EGNiYA Technical Report on A New Metric for Stochastic Models and Dynamic Systems

1. A Metric in Angularly
Quantized Polar Domain for
Pole-Based Classification of
Stochastic Models and
Dynamic Systems – EGNIYA
Technical Report
EGNIYA
August 2016

2. 1
A Metric in Angularly
Quantized Polar Domain for
Pole-Based Classification of
Stochastic Models and
Dynamic Systems
Abstract – Autoregressive (AR) models are
used in a variety of applications and an AR
model can be represented by the poles
corresponding to describing AR coefficients.
Similarly, the behavior of linear dynamic single-
input-single-output (SISO) systems can be
described as a function of the system poles,
which are directly estimated from the given data
and represent a system as a set of poles without
any identities, which is analogous to the nature
of association-free multi-target tracking and
corresponding application of set distances
known as optimal subpattern assignment
(OSPA) distance. In this work, we define a new
metric in terms of sums and differences of
magnitude and angles of poles in pole-space and
provide a measure of the “distance” between AR
processes, or linear dynamic systems
represented by poles.
Index Terms—Autoregressive (AR) models,
classification, stochastic models, metric, distance
measures, poles, linear dynamic systems
I. INTRODUCTION
In many applications, autoregressive (AR)
models are increasingly popular. In radar signal
processing, the properties of the Gaussian clutter
can be analyzed by modeling it by an AR process.
In biomedical applications, AR models can be
used to classify signals of patients with a specific
pathology from signals recorded with healthy
people. In speech processing such as speech
analysis and Kalman-filter based enhancement,
𝑴 sets of 𝒑 AR parameters {𝒂𝒊} estimated by
different methods, are often compared one
another. [10] Another area is classification of
systems described by poles which includes
several similar but slightly different topics.
Change detection considers the problem of
finding modifications to the system based on
observations. Anomaly detection is concerned
with finding elements or behavioral patterns
deviating from elements or patterns that are
defined as normal. Outliers detection addresses
the problem of finding elements of a set that
deviate markedly from other set members. Fault
detection is concerned with the problem of
recognizing system behavior or states that
exceed a permitted region of operation. Concept
drift addresses the very slow and gradual
changes in processes. Novelty detection emerged
from the field of machine learning and has its
focus on discerning incoming information with
respect to its coverage by already learned
models and pattern. In all those areas, a form of
model/system comparison (based on either
dissimilarity/distance/divergence or contrast)
and consequent classification is required. [9]
Commonly used dissimilarity measures between
AR-model pairs include the Itakura divergence,
the Itakura-Saito divergence, the log-spectral
distance or Jeffrey’s divergence (JD) which are
based on spectral characteristics of the
processes. [9],[6] One common distance
measure between systems is based on the
spectral characteristics, namely a distance
between the cepstra, the inverse Fourier
transform of the logarithm of the power
spectrum. The ARMA distance, a metric for
ARMA processes as the distance of two cepstra,
can be calculated by a function of the system
poles. [4] Another distance measure between
systems is association-free distance measure,
namely the optimal subpattern assignment
(OSPA) distance (and a variant of it called as
MAX-OSPA), the system distance as a distance
between their sets of poles representing the
spectral characteristics of the system. [1], [2], [9]
The key to comparison and classification of
models or systems represented by poles is a set
of representative and intuitive parameters, (in
order to define a distance measure as a basis for
decision-making) which can be the set of the AR
parameters or system coefficients, the set of
poles or the corresponding PSDs. [9], [10] In this
work, the main contribution is a new approach
and derivation of a novel metric based on poles
which can be used for classification of AR
processes, or classification, identification or
change detection in linear dynamic systems
represented by system poles.
The remainder of this paper is structured as
follows. First, in Sec. II we give a formal
description of the problem along with
parametric models, pole estimation and existing
approaches. Sec. III introduces the classification
of stochastic models and linear dynamic systems
in pole-space. We derive an efficient novel
metric in pole-space in Sec. IV. A numerical
example along with a comparison of the
proposed metric is presented in Sec. V. Finally,
Sec. VI concludes the work.

3. 2
II. PROBLEM STATEMENT
A. Mathematical Model
The distance 𝒅 𝒌 between a reference model
𝑺 𝒌
(𝒓)
and a test model 𝑺 𝒌
(𝒕)
calculated on basis of
their parameter vectors 𝜽 𝒌
(𝒓)
and 𝜽 𝒌
(𝒕)
, which are
usually estimated from the input and observed
output data,
𝒅 𝒌 = 𝑫(𝑺 𝒌
(𝒓)
, 𝑺 𝒌
(𝒕)
) = 𝑫(𝜽 𝒌
(𝒓)
, 𝜽 𝒌
(𝒕)
) (1)
where the superscripts (𝒓) and (𝒕) represents
the reference and test models respectively,
which are either two models in AR models space
or two systems in linear time-varying systems
space. [9]
B. Parametric Models
Autoregressive (AR), moving-average (MA) and
autoregressive moving average (ARMA) models
are useful time-domain models for the
representation of discrete-time signals. The time
series 𝒙 𝒏 is called as an ARMA(𝒑, 𝒒) model if
𝒙 𝒏 = − ∑ 𝒂𝒋 𝒙 𝒏−𝒋
𝒑
𝒋=𝟎 + ∑ 𝒃𝒋 𝒗 𝒏−𝒋
𝒒
𝒋=𝟎 (2)
where 𝒗 𝒏 is uncorrelated noise of variance 𝝈 𝒗
𝟐
. In
𝒛 −domain the system function is
𝑯(𝒛) = 𝝈 𝒗
𝟐
∑ 𝒃 𝒋 𝒛−𝒋𝒒
𝒋=𝟎
∑ 𝒂 𝒋 𝒛−𝒋𝒑
𝒋=𝟎
= 𝝈 𝒗
𝟐 ∏ (𝟏−𝜷 𝒊/𝒛)
𝒒
𝒊=𝟏
∏ (𝟏−𝝀 𝒊/𝒛)
𝒑
𝒊=𝟏
(3)
where 𝒂 𝟎 = 𝒃 𝟎 = 𝟏, the 𝒂𝒋’s are the AR
coefficients, 𝒃𝒋’s are the MA coefficients, 𝝀𝒊 and
𝜷𝒊 are the poles and zeros of the model
respectively. [4]
C. Pole Estimation
There are different approaches for the
estimation of the poles of a system but our main
focus in this work is providing a metric for a
given set of system or model poles rather than
estimation or statistical properties of the
estimator, so we assume we have different sets
of poles obtained using a proper pole estimation
method.
However, in systems with real valued output,
poles occur either real or in complex conjugate
pairs and a known proportion of real and
complex poles can be assumed. Since an a prior
estimate is easy to attain from a small number of
data samples, this assumption is not unrealistic.
D. Existing Approaches
There are several proposals in literature using
information on the positions of poles,
amplitudes of characteristic frequencies and
subspace formulations [10], which constitutes
different metrics and distance measures used for
classification of models and systems. We are
briefing some popular ones and shortly
mentioning about similarities and limitations of
others next in Section III.
III. POLE-BASED DISTANCE MEASURES
A. Base Distance of Poles
The base distance 𝒃(. , . ), with 𝒃: ℂ𝒙ℂ → 𝑹 𝟎
+
as a
function satisfying identity, symmetry, and the
triangle inequality. The base distance is the
distance between two complex poles 𝝀𝒊
(𝒓)
and
𝝀𝒊
(𝒕)
The 𝒛 −plane, in which the poles of a
discrete-time system are defined, can be
interpreted as a specific Mobius transformation
from the more intuitive 𝒔 −plane of a
continuous-time system that transforms the
Cartesian coordinate system of the 𝒔 −plane into
a polar coordinate system.[9] Let 𝒔 and 𝒔̃ be two
poles of a continuous-time system in the
𝒔 −plane. The base distance 𝒃(𝒔, 𝒔̃) between 𝒔
and 𝒔̃ is defined as
𝒃(𝒔, 𝒔̃)
= √[𝑹𝒆(𝒔) − 𝑹𝒆(𝒔̃)] 𝟐 + [𝑰𝒎(𝒔) − 𝑰𝒎(𝒔̃)] 𝟐
(4)
which is the Euclidian distance between 𝒔 and 𝒔̃
in the complex plane. Let 𝒛 and 𝒛̃ be two poles of
a discrete-time system in the 𝒛 −plane. The base
distance 𝒃(𝒛, 𝒛̃) between 𝒛 and 𝒛̃ is defined as
𝒃(𝒛, 𝒛̃) =
𝟏
𝑻
√[𝑰𝒏(𝒓) − 𝑰𝒏(𝒓̃)] 𝟐 + [𝝆 − 𝝆̃] 𝟐 (5)
where 𝑻 is the sampling period, 𝒓, 𝝆 and 𝒓̃, 𝝆̃ are
polar coordinates of 𝒛 and 𝒛̃ respectively. [9]
B. OSPA and MAX-OSPA Distance
The optimal subpattern assignment (OSPA)
distance between two sets of poles
{𝝀 𝟏
(𝒓)
, 𝝀 𝟐
(𝒓)
, … , 𝝀 𝒑
(𝒓)
} and {𝝀 𝟏
(𝒕)
, 𝝀 𝟐
(𝒕)
, … , 𝝀 𝒑
(𝒕)
} each
comprising 𝒑 elements and given by the ordered
vectors 𝜽 𝒌
(𝒓)
= [𝝀 𝟏
(𝒓)
, 𝝀 𝟐
(𝒓)
, … , 𝝀 𝒑
(𝒓)
]
𝑻
and
𝜽 𝒌
(𝒕)
= [𝝀 𝟏
(𝒕)
, 𝝀 𝟐
(𝒕)
, … , 𝝀 𝒑
(𝒕)
]
𝑻
is defined by

4. 3
𝒅 𝑶𝑺𝑷𝑨(𝜽 𝒌
(𝒓)
, 𝜽 𝒌
(𝒕)
)
= [
𝟏
𝒑
𝐦𝐢𝐧
𝝅∈𝚷 𝒑
(∑ 𝒃(𝝀𝒊
(𝒓)
, 𝝀 𝝅(𝒊)
(𝒕)
)
𝒒
𝒑
𝒊=𝟏
)]
𝟏
𝒒
(𝟔)
with ≥ 𝟏 , 𝒃 (𝝀𝒊
(𝒕)
, 𝝀𝒋
(𝒕)
) is the base distance
between the two elements 𝝀𝒊
(𝒓)
and 𝝀𝒋
(𝒕)
and 𝚷 𝒑
describes all permutations of the set {𝟏, 𝟐, … , 𝒑}.
The notation 𝝀 𝝅(𝒊)
(𝒕)
represents the 𝒊-th element of
the permutation 𝝅 of 𝜽 𝒌
(𝒕)
which is generated by
reordering the vector. Given two sets of poles by
the two vectors 𝜽 𝒌
(𝒓)
and 𝜽 𝒌
(𝒕)
, the MAX-OSPA
distance is defined by
𝒅 𝒎𝑶𝑺𝑷𝑨(𝜽 𝒌
(𝒓)
, 𝜽 𝒌
(𝒕)
) = 𝐦𝐚𝐱 𝒊=𝟏,𝟐,..𝒑 (𝒃 (𝝀𝒊
(𝒓)
, 𝝀 𝝅∗(𝒊)
(𝒕)
))
(7)
where 𝝅∗
is the optimal OSPA permutation given
by
𝝅∗
= 𝒂𝒓𝒈𝐦𝐢𝐧
𝝅∈𝚷 𝒑
[𝐦𝐚𝐱 𝒊=𝟏,𝟐,..𝒑 (𝒃 (𝝀𝒊
(𝒓)
, 𝝀 𝝅∗(𝒊)
(𝒕)
))]
(8)
So, the maximum base distance between two
single elements of a pair 𝜽 𝒌
(𝒓)
and 𝜽 𝒌
(𝒕)
is chosen,
while using the optimal subpattern assignment
over the maximum norm (𝒒 → ∞) resulted in
minimized maximal base distance.
The OSPA distance finds the optimal assignment
by minimizing over the sum of distances
between poles. The MAX-OSPA chooses an
optimal assignment by minimizing the maximum
distance between two poles.
In OSPA and MAX-OSPA a weighting scheme is
generally used (usually as the exponential
weighting) in order to take closeness of poles to
unit circle (or imaginary axis in 𝒔 −domain) into
account [9]
C. ARMA Metric
ARMA metric is one of the popular metrics
whose derivation is based on cepstrum domain
processing or homomorphic processing. For two
stable AR models a reference model 𝑺 𝒌
(𝒓)
and a
test model 𝑺 𝒌
(𝒕)
with orders 𝒑 and two sets of
poles {𝝀 𝟏
(𝒓)
, 𝝀 𝟐
(𝒓)
, … , 𝝀 𝒑
(𝒓)
} and {𝝀 𝟏
(𝒕)
, 𝝀 𝟐
(𝒕)
, … , 𝝀 𝒑
(𝒕)
}
the ARMA metric is defined as;
𝒅 𝑨𝑹𝑴𝑨( 𝜽 𝒌
(𝒓)
, 𝜽 𝒌
(𝒕)
)
= [ 𝑰𝒏
∏ ∏ (𝟏 − 𝝀𝒊
(𝒓)
𝝀𝒋
∗(𝒕)
) ∏ ∏ (𝟏 − 𝝀𝒊
(𝒕)
𝝀𝒋
∗(𝒓)
)
𝒑
𝒋=𝟏
𝒑
𝒊=𝟏
𝒑
𝒋=𝟏
𝒑
𝒊=𝟏
∏ ∏ (𝟏 − 𝝀𝒊
(𝒓)
𝝀𝒋
∗(𝒓)
) ∏ ∏ (𝟏 − 𝝀𝒊
(𝒕)
𝝀𝒋
∗(𝒕)
)
𝒑
𝒋=𝟏
𝒑
𝒊=𝟏
𝒑
𝒋=𝟏
𝒑
𝒊=𝟏
]
𝟏/𝟐
(9)
which is in the form of finite products in pole
domain. [4]
E. Limitations & Requirements
Most of the existing or proposed metrics or
measures use either 𝑳 norms for distances
between poles in 𝒔 −domain or transformed
poles in 𝒛 −domain assisted with some
weighting schemes. The pole representation in
either in 𝒔 −domain or in 𝒛 −domain is a two
dimensional representation and any distance
measure based solely on this two dimensional
representation (which corresponds to a pattern
with two features forming a two dimensional
feature vector) is far from measuring the
distance between corresponding
models/systems. Besides a distance measure in
pole-space must consider not only the distance
between poles of reference and test models but
also the distance among the poles (intra-pole
distance) of each model/system. So the distance
among poles of a model is a factor as well, which
might be as important as the closeness of a pole
to the unit circle. In other words, the metrics or
distance measures based solely on two
dimensional representation might suffer from
loss of available information. ARMA metric can
be thought as a nonlinear transformation of two
dimensional space into a higher dimensional
space and might not have suffering from loss of
information.
The main requirement can be defined as reliable,
analytically tractable, physically meaningful and
computationally efficient metric to measure
distance between models or systems in their
associated spaces. We are outlining a new
approach and a new metric based on this
requirements avoiding majority of the
limitations next in Section IV.
IV. PROPOSED APPROACH & METRIC
A. Preliminaries
A model/system can be represented by poles,
coefficients or PSD. Although there is not a one-
to-one correspondence, pole representation can
be thought as a vector quantized form, or low
rank approximation or projection onto a lower
dimensional space, in other words it contains all

5. 4
the information contained by PSD in a lower
dimensional space. Our aim is extracting this
available information contained in poles to
measure the distance between two
models/systems.
B. Approach
Consider two 𝒑𝒙𝟏 (as pole vectors, consisting of
poles as components of the vector)
𝜽 𝒌
(𝒓)
= [𝝀 𝟏
(𝒓)
, 𝝀 𝟐
(𝒓)
, … , 𝝀 𝒑
(𝒓)
]
𝑻
𝜽 𝒌
(𝒕)
= [𝝀 𝟏
(𝒕)
, 𝝀 𝟐
(𝒕)
, … , 𝝀 𝒑
(𝒕)
]
𝑻
(10)
where 𝜽 𝒌
(𝒓)
and 𝜽 𝒌
(𝒕)
represents the pole vectors
of reference and test models/systems
respectively. 𝝀𝒊
(𝒓)
and 𝝀𝒊
(𝒕)
are the poles which
can be written as
𝝀𝒊
(𝒓)
= 𝒓𝒊
(𝒓)
𝒆𝒋 ∅ 𝒊
(𝒓)
𝝀𝒊
(𝒕)
= 𝒓𝒊
(𝒕)
𝒆𝒋 ∅ 𝒊
(𝒕)
(11)
in polar domain. The key idea is extracting
information contained in poles by using a
transformation to a higher dimensional space in
such a way that each pole (actually magnitude of
each complex pole) corresponds to a coordinate
in the new transformed space and each
model/system can be represented as a point
(which is defined by the 𝒑 coordinates of the
model/system) in this high dimensional space,
(which can be thought as a model/system space
where magnitudes of poles forms the
coordinates as aligned vectors with a subgroup
of the axes of new space) .
Assume that polar domain is quantized angularly
(with 𝒏 angular sectors, each angular sector
width is 𝟐𝝅/𝒏, 𝒏 = 𝟐𝝅/∆∅ 𝒎𝒊𝒏 and they are
numbered from 1 to 𝒏 in counterclockwise
direction starting from (0,1) point of complex
plane) as shown in Fig.1, each non-empty (not
containing any poles) angular sector contains
either one pole of reference model or one pole of
test model.
If we think each angular sector as an axis of the
new space (which is the high dimensional
transform space with dimension 𝒏), then each
model/system can be uniquely represented as
𝒏 −dimensional vectors 𝜽̃
𝒌
(𝒓)
and 𝜽̃
𝒌
(𝒕)
which are
𝜽̃
𝒌
(𝒓)
= [𝟎, 𝟎, . . 𝒓 𝟏
(𝒓)
… , 𝒓 𝟐
(𝒓)
, … , 𝒓 𝒑
(𝒓)
… , 𝟎, 𝟎]
𝑻
𝜽̃
𝒌
(𝒕)
= [𝟎, 𝟎, . . 𝒓 𝟏
(𝒕)
… , 𝒓 𝟐
(𝒕)
, … , 𝒓 𝒑
(𝒕)
… , 𝟎, 𝟎]
𝑻
(12)
now model/system vectors for reference and
test models/systems respectively.(note that, the
positions of both 𝒓𝒊
(𝒓)
and 𝒓𝒊
(𝒕)
components in 𝜽̃
𝒌
(𝒓)
and 𝜽̃
𝒌
(𝒕)
is based on angular quantization and the
value of 𝒏).
This operation (angular quantization and
obtaining model/system vectors) can be thought
as transforming the original 𝒑𝒙𝟏pole vectors to
𝒏𝒙𝟏 vectors by a rank 𝒑 matrix which contains
𝒆−𝒋 ∅ 𝒊
(𝒓)
(or 𝒆−𝒋 ∅ 𝒊
(𝒕)
) as components in the
respective diagonal elements. (and also provides
a link to cepstrum, FFT and PSD domains)
Although we will not use angular sectorization
and high dimensional space in final metric
definition, it is useful in derivation. Selection of 𝒏
is important and one condition on that comes
from representability of each model/system as a
point in the new high dimensional space, which
means each angular sector must contain only
one pole of either reference or test model
otherwise none. This can be satisfied by
selecting 𝒏 considering the minimum angular
separation ∆∅ 𝒎𝒊𝒏 between poles of either test or
reference model, whichever is smaller. (This
operation takes the distance among poles of
model/system (intra-poles distance) argument
in previous section into account)
C. Proposed Metric
Since we have 𝒏-dimensional vector space and
models/systems represented as points in this
space as
𝜽̃
𝒌
(𝒓)
= [𝟎, 𝟎, . . 𝒓 𝟏
(𝒓)
… , 𝒓 𝟐
(𝒓)
, … , 𝒓 𝒑
(𝒓)
… , 𝟎, 𝟎]
𝑻
𝜽̃
𝒌
(𝒕)
= [𝟎, 𝟎, . . 𝒓 𝟏
(𝒕)
… , 𝒓 𝟐
(𝒕)
, … , 𝒓 𝒑
(𝒕)
… , 𝟎, 𝟎]
𝑻
(13)
Figure 1. Selection of angular
quantization parameter 𝒏 (number of
angular sectors) based on minimum
angular separation between poles

6. 5
then we can use familiar vector space properties
and define the metrics to measure distance
between the models/systems.
Although usual vector space metrics (𝑳 𝟏 and 𝑳 𝟐
or 𝒑-norms for example) can be used to measure
distance between two models represented by
vectors 𝜽̃
𝒌
(𝒓)
and 𝜽̃
𝒌
(𝒕)
, this would be a coarse
distance measure (does not take angular sector
index or angular separation into account, and
results in virtual equidistant models) and
requires formation of those 𝒏-dimensional
vectors implying realization of angular
sectorization by search methods or 𝒏𝒙𝒏 matrix
multiplication which we avoid due to
computational efficiency considerations. To be
able to take effect of angular separation into
account and in fact to be able to provide a solid
link to cepstrum, FFT and PSD domains a form
of weighting should be applied as well.
Weighting can be applied either in complex
domain as pre-weighting (which is a transform
of 𝒑𝒙𝟏vectors 𝜽 𝒌
(𝒓)
and 𝜽 𝒌
(𝒕)
by a real valued 𝒑𝒙𝒑
diagonal weighting matrix) or in high
dimensional transform space as post weighting
(which is a transform of 𝒏𝒙𝟏 vectors 𝜽̃
𝒌
(𝒓)
and
𝜽̃
𝒌
(𝒕)
by a real valued 𝒏𝒙𝒏 diagonal weighting
matrix). In first case since original poles move
through radial directions the distances between
models/systems are preserved to a possible
scale factor. In latter case, the post weighting
matrix transforms 𝒏 −dimensional linear vector
space and distances are defined on
hyperellipsoids, in other words the distances can
be called as well-known Mahalonabis distances.
Before going back to 𝒑 −dimensional pole space
and defining the metric in that space we
continue with derivations in 𝒏 −dimensional
space.
When the minimum angular separation ∆∅ 𝒎𝒊𝒏 is
chosen as minimum angular separation between
the angularly nearest two poles, the respective
positions of both 𝒓𝒊
(𝒓)
and 𝒓𝒊
(𝒕)
components in 𝜽̃
𝒌
(𝒓)
and 𝜽̃
𝒌
(𝒕)
is different and absolute value metric
can be written as sum of components 𝒓𝒊
(𝒓)
and
𝒓𝒊
(𝒕)
.
𝒅 𝟏 = 𝒅 𝟏(𝜽̃
𝒌
(𝒓)
, 𝜽̃
𝒌
(𝒕)
) = [∑|𝜽̃
𝒌
(𝒓)
(𝒊) − 𝜽̃
𝒌
(𝒕)
(𝒊)|
𝒏
𝒊=𝟏
]
= ∑(𝒓𝒊
(𝒓)
+ 𝒓𝒊
(𝒕)
)
𝒑
𝒊=𝟏
(14)
As mentioned, this would be a coarse distance
measure producing virtual equidistant models
and to take effect of angular separation a form of
weighting is applied as each weight defined as
the angular separation between corresponding
poles,
𝒘𝒊𝒊 = |∅𝒊
(𝒓)
| − |∅𝒊
(𝒕)
| (15)
Actually this weighting scheme can be
considered as the corollary of angular
sectorization as radial quantization, for each
radial quanta each pole angle represents one
coordinate in the axes of the high dimensional
space formed by radial quantization and weights
as pole angle differences are the absolute value
metrics on this new high dimensional space
constructed by radial quantization. Then
weighted distance measure can be written as;
𝒅 𝟏𝑾 = 𝒅 𝟏𝑾(𝜽̃
𝒌
(𝒓)
, 𝜽̃
𝒌
(𝒕)
) = ∑ 𝒘𝒊𝒊|𝜽̃
𝒌
(𝒓)
(𝒊) − 𝜽̃
𝒌
(𝒕)
(𝒊)|
𝒏
𝒊=𝟏
(16)
where 𝑾 is the 𝒏𝒙𝒏 diagonal weighting matrix
with each diagonal element 𝒘𝒊𝒊 is in the form of
multiples of ∆∅ 𝒎𝒊𝒏,
𝑾 = [
∆∅ 𝒎𝒊𝒏 𝟎 ⋯ 𝟎
𝟎 𝟐∆∅ 𝒎𝒊𝒏 . .
⋮ . ⋱ ⋮
𝟎 . ⋯ 𝒏∆∅ 𝒎𝒊𝒏
] (17)
In limiting case when any two poles are too
close, ∆∅ 𝒎𝒊𝒏 → 𝟎 and 𝒏 → ∞, which allows us to
write Eq.16 as
𝒅 𝟏𝑾(𝜽̃
𝒌
(𝒓)
, 𝜽̃
𝒌
(𝒕)
) = ∑(|∅𝒊
(𝒓)
| − |∅𝒊
(𝒕)
|)(𝒓𝒊
(𝒓)
+ 𝒓𝒊
(𝒕)
)
𝒑
𝒊=𝟏
(18)
However, although defined for ∆∅ 𝒎𝒊𝒏 → 𝟎 , the
distance 𝒅 𝟏 is valid for ∅𝒊
(𝒓)
≠ ∅𝒊
(𝒕)
to satisfy
initial assumptions that each angular sector
contains only one pole of either test or reference
model or not any poles. Using similar arguments
for radial quantization as used in angular
sectorization gives the final form of the metric in
pole space which we call as 𝒅 𝑷𝑺𝑴 (PSM stands
for pole space metric) and defined as;
𝒅 𝑷𝑺𝑴 = 𝒅 𝑷𝑺𝑴(𝜽 𝒌
(𝒓)
, 𝜽 𝒌
(𝒕)
)
𝒅 𝑷𝑺𝑴 = |∑(𝒓𝒊
(𝒓)
− 𝒓𝒊
(𝒕)
)(|∅𝒊
(𝒓)
| + |∅𝒊
(𝒕)
|)
𝒑
𝒊=𝟏
+ ∑(|∅𝒊
(𝒓)
| − |∅𝒊
(𝒕)
|)(𝒓𝒊
(𝒓)
+ 𝒓𝒊
(𝒕)
)
𝒑
𝒊=𝟏
|
(19)
In vector notation, it can be rewritten as;
𝒅 𝑷𝑺𝑴(𝜽 𝒌
(𝒓)
, 𝜽 𝒌
(𝒕)
) = |𝐫∆
𝑻
𝚽 𝚺 + 𝚽∆
𝑻
𝐫𝚺|
where
𝐫∆ = [(𝒓 𝟏
(𝒓)
− 𝒓 𝟏
(𝒕)
), … , (𝒓 𝒑
(𝒓)
− 𝒓 𝒑
(𝒕)
)]
𝑻

7. 6
Figure 2. Behavior of Proposed Metric PSM
and comparison with ARMA & OSPA metrics
for radially moving complex pole pair
𝐫𝚺 = [(𝒓 𝟏
(𝒓)
+ 𝒓 𝟏
(𝒕)
), … , (𝒓 𝒑
(𝒓)
+ 𝒓 𝒑
(𝒕)
)]
𝑻
𝚽∆ = [(|∅ 𝟏
(𝒓)
| − |∅ 𝟏
(𝒕)
|), … , (|∅ 𝒑
(𝒓)
| − |∅ 𝒑
(𝒕)
|)]
𝑻
𝚽 𝚺 = [[(|∅ 𝟏
(𝒓)
| + |∅ 𝟏
(𝒕)
|), … , (|∅ 𝒑
(𝒓)
| + |∅ 𝒑
(𝒕)
|)]]
𝑻
(20)
It should be noted that this form of the metric is
association-free or valid for unlabeled or non-
ordered pole pairs of test and reference models
which avoids permutations and also performs
well for both higher order models/systems or
applications requiring more than two model
classification.
As seen from Eq.19, computational load is
relatively low, can be computed by 𝟐𝒑
multiplications and (𝟒𝒑 + 𝟏) additions, and can
be used for high order model classification
problems as well. We are dealing with a
numerical example next in Section V.
V. NUMERICAL ANALYSIS
There might be some applications where pole
dominancy or movement of some poles might be
important as in change detection or system
identification problems. We use pole movement
to illustrate the numerical evaluation of the
metric and provide a comparison with ARMA
and OSPA metrics.
The ARMA metric 𝒅 𝑨𝑹𝑴𝑨( 𝜽 𝒌
(𝒓)
, 𝜽 𝒌
(𝒕)
) used in
comparison simulations and graphs are actually
squared form of the ARMA metric given in Eq (9)
so can be written as;
𝒅 𝑨𝑹𝑴𝑨( 𝜽 𝒌
(𝒓)
, 𝜽 𝒌
(𝒕)
)
= [ 𝑰𝒏
∏ ∏ (𝟏 − 𝝀𝒊
(𝒓)
𝝀𝒋
∗(𝒕)
) ∏ ∏ (𝟏 − 𝝀𝒊
(𝒕)
𝝀𝒋
∗(𝒓)
)
𝒑
𝒋=𝟏
𝒑
𝒊=𝟏
𝒑
𝒋=𝟏
𝒑
𝒊=𝟏
∏ ∏ (𝟏 − 𝝀𝒊
(𝒓)
𝝀𝒋
∗(𝒓)
) ∏ ∏ (𝟏 − 𝝀𝒊
(𝒕)
𝝀𝒋
∗(𝒕)
)
𝒑
𝒋=𝟏
𝒑
𝒊=𝟏
𝒑
𝒋=𝟏
𝒑
𝒊=𝟏
]
The PSM pole space metric 𝒅 𝑷𝑺𝑴( 𝜽 𝒌
(𝒓)
, 𝜽 𝒌
(𝒕)
) is
used as it is;
𝒅 𝑷𝑺𝑴( 𝜽 𝒌
(𝒓)
, 𝜽 𝒌
(𝒕)
) = |𝐫∆
𝑻
𝚽 𝚺 + 𝚽∆
𝑻
𝐫𝚺|
and the OSPA metric is used without number of
poles normalization with 𝒒 = 𝟏 as;
𝒅 𝑶𝑺𝑷𝑨( 𝜽 𝒌
(𝒓)
, 𝜽 𝒌
(𝒕)
) = [𝐦𝐢𝐧
𝝅∈𝚷 𝒑
(∑ 𝒃(𝝀𝒊
(𝒓)
, 𝝀 𝝅(𝒊)
(𝒕)
)
𝒑
𝒊=𝟏
)]
Consider a reference model with
𝝀 𝟏
(𝒓)
= 𝟎. 𝟏𝒆𝒋𝝅/𝟑
𝝀 𝟓
(𝒓)
= 𝟎. 𝟑𝒆𝒋𝟗𝝅/𝟓
𝝀 𝟐
(𝒓)
= 𝟎. 𝟑𝒆𝒋𝝅/𝟓
𝝀 𝟔
(𝒓)
= 𝟎. 𝟕𝒆𝒋𝟏𝟏𝝅/𝟔
𝝀 𝟑
(𝒓)
= 𝟎. 𝟕𝒆𝒋𝝅/𝟔
𝝀 𝟕
(𝒓)
= −𝟎. 𝟒
𝝀 𝟒
(𝒓)
= 𝟎. 𝟏𝒆𝒋𝟓𝝅/𝟑
𝝀 𝟖
(𝒓)
= −𝟎. 𝟕
and a test model with same poles except 𝝀 𝟑
(𝒕)
and
𝝀 𝟔
(𝒕)
complex pole pair, which are the only
difference between reference and test models
and those two complex pole pair move radially
(while phase kept same, magnitudes 𝒓 𝟑
(𝒕)
and 𝒓 𝟔
(𝒕)
changes between 0 and 1) in first scenario and
moves angularly (while magnitude kept same,
phases ∅ 𝟑
(𝒕)
and ∅ 𝟔
(𝒕)
changes between 0 and 2𝝅)
in second scenario.
In first scenario, since the reference model has
poles at 𝝀 𝟑
(𝒓)
= 𝟎. 𝟕𝒆𝒋𝝅/𝟔
and 𝝀 𝟔
(𝒓)
= 𝟎. 𝟕𝒆𝒋𝟏𝟏𝝅/𝟔
when the radially moving complex pole pair of
test model coincides with those then the
distance becomes zero, as the complex poles pair
moves away from that point the distance
between models increase as expected, as shown
in Fig 2.
In second scenario, since the reference model
has poles at 𝝀 𝟑
(𝒓)
= 𝟎. 𝟕𝒆𝒋𝝅/𝟔
and 𝝀 𝟔
(𝒓)
=
𝟎. 𝟕𝒆𝒋𝟏𝟏𝝅/𝟔
when the angularly moving complex
pole pair of test model coincides with those then
the distance becomes zero, as the complex poles
pair moves away from that point the distance
between models increase as expected, as shown
in Fig 3.

8. 7
In both scenarios, the proposed metric 𝒅 𝑷𝑺𝑴
behaves properly as expected and this behavior
is very similar to ARMA metric, those two
scenarios summarize the overall picture of
classification of models or systems based on
their pole characterization, test model/system
poles move away or get closer to reference
model/system poles in time varying systems,
and at a distance in static model/system
classification, so the PSM metric 𝒅 𝑷𝑺𝑴 can be
used in a great range of problems, especially for
the ones with computational sensitivities.
The computational comparison in terms of
number of multiplications and number of
additions are given in Figure 4 and Figure 5 for
PSM and ARMA metrics. The reason OSPA metric
is not included in that plots is due to huge
number of both multiplications and additions
required by OSPA metric due to permutation
operation which makes computational load
proportional to ! . Just to show effect of this
permutation operation on computational load
we provided another plot which shows the
number of multiplications required by each
metric for number of poles up to ten in Figure 6.
In all those three graphs the axis showing the
required number of computations is logarithmic.
As seen from the graphs, the PSM metric is the
least demanding in terms of both multiplications
and additions and does not require
permutations for non-ordered set of poles,
which makes it suitable for
classification/identification of state changes in
Figure 3. Behavior of Proposed Metric PSM
and comparison with ARMA & OSPA metrics
for an angularly moving complex pole pair
Figure 4. Computational requirements of
ARMA and PSM metrics in terms of
multiplications
Figure 5. Computational requirements of
ARMA and PSM metrics in terms of additions
Figure 6. Computational requirements of
ARMA,OSPA and PSM metrics in terms of
multiplications

9. 8
linear dynamic systems as well as applications
involving high model orders and more than two
model classification requirements.
VI. CONCLUSIONS
We have described a new approach for pole
based distance measure and derived a
computationally efficient (𝟐𝒑 multiplications
and ((𝟒𝒑 + 𝟏) additions), analytically tractable
and physically meaningful pole-based metric.
The metric can be used in classification of
models and systems which has a pole-space
representation and also in detection, estimation
and classification of change in model parameters
or system state changes in time varying AR
models or linear time varying dynamic systems.
Our main focus was on stable AR models and
systems represented by poles, however the
approach can be extended to involve other
models and might work for non-stable non-
minimum phase systems or the systems
represented by entities both inside and outside
of the unit circle of complex plane.
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