Robust Estimators for Missing Observations in Linear Discrete-Time
Stochastic Systems with Uncertainties
SEIICHI NAKAMORI
Professor Emeritus, Faculty of Education,
Kagoshima University,
1-20-6, Korimoto, Kagoshima, 890-0065,
JAPAN
Abstract: - As a first approach to estimating the signal and the state, Theorem 1 proposes recursive least-
squares (RLS) Wiener fixed-point smoothing and filtering algorithms that are robust to missing measurements
in linear discrete-time stochastic systems with uncertainties. The degraded quantity is given by multiplying the
Bernoulli random variable by the degraded signal caused by the uncertainties in the system and observation
matrices. The degraded quantity is observed with additional white observation noise. The probability that the
degraded signal is present in the observation equation is assumed to be known. The design feature of the
proposed robust estimators is the fitting of the degraded signal to a finite-order autoregressive (AR) model.
Theorem 1 is transformed into Corollary 1, which expresses the covariance information in a semi-degenerate
kernel form. The autocovariance function of the degraded state and the cross-covariance function between the
nominal state and the degraded state is expressed in semi-degenerate kernel forms. Theorem 2 shows the robust
RLS Wiener fixed-point and filtering algorithms for estimating the signal and state from degraded observations
in the second method. The robust estimation algorithm of Theorem 2 has the advantage that, unlike Theorem 1
and the usual studies, it does not use information on the existence probability of the degraded signal. This is a
unique feature of Theorem 2.
Key-Words: - Robust RLS Wiener fixed-point smoother, robust RLS Wiener filter, missing observations,
discrete-time stochastic systems, uncertain parameters, degraded signal.
Received: July 12, 2022. Revised: October 19, 2023. Accepted: November 22, 2023. Published: December 29, 2023.
1 Introduction
1.1 Brief Literature Review
In sensor network systems, missing measurements
are often due to the limited bandwidth of the
network. Missing measurements occur at random
rates. The presence of the degraded signal in the
observation equation is described by the Bernoulli
random variable. It takes the value 1 or 0 with a
known probability. This study aims to develop a
new robust estimation technique for missing
measurements.
A variety of estimation problems for systems
with missing measurements have been studied in
detail, [1], [2], [3], [4], [5], [6], [7], [8], [9], [10],
[11], [12], [13], [14], [15], [16], [17], [18]. For
nonlinear stochastic systems without uncertainties,
estimators for missing measurements have been
developed, [10], [11]. A robust filter for nonlinear
time-delayed stochastic systems with uncertainties
was developed in, [12]. In, [13], a robust finite-
horizon Kalman filter was presented for systems
with norm-bounded parameter uncertainty and
missing measurements with a random N-step
observation delay. Missing probabilities are used for
the robust Kalman filter. Robust fusion estimation
problems with missing measurements have been
studied in multisensor network systems, [2], [7],
[14], [15], [16], [17], [18]. In, [7], [16], robust
centralized fusion (CF) and weighted measurement
fusion (WMF) Kalman estimators were designed for
missing measurements in linear stochastic systems
with uncertainties. In, [2], robust fusion Kalman
estimators were proposed for stochastic systems
with uncertainties in the system and input matrices.
Observations are delayed by one randomly
occurring step and missing observations occur.
In linear discrete-time stochastic systems with
uncertainties, robust recursive least-squares (RLS)
Wiener estimators are developed as follows: (1)
Robust RLS Wiener fixed-point smoother and filter
for signal estimation, [19], (2) Robust RLS Wiener
finite impulse response (FIR) predictor, [20], (3)
Centralized multisensor robust Chandrasekhar-type
RLS Wiener filter, [21]. In, [22], robust RLS
Wiener estimators, [19], were applied to the
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estimation problem for random delays, packet
dropouts, and out-of-order packets in observed data
when the system matrix and observation vector have
uncertain parameters.
1.2 Current Study
As the first approach for estimating the signal and
state, Theorem 1 proposes RLS Wiener fixed-point
smoothing and filtering algorithms that are robust to
missing measurements in linear discrete-time
stochastic systems with uncertainties. The Bernoulli
random variable is multiplied by the degraded signal
due to the uncertainties in the system and
observation matrices. It is assumed that the
probability of the degraded signal being present in
the observation equation is known. Theorem 1
presents the RLS Wiener fixed-point smoothing and
filtering algorithms that estimate the signal and the
state from the missing measurements using
information on the probability and without using
any information on the uncertainties. The design
feature of the proposed robust estimators is to fit the
degraded signal to an autoregressive (AR) model of
a finite order. Theorem 1 is transformed into
Corollary 1, which expresses the covariance
information in a semi-degenerate kernel form. The
autocovariance function of the degraded state and
the cross-covariance function between the nominal
state and the degraded state is expressed in semi-
degenerate kernel forms. Theorem 2 shows the
robust RLS Wiener fixed-point smoothing and
filtering algorithms, [19], or estimating the signal
and the state from degraded observations in the
second method. A degraded quantity is given by
multiplying the Bernoulli random variable by the
degraded signal caused by the uncertainties in the
system and observation matrices. The degraded
quantity is observed with additional white
observation noise. In contrast to Theorem 1 and
other studies, e.g., [10], [11], [12], [13], the robust
estimation algorithms of Theorem 2 have the
advantage of not using information on the existence
probability of the degraded signal. This is a unique
feature of Theorem 2.
By the way, a combination of an unscented
Kalman filter and a back-propagation neural
network has been applied to GPS/SINS integrated
navigation, [23]. In order to predict the traffic state
of the entire network by modeling the dependencies
of individual self and neighbors, a deep learning
framework called Deep Kalman Filtering Network
has been studied, [24]. In, [25], the extended
estimators using covariance information are
presented. Its estimation accuracy is superior to the
Kalman filter neuro-computing and maximum a
posteriori (MAP) estimation methods.
The numerical simulation example in Section 4
compares the estimation accuracies of the robust
RLS Wiener estimators in Theorem 1 with those of
the robust RLS Wiener estimators in Theorem 2.
The estimation accuracies of the robust RLS Wiener
estimators in Theorem 2 are superior to those of the
robust RLS Wiener estimators in Theorem 1.
2 Recursive Least-Squares Fixed-Point
Smoothing Problem
Let the state-space model for the signal be
given by (1).
: observation vector; :
signal vector; : state vector;
: white observation noise with
mean zero; : input noise vector
with mean zero; : system matrix; :
observation matrix.
(1)
is the signal to be estimated. Here, the
following assumptions are introduced.
(1) is the white observation noise with the
variance . is the white input noise with
the variance denotes the
Kronecker delta function. and have
mean zeros.
(2) The state , the observation noise , and
the input noise are mutually independent.
Consider the degraded state-space model (2) with
uncertainties in the system and observation matrices
for the system (1).
(2)
: degraded observation vector; :
degraded signal vector; : degraded
state vector;
: degraded system matrix;
: degraded observation matrix;
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: uncertain matrix; :
uncertain matrix
The degraded observed value is given as
the sum of the degraded quantity and the
observation noise . The Bernoulli random
variable in the observation equation has the
probabilities and
. The probability that is
. For , the observation equation is
given by . The probability that
is . For, the observed
value consists only of the observation noise
. The degraded system matrix
is given as
a sum of the system matrix and the uncertain
matrix . The degraded observation matrix
is given as a sum of the system matrix and
the uncertain matrix . Assume that
and contain uncertain parameters,
respectively.
The first objective of this study is to design the
RLS Wiener fixed-point smoothing and filtering
algorithms that estimate the signal from the
observed value using information such as the
probability and without using any information
on the uncertain matrices and .
Let the sequence of the degraded signal be
fitted to a th-order AR model.
(3)
Let be expressed as
(4)
From (3) and (4), the state equation for is
given by
(5)
Let
be the autocovariance function of the
state .
has the wide-sense stationarity
(WSS)
[26].
is
expressed in the semi-degenerate functional form as
follows:
.
(6)
Here
denotes the system matrix of the state
equation (5).
is given by
(7)
From the relation
in wide-sense stationary stochastic
systems, [26], and (4), the autovariance function
of the state becomes
(8)
Then
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The Yule-Walker equations for the AR parameters
are the following:
(9)
Here,
Let
denote the cross-covariance function
between the state and the observed value .
satisfies the relation
in wide-sense stationary
stochastic systems, [26]. Let
denote the
cross-covariance function between the state
and the degraded state .
is expressed
in the following functional form:
.
(10)
From (1), represents the system matrix for the
state .
Let the fixed-point smoothing estimate of
the state at the fixed point be given by
,
(11)
using the observed values . In (11),
denotes a time-varying impulse response
function. Consider the estimation problem of
minimizing the mean square value (MSV)
(12)
of fixed-point smoothing errors. By the orthogonal
projection lemma, [26],
(13)
the impulse response function satisfies the Wiener-
Hopf equation
(14)
Here, ‘’ denotes the orthogonality notation. From
(1), (2), (4), (8), and the relation
, we get
(15)
Here,
is the cross-
covariance function between the state and the
degraded signal . Clearly,
.
3 Robust RLS Wiener Fixed-Point
Smoothing and Filtering
Algorithms
In (2), the degraded observation is given as the
sum of the degraded quantity and the
observation noise . The Bernoulli random
variable in the observation equation has the
probabilities and
. Theorem 1 assumes that is
known. The degraded signal in (2) is affected
by the uncertain matrices and . The
sequence of is fitted to the th-order AR
model (3). The AR model corresponds to the state-
space model (5) for . The model parameters are
calculated using the Yule-Walker equation (9). The
observation matrix
and the system matrix
do
not use any information about and .
and
are used in the robust RLS Wiener algorithms
for the fixed-point smoothing estimate at the
fixed point and the filtering estimate of the
signal in Theorem 1.
Based on the linear estimation problem for the
state in Section 2, Theorem 1 proposes the
robust RLS Wiener fixed-point smoothing and
filtering algorithms.
Theorem 1 Let and denote the system and
observation matrices, respectively, for the signal
in the state-space model (1). In the state-space
model (2), the system and observation matrices
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and
contain the uncertain matrices
and , respectively.
and
denote the system
and observation matrices, respectively, when the
degraded signal process in is fitted to the AR
model (3) of order . Let
be the variance of
the state for the degraded signal and
the cross-variance function between the
state and the degraded state . In the
observation equation (2) for , the presence of
the degraded signal depends on the values of
the Bernoulli random variable . Let be the
variance of the white observation noise . Then
the robust RLS Wiener algorithms for the fixed-
point smoothing estimate at the fixed point
and the filtering estimate of the signal
consist of (16)-(26) in linear discrete-time stochastic
systems with uncertainties.
Fixed-point smoothing estimate of the signal at
the fixed point :
(16)
Fixed-point smoothing estimate of the state at
the fixed point :
(17)
Smoother gain for in (17):
(18)
(19)
Filtering estimate of the signal :
(20)
Filtering estimate of the state :
(21)
Filter gain for in (21):
(22)
Filtering estimate of the degraded state :
(23)
Filter gain for
in (23):
(24)
Autovariance function of
:
(25)
Cross-variance function between and
:
(26)
Proof of Theorem 1 is deferred to the Appendix.
The conditions for the stability of the fixed-
point smoothing and filtering algorithms of
Theorem 1 are as follows:
1. All the eigenvalues of the matrix lie
within the unit circle.
2. All the eigenvalues of the matrix
lie within the unit circle.
3.
is a positive definite matrix,
and its inverse exists.
Instead of Theorem 1, Corollary 1 presents
robust RLS Wiener fixed-point smoothing and
filtering algorithms using covariance information.
Corollary 1 Let the autocovariance function
of the state be given by (6). Let the cross-
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covariance function between the state and the
degraded state be given by (10). Let the state-
space model for the signal be given by (1). Let
the degraded state and observation equations
containing the uncertain matrices and be
given by (2). In the observation equation (2) for
, the presence of the degraded signal
depends on the values of the Bernoulli random
variable . Let the variance of the white
observation noise be . Using the covariance
information, the robust RLS Wiener algorithms for
the fixed-point smoothing estimate at the
fixed point and the filtering estimate of the
signal consist of (27)-(39) in linear discrete-
time stochastic systems with uncertainties.
Fixed-point smoothing estimate of the signal at
the fixed point :
(27)
Fixed-point smoothing estimate of the state at
the fixed point :
(28)
Smoother gain for in (28):
(29)
(30)
Filtering estimate of :
(31)
Filtering estimate of :
(32)
(33)
(34)
(35)
Filtering estimate of :
(36)
(37)
(38)
(39)
Proof
(28) is obtained from (6), (A-30), and (A-48). (29) is
obtained from (6), (10), (A-16), (A-41), and (A-42).
(30) is obtained from (A-42). From (10), (A-9), (A-
13), and (A-40), is obtained. From (A-27)
and (A-28), (32) is obtained. From (6), (A-30), and
(A-31) we get (33). The initial condition is
clear from (A-28). From (6), (10), and (A-18) we
get (34). From (6) and (A-17) we get (35). From (A-
13) the initial condition is clear. From (6)
and (A-32) we get (36). From (6), (A-32), and (A-
33) we get (37). From (6), (A-19), and (A-23), (38)
is obtained. (A-21) is equivalent to (39). The initial
condition is clear from (A-16).
(Q.E.D.)
Note that the robust fixed-point smoother and the
filter in Theorem 1 use information about the
existence probability of and the degraded
signal . Suppose that the degraded quantity
is defined as the multiplication of the
Bernoulli random variable by the degraded
signal . The observation equation for and
the state equation for in (2) are rewritten as
(40)
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in linear discrete-time stochastic systems with
uncertainties. Assume that the sequence of the
degraded signal
is fitted to the th-order AR
model.
(41)
Suppose that
is represented by
(42)
From (41) and (42), the state equation for the
degraded state
becomes
(43)
The relation
holds for the
autocovariance function of the state
in wide-
sense stationary stochastic systems, [26]. Let
be expressed in the following semi-
degenerate functional form:
.
(44)
Here,
is the state transition matrix for the state
. The system matrix
in the state equation
(43) is given by
(45)
By letting
, the autovariance function
of the state
is given by
(46)
Then
.
Using
, the Yule-Walker equations for the
AR parameters are given by
.
(47)
Here,
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.
Let
represent the cross-covariance
function between the state and the degraded
state
in wide-sense stationary stochastic
systems. Assume that
has the following
functional form:
.
(48)
Here, is the system matrix for the state .
Theorem 2 Let the state-space model containing the
uncertain matrices and be given by (40). Let
and be the system and observation matrices for
the signal process in (1) for , respectively.
When the sequence of the degraded signal
is
fitted to the AR model (41) of order and
represented in the state-space model,
and
stand
for the system matrix and the observation matrix,
respectively. Let the variance
of the state
for the degraded signal
and the cross-
variance function
between the state
for the signal and the state
for the
degraded signal
be given. Let the variance of
the white observation noise be . Then, the
robust RLS Wiener algorithms for the fixed-point
smoothing estimate at the fixed point and
the filtering estimate of the signal
consist of (49)-(59) in linear discrete-time stochastic
systems with uncertainties.
Fixed-point smoothing estimate of the signal at
the fixed point :
(49)
Fixed-point smoothing estimate of the state at
the fixed point :
(50)
Smoother gain for in (50):
(51)
(52)
Filtering estimate of the signal :
(53)
Filtering estimate of the state :
(54)
Filter gain for in (54):
(55)
represents the cross-variance function
between and the degraded signal
.
Filtering estimate of
:
(56)
Filter gain for
in (56):
(57)
Autovariance function of
:
(58)
Cross-variance function of with
:
(59)
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See, [19], for the proof of Theorem 2.
The conditions for the stability of the fixed-
point smoothing and filtering and algorithms of
Theorem 2 are as follows:
1 All eigenvalues of the matrix lie
within the unit circle.
2 All eigenvalues of the matrix
are inside the unit circle.
3
is a positive definite matrix, and its inverse
exists.
4 Filtering Error Variance Function
of Signal in Theorem 1
This section presents the filtering error variance
function
for the filtering estimate in
Theorem 1. Let the autocovariance function
of the state be expressed by
(60)
The filtering error variance function for the filtering
estimate is given by
(61)
From (10) and (A-27), and introducing a function,
(62)
(61) is rewritten as
(63)
Subtracting from , using (A-11) and
introducing a function
(64)
we have
(65)
Subtracting from and using (A-5),
we have
(66)
Therefore, the filtering error variance function
is calculated by (63) with (34), (35), (38), (39), (65),
and (66) recursively.
Since
is the semidefinite function, the
filtering variance function
is
upper bounded by and lower bounded
by the zero matrix as follows:
(67)
This shows the existence of a robust filtering
estimate of the signal .
5 A Numerical Simulation Example
Suppose that the scalar observation and state
equations for the state are given by
(68)
In (68), the signal process for is represented by
a second-order AR model. Suppose that the state-
space model containing the uncertain matrices
and is given by
(69)
In linear discrete-time stochastic systems. The
observed value is given by the sum of the
degraded quantity and the observation
noise . It should be noted that the matrices
and are uncertain. ζ(k) in (69)
denotes the random variable generated by the “rand"
command in MATLAB or Octave. Let the
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probability that be . ,
, and consist of the mean values and
zero mean stochastic variables, respectively. The
task is to recursively estimate the signal from
the observed value . Suppose that is fitted
to the th -order AR model:
.
(70)
From (4), for the scalar observation equation in (69),
is given by
: vector.
(71)
The state equation for is given by (5). In this
example, this equation corresponds to the case
. The autocovariance function
of the state
is expressed in the form of the semi-
degenerate function in (6) and has the property
in the wide sense of stationary
stochastic systems. In (6),
is the system matrix for
the state .
is given by (7).
of the state
is described as follows:
(72)
Suppose that
represents
the cross-covariance function between the signal
and the degraded signal . From (4) and
(68), the cross-covariance function
is
given by
=
(73)
The Yule-Walker equations (9) calculate the AR
parameters in (70). Substituting
,
, ,
,
,
, and
into the robust RLS Wiener estimation
algorithms of Theorem 1, the fixed-point smoothing
and filtering estimates are recursively computed. In
evaluating
in (7),
in (72), and
in (73), 2,000 data sets of signal and degraded signal
are used, respectively. The computation of
in (22) uses 2,000 data sets of signal and
observation, respectively. Figure 1 illustrates the
fixed-point smoothing estimate and the
filtering estimate of the signal by
Theorem 1 vs. for the white Gaussian observation
noise in the case of the AR model order
. Figure 2 illustrates the mean square values
of the filtering errors and the fixed-
point smoothing errors vs.
, , by Theorem 1 for the white
Gaussian observation noises , ,
, and in the case of the AR
model order Figure 3 illustrates the fixed-
point smoothing estimate and the
filtering estimate of the signal by
Theorem 2, [19], vs. for the white Gaussian
observation noise in the case of the AR
model order . Figure 4 illustrates the mean
square values of the filtering errors
and the fixed-point smoothing errors
vs. , , by Theorem
2 for the white Gaussian observation noises
, , , and
in the case of the AR model order As
shown in Figure 2 and Figure 4, the estimation
accuracies of the RLS Wiener filter and fixed-point
smoother of Theorem 2 are superior to those of
Theorem 1 for each observation noise. In Figure 4,
the MSV decreases as increases. This shows
the smoothing effect of the RLS Wiener fixed-point
smoother by Theorem 2. Figure 2 shows the
smoothing effect by Theorem 1 only for the
observation noise . In Figure 2 and Figure
4, the MSVs of the fixed-point smoothing and
filtering errors are evaluated by
and
, respectively.
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Fig. 1: Fixed-point smoothing estimate
and filtering estimate of the signal by
Theorem 1 vs. for the white Gaussian observation
noise in the case of the AR model order
.
Fig. 2: MSVs of the filtering errors
and the fixed-point smoothing errors
vs. , , by Theorem
1 for the white Gaussian observation noises
, , , and
in the case of the AR model order
Fig. 3: Fixed-point smoothing estimate
and filtering estimate of the signal by
Theorem 2, [19], vs. for the white Gaussian
observation noise in the case of the AR
model order .
Fig. 4: MSVs of the filtering errors
and the fixed-point smoothing errors
vs. , , by Theorem
2, [19], for the white Gaussian observation noises
, , and in
the case of the AR model order
6 Conclusion
Theorem 1 proposed the robust RLS Wiener fixed-
point smoother and filter for missing measurements
in linear discrete-time stochastic systems with
uncertainties. In (2), the degraded observation
is given as the sum of the degraded quantity
and the observation noise . The
Bernoulli random variable in the observation
equation has the probabilities
and . Theorem 1 assumes
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that is known. The degraded signal in (2)
is affected by the uncertain matrices and
. The sequence of is fitted to the th-
order AR model (3). The AR model corresponds to
the state-space model (5) for . The model
parameters are calculated using the Yule-Walker
equation (9). The observation matrix
and the
system matrix
do not use any information about
and .
and
are used in the robust
RLS Wiener algorithms for the fixed-point
smoothing estimate at the fixed point and
the filtering estimate of the signal in
Theorem 1. The design feature of the proposed
robust estimators is to fit the degraded signal to a
finite-order AR model. Theorem 1 is transformed
into Corollary 1, which expresses the covariance
information in the semi-degenerate kernel form.
Second, Theorem 2 showed the robust RLS Wiener
fixed-point smoother and filter, [19]. The robust
estimation algorithm of Theorem 2 has the
advantage that, unlike Theorem 1 and conventional
studies, it does not use information on the existence
probability of the degraded signal.
As shown in Figure 2 and Figure 4, the
estimation accuracies of the RLS Wiener filter and
fixed-point smoother of Theorem 2 are superior to
those of Theorem 1 for each observation noise. As
shown in Figure 4, the MSV decreases as the
increases. This shows the smoothing effect of the
RLS Wiener fixed-point smoother by Theorem 2.
Extending this study to robust fusion estimation
problems with missing measurements is future work
in multisensor network systems. The current study is
based on the least-squares estimation method. The
neural network-aided Kalman filter is known. A
combination of the current study with neural
networks is also left for future work at this time.
APPENDIX
proof of Theorem 1
The impulse response function satisfies
(15). Subtracting from ,
we have
(A-1)
Introducing
(A-2)
we obtain
(A-3)
Subtracting from , we get
(A-4)
From (A-2) and (A-4), we obtain
(A-5)
The filtering estimate is given by
(A-6)
From (15), the impulse response function
satisfies
(A-7)
Introducing
(A-8)
we obtain
(A-9)
Subtracting from , we have
(A-10)
From (A-2) and (A-10), we obtain
(A-11)
From (A-8), satisfies
(A-12)
Using (6) and introducing
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(A-13)
we obtain
(A-14)
Subtracting from , we have
(A-15)
Introducing
(A-16)
from (A-11), we obtain
(A-17)
Substituting (A-17) into (A-14), we have
(A-18)
Introducing a function
(A-19)
we rewrite (A-18) as
(A-20)
Subtracting from and using (A-5),
we obtain
(A-21)
From (A-2), satisfies
From (6) and (A-16), it follows that
(A-22)
Substituting (A-21) into (A-22), we obtain an
expression for as
(A-23)
From (A-19) and (A-21), it follows that
(A-24)
Let us introduce a function
(A-25)
From (A-23) and (A-25), it follows that
(A-26)
Now, from (A-6) and (A-9), the filtering estimate
of is given by
(A-27)
Introducing a function
(A-28)
the filtering estimate is expressed as
(A-29)
Subtracting from , using (A-5) and
(A-11), and introducing a function
(A-30)
we obtain
(A-31)
Let us introduce a function
(A-32)
which represents the filtering estimate of .
Subtracting from and using (A-5),
we obtain
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(A-33)
Substituting (A-31) into (A-29), we have
(A-34)
From (A-18), and by introducing a function
(A-35)
is expressed as
(A-36)
From (A-32) and (A-33), it follows that
(A-37)
From (A-17) and (A-35), it follows that
(A-38)
From (15), satisfies
(A-39)
Introducing a function
(A-40)
we have an expression for as
(A-41)
Subtracting from , from (A-3)
and (A-16), we have
(A-42)
Let us introduce a function
(A-43)
From (A-19), (A-42), and (A-43), it follows that
(A-44)
From (A-41), it is clear that
(A-45)
Substituting (A-44) into (A-45), we have
(A-46)
From (A-9), (A-13), (A-35), (A-40), and (A-43), the
initial condition in (A-44) for at
is given by
(A-47)
The fixed-point smoothing estimate is given by (11).
Subtracting from and using (A-
3), (A-30), and (A-32), we have
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(A-48)
(Q.E.D.)
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stages from the formulation of the problem to the
final findings and solution.
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Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare.
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