Estimation-free Prediction Algorithms
NICHOLAS ASSIMAKIS1, MARIA ADAM2, CHRISTOS TSINOS1, ATHANASIOS POLYZOS1
1Department of Digital Industry Technologies,
National and Kapodistrian University of Athens,
34400 Psachna Evias,
GREECE
2Department of Computer Science and Biomedical Informatics,
University of Thessaly,
2-4 Papasiopoulou Str., 35131, Lamia,
GREECE
Abstract: - For Time-varying, Time-invariant, and steady-state systems, Kalman Filter can be implemented as a
prediction algorithm, since it produces the state prediction and the corresponding prediction error covariance
matrix via the state estimation and the corresponding estimation error covariance matrix. Lainiotis Filter is
equivalent to Kalman Filter and can be used to compute the prediction. In this paper, for Time-varying, Time-
invariant and steady state systems, estimation-free Prediction Algorithms are derived via Kalman and Lainiotis
filters; they are equivalent and compute iteratively the prediction and the corresponding prediction error
covariance matrix. The estimation and the corresponding estimation error covariance matrix are not needed and
are not computed. The proposed estimation-free prediction algorithms are faster than the Kalman filter.
Key-Words: - Kalman filter, Lainiotis Filter, Time-varying System, Time-invariant System, Estimation,
Prediction.
Received: March 23, 2023. Revised: December 5, 2023. Accepted: December 23, 2023. Published: December 31, 2023.
1 Introduction
Prediction and estimation play an important role in
many fields of science: applications to the aerospace
industry, chemical process, communication systems
design, control, civil engineering, filtering noise
from 2-dimensional images, pollution prediction,
and power systems are mentioned in [1]. The
estimation problem arises in linear estimation and is
associated with discrete-time systems described by
the following state space equations:
(1)
where is the state vector, is the
measurement vector, is the
transition matrix, is the output matrix,
is the state noise and is the
measurement noise at time.
The statistical model expresses the nature of the
state and the measurements. The basic assumption is
that the state noise and the measurement
noise are white noises, i.e. a stochastic
process with uncorrelated successive values:
is a zero mean, Gaussian process with known
covariance of dimension and is a
zero mean, Gaussian process with known
covariance of dimension . The
following assumptions also hold: (a) the initial value
of the state is a Gaussian random variable with
mean and covariance ; (b) the stochastic
processes , and the random variable
are independent.
The discrete-time Kalman filter, [1] and Lainiotis
filter, [2] are well-known algorithms that solve the
filtering problem, producing the state estimation
and the corresponding estimation error
covariance matrix. The filters can be Time-
Varying(TV), Time-invariant (TI) or Steady State
(SS). Kalman filter can be seen as a prediction
algorithm as well, because it produces the state
prediction and the corresponding
prediction error covariance matrix.
The importance of filtering algorithms is without
doubt: Kalman filter has been used in electric load
estimation, [3], power generation prediction, [4],
weather forecasts, [5], cases and deaths prediction
of Covid-19, [6], satellite orbit determination, [7],
multi-observation fusion applications related to
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timescale, [8] which is widely used in satellite
navigation, [9].
In this paper estimation-free Prediction
Algorithms are derived via Kalman and Lainiotis
filters, for Time-varying, Time-invariant, and steady
state cases. The proposed algorithms can be applied
in many applications that require computation of
prediction: short-term electric load forecasting, [10],
weather prediction, [11], prediction of air pollution
levels, [12], stock price prediction, [13], [14],
prediction of the control effectiveness of the
actuator on behalf of an actuator stuck fault incident
occurring on airplanes, [15], Kalman filter
prediction that accounts for measurement
differences, for the case of time-correlated
measurement errors, [16], Global Positioning
System (GPS) and Inertial Navigation System (INS)
integration during GPS outages using machine
learning augmented with Kalman filter, [17].
The paper is organized as follows: Time-varying,
Time-invariant and steady state Kalman and
Lainiotis filters are summarized in section 2. Time-
varying, Time-invariant, and steady state
estimation-free Prediction Algorithms are derived
via Kalman filter in section 3. Time-varying, Time-
invariant and steady state estimation-free Prediction
Algorithms are derived via Lainiotis filter in section
4. It is established that the Kalman filter and the
Lainiotis filter based prediction algorithms are
equivalent concerning their behavior, since they
produce the same predictions. In section 5 the FIR
form of the steady state estimation-free prediction
algorithms is presented. In section 6 the multiple
steps prediction algorithms are derived. The
computational requirements of estimation-free
prediction algorithms are determined in section 7. It
is shown that the estimation-free prediction
algorithms are faster than Kalman filter. Finally,
Section 8 summarizes the conclusions.
2 Kalman and Lainiotis Filters
Time-varying, Time-invariant and steady state
Kalman and Lainiotis filters are summarized in this
section.
2.1 Kalman Filter
Kalman filter produces the state estimation and the
estimation error covariance, as well as the state
prediction and the corresponding prediction error
covariance matrix.
For Time-varying systems, the Time-varying
Kalman Filter is derived:
Time-varying Kalman Filter (TVKF)
(2)
for , with initial conditions ,
.
The notation is used for the transpose matrix of
matrix .
The notation is used for the identity matrix.
Note that these initial conditions are connected to
classically used initial conditions, [2]
, through the
following equations:
The choice of these initial conditions is due to
reasons of uniformity concerning all algorithms of
this paper.
For Time-invariant systems, where the transition
matrix , the output matrix ,
as well as the plant and measurement noise
covariance matrices and are
constant matrices, the Time-invariant Kalman Filter
is derived:
Time-invariant Kalman Filter (TIKF)
(3)
for , with initial conditions ,
Note that these initial conditions are connected to
classically used initial conditions, [2]
, through the
following equations:
For Time-invariant systems, it is well known,
[1], that if the signal process model is
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asymptotically stable, then there exists a steady state
value of the prediction error covariance matrix.
This value remains constant after the steady state
time is reached. The Steady State Kalman Filter is
derived:
Steady State Kalman Filter (SSKF)
(4)
for , with initial condition
where
(5)
and
(6)
is the steady state Kalman Filter gain
and is the solution of the Riccati equation
(7)
The steady state coefficients in (5) are
calculated offline by first solving the corresponding
discrete-time Riccati equation emanating from the
Kalman filter, [1].
Steady State Kalman Filter can be seen as a
prediction algorithm as well by computing the
prediction .
Note that this initial condition is connected to
classically used initial conditions, [2]
, through the
following equations:
2.2 Lainiotis Filter
Lainiotis filter produces the state estimation and the
estimation error covariance. It can be used to
compute the state prediction and the prediction error
covariance, using Kalman filter equations.
For Time-varying systems, the Time-varying
Lainiotis Filter is derived:
Time-varying Lainiotis Filter (TVLF)
(8)
for , with initial conditions ,
,
where
(9)
Note that these initial conditions are connected
to classically used initial conditions, [2]
, through the
following equations:
The choice of these initial conditions is due to
reasons of uniformity concerning all algorithms of
this paper.
Time-varying Kalman and Lainiotis filters are
equivalent with respect to their behavior, since they
produce the same estimations and the same
estimation error covariance matrices, [2].
Time-varying Lainiotis Filter can be used to
compute the state prediction and the prediction error
covariance, using Kalman filter equations.
Remark 1.
Time-varying Kalman and Lainiotis filters have the
same structure:
From (2) we get:
(10)
while from (8) we get:
(11)
Due to the fact that the two filters are equivalent, we
obtain:
(12)
For Time-invariant systems, where the
transition matrix , the output matrix
, as well as the plant and measurement
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noise covariance matrices and
are constant matrices, the Time-invariant Lainiotis
Filter is derived:
Time-invariant Lainiotis Filter (TILF)
(13)
for
with initial conditions , .
W
where
(14)
Obviously, the constant matrices in (14) are
computed off-line.
Note that these initial conditions are connected
to classically used initial conditions, [2]
, through the
following equations:
Time-invariant Kalman and Lainiotis filters are
equivalent concerning their behavior, since they
produce the same estimations and the same
estimation error covariance matrices, [2].
Time-invariant Lainiotis Filter can be used to
compute the state prediction and the prediction error
covariance, using Kalman filter equations.
Remark 2.
Time-invariant Kalman and Lainiotis filters have the
same structure.
From (3) we get:
(15)
while from (13) we get:
(16)
Because the two filters are equivalent, we obtain:
(17)
For Time-invariant systems, it is well known,
[1], that if the signal process model is
asymptotically stable, then there exists a steady state
value of the estimation error covariance matrix.
This value remains constant after the steady state
time is reached. The Steady State Lainiotis Filter is
derived:
Steady State Lainiotis Filter (SSLF)
(18)
for , with initial condition ,
where
(19)
and is the solution of the Riccati equation
(20)
The steady state coefficients in (19) are
calculated off-line by first solving the corresponding
discrete-time Riccati equation emanating from the
Lainiotis filter, [18].
Steady State Lainiotis filter can be seen as a
prediction algorithm as well by computing the
prediction . The steady state
prediction error covariance can be computed by the
steady state estimation error covariance:
Note that this initial condition is connected to
classically used initial conditions, [2]
, through the
following equations:
The Steady State Kalman Filter (SSKF) and the
Steady State Lainiotis Filter (SSLF) are equivalent
since they produce the same estimations, [2].
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Remark 3.
Steady State Kalman and Lainiotis filters have the
same structure.
Due to the fact that the two filters are equivalent, we
obtain, [2]:
(21)
3 Estimation-free Prediction
Algorithms via Kalman Filter
Estimation-free Prediction Algorithms are derived
via Kalman filter for Time-varying, Time-invariant
and steady state systems. The prediction and the
corresponding prediction error covariance are
computed iteratively; the estimation and the
corresponding estimation error covariance are not
needed and are not computed.
3.1 Time-varying Prediction Algorithm via
KF
The Time-varying Prediction Algorithm via KF is
derived from Time-varying Kalman Filter equations.
Time-varying Prediction Algorithm via KF
(TVPAKF)
(22)
for , with initial conditions
, .
Proof.
From Time-varying Kalman Filter equations we can
write the Kalman Filter gain as:
Concerning the prediction, from (2) we have:
Setting
we get
Concerning the prediction error covariance, from (2)
we have:
3.2 Time-invariant Prediction Algorithm via
KF
For Time-invariant systems, where the transition
matrix , the output matrix ,
as well as the plant and measurement noise
covariance matrices and are
constant matrices, the Time-invariant Prediction
Algorithm via KF is derived.
Time-invariant Prediction Algorithm via KF
(TIPAKF)
(23)
for , with initial conditions
, .
3.3 Steady State Prediction Algorithm via
KF
For Time-invariant systems, it is well known, [1],
that if the signal process model is asymptotically
stable, then there exists a steady state value of the
prediction error covariance matrix. This value
remains constant after the steady state time is
reached. The Steady State Prediction Algorithm via
KF is derived.
Steady State Prediction Algorithm via KF
(SSPAKF)
(24)
for , with initial condition
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where
(25)
and
is the steady state
Kalman Filter gain and is the solution of the
Riccati equation:
.
The steady state coefficients in (25) are
calculated off-line by first solving the corresponding
discrete time Riccati equation emanating from
Kalman filter [1].
4 Estimation-free Prediction
Algorithms via Lainiotis Filter
Estimation-free Prediction Algorithms are derived
via Lainiotis filter for Time-varying, Time-invariant
and steady state systems. The prediction and the
corresponding prediction error covariance are
computed iteratively; the estimation and the
corresponding estimation error covariance are not
needed and are not computed.
4.1 Time-varying Prediction Algorithm via
LF
The Time-varying Prediction Algorithm via LF is
derived from Time-varying Lainiotis Filter
equations.
Time-varying Prediction Algorithm via LF
(TVPALF)
(26)
for , with initial conditions
, .
Proof.
Concerning the prediction, due to the fact that
Kalman and Lainiotis filters are equivalent, [2],
from Time-varying Kalman Filter equations we
have:
x
with the assumption that the matrices
are nonsigular.
Then from (8) we have:
Thus
where
and
Thus
Furthermore, recall (12) and hence:
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Concerning the prediction error covariance, from (8)
we have:
From (2) we have:
But
Then
Remark 4.
Time-varying Prediction Algorithm via KF and
Time-varying Prediction Algorithm via LF have the
same structure:
From (22) we get:
while from (26) we get:
Due to the fact that the two filters are equivalent, we
obtain:
(27)
Remark 5.
The following relations between estimation and
prediction coefficients hold:
(28)
4.2 Time-invariant Prediction Algorithm via
LF
For Time-invariant systems, where the transition
matrix , the output matrix ,
as well as the plant and measurement noise
covariance matrices and are
constant matrices, the Time-invariant Prediction
Algorithm via LF is derived.
Time-invariant Prediction Algorithm via LF
(TIPALF)
(29)
for , with initial conditions
, .
Remark 6.
Time-invariant Prediction Algorithm via KF and
Time-invariant Prediction Algorithm via LF have
the same structure:
From (23) we get:
while from (29) we get:
Due to the fact that the two filters are equivalent, we
obtain:
(30)
4.3 Steady State Prediction Algorithm via LF
For Time-invariant systems, it is well known [1]
that if the signal process model is asymptotically
stable, then there exists a steady state value of the
prediction error covariance matrix. This value
remains constant after the steady state time is
reached. The Steady State Prediction Algorithm via
LF is derived.
Steady State Prediction Algorithm via LF
(SSPALF)
(31)
for , with initial condition
.
where
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(32)
and is the solution of the Riccati equation:
.
The steady state coefficients in (32) are
calculated off-line by first solving the corresponding
discrete time Riccati equation emanating from
Kalman filter, [1].
Remark 7.
Steady State Prediction Algorithm via KF and
Steady State Prediction Algorithm via LF have the
same structure:
From (24) we get:
while from (31) we get:
Due to the fact that the two filters are equivalent, we
obtain:
(33)
Remark 8.
The following relations between steady state
estimation and prediction coefficients hold:
(34)
5 FIR Form of the Steady State
Prediction Algorithms
The FIR form of the steady state estimation-free
prediction algorithms is presented in the following.
From (24) we take:
Then
…
If , then
, i.e there exists
:
,
Thus, for we take:
Hence we derive the following FIR form of the
Steady State Prediction Algorithm via KF:
FIR form of Steady State Prediction Algorithm
via KF
(35)
Similarly, we derive the following FIR form of
the Steady State Prediction Algorithm via LF:
FIR form of Steady State Prediction Algorithm
via LF
(36)
Remark 9.
The FIR Steady State Prediction Algorithm
coefficients are calculated a-priori.
Remark 10.
The prediction depends only on a well-defined
set of measurements.
6 Multiple Steps Prediction
Algorithms
All the presented estimation-free prediction
algorithms compute the one step prediction
and the corresponding one step prediction
error covariance and can be used to
compute multiple steps prediction and the
corresponding multiple step prediction error
covariance.
For Time-varying systems, we derive:
(37)
where
(38)
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For Time-invariant systems, we derive:
(39)
For steady state systems, we derive:
(40)
7 Computational Requirements
Kalman Filter is the classical prediction algorithm:
it uses estimation on order to compute estimation. It
is established that Kalman and Lainiotis filters are
equivalent and can be used to compute the
prediction.
Estimation-free prediction algorithms were
derived by Kalman and Lainiotis filters; they are
equivalent and compute the prediction.
In order to investigate possible computational
advantages of estimation-free prediction algorithms
versus classical Kalman filters, we are going to
compare estimation-free Prediction Algorithm via
KF to Kalman Filter, for Time-varying, Time-
invariant and steady state systems. All algorithms
are iterative. Then, it is reasonable to assume that
they compute the prediction and the prediction error
covariance executing the same number of iterations.
Thus, in order to compare the algorithms with
respect to their computational time, we have to
compare their per step (iteration) calculation burden
(CB) required for the on-line calculations; the
calculation burden of the off-line calculations
(initialization process for Time-invariant and steady
state algorithms) is not taken into account.
Scalar operations are involved in matrix
manipulation operations, which are needed for the
implementation of the filtering algorithms. Table 1
summarizes the calculation burden of needed matrix
operations. Note that a symmetric matrix is denoted
by . The details are given in [2].
Table 1. Calculation burden of matrix operations
Matrix
Operation
Matrix
Dimensions
Calculation
Burden
The per iteration calculation burdens of the
classical prediction algorithm Kalman Filter (KF)
and the proposed prediction algorithm estimation-
free Prediction Algorithm via Kalman Filter (PAKF)
are analytically calculated in the Appendix and
summarized in Table 2.
Table 2. Per iteration calculation burden of
prediction algorithms:
Kalman Filter (KF) and estimation-free Prediction
Algorithm via Kalman Filter (PAKF)
System
Algorithm
Calculation Burden
Time
Varying
KF
Time
Varying
PAKF
Time
Invariant
KF
Time
Invariant
PAKF
Steady
State
KF
Steady
State
PAKF
From Table 2, it is clear that:
for Time-varying and Time-invariant systems,
the estimation-free prediction algorithms are
faster than Kalman filter, since
(41)
for steady state systems, the estimation-free
prediction algorithm is faster than Kalman
filter, since Steady State Kalman Filter can be
seen as a prediction algorithm as well by
additionally computing the prediction
.
8 Conclusion
Many applications require computation of
prediction instead of estimation. For Time-varying,
Time-invariant and steady state systems, Kalman
Filter can be implemented as a classical prediction
algorithm, since it produces the state prediction and
the corresponding prediction error covariance
matrix via the state estimation and the
corresponding estimation covariance matrix.
Lainiotis Filter is equivalent to Kalman Filter and
can be used to compute the prediction.
In this paper, for Time-varying, Time-invariant
and steady state systems, estimation-free Prediction
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Algorithms are derived via Kalman and Lainiotis
filters; they are equivalent and compute iteratively
the prediction and the corresponding prediction
error covariance matrix. The estimation and the
corresponding estimation error covariance are not
needed and are not computed.
The FIR form of the steady state estimation-free
prediction algorithms is derived.
The multiple steps prediction algorithms are
derived.
The computational requirements of estimation-
free prediction algorithms are determined and it
shown that the proposed estimation-free prediction
algorithms are faster than Kalman filter; this is the
main advantage of the proposed algorithms over the
classical Kalman filter.
A subject of future research is to investigate the
application the proposed estimation free prediction
algorithms to dynamical continuous-time systems,
[19], to Linear Quadratic Regulator (LQR), [20].
Another area of future research may be the use of
the proposed algorithms in the derivation of Time-
varying and Time-invariant information filters,
using the inverse of the prediction error covariance
matrix.
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APPENDIX
A. Calculation Burden of Kalman Filter
1. Time-varying Kalman Filter
The computation burden of Time-varying Kalman
Filter (eq. 2) is analytically calculated:
Matrix Operation
Calculation Burden
2. Time-invariant Kalman Filter
The computation burden of Time-invariant Kalman
Filter (eq. 3) is equal to the computation burden of
Time-varying Kalman Filter (eq. 2):
3. Steady State Kalman Filter
The computation burden of Steady State Kalman
Filter (eq. 4) is analytically calculated:
Matrix Operation
Calculation Burden
B. Calculation Burden of Estimation-Free
Prediction Algorithms via Kalman Filter
1. Time-varying Prediction Algorithm via
Kalman Filter
The computation burden of Time-varying Prediction
Algorithm via Kalman Filter (TVPAKF) Kalman
Filter (eq. 22) is analytically calculated:
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2023.18.59
Nicholas Assimakis, Maria Adam,
Christos Tsinos, Athanasios Polyzos
E-ISSN: 2224-2856
579
Volume 18, 2023
Matrix Operation
Calculation Burden
2. Time-invariant Prediction Algorithm via
Kalman Filter
The computation burden of Time-invariant
Prediction Algorithm via Kalman Filter (eq. 23) is
equal to the computation burden of Time-varying
Prediction Algorithm via Kalman Filter (eq. 22):
3. Steady State Prediction Algorithm via Kalman
Filter
The computation burden of the Steady State
Prediction Algorithm via Kalman Filter (eq. 24) is
analytically calculated:
Matrix Operation
Calculation Burden
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Policy)
The authors equally contributed to the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
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.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2023.18.59
Nicholas Assimakis, Maria Adam,
Christos Tsinos, Athanasios Polyzos
E-ISSN: 2224-2856
580
Volume 18, 2023
