+⁢(Xs,Rs)⁢𝑑s+∫0tlog⁡(λ-⁢(Xs,Rs))⁢𝑑Ds--∫0tλ-⁢(Xs,Rs)⁢𝑑s+R⁢(θ′).

Note that we can write

λ+⁢(Xs,Rs)=∑i=1K∑j=1Kλ+,i⁢j⁢1{Rs=ej}⁢〈Xs,ei〉.

Hence

L⁢(λ+,λ+⁣′)=∫0t∑k=1K∑j=1Klog⁡(λ+,j⁢k)⁢1{Rs=ej}⁢〈Xs,ek〉⁢d⁢Ds+-∫0t∑k=1K∑j=1Kλ+,j⁢k⁢1{Rs=ej}⁢〈Xs,ek〉⁢d⁢s+R⁢(λ+⁣′).

Thus we have

L⁢(λ+,λ+⁣′)=∑k=1K∑j=1Klog⁡(λ+,j⁢k)⁢Bt+,j⁢k-∑k=1K∑j=1Kλ+,j⁢k⁢Ctj⁢k+R⁢(λ+⁣′).

Next we write the filtered estimate of the log-likelihood function:

L⁢(λ+,λ+⁣′)^=∑k=1K∑j=1Klog⁡(λ+,j⁢k)⁢Bt+,j⁢k^-∑k=1K∑j=1Kλ+,j⁢k⁢Ctj⁢k^+R⁢(λ+⁣′).

Hence, we have what is needed for the E-Step. Let us now use the parametrization

λ+,j⁢k=λ1+⁢1{k<j}+λ2+⁢1{k=j}+λ3+⁢1{k>j},1≤j,k≤K,k>1.

Hence

L⁢(λ+,λ+⁣′)^=∑k=2K∑j=1Klog⁡(λ1+⁢1{k<j}+λ2+⁢1{k=j}+λ3+⁢1{k>j})⁢Bt+,j⁢k^

-∑k=2K∑j=1K(λ1+⁢1{k<j}+λ2+⁢1{k=j}+λ3+⁢1{k>j})⁢Ctj⁢k^+R⁢(λ+⁣′).

From the first order conditions we then obtain the following estimates:

λ1+^=∑j=1K∑1<k<jKBt+,j⁢k^∑j=1K∑1<k<jKCtj⁢k^,λ2+^=∑k=2KBt+,k⁢k^∑k=2KCtk⁢k^,λ3+^=∑j=1K∑k>jKBt+,j⁢k^∑j=1K∑k>jKCtj⁢k^.

To apply the algorithm, we need to obtain the filtered estimates for the quantities Ctj⁢k and Btj⁢k, and their robust version. These are computed exactly as in Section 3.

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Received: 2017-6-26

Revised: 2017-10-22

Accepted: 2017-10-23

Published Online: 2017-11-21

Published in Print: 2018-1-1

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Schlagwörter für diesen Artikel

Expectation maximization (EM) algorithm; hidden Markov models; point processes; nonlinear filtering; goodness-of-fit tests; credit risk ratings