APPROACHES IN USING EXPECTATIONMAXIMIZATION ALGORITHM FOR MAXIMUM LIKELIHOOD ESTIMATION OF THE PARAMETERS OF A CONSTRAINED STATE SPACE MODEL WITH AN EXTERNAL INPUT SERIES

David C. Wyld et al. (Eds) : DBDM, CICS, CSIP, AI&FL, SCOM, CSE, CCNET-2016
pp. 57–66, 2016. © CS & IT-CSCP 2016 DOI : 10.5121/csit.2016.60506
APPROACHES IN USING EXPECTATION-
MAXIMIZATION ALGORITHM FOR
MAXIMUM LIKELIHOOD ESTIMATION OF
THE PARAMETERS OF A CONSTRAINED
STATE SPACE MODEL WITH AN
EXTERNAL INPUT SERIES
Chengliang Huang1
, Xiao-Ping Zhang1
and Fang Wang2
1
Department of Electrical and Computer Engineering, Ryerson University
Toronto, On, Canada
c28huang@ryerson.ca, xzhang@ee.ryerson.ca
2
School of Business and Economics, Wilfrid Laurier University
Waterloo, On, Canada
fwang@wlu.ca
ABSTRACT
EM algorithm is popular in maximum likelihood estimation of parameters for state-space
models. However, extant approaches for the realization of EM algorithm are still not able to
fulfill the task of identification systems, which have external inputs and constrained parameters.
In this paper, we propose new approaches for both initial guessing and MLE of the parameters
of a constrained state-space model with an external input. Using weighted least square for the
initial guess and the partial differentiation of the joint log-likelihood function for the EM
algorithm, we estimate the parameters and compare the estimated values with the “actual”
values, which are set to generate simulation data. Moreover, asymptotic variances of the
estimated parameters are calculated when the sample size is large, while statistics of the
estimated parameters are obtained through bootstrapping when the sample size issmall. The
results demonstrate that the estimated values are close to the “actual” values.Consequently, our
approaches are promising and can applied in future research.
KEYWORDS
State-Space Model, Maximum Likelihood Estimation, Expectation Maximization Algorithm,
Kalman filtering and smoothing, Asymptotic variances, Bootstrapping
1. INTRODUCTION
One of the advantages of the state-space models for linear dynamic systems is their ability to fit
more parsimonious structures with fewer parameters to describe a multivariate time series. As a
result, the application of state-space models is not limited to engineering practice.
For example, in certain scenarios, the dynamics of the brand equity of a firm can be represented
by a state-space model in the form of Kalman filter [1],

58 Computer Science & Information Technology (CS & IT)
൜
‫ݔ‬௧ = α‫ݔ‬௧ିଵ + ߛ‫ݑ‬୲ + ݁୲
‫ݕ‬௧ = ℎ‫ݔ‬௧ + ‫ݓ‬௧
, (1)
where ‫ݔ‬௧ is the invisible brand equity, ‫ݑ‬୲ is the investment, the external input, ‫ݕ‬௧ is the brand
performance, the output, at a certain step ‫.ݐ‬ The transition coefficient, α(|ߙ| < 1), and the input
coefficient, ߛ, are parameters of interest. The observation coefficient, ℎ, is constrained to be a
constant, i.e., ℎ=1. The observation noise, ‫ݓ‬௧, the process noise,݁௧, and the initial state,‫ݔ‬଴, at each
step, {x0, ݁ଵ, ..., ݁୲, ‫ݓ‬ଵ ... ‫ݓ‬௧}, are all assumed to be mutually independent, where
‫ݓ‬௧~ܰ(0, ߪ௪
ଶ
),݁௧~ܰ(0, ߪ௘
ଶ
), and ‫ݔ‬଴~ܰ(ߤ଴, ߪ଴
ଶ
).
As exemplified in Equation (1) where ℎ=1, a constrained state-space modemeans that some
parameters, some elements in the parameter matrix of a Kalman filter, are fixed or shared thus not
all of them have to be estimated. Conversely, if the model is unconstrained, any parameters or any
elements of each parameter matrix will be estimated. Moreover, the model has an time-variant
external input series, ‫ݑ‬୲. By the way, parameter estimation of state space models with external
inputs can be seen as supervised problems while that of state space models without external
inputs can be treated as unsupervised problems.
Maximum likelihood estimation (MLE) [2] is used to obtain the time-invariant parameters of the
Kalman filter from input, ‫ݑ‬୲, and the output, ‫ݕ‬௧. Because of the existence of hidden variables, ‫ݔ‬௧,
in the formulated likelihood function, expectation-maximization (EM) algorithm [3], an iterative
method, is used to complete MLE of parameters in state-space models.
The approaches for the implementation of EM algorithm in the MLE of unconstrained state-space
model are delivered in [4]-[7]. The approaches for certain constrained state-space models are
covered in [7]-[10]. Since those constraints are quite specific, the approaches proposed are lack of
generalicity. On the other hand, approaches for state-space models with external inputs are
usually not provided except in [7]-[9]. However, the state space mode in [7]-[9] is not typical
because the inputs are constants or do not affect the hidden variable. Especially in [5], the
external input is actually a constant, which is a parameter to be estimated. Therefore, as we know
from extant literature, no innovative methods are explored to estimate the unknown parameters of
such constrained state-space modelsin the recent years.
Our research is to find appropriate methods for the supervised problem: using EM algorithm for
the MLE of those constrained state-space models who have external inputs. In this paper, we use
the model represented by Equation (1) as an example. Our target is to estimate the parameters,α,
ߛ, ߪ௘, ߪ௪, ߤ଴ and ߪ଴. Our work is carried out mainly in two phases: (i) initial guessing using our
innovative approach, and (ii) implementation of MLE using EM algorithm with approaches
different from those in extant literature. In addition, if the sample size is large, we use the
asymptotic variances of the estimated parameters to check the accuracy of the estimation. If the
sample size is small, we introduced bootstrapping to examine the distribution of the estimated
parameters.
2. INITIAL GUESSING
The initial guess is performed through two steps. Firstly, the system parameter ߙ and the
variances ߪ௘, ߪ௪ and ߪ଴ are guessed using autocovariance of the observations. Then the initial
state mean ߤ଴ and input parameter γ are guessed using weighted linear square.
2.1.GuessingSystem Parameter and the Variances
Denote ߟ௧ = ∑ ߙ௜
݁௧ି௜
ஶ
௜ୀ଴ , from (1) it can be proved that

Computer Science & Information Technology (CS & IT) 59
‫ݔ‬௧ =
ఊ௨౪
ଵିఈ
+ ߟ௧ . (2)
For ℎ = 0,1,2 … …, we can obtain the covariance and autocovariance of ߟ௧ as
ߛఎ(ℎ) =
஢౛
మఈ೓
ଵିఈమ, (3)
while it is obvious that
σ଴
ଶ
=
஢౛
మ
ଵି஑మ. (4)
which means that we don’t have to estimate ߪ௘ and ߪ଴ separately, as was performed in extant
literature.
Moreover, we have the variance of ‫ݕ‬୲
ߛ௬(0) = ‫ݒ‬଴ + ‫ݒ‬୵, (5)
and the covariance of ‫ݕ‬୲ when ℎ = 1, 2, 3 ⋯,
ߛ௬(ℎ) = ߛఎ(ℎ). (6)
Hence we can obtain the guessed initial values for system parameter,ߙ, and standard deviations,
ߪ௘ and ߪ௪, from
‫ە‬
ۖ
‫۔‬
ۖ
‫ۓ‬ ߙ =
ఊ೤(ଶ)
ఊ೤(ଵ)
ߪ௘
ଶ
=
൫ଵି஑మ൯ఊ೤(ଵ)
஑
ߪ௪
ଶ
= ߛ௬(0) −
ఙ೐
మ
ଵି஑మ
. (7)
2.2. Guessing Initial State Mean and Input Parameter
Denoting
‫ݖ‬௧ = ∑ α୲ି୧
‫ݑ‬୧
௧
௜ୀଵ , (8)
and
ߞ௧ = ∑ α୲ି୧
݁୧
௧
௜ୀଵ + ‫ݓ‬௧, (9)
we have,
‫ݕ‬௧ = α୲
ߤ଴ + ߛ‫ݖ‬௧ + ߞ௧, (10)
where ߞ௧~ܰ൫0, ߪ఍
ଶ
൯ and
ߪ఍
ଶ
=
ଵି஑మ౪
ଵି஑మ ߪ௘
ଶ
+ ߪ௪
ଶ
(11)
In order to estimate ߤ଴ and ߛ, we perform linear regression between ‫ݕ‬௧, as dependent variable,
and α୲
and ‫ݖ‬௧, as independent variables, using T samples of ‫ݑ‬௧ and ‫ݕ‬௧. Since ߞ௧ is
heteroscadestical, we apply weighted least square (WLS). WLS finds its optimum when the
weighted sum, S, of squared residuals is minimized where
ܵ = ∑
൫௬೟ି஑౪ఓబିఊ௭೟൯
మ
భషಉమ౪
భషಉమ ఙ೐
మାఙೢ
మ
்
௧ୀଵ . (12)
Denote that ‫ݒ‬௧ = (1 − αଶ୲)ߪ௘
ଶ + (1 − αଶ)ߪ௪
ଶ , we solve the gradient equation (regarding ߤ଴ and ߛ
respectively) for the sum of squares

൞
∑
஑౪൫௬೟ି஑౪ఓబିఊ௭೟൯൫ଵି஑మ൯
௩೟
்
௧ୀଵ = 0
∑
௭೟൫௬೟ି஑౪ఓబିఊ௭೟൯൫ଵି஑మ൯
௩೟
்
௧ୀଵ = 0
. (13)
Therefore, we will have initial guess about as below:
‫ە‬
ۖ
‫۔‬
ۖ
‫ۓ‬ߛ =
∑
ಉమ౪
ೡ೟
೅
೟సభ ∑
೥೟೤೟
ೡ೟
೅
೟సభ ି∑
ಉ౪೤೟
ೡ೟
೅
೟సభ ∑
ಉ౪೥೟
ೡ೟
೅
೟సభ
∑
ಉమ౪
ೡ೟
೅
೟సభ ∑
೥೟
మ
ೡ೟
೅
೟సభ ି∑
ಉ౪೥೟
ೡ೟
೅
೟సభ ∑
ಉ౪೥೟
ೡ೟
೅
೟సభ
ߤ଴ =
∑
ಉ౪೤೟
ೡ೟
೅
೟సభ ∑
೥೟
మ
ೡ೟
೅
೟సభ ି∑
ಉ౪೥೟
ೡ೟
೅
೟సభ ∑
೤೟೥೟
ೡ೟
೅
೟సభ
∑
ಉమ౪
ೡ೟
೅
೟సభ ∑
೥೟
మ
ೡ೟
೅
೟సభ ି∑
ಉ౪೥೟
ೡ೟
೅
೟సభ ∑
ಉ౪೥೟
ೡ೟
೅
೟సభ
. (14)
3. ESTIMATION USING EM ALGORITHM
The conditional density for the states and outputs are,
ܲ(‫ݔ‬௧|‫ݔ‬௧ିଵ) =
ଵ
ఙ೐√ଶగ
݁‫݌ݔ‬ ቂ−
(௫೟ି஑௫೟షభିఊ௨೟)మ
ଶఙ೐
మ ቃ, (15)
ܲ(‫ݕ‬௧|‫ݔ‬௧) =
ଵ
ఙೢ√ଶగ
݁‫݌ݔ‬ ቂ−
(௬೟ି௫೟)మ
ଶఙೢ
మ ቃ. (16)
Assuming a Gaussian initial state density
ܲ(‫ݔ‬଴) =
ଵ
ఙబ√ଶగ
݁‫݌ݔ‬ ቂ−
(௫బିఓబ)మ
ଶఙబ
మ ቃ, (17)
By the Markov property implicit in this model, we calculate the joint probability, not the partial
probability used by Shumway (2011), regarding all T samples of ‫ݔ‬௧ and ‫ݕ‬௧, denoted as ሼ‫ݔ‬ሽ and
ሼ‫ݕ‬ሽ respectively:
ܲ(ሼ‫ݔ‬ሽ, ሼ‫ݕ‬ሽ) = ܲ(‫ݔ‬଴) ∏ ܲ(‫ݔ‬௧|‫ݔ‬௧ିଵ)்
௧ୀଵ ∏ ܲ(‫ݕ‬௧|‫ݕ‬௧ିଵ)்
௧ୀଵ (18)
We denote the joint log probability as
ߗ = logܲ(ሼ‫ݔ‬ሽ, ሼ‫ݕ‬ሽ). (19)
According to Equation (4), we only need to estimate the parameter set, ߖ = ሼα, ߛ, ߪ௘, ߪ௪, ߤ଴ሽ,
through maximizing the objective function:
ߗ(α, ߛ, ߪ௘, ߪ௪, ߤ଴) = −
൫ଵି஑మ൯(௫బିఓబ)మ
ଶఙ೐
మ −
ଵ
ଶ
log
ఙ೐
మ
ଵି஑మ − ∑
(௫೟ି஑௫೟షభିఊ௨೟)మ
ଶఙ೐
మ
்
௧ୀଵ −
ଵ
ଶ
ܶlogߪ௘
ଶ
−
∑
(௬೟ି௫೟)మ
ଶఙೢ
మ
்
௧ୀଵ −
ଵ
ଶ
ܶlogߪ௪
ଶ
−
ଶ்ାଵ
ଶ
log(2ߨ). (20)
3.1. EM Algorithm
Since the objective function expressed by Equation(20) depends on the unobserved data
series,‫ݔ‬௧(‫ݐ‬ = 1, 2, … T), we consider applying the EM algorithm conditionally with respect to the
observed output series ‫ݕ‬ଵ, ‫ݕ‬ଶ, …, ‫ݕ‬୘. The objective function above has an input series.
Accordingly, the input coefficient has to be estimated. Consequently, our approaches are unlike
the approaches[6] used inthe implementation ofEM algorithm for linear dynamic systems.

The EM algorithm mainly has two steps: the E-STEP and the M-Step. During the E-Step, the
parameters are assumed known, the hidden states and their variance are estimated over all the
samples, and then the likelihood function constructed from joint probability are calculated.
During the M step, we have to find the parameter set ߖ෡(݇) = ሼα(݇), ߛ(݇), ߪ௘(݇), ߪ௪(݇), ߤ଴(݇)ሽ
for the kth counts of the recursions by maximizing the conditional expectation, or the above
objective function.
The overall procedure for the implementation of EM algorithm is as below:
(i) Initialize the procedure by selecting the guessed values as starting values for the
parameters.
On iteration k, (k=1,2,……)
(ii) Compute the log-likelihood (optional),
(iii) Use the parameters to obtain the smoothed values of the hidden states and their
correlations, for t= 1,2, …...,T.
(iv) Use the smoothed values to calculate the updated parameters.
(v) Repeat Steps (ii) – (iv) to convergence.
We mainly perform two sub-steps in the E-step of EM algorithm: Kalman filtering and Kalman
smoothing.
3.2. Kalman Filtering and Smoothing
Assuming thatwe already know the parameter set { α, ߛ, ߪ௘,ߪ௪,ߤ଴,ߪ଴}(‫ݔ‬଴~ܰ(ߤ଴, ߪ଴
ଶ
)), and the
observations ‫ݕ‬௧ and ‫ݑ‬௧, we have the estimation of the hidden state, as well as the variances
estimated based on the observations for the period 1 to t.
‫ݔ‬௧|௧ିଵ = α‫ݔ‬௧ିଵ|௧ିଵ + ߛ‫ݑ‬௧, (21a)
‫ݒ‬௧|௧ିଵ = αଶ
‫ݒ‬௧ିଵ|௧ିଵ + ߪ௘
ଶ
, (21b)
‫ݕ‬௧෥ = ‫ݕ‬௧ − ‫ݔ‬௧|௧ିଵ, (21c)
‫ݏ‬௧ = ‫ݒ‬௧|௧ିଵ + ‫ݒ‬௪, (21d)
݇௧ = ‫ݒ‬௧|௧ିଵ‫ݏ‬௧
ିଵ
, (21e)
‫ݔ‬௧
௧
= ‫ݔ‬௧|௧ିଵ + ݇௧‫ݕ‬௧෥ , (21f)
‫ݒ‬௧
௧
= (1 − ݇௧)‫ݒ‬௧|௧ିଵ, (21g)
where ‫ݔ‬଴|଴ = ߤ଴ and ‫ݒ‬଴|଴ = ߪ଴
ଶ
.
According to [6],to compute‫ܧ‬ሾ‫ݔ‬௧|ሼ‫,ݕ‬ ‫ݑ‬ሽሿ ≡ ‫ݔ‬௧
்
and the correlation matrices ‫݌‬௧ ≡ ‫ݒ‬௧
்
+ ‫ݔ‬௧
்(‫ݔ‬௧
்)ᇱ
one performs a set of backward recursion using
݆௧ିଵ = α
௩೟షభ|೟షభ
௩೟|೟షభ
, (22)

‫ݔ‬௧ିଵ
்
= ‫ݔ‬௧ିଵ|௧ିଵ + ݆௧ିଵ൫‫ݔ‬௧
்
− α‫ݔ‬௧ିଵ|௧ିଵ − ߛ‫ݑ‬௧൯, (23)
‫ݒ‬௧ିଵ
்
= ‫ݒ‬௧ିଵ|௧ିଵ + ݆௧ିଵ൫‫ݒ‬௧
்
− ‫ݒ‬௧|௧ିଵ൯݆௧ିଵ
ᇱ
, (24)
where ‫ݔ‬்
்
= ‫ݔ‬்|் and ‫ݒ‬்
்
= ‫ݒ‬்|். We also have ‫݌‬௧,௧ିଵ ≡ ܸ௧,௧ିଵ
்
+ ‫ݔ‬௧
்(‫ݔ‬௧ିଵ
் )ᇱ, where ‫ݒ‬௧,௧ିଵ
்
can be
obtained through the backward recursions
‫ݒ‬௧ିଵ,௧ିଶ
்
= ‫ݒ‬௧ିଵ|௧ିଵ݆௧ିଶ
ᇱ
+ ݆௧ିଵ൫‫ݒ‬௧,௧ିଵ
்
− α‫ݒ‬௧ିଵ|௧ିଵ൯݆௧ିଶ
ᇱ
, (25)
which is initialized using ‫ݒ‬்,்ିଵ
்
= α(1 − ்݇)‫ݒ‬்ିଵ
்ିଵ
.
Note that the state estimate, ‫ݔ‬௧
்
, differs from the one computed in a Kalman filter in that it is the
smoothed estimator of ‫ݔ‬௧ based on all of the observed data (and input data), i.e. it depends on past
and future observations; the Kalman filter estimates‫ܧ‬ሾ‫ݔ‬௧|ሼ‫ݕ‬ሽଵ
௧ ሿ is the usual Kalman filter
estimator based on the obsearved data up to the current time instant ࢚.
3.3. Expected Log-Likelihood Formulation
After we have got the expected values for ‫ݔ‬଴and‫ݔ‬௧ as ‫ݔ‬଴
்
≡ ‫ܧ‬ሾ‫ݔ‬଴|ሼ‫,ݕ‬ ‫ݑ‬ሽሿ and ‫ݔ‬௧
்
≡ ‫ܧ‬ሾ‫ݔ‬௧|ሼ‫,ݕ‬ ‫ݑ‬ሽሿ
respectively, we can calculate the expectation of the log-likelihood
‫)ߗ(ܧ‬ = ‫ܧ‬ሾlogܲ(ሼ‫ݔ‬ሽ, ሼ‫ݕ‬ሽ)ሿ. (26)
Denote
ቐ
ܲ௧ିଵ
்
= ∑ ‫ܧ‬ሾ‫ݔ‬௧ିଵ‫ݔ‬௧ିଵ
ᇱ
|ሼ‫ݕ‬ሽሿ்
௧ୀଵ
ܲ௧
்
= ∑ ‫ܧ‬ሾ‫ݔ‬௧‫ݔ‬௧
ᇱ
௧ୀଵ
ܲ௧,௧ିଵ
்
= ∑ ‫ܧ‬ሾ‫ݔ‬௧‫ݔ‬௧ିଵ
ᇱ
௧ୀଵ
, ቐ
‫ܯ‬௧ିଵ
்
= ∑ ‫ݔ(ܧ‬௧ିଵ‫ݑ‬௧)்
௧ୀଵ
‫ܯ‬௧
் = ∑ ‫ݔ(ܧ‬௧‫ݑ‬௧)்
௧ୀଵ
ܷ௧
்
= ∑ ‫ݑ‬௧
ଶ்
௧ୀଵ
, and ቐ
ܹ௧ିଵ
்
= ∑ ‫ݔ(ܧ‬௧ିଵ‫ݕ‬௧)்
௧ୀଵ
ܹ௧
் = ∑ ‫ݔ(ܧ‬௧‫ݕ‬௧)்
௧ୀଵ
ܻ௧
்
= ∑ ‫ݕ‬௧
ଶ்
௧ୀଵ
,
we have
‫)ߗ(ܧ‬ = −
ଵ
ଶ
‫ߗ(ܧ‬଴) −
ଵ
ଶ
‫ߗ(ܧ‬ଵ) −
ଵ
ଶ
‫ߗ(ܧ‬ଶ) −
ଶ்ାଵ
ଶ
log(2ߨ) (27)
where
‫ߗ(ܧ‬଴) =
ଵି஑మ
ఙ೐
మ ሾ‫ݒ‬଴
்
+ (‫ݔ‬଴
்
− ߤ଴)ଶሿ + log
ఙ೐
మ
ଵି஑మ (27a)
‫ߗ(ܧ‬ଵ) = ߪ௘
ିଶ
൫ܲ௧
்
+ αଶ
ܲ௧ିଵ
்
+ ߛଶ
ܷ௧
்
− 2αܲ௧,௧ିଵ
்
− 2ߛ‫ܯ‬௧
்
+ 2αߛ‫ܯ‬௧ିଵ
்
൯ + ܶlogߪ௘
ଶ
(27b)
‫ߗ(ܧ‬ଶ) = ߪ௪
ିଶ(ܻ௧
்
+ ܲ௧
்
− 2ܹ௧
்) + ܶlogߪ௪
ଶ
(27c)
3.4. Estimation of the Parameters
We use the first order condition on partial derivatives of ‫)ߗ(ܧ‬ to individual parameters to obtain
the gradient and then the values of individual parameters. This method is not the multivariate
regression approach [6]. The parameters are chosen when the objective function is maximized,
i.e., the gradients are all zero. The estimates of α, ߛ, ߪ௘
ଶ
, ߪ௪
ଶ
and ߤ଴are from below five equations:
α(1 − αଶ)‫ݒ‬଴
்
− αߪ௘
ଶ
− α(1 − αଶ)ܲ௧ିଵ
்
+ (1 − αଶ)ܲ௧,௧ିଵ
்
− ߛ(1 − αଶ)‫ܯ‬௧ିଵ
்
= 0 (28a)
ߛܷ௧
்
− ‫ܯ‬௧
்
+ α‫ܯ‬௧ିଵ
்
= 0 (28b)
(1 − αଶ)‫ݒ‬଴
்
− (1 + ܶ)ߪ௘
ଶ
+ ܲ௧
்
+ αଶ
ܲ௧ିଵ
்
+ ߛଶ
ܷ௧
்
− 2αܲ௧,௧ିଵ
்
− 2ߛ‫ܯ‬௧
்
+ 2αߛ‫ܯ‬௧ିଵ
்
= 0 (28c)

ߪ௪
ଶ
=
ଵ
்
(ܻ௧
்
+ ܲ௧
்
− 2ܹ௧
்) (28d)
ߤ଴ = ‫ݔ‬଴
்
(28e)
Moreover, we use the second orders of the derivatives of ‫)ߗ(ܧ‬ to calculate the second derivatives
and then the information matrix. Most of the second order derivatives of ‫)ߗ(ܧ‬ are zero except
those listed below:
డమா(ఆ)
డఈమ =
௩బ
೅ା൫௫బ
೅ିఓబ൯
మ
ି௉೟షభ
೅
ఙ೐
మ −
ଵା஑మ
(ଵି஑మ)మ, (29)
డమா(ఆ)
డఊడఈ
=
డమா(ఆ)
డఈడఊ
= −
ெ೟షభ
೅
ఙ೐
మ , (30)
డమா(ఆ)
డఙ೐డఈ
=
డమா(ఆ)
డఈడఙ೐
=
ଶ൫஑௉೟షభ
೅ ି௉೟,೟షభ
೅ ାఊெ೟షభ
೅
൯ିଶ஑ቂ௩బ
೅ା൫௫బ
೅ିఓబ൯
మ
ቃ
ఙ೐
య , (31)
డమா(ఆ)
డఊమ = −
௎೟
೅
ఙ೐
మ (32)
డమா(ఆ)
డఙ೐డఊ
=
డమா(ఆ)
డఊడఙ೐
=
ଶ൫ఊ௎೟
೅
ିெ೟
೅
ା஑ெ೟షభ
೅
൯
ఙ೐
య (33)
డమா(ఆ)
డఙ೐
మ =
்ାଵ
ఙ೐
మ −
ଷ൫ଵି஑మ൯ቂ௩బ
೅
ା൫௫బ
೅
ିఓబ൯
మ
ቃ
ఙ೐
ర −
ଷ൫௉೟
೅
ା஑మ௉೟షభ
೅
ାఊమ௎೟
೅
ିଶ஑௉೟,೟షభ
೅
ିଶఊெ೟
೅
ାଶ஑ఊெ೟షభ
೅
൯
ఙ೐
ర (34)
డమா(ఆ)
డఙೢ
మ =
்
ఙೢ
మ −
ଷ൫௒೟
೅ା௉೟
೅ିଶௐ೟
೅
൯
ఙೢ
ర (35)
డమா(ఆ)
డఓబ
మ = −
ଵି஑మ
ఙ೐
మ (36)
According to Cramer-Rao Theorem, the MLE is an efficient estimate. When the sample size is
large enough, the asymptotic variances of the estimates can be considered as the metric of the
accuracy of the estimation. The asymptotic variances are calculated using the inverse of the
information matrix, which is the inverse of the negative of the expected value of the Hessian
matrix. The vector of the asymptotic variances of the estimates is
‫ۏ‬
‫ێ‬
‫ێ‬
‫ێ‬
‫ێ‬
‫ێ‬
‫ێ‬
‫ێ‬
‫ێ‬
‫ێ‬
‫ۍ‬
(ܶ + 1)(αଶ
− 1)ଶ
ܷߪ௘
ଶ
(ܶ + 1)(1 − αଶ)ଶ(P଴ܷ − ܷߪ଴
ଶ − M଴
ଶ) + ( 1 + ܶ − αଶ + ܶαଶ)ܷߪ௘
ଶ
ሾ(1 + ܶ)(1 − αଶ)ଶ(P଴ − ߪ଴
ଶ) + (1 + ܶ − αଶ
+ ܶαଶ)ߪ௘
ଶሿߪ௘
ଶ
(ܶ + 1)(1 − αଶ)ଶ(P଴ܷ − ܷߪ଴
ଶ
− M଴
ଶ) + ( 1 + ܶ − αଶ + ܶαଶ)ܷߪ௘
ଶ
ሾ(1 − αଶ)ଶ(P଴ܷ − M଴
ଶ
− ܷߪ଴
ଶ) + ( 1 + αଶ)ܷߪ௘
ଶሿߪ௘
ଶ
(ܶ + 1)(1 − αଶ)ଶ(P଴ܷ − ܷߪ଴
ଶ
− M଴
ଶ) + ( 1 + ܶ − αଶ + ܶαଶ)ܷߪ௘
ଶ
ߪ௪
ଶ
2ܶ
ߪ଴
ଶ
‫ے‬
‫ۑ‬
‫ۑ‬
‫ۑ‬
‫ۑ‬
‫ۑ‬
‫ۑ‬
‫ۑ‬
‫ۑ‬
‫ۑ‬
‫ې‬
If the sample size is small, we introduce boot-strapping procedure where the estimates are
obtained from likelihood constructed from re-sampled standardized innovation, ‫ݕ‬௧෥ , in Equation
(21c). Moreover, the mean squared errors (MSE) of the state variables which is estimated from
Equations (21a-g) using estimated parameters are also estimated.

4. SIMULATION AND RESULTS
The output data is generated through presetting the input series and the values of the parameters.
We implement the initial guessing, EM iteration, and finally obtain the parameter estimates. This
makes it easier to evaluate our work by comparing the actual values with the estimated values, or
by checking the standard deviation of the estimates.
4.1. Data Generation
We generate data from the state-space model described as Equation (1). We assume α = 0.8,
γ = 1.5, and ߤ଴ = 0. Moreover, the process noises, ݁୲, and observation noise, ‫ݓ‬௧, are generated
independently where ݁୲~ܰ(0, 1.1ଶ) and ‫ݓ‬୲~ܰ(0, 0.9ଶ). We assume that the input,‫ݑ‬௧, is a slow
changing periodical square wave signal whose period is 10 time unit. The standard deviation of
initial state, ߪ଴, is not needed during the data generation but can be calculated according to
Equation (2). The expected log-likelihood can be calculated using Equation (27). Both are treated
as “actual” values to be compared with guessed values and estimated values.
We performed our simulation using two different sample sizes: the large size of 1000 and the
small size of 50. When the sample size is small, we applied bootstrapping method to estimate the
accuracy of the estimate.
4.2. Results
We provide the results of the simulation with small sample size of 50 in Table 1, and the results
of the simulation with large sample size of 1000 in Table 2.
Table 1. The parameters estimate with small sample size
Parameters Actual Guessed Estimated Std. Dev.
α 0.8 0.879 0.801 0.023
γ 1.5 1.075 1.438 0.104
σe 1.1 1.400 0.708 0.158
σw 0.9 0.922 0.862 0.193
µ0 0 0.992 2.143 1.231
σ0 1.83 3.811 1.183 0.252
In Table 1and Table 2, we displayed the actual values, the guessed values, the estimated values
and the standard deviations of the estimated values for transition coefficient, α, input coefficient,
γ, standard deviation of process errors, σe, standard deviation of observation errors, σw, mean of
initial state, µ0, and standard deviation of initial state, σ0. In general, the guessed value is near the
actual value while the estimated value is much more close to the actual value than the guessed
ones for the parameters of the most interest: α and γ. The standard deviations in Table 1 are from
bootstrapped distribution while the standard deviations in Table 2 are from the asymptotic
variances.
Table 2: The parameters estimated with large sample size
Parameters Actual Guessed Estimated Std. Dev.
α 0.8 0.879 0.800 0.0001
γ 1.5 1.276 1.424 0.0012
σe 1.1 2.024 1.033 0.0011
σw 0.9 0.533 0.971 0.0005
µ0 0 2.459 1.667 2.9614
σ0 1.83 4.250 1.721 NA

In general, the guessed value is near the ‘actual” value while the estimated value is much closer to
the “actual” value than the guessed ones, especially for the parameters of interest: ߙ and ߛ. It is
worth noting that the deviations of the estimated values are larger than the asymptotic ones due to
the imperfectly generated data.
5. CONCLUSIONS
The research is to validate our innovative approaches in the application of EM algorithm in the
MLE of a constrained dynamic linear system with external input. There are two main
contributions in this research. Firstly, we realized that σ0 and σe has the relationship expressed by
Equation (4) thus we don’t have to estimate both of them during the implementation of EM
algorithm. Accordingly, the likelihood function in Equation (27) is not similar with those
researchers who ignored the relationship. Secondly, in initial guessing of the value of input
coefficient and the mean of initial state, we introduce weighted least square for the guessing of
input coefficient, γ, and the mean of the initial state, µ0.
It is obvious that more techniques have to be discovered for the initial guessing, and the
estimating based on the guessed initial guessing of, the parameter values, especially for the
implementation of the M-step of the EM algorithm. The approaches we proposed can be a new
start point for the future research on the estimation of dynamic systems with higher dimensions of
external inputs, hidden states and observations.
REFERENCES
[1] Kalman,R. E.,(1960) "A new approach to linear filtering and prediction problems",Journal of Basic
Engineering, 82 (1), pp. 35–45.
[2] James D. Hamilton, Time Series Analysis, Princeton University Press, Princeton, New Jersey, 1996.
[3] Dempster,A.P., Laird, N.M. and Rubin, D.B. (1977) "Maximum likelihood from incomplete data via
the EM algorithm",Journal of the Royal Statistical Society, Series B,Vol. 39, No.1, pp. 1–38.
[4] Ghahramani,Z. and Hinton,G. E. (1996) “Parameter estimation for linear dynamic systems”,Technical
reportCRC-TR-96-2, Department of Computer Science, University of Toronto.
[5] Deng, L. and Shen, X. (1997) “Maximum likelihood in statistical estimation of dynamic systems:
decomposition algorithm and simulation results”,Signal Processing, Vol. 57, No. 1.
[6] Shumway,Robert H. and Stoffer,David S. (2011) Time Series Analysis and Its Applications: With R
Examples, New York, NY, Springer, New York.
[7] Holmes, Elizabeth Eli. (2013)“Derivation of an EM algorithm for constrained and unconstrained
multivariate autoregressive state-space (MARSS) models”,Technical Report, Northwest Fisheries
Center, NOAA Fisheries.
[8] Wu,Lilian Shiao-Yen, Pai,Jeffrey S. and Hosking,J. R. M.(1996) “An algorithm for estimating
parameters of state-space models”,Statistics & Probability Letters, Vol. 28, pp. 99-106.
[9] Zuur, A. F.,Fryer, R. J.,Jolliffe,I. T.,Dekker, R. and Beukema, J. J. (2003) “Estimating common trends
in multivariate time series using dynamic factor analysis”,Environmetrics, Vol. 14, No. 7, pp. 665-
685.
[10] Dewar,M. Scerri, K. and Kadirkamanathan, V. (2009) “Data-driven spatio-temporal modeling using
the integro-difference equation”,IEEE Transactions on Signal Processing,Vol. 57, No. 1, pp.83-91.

AUTHORS
Chengliang Huang is currently a Ph. D. candidate at Ryerson University. He had been a
senior engineer and project manager at Huawei and Intel for four years respectively. He
graduated as MASc in Electrical and Computer Engineering, Ryerson University in
2009, MBA in Economic Management, Beijing Jiaotong University and B. Sci. in
Electronic Engineering, Fudan University in 1990.He has published papers on peer-
reviewed academic journals, including IEEE Sensor Journal and Wireless Personal
Communications. His research interest includes wireless communications, random signal
processing, and system identification, time series analysis, marketing data mining/analysis, and marketing
productivity.
Xiao-Ping Zhang (M'97, SM'02) received the B.S. and Ph.D. degrees from Tsinghua
University, in 1992 and 1996, respectively, all in electronic engineering. He holds an
MBA in Finance, Economics and Entrepreneurship with Honors from Booth School of
Business, the University of Chicago. Since Fall 2000, he has been with the Department
of Electrical and Computer Engineering, Ryerson University, where he is now Professor,
Director of Communication and Signal Processing Applications Laboratory (CASPAL)
and Program Director of Graduate Studies. His research interests include multimedia
communications and signal processing, multimedia retrieval and video content analysis,
sensor networks and electronic systems, computational intelligence, and applications in bioinformatics,
finance, and marketing. He is a frequent consultant for biotech companies and investment firms. He is
cofounder and CEO for EidoSearch, Inc., which offers a content-based search and analysis engine for
financial data. Dr. Zhang is a registered Professional Engineer in Ontario, Canada, a Senior Member of
IEEE and a member of Beta Gamma Sigma Honor Society. He is the publicity co-chair for ICME'06 and
program co-chair for ICIC'05. He is currently an Associate Editor for IEEE Transactions on Signal
Processing, IEEE Signal Processing Letters and Journal of Multimedia.
Dr. Wang’s research focuses on brand equity assessment, marketing data
mining/analysis, long-term marketing productivity, and marketing strategy. She also
works on e-commerce and information systems related topics. Her research on brand
equity assessment is funded by the Social Sciences and Humanities Research Council
of Canada (SSHRC). She is a key team member for a NSERC (Natural Sciences and
Engineering Research Council of Canada) supported project on financial market
information analysis. Her work has appeared in many academic journals, including
Journal of Marketing, Journal of the Academy of Marketing Science, Information & Management, Journal
of Consumer Marketing, among others.

APPROACHES IN USING EXPECTATIONMAXIMIZATION ALGORITHM FOR MAXIMUM LIKELIHOOD ESTIMATION OF THE PARAMETERS OF A CONSTRAINED STATE SPACE MODEL WITH AN EXTERNAL INPUT SERIES

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