International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056
Volume: 08 Issue: 07 | July 2021 www.irjet.net p-ISSN: 2395-0072
© 2021, IRJET | Impact Factor value: 7.529 | ISO 9001:2008 Certified Journal | Page 2057
Fitting Sum Of Exponentials Model Using Differential Linear Regression
Molise Mokhomo
1
, Naleli Jubert Matjelo
2
1
Mukuru, Cape Town, Western Cape, South Africa.
2
National University of Lesotho, Department of Physics and Electronics, P. O. Roma 180, Lesotho
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Abstract – In this paper, we present the problem of fitting
a sum of exponentials (growth or decay) to some data using
linear regression. We start by transforming the problem
from the commonly adopted nonlinear regression
formulation and present it as a linear regression problem
using differential formulation. This allows the problem to be
solved in closed form. We demonstrate here by fitting a
dataset to a two-component sum of exponentials and show
its performance in both noisy and noise-free datasets. The
method is shown to be sensitive to noise and this due to the
noise amplification inherent in any differential formulation.
Key Words: Exponential Growth, Exponential Decay,
Regression, Noise Amplification, Model Fitting,
Parameter Estimation.
1. INTRODUCTION
Fitting models to data comes frequently in data-based
research. Model fitting makes extensive use of parameter
estimation methods to obtain the optimal model
parameters associated with the given data set. Some of the
disciplines making use of parameter estimation include
system identification, characterization, behavioral
analysis, model-based control, state estimation,
forecasting, smoothing, and filtering [1], [2], [3]. One of the
models used to estimate almost many smooth continuous
functions is the sum of the exponentials model (SEM),
composed of the weighted average of exponentials [4].
Fitting SEM to data is often considered as a nonlinear
regression problem from which there exists no closed-
form solution to the problem hence the need for iterative
optimization algorithms for solving this problem [4], [5].
In this paper, we adopt a differential formulation of the
SEM which allows the parameter estimation problem to
posed as a linear regression problem, solvable in closed-
form. The approach is based on exploiting the differential
equation satisfied by the SEM when formulating the
regression problem. In this work we consider the two-
component SEM to demonstrate the differential-based
linear regression formulation of the SEM fitting problem.
This method is related to the method based on integral
equations in [6].
The rest of this paper is organized as follows. Section 2
presents a two-component SEM and the resulting
differential linear regression problem formulation. Section
3 presents model-fitting simulation results and discussion.
Section 4 concludes this work with a summary of major
findings and some remarks.
2. LINEAR REGRESSION MODEL FOR SEM
2.1 Sum of Exponentials Model
A general one-dimensional -component SEM can be
represented as follows,
∑
(1)
with
as the
component weight, and
being related
to the
component decay or growth rate. We restrict
ourselves to however, the same concept applies to
higher values of . With we have the following
model with four unknown parameters to be determined
from the data,
(2)
2.2 Sum of Exponentials Model In Differential Form
Differentiating equation (2) twice and relating it with its
first two derivatives we obtain the following
homogeneous ordinary differential equation with constant
coefficients,
(3)
with parameters
and
given by,
(4)
(5)
2.3 Differential Linear Regression Model
Given a data
of size (i.e. ) to fit an
SEM, we can formulate the linear regression problem
using equations (3-5) as shown in the cost function below,
∑
(6)
The resulting solution to this linear regression problem is
given by the following matrix equation,