Universal Fitting Function for Quantitative Description of...

Computer Communication & Collaboration (Vol. 5, Issue 2, 2017)

ISSN 2292-1028(Print) 2292-1036(Online)

Submitted on Feb. 15, 2017

DOIC: 2292-1036-2017-02-002-83

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"Universal" Fitting Function for Quantitative Description of

Quasi-Reproducible Measurements

Raoul R. Nigmatullin1,3

(Correspondence author)

1 The Radioelectronic and Informative -Measurements Technics (R&IMT) Department,

Kazan National Research Technical University (KNRTU-KAI), 10 Karl Marx str., 420111,

Kazan, Tatarstan, Russian Federation

E-mail: [email protected]

Wei Zhang 2,3

, Renhuan Yang 2 , Yaosheng Lu

2

2 Department of Electronic Engineering, Jinan University, Guangzhou 510632, China

3 JNU-KNRTU(KAI) Joint Lab. of FracDynamics and Signal Processing, JNU, Guangzhou, China


Guido Maione 4

4 Department of Electrical and Information Engineering (DEI), Polytechnic University of Bari

Via E. Orabona 4, 70125, Bari, Italy


Abstract: In this paper the authors are trying to justify the following statement: is there a general

platform for description of a wide class of quasi-reproducible (QR) experiments? Under QR

experiments we understand the set of measurements when during a measurement cycle they can be

varied significantly under the influence of uncontrollable factors. If this general platform exists then it

facilities the analysis of experiments associated with properties of complex systems, when the

"best-fit function" that follows from the microscopic model is absent. This platform will bring a

certain profit for a competitive/alternative theory. If the fitting function obtained from the theory

looks very complicated then it can be expressed in terms of the IM and the parameters of this IM are

expressed in terms of the generalized Prony spectrum. The physical meaning of the Fourier transform

that used as the fitting function is explained also. It corresponds to an "ideal experiment" when all

measurements during the experimental cycle are fully identical to each other. The new formulae for

elimination of the apparatus functions are suggested. They help to evaluate the influence of

uncontrollable factors and at certain conditions reduce the measured data to an "ideal experiment". As

a nontrivial example we consider the fit of the single heartbeat (HB) recorded from anonymous

persons. We took the ECG data from the reliable Internet resource PhysioNet Bank

(http://physionet.org/cgi-bin/atm/ATM). The obtained results of its nontrivial application

demonstrate its effectiveness in quantitative description of the ECG data in the frame of the suggested

theory. We do hope that the quantitative description of the single heartbeat will find an application in

the modern cardiology.

mailto:[email protected]

Computer Communication & Collaboration (2017) Vol. 5, Issue 2: 8-34

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Keywords: General Theory of the QR measurements, Intermediate Model, The Generalized Prony

Spectrum, The Functional Least Square Method, The Fit of the Single Heartbeat.

List of abbreviations:

BLC – Bell-Like Curves

ECG – Electrocardiogram

(F)LLSM – (Functional) Linear Least Square Method

GPS –Generalized Prony Spectrum

HBs – Heartbeat(s)

IM – Intermediate Model

QR – Quasi-Reproducible

1. Introduction and the formulation of the problem

Is it possible to create a general theory that proposes a "universal" fitting function for a quantitative

description of a set of quasi-reproducible (QR) measurements without attraction of a priori or model

hypothesis? The first reaction of any researcher working in the region of data treatment will be

definitely negative. There are a giant collection of different systems that are needed to be measured by

various equipments and a lot of competitive theories/hypothesis that pretend to describe these

metadata. There are a lot of excellent books [1-9], countless number of papers (that cannot be cited

here) that create the clearly expressed trend in the data/signal processing analysis and mathematical

statistics regions. The conventional paradigm is clear and understandable. The pretending

(alternative) theory suggests a model and the parameters of this model including the desired fitting

function can be compared with accurately measured data, where the influence of other uncontrollable

factors and the apparatus function of the used equipment are supposed to be negligible. In the case of

the coincidence of the theoretical curve with the "purified" data we obtain new deterministic

evidences in deeper understanding of the existing reality. But this "ideal" scheme in many cases does

not work. The proposed model has narrow limit of their applicability, the experimental data are

"noisy" and the reliable comparison of the complicated theoretical curve with data in many practical

and important cases becomes unreliable and questionable.

Let us reformulate the first question posed above by another way.

Is it possible to propose an intermediate model (IM) that enables to conciliate the measured data

with the "best fit" curve that follows from the microscopic or phenomenological theory?

It means that the "true" theory being expressed in the unified parameters belonging to the IM will

find a common platform, where the reliable data (being cleaned from the influence of uncontrollable

factors) will be expressed quantitatively by means of the limited number of the fitting parameters that

follow from the IM. It implies that any alternative theory should be expressed also in the same set of

the fitting parameters that were proposed by the IM. The authors are going to prove that a "bridge",

where an experiment and a "true" theory can "meet" with each other, does exist and the desired fitting

function that follows from the IM can be presented by the segment of the generalized Prony spectrum

(GPS). This paper can be considered as a logic continuation of the results obtained earlier in papers

[10-13], where it was proved that the reproducible or quasi-periodic measurements have a memory

and mathematically this observation (confirmed on available data) can be expressed as

© 2017 Academic Research Centre of Canada

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1

0

( ) ( )L

L l l

l

F x a F x b

. (1)

Here Fm(x) F(x + mT) determines a response function that coincides approximately with current

measurement ym(x) i.e. ( ) ( )m mF x y x

, where (m = 0,…,M–1) defines the total number of

measurements, L is a parameter that determines the specific length of the "memory" (or partial

correlations) that exists between reproducible measurements, x is a input variable (it can be associated

with time (t), length of wave (), frequency (), tension of electromagnetic field (E, H), power (P)

and etc.), T is a mean period of measurements associated with variable x. In expression (1) we made a

basic assumption that the set of the constants al (which are found easily by the LLSM) are not

changed during the whole cycle of measurements. If we want to generalize expression (1) and

increase the limits of its applicability then it is necessary to consider the case, when the set of the

constants should be replaced by a set of functions <al(x)> (l = 0,1,…,L–1; L<M). If this set of the

functions would be found and the solution of the functional equation

1

0

( ) ( ) ( )L

L l l

l

F x a x F x

, (2)

for a set of the functions Fm(x) F(x + mT) could be obtained then one can apply equation (2) for

description of the quasi-reproducible (QR) experiments, when the influence of different

uncontrollable factors becomes significant and the so-called successive/reproducible measurements

during the measurement process can be differed from each other. This paper contains the justified

arguments on the questions posed above and pretends on description of the QR experiments in the

frame of expression (2) using one assumption only. We assume that the set of the functions

l la x T a x are periodical towards mean period T and in other aspects they can be

arbitrary. We should stress also that the proposed theory is self-consistent. It implies that we do not

use any a priori hypothesis and a "universal" fitting function is derived from random functions

completely associated with the measured data.

The content of the paper is organized as follows. In the second chapter we give the basics of the

general theory related to derivation of the desired fitting function and quantitative description of the

QR data. The third chapter is associated with description of the single heartbeat (the desired ECG data

are taken from the reliable open sources) and the final section is related to discussion of the obtained

results and perspectives of the further research.

2. The general theory of the QR experiment

2.1. Self-consistent solutions of the functional equation (2).

Let us consider the solution of the functional equation (2) when the length of the memory L is

supposed to be known. We assume that all successive measurements are satisfied to the functional

equation

1

0

( ) ( ) ( ), 0,1,..., 1.L

L m l l m

l

F x a x F x m M

(3)

In order to find the unknown functions <al(x)> (l = 0,1,…,L; L<M) we generalize the well-known


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LLSM and require that the functional dispersion should accept the minimal value

21

0

( ) ( ) ( ) ( ) minL

L m l l m

l

x F x a x F x

. (4)

Taking the functional derivatives with respect to unknown functions <al(x)>, we obtain

1 1

0 0

( ) 1( ) ( ) ( ) ( ) 0

( )

M L L

l m L m s s m

m sl

xF x F x a x F x

a x M L

(5)

Here we add the averaging procedure towards to all set of measurements assuming that the set of the

functions <al(x)> (l = 0,1,…,L; L<M) do not depend on index m. Introducing the definition of the pair

correlations functions

1 1

, ,

0 0

1 1( ) ( ), ( ) ( ),

, 0,1,..., 1,

M L M L

L l L m l m s l s m l m

m m

K F x F x K F x F xM L M L

s l L

(6)

we obtain the system of linear equations for the finding of unknown functions <al(x)>

1

, ,

0

( ) ( ) ( ), 0,1,..., 1,L

s l s L l

s

K x a x K x for l L

. (7)

It has a sense to define this approach as the functional least square method (FLSM), which includes

the conventional LLSM as a partial case. Let us come back to solution of the functional equation (3).

We are looking for the solution of this equation in the form

/ /

0( ) ( ) Pr( ), ( ) ( ) Pr( ).x T m x T

mF x x x F x x x

(8)

Here the functions ( ) ( ), Pr Pr( )x T x x T x

in accordance with supposition

l la x T a x made above are periodic functions and can be presented by the segment of the

Fourier series

1

0

1

( ) cos 2 sin 2K

k k

k

x xx A Ac k As k

T T

. (9)

Naturally, the decomposition coefficients Ack, Ask (k = 1,2,…,K) depend on the type of the function

(x). Inserting the trial solutions (8) into (3) we obtain the equation for calculation of the unknown

functions (x)

1

0

( ) ( ) ( ) 0L

L l

l

l

x a x x

. (10)


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If the "roots"( ), 1,2,...,q x q L

of equation (10) can be found then the general solution for the

function Fm(x) is written in the form.

/ ( / )

0

1 1

( ) ( ) Pr ( ), ( ) ( ) Pr ( ),

0,1,..., 1.

L Lx T m x T

q q m q q

q q

F x x x F x x x

m M

(11)

The set of periodic functions Prq(x) should coincide with the number of the functions that are

calculated from equation (10). So, the last relationships (4)-(11) demonstrate the general solution of

the functional equation (3). The physical interpretation of the basic equation (3) is the following. If

the successive measurements "remember" each other and can be varied during the mean period of

measurement T then a "universal" fitting function for description of these measurements can be found

self-consistently and it is derived totally by all set of measurements participating in this process. In

contrast with conventional approach we do not need a priori hypothesis and evaluation its fitting

parameters. This hypothesis can be considered as a competitive model and can be estimated alongside

with other ones in terms of the fitting parameters that are given by the IM. It is obvious that these

results generalize the previous results [10] obtained for the case when the functions <al(x)> are

replaced by the constants al. In comparison with the previous case we define these experiments as the

quasi-reproducible ones having in mind the dependence of the functions <al(x)> towards input

variable x. It would be important to receive the solutions of equation (3) when the functions <al(x)>

are not completely periodic. But, as far as we know, the theory of solutions of the functional equations

of an arbitrary order is not well-developed section of mathematics [14] in comparison with the theory

of differential equations, for example. So, general theory of experiments determines a new direction

of research for mathematicians working in the theory of the functional analysis aimed for applications

in physics, chemistry and technics.

Let us consider in detail the most probable case with minimal "memory" length L=2, when the

number of the fitting parameters entering to the IM is minimal. This case will be used for the fitting

purposes in the next section. For L = 2 we obtain

2 1 1 0( ) ( ) ( )

0,1,..., 1.

m m mF x a x F a x F

m M

(12)

Equation (7) for this case accepts the form

00 0 10 1 20

10 0 11 1 21

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

K x a x K x a x K x

K x a x K x a x K x

(13)

The solution of equation (12) is written as

/ /

0 1 1 2 2

2

1 1

1,2 0

( ) ( ) Pr ( ) ( ) Pr ( ),

( ) ( )( ) ( ) .

2 2

x T x TF x x x x x

a x a xx a x

(14)


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If one of the roots in (14) is negative (for example, 2( ) 0x ) then the general solution can be

presented approximately as

//

0 1 1 2 2( ) ( ) Pr ( ) ( ) cos Pr ( )x Tx T x

F x x x x xT

. (15)

If the order of measurements is important the proposed theory allows to restore all other

measurements in accordance with relationships

/ /

1 1 2 2( ) ( ) Pr ( ) ( ) cos Pr ( ),

0,1,..., 1.

m x T m x T

m

xF x x x x m x

T

m M

(16)

Here the periodical functions Pr1,2(xT) = Pr1,2(x) keep their periodicity but the decomposition

coefficients Ack, Ask (k=1,2,…,K) entering into decomposition (9) can be different from the initial

case) m=0 and reflect possible instability during the whole measurement process. How to define the

minimal limit of measurements (minimum of the parameter M) in order to keep the validity of this

approach? This question is important for the case when the cycling measurements are realized and we

should know the minimal number of M for realization of the first quantitative description of an

"initial" measurement. We will find the answer for the minimal memory length L=2. From equation

(12) for m=0 we have

2 1 1 0 0( ) ( ) ( )F x a x F a x F (17)

Equation (17) connects only two independent measurements (m = 0,1; M=3) but in this case the

determinant of the system (13)

2

00 11 10

22 2

0 1 0 1

( ) ( ) ( )

( ) ( ) ( ) ( ) 0,

K x K x K x

F x F x F x F x

(18)

equals zero. So, for this case the functions <a1,0(x)> are reduced to the constants a1,0 that can be found

by the LLSM. The functions <a1,0(x)> are not degenerated when M = 4. For this case the pair

correlation functions (6) accept the form 1 1

2 2

0 0

1 1( ) ( ), ( ) ( ),

2 2

, 0,1.

l m l m sl s m l m

m m

K F x F x K F x F x

s l

(19)

Equation (12) allows to find the functional dependence of each measurement (m=0,1,…,M–1) if two

of them are independent F0,1(x) and known. But the quantitative description of these measurements in

the frame of proposed theory will start from initial four measurements (M=4). This result can be

generalized for any L. One can notice that for more general case we have M – L = 2 and this simple

relationship connects the given value of L with the minimal number of measurements as M = L+2.

If the true sequence of measurements is not important and the results of the measurements remain

invariant relatively their permutations then one can combine them into three clusters and further

consideration is reduced only to analysis of these three mean measurements. This simple procedure


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allows to decrease significantly the number of the fitting parameters required for the further analysis.

This reduction procedure to three clusters is described in the next section.

2.2. The clusterization procedure and reduction to an "ideal experiment".

As it has been stressed in paper [10] the evaluation of the "true" value of L in general case represents

itself unsolved problem. If we assume that the true sequence of measurements (m = 0,1,2…, M–1) is

remained invariant relatively their permutations during the experimental cycle then one suggests the

following procedure for their clusterization. We consider the distribution of the slopes with respect to

mean measurement

1

0 1

( , ) ,

1, .

m

m m

M N

m j j

m j

y ySl slope y y

y y

y y A B A BM

(20)

The parenthesis determines the scalar product between two functions having j=1,2,…,N

measured data points. Here we suppose that the initial measurements ym(x), for m = 0,1,…, M–1,

coincide approximately with the functions Fm(x) (( ) ( )m my x F x

) figuring in the functional equation

(2). If we construct the plot Slm with respect to successive measurement m and then rearrange all

measurements in the descending order SL0>SL1>…>SLM-1, then all measurements can be divided in

three groups. The "up" group has the slopes located in the interval (1+, SL0); the mean group

(denoted by "mn") with the slopes in (1–, 1+); the down group (denoted by "dn") with the slopes in

(1–, SLM-1). The value is chosen for each set of the QR-measurements separately. This curve has a

great importance and reflects the quality of the realized successive measurements and the used

equipment. Preliminary analysis realized on many available data allows to select three different cases.

They are shown on Figs. 1(a,b,c), correspondingly. The bell-like curve (BLC) (that can be fitted with

the help of four fitting parameters α, β, A, B) is obtained after elimination of the corresponding mean

value and subsequent integration can be described by the beta-function

( ; , , , ) 1Bd m A B A m M m B

, (21)

and reflects the quality of the realized measurements. This presentation is very convenient and

contains additional information about the process of measurement that before was not taken into

account. The straight line (it can have a slope not coinciding with horizontal line) divides all

measurements in three groups: (a) the beginning point of a BLC up to the first intersection point

determines the number Nup of measurements ( ) ( )up

my x (m=1,2,…, Nup) entering in the “up” group

and is characterized by the mean Yup(x) curve; (b) the region between the two intersection points

determines the number Nmn of measurements ( ) ( )mn

my x (m=1,2,…,Nmn) in the “mn” group with

slope close to one and characterized by the set of measurements forming the mean curve Ymn(x) and,

finally, (c) the rest of the measurements Ndn in the “dn” group is covered by the curve Ydn(x). If the

number of measurements Nmn > Nup+Ndn then this cycle of measurements is characterized as

"good" (stable), in the case when NmnNdnNup the measurements (and the corresponding

equipment) are characterized as "acceptable", and the case when Nmn < Nup + Ndn is


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characterized as "bad" (very unstable). Quantitatively, all three cases can be characterized by the ratio

100% 100%Nmn Nmn

RtNup Ndn Nmn M

. (22)

In the last expression (22), M determines the total number of measurements. Based on this ratio

one can determine easily three classes of measurements: "good" when 60% <Rt< 100%, "acceptable"

when 30% <Rt< 60%, and "bad" when 0 <Rt< 30%. This preliminary analysis is supported by

Figs.1(a, b, c). So, if this clusterization will be realized then instead of equation (12) we have

approximately

2 1 1 0 0

1( )

2 0

0

1( )

1

0

1( )

0

0

( ) ( ) ( ) ( ) ( ),

1( ) ( ) ( ), 1 ,

1( ) ( ) ( ), 1 ,

1( ) ( ) ( ), 1 1 .

Nupup

m m

m

Ndndn

m M m

m

Nmnmn

m m

m

F x a x F x a x F x

F x Yup x y x Sl SLNup

F x Ydn x y x SL SlNdn

F x Ymn x y x SlNmn

. (23)

Here the function SLm determines the slopes located in the descending order and the parameter

associated with the value of the confidence interval is selected for each specific set of measurements

separately. We added these three "artificially" created measurements F2,1,0(x) to the previous set ym(x)

and in the result of this procedure the functions <a1,2(x)> do not depend on the index m and remain

almost the same in comparison with functions where this reduction procedure was not used. We

suppose also that the mean function Ymn(x) is identified with initial measurement F0(x), while two

other measurements F2,1(x) coincide with the functions Yup(x), Ydn(x), correspondingly. The solution

of equation (23) is defined by expression (15). This reduction procedure will be used for the fitting of

a single heartbeat (HB) from available ECG data, described below.

The next question that should be considered in this section is associated with reduction of the

measurements to an "ideal" experiment. In accordance with definition given in [10] under "ideal"

experiment we imply the situation

1( ) ( ) ( ) 1m mF x F x mT F x F x m T , (24)

when the results of measurements remain the same for all measurements associated with one cycle. It

means that the IM for this ideal case coincides with the segment of the Fourier-series (9). So, if it

becomes possible to extract the F-components Prq(x) (q=1,2,…,L) from the general solution (11) then

one can eliminate the influence of uncontrollable factors (associated with the functions q(x) and the

apparatus function) and "present" to theoreticians for further analysis the refined function that can be

compared with the theory pretending to description of the experimental results from the microscopic

point of view. We will demonstrate the desired expression only for the simple case (L=2) having in

mind that this situation is the most probable in practical applications and generalizations of these

expressions for any L are relatively simple.


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-10 0 10 20 30 40 50 60 70 80 90 100 110

0

1

2

3

-10 0 10 20 30 40 50 60 70 80 90 100 110

-2

0

2

4

6

8

10

12

14

16

BL

cu

rve

1 < m < 100

Bell-like curve

LineRt=71% ("good" experiment)

Nup=15

Nmn=71

Ndn=14

Slo

pe

s a

nd

SR

A

1 < m <100

slopes

SL - ordered slopes

Figure 1(a) The distribution of the slopes corresponding to "good" experiment. The most of the

measurements are located in the vicinity of the unit slope. The value of the ratio Rt = 71%. The

bell-like curve shown above explains the idea of clusterization. Other details are explained in the text.

-10 0 10 20 30 40 50 60 70 80 90 100 110

0.0

0.5

1.0

1.5

2.0

2.5

-10 0 10 20 30 40 50 60 70 80 90 100 110

-2

0

2

4

6

8

10

12

14

16

18

20

BL

cu

rve

1 < m < 100

BL curve

L ine

N up=19

N m n=51

N dn=30

R t=51% "acceptab le" experim ent

Slo

pes

and

SR

A

1 < m < 100

Sl

OdrSL

SL

Figure 1(b) The distribution of the slopes corresponding to "acceptable" experiment. All slopes are

distributed in three intervals in accordance with proportion Nup + Ndn Nmn. The value of the ratio

Rt = 51%. The intersection points obtained with the slope line give the following values of

measurements distributed over all slopes: Nup=19, Nmn=51, Ndn=30.


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-10 0 10 20 30 40 50 60 70 80 90 100 110

0.0

0.5

1.0

1.5

2.0

-10 0 10 20 30 40 50 60 70 80 90 100 110

-2

0

2

4

6

8

10

12

14

16

18

20

22

Slo

pe

s a

nd

SR

A

1 < m < 100

Slopes

SRA

Be

ll-lik

e c

urv

e

1 < m <100

BL curve

Line

Nup=29Nmn=18

Ndn=53

Rt=18%, "bad" experiment

Figure 1(c) The distribution of the slopes corresponding to "bad" experiment. All slopes are

distributed presumably in "up" and "dn" intervals in accordance with proportion Nup + Ndn > Nmn.

The value of the ratio Rt = 18%. The intersection points obtained with the slope line give the

following values of measurements distributed over all slopes: Nup=29, Nmn=18, Ndn=53.

1. L=2, the case, when 1,2(x) > 0

/ /

0 1 1 2 2

1 / 1 /

1 1 1 2 2

( ) ( ) Pr ( ) ( ) Pr ( )

( ) ( ) Pr ( ) ( ) Pr ( )

x T x T

x T x T

F x x x x x

F x x x x x

(25)

From this system of equations one can find easily the periodic function Pr(x) that presents the linear

combination of the functions Pr1,2(x)

/ 0 2 11 1

2 1

/ 1 0 12 2

2 1

1 1 2 2

( ) ( ) ( )Pr ( ) ( ) ,

( ) ( )

( ) ( ) ( )Pr ( ) ( ) ,

( ) ( )

Pr( ) Pr ( ) Pr ( ).

x T

x T

F x x F xx x

x x

F x F x xx x

x x

x w x w x

(26)

Here we introduce the unknown constants w1,2 that can be used in the final stage of comparison of the

microscopic theory with "pure" data. It is obvious also that the zeros of these functions do not define

the desired periodic functions and the case 2 1( ) ( ) 0x x should be considered separately.

2. L=2, The case, when 1 (x)>0, 2(x)<0


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//

0 1 1 2 2

1 /1 /

1 1 1 2 2

( ) ( ) Pr ( ) ( ) cos Pr ( )

( ) ( ) Pr ( ) ( ) cos Pr ( ),

x Tx T

x Tx T

xF x x x x x

T

xF x x x x x

T

(27)

The solution for this case is written in the form

/ 1 2 0

1 1

1 2

/0 1 1

2 2

1 2

1 1 2 2

( ) ( ) ( )Pr ( ) ( ) ,

( ) ( )

( ) ( ) ( )Pr ( ) ( )

( ) ( )

Pr( ) Pr ( ) Pr ( ).

x T

x T

F x x F xx x

x x

F x x F xx x

x x

x w x w x

(28)

The cases of the degenerated "roots" when the functions 1(x)2(x) coincides identically with other

and the complex-conjugated "roots" ( 1,2( ) Re ( ) Im ( )x x i x

) are not considered here

because they have low chance to be realized in real measurements.

3. The description of the algorithm and treatment of real data

The basic aim of this paper is to show the application of the general approach to the treatment of real

data related to the complex system where the "best-fit" function is not known. We took the ECG data

from the reliable Internet resource PhysioNet Bank (http://physionet.org/cgi-bin/atm/ATM) We select

randomly the ECG data from the section (ECG-ID Database(ecgiddb)) and prepared approximately

100-104 single heartbeats (HBs) associated with each randomly selected person. Each file occupies

approximately 600 data points and intensity of a heartbeat (depolarization voltage) is measured in

(mV). The basic problem is to fit the single heartbeat (HB) and express its behavior in terms of the

fitting parameters belonging to the GPS. We consider a human heart as a biologic "pump" and the

knowledge of the distribution of the amplitudes describing the individual heartbeat is important

problem for the whole cardiology. We want to show here how to solve this problem in the frame of the

proposed theory. We selected for the fitting purposes five anonymous patients only (s0020are(1),

s0022lre(2), s0026lre(3), s0039lre(4), s0043lre(5)) from ECG-ID Database(ecgiddb) keeping the

designation of the anonymous patients as it was accepted in the base. The detailed analysis of the

single HBs associated with different diseases can be a subject of the separate research (ECG-ID

Database(ecgiddb)). The proposed algorithm can be divided on three basic stages.

Stage 1. The separation of all measurements in three mean curves

We illustrate this stage by the Fig. 2 where some HBs for the first person s0020are(1) are shown.

Here we show some randomly selected HBs (HB1, HB10, HB25, HB50, HB75, HB100, <HB>)

representing all set (M=104) as the function of time in the interval [0, 600] in some arbitrary units

(a.u) and with respect to their mean function <HB>. The last plot shown above in Fig. 2 demonstrates

strong correlations between measurements. The distribution of the slopes (see equation (20)) for all

measurements m (m = 1,2,…,M) is shown on Fig. 3(a). We select approximately the value of = 1/3

and divide the span between the ordered measurements (the curve SL) in three parts: "up" with

(1+up, max(SL)), "mn" with (1–dn, 1+up), "dn" with (min(SL),1 –dn,), where up =

[max(SL)-1]/3and dn = [1-min(SL)]/3. The number of measurements that is located in each selected

http://physionet.org/cgi-bin/atm/ATM


~ 19 ~

interval are: Nup = 25, Nmn = 63, Ndn = 16. After subtraction of the unit value and the subsequent

integration, we obtain the bell-like curve shown above in Fig.3(b). The quality of measurements

calculated in accordance with expression (13) equals Rt(Nmn=63)/(M=104) =60.58%. This value of

Rt> 60% can be interpreted as a relatively stable work of the human heart. The most beats are located

in the vicinity of the slopes equaled one. Clusterization realized with the help of expression (23) helps

to receive only three mean curves Yup, Ydn, Ymn shown in Fig.4. These averaged curves can be added

to the previous set of measurements. The realization of the procedure described by expression (6)

shows that the values of the averaged constants <a0,1(t)> are remained practically unchanged. These

functions together with the functional "roots" 1,2(t) (see expression (14)) are shown in Fig.5.

0 100 200 300 400 500 600

-500

0

500

1000

1500

-500 0 500 1000 1500

-500

0

500

1000

1500

Heart

Beats

tow

ard

s m

ean v

alu

e

<HB>

HB1

HB10

HB25

HB50

HB75

HB100

He

art

Be

ats

(in

mV

)

t (arbitrary units)

HB1

HB10

HB25

HB50

HB75

HB100

<HB>

Figure 2. The typical behavior of the single HBs for the person s0020are. We selected for analysis of

M=104 HBs and here we demonstrate 6 randomly taken HBs. The mean HB is bolded by dark grey

line. On the figure shown above one can notice that all signals occupy a band and can be presented

approximately by a set of straight lines.

Stage 2. Reduction to three incident points

This procedure is very important for justifying the scaling properties of the curves that are

subjected to the fitting procedure. The general solution (11) depends on the ratio x/T and so it remains

invariant after the scaling transformation

/ '

/ '

x x x

T T T

, (29)

where is an arbitrary scaling parameter. So, this transformation helps to decrease the number of the

modes figuring in the GPS keeping the same information in the shorten/scaled data.


~ 20 ~

-10 0 10 20 30 40 50 60 70 80 90 100 110

0.90

0.95

1.00

1.05

1.10

Slo

pe

s a

nd

SR

A

1 < m < 104

Slopes

SRA(SL)

SL

1+up

=1.02589

1-dn

=0.96741

Figure 3(a). This figure demonstrates the distribution of the slopes for 104 HBs. The horizontal lines

show the limits between three clusters. The values of these lines are calculated in accordance with

procedure described in the text.

-10 0 10 20 30 40 50 60 70 80 90 100 110

0.0

0.5

1.0

1.5

Bell-like curve

The fit to beta-function

B

1< m < 104

Nup=25

Nmn=63

Ndn=16

Rt=60.58% - "good" HBs

Figure 3(b). The bell-like curve demonstrates clearly the number of measurements that enter in each

cluster and are used for calculations of the three mean curves Yup(Nup=25), Ymn(Nmn=63) and

Ydn(Ndn=16). In accordance with criterion described in the text the most HBs are located in the band

(1 – dn,1 + up) and the work of the "human pump" is characterized as "good". This new source of

information will be used as an additional information in cardiology. The solid line shows the fit of this

curve to the beta-distribution function (21). The necessary fitting parameters are collected in Table 2.


~ 21 ~

0 100 200 300 400 500 600

-500

0

500

1000

1500

-500 0 500 1000 1500

-500

0

500

1000

1500

mean H

Bs

<HB>

HBup

HBdn

HBmn

HB

up

, H

Bm

n a

nd

HB

dn

(m

V)

time in (a.e.)

HBup

HBdn

HBmn

Figure 4. This figure demonstrates three mean HBs obtained after clusterization procedure. In order

to compare these curves with mean <HB> we place another plot shown above. This plot shows that

mean curves are close to each other. All specific extreme points defined conventionally as PQRST are

clearly seen.

0 100 200 300 400 500 600

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

0 100 200 300 400 500 600

-0.5

0.0

0.5

1.0

Th

e c

alc

ula

ted

ro

ots

1,2(t

)

t (a.u.)

x)

2(x)

Th

e b

eh

avio

r o

f th

e c

on

sts

nts

<a

0,1(x

)>, x=

t.

time (a.u)

<a0(t)>

<a1(t)>

Figure 5. This figure shows the behavior of the functions <a0,1(t)> and the "roots" 1,2(t) shown

above. As one can notice from this figure the function 1(t) > 0, while 2(t) < 0.


~ 22 ~

0 100 200 300 400 500 600

-500

0

500

1000

1500

0 100 200 300 400 500 600

-500

0

500

1000

1500

Ym

n

t (a.u)

Ymn

Re

du

ce

d y

up

, ym

n,y

dn

tmn

yup

ydn

ymn

P

Q

R

S

T

Figure 6. The result of application of the reduction procedure to three incident points. On the small

figure above we show the curve Ymn(t). On the central figure we show three reduced curves that

occupy the length R = 100. From analysis of these curves one concludes that the fit of the curve

ymn(tmn) is sufficient, because other two curves yup(tmn) and ydn(tmn) are similar to the central one.

0 100 200 300 400 500 600

-0.5

0.0

0.5

1.0

1(t)

2(t)

rsm1(t)

rsm2(t)

Th

e r

oo

ts

1,2(t

) a

nd

the

ir r

ep

lica

s rs

m1

,2(t

)

t(a.u)

Figure 7(a). Here we show the smoothed functions (shown by bold lines) that can be reduced to three

incident points (we keep the same value b = 6).


~ 23 ~

0 100 200 300 400 500 600

-0.5

0.0

0.5

1.0T

he

re

du

ced

ro

ots

r1 a

nd

r2

tmn

r1

r2

Figure 7(b). Here we demonstrate the self-similar properties of the reduced roots r1,2 (tmn) that are

obtained in the result of reduction procedure to three incident points. Compare this figure with the

previous figure 7(a).

0 100 200 300 400 500 600

-500

0

500

1000

1500

y(x)

Fit_y(x)

y(x)

x=tmn

P

Q

R

S

T

Figure 8. Here we demonstrate the final fit of the single HB (mean reduced function) realized with

the help of the fitting function (30). The value of the fitting error is less than 1.7%. The distribution of

the amplitudes is shown below. Additional fitting parameters are collected in Table 1.


~ 24 ~

Table 1. Additional quantitative parameters that enter into the fitting function (30)

Person Period T ln(mean(r1)) ln(mean(r2)) A0 Range(Ampl) RelErr(%) K

s0020are 408 -0.28498 -0.83963 3.2843 708.651 1.66194 18

s0022lre 480 -0.06952 -0.86644 82.7063 788.482 2.02615 18

s0026lre 462 -0.20376 -0.91514 11.5059 349.228 4.10437 18

s0039lre 426 -0.1709 -0.8792 273.108 2519.02 4.50235 18

s0043lre 462 -0.1709 -0.8792 -212.449 1851.55 9.12826 18

The value of the sixth column is determined as Range (f) = max(f) – min(f) and shows the range of all

amplitudes that enter into the final function (30).

Table 2. The fitting parameters that enter into the beta-distribution function (21).

Person A B xmx, ymx As(%) RelErr(%)

s0020are 0.00302 0.76799 1.55191 6.46622E-4 52; 1.3897 -0.48544 0.09681

s0022lre 7.31894E-4 0.94533 1.90262 0.00462 51; 1.3004 -0.49505 0.07439

s0026lre 0.00151 1.16655 1.9679 0.03408 61; 3.9877 7.69231 0.21553

s0039lre 0.00175 0.85788 1.66406 -0.0054 56; 1.223 3.92157 0.04207

s0043lre 0.00114 0.8368 1.70635 0.00318 52; 0.9138 1.51515 0.17451

In Table 2 the seventh column determines the quantitative measure of the horizontal asymmetry:

0

0

2 100%

Nmx

N

x xx

Asx x

.

This parameter characterizes the asymmetry of the beta-distribution towards to the central point. The

parameters x0 = 0 and xN = M-1 define the limiting points of the distribution.

This procedure was successfully applied to many random functions [15] proving their

self-similar (fractal) properties. We choose s = 1,2,…,b=6 points (Y1,Y2,…,Yb) and reduce them to

three incident points (max(Y), mean(Y), min(Y)) that are invariant relatively to all permutations inside

the chosen b points. Having in mind the total number of data points N=600 and the length of a small

"cloud" of points b = 6, we obtain the reduced number R of data points calculated as the integer part of

the ratio [N/b] (R=100), by keeping the form of the initial curve almost unchanged relatively to this

transformation. As a new variable t we take the value of tmn averaged over b points in each chosen

interval. The result of the reduction procedure applied to the curve Ymn(tmn) is shown on Fig.6.

Because of the strong correlations two other curves Yup(tmn) and Ydn(tmn) are very similar to

Ymn(tmn) and are not considered. If one compares the central curve Ymn(t) depicted on Fig.6 above

with its reduced replicas ymn(tmn), yup(tmn) and ydn(tmn) depicted on the central plot of the same

Fig.6, then one can notice that they are similar to each other. Other two reduced curves yup(tmn) and

ydn(tmn) are not considered because this reduction (b=6) makes them practically identical to the


~ 25 ~

curve ymn(tmn). The calculations of the functions 1,2(t) by the formulae (14) are shown in Fig.5.

Direct application of the reduction procedure to these functions is impossible because the HF

fluctuations destroy completely the self-similar property [16]. In order to restore this property and

then apply the reduction procedure again, we should smooth preliminary the functions 1,2(t) with the

value of the correlation between initial and smoothed curves equaled 0.9. They are shown on Fig. 7(a)

by bold lines. These smoothed functions can be reduced again and after reduction we obtain the

reduced functions r1,2(x) from the smoothed roots. These functions are shown on Fig.7(b). For

comparison we placed the original smoothed roots on the small figure above.

Stage 3.The fitting of the mean reduced function

The previous stages have a preparatory character. The basic result will be obtained when we fit

the reduced function y(x) (y = Ymn, x = tmn) to function (15). For convenience we present this

function in the form

(1) (1) (1) (1) (1) (2) (2) (2) (2)

0 0

1 1

//

0 1 2

/ /(1) (1)

1 1

( ) ( ; , ) ( ) ( ) ( ) ,

( ) ( ) ( ) cos ,

( ) cos 2 , ( ) sin 2 ,

xx

x x

K K

x k k k k k k k k

k k

x Tx T

x T x T

k k

x x

y x F x K T A E x Ac Ec x As Es Ac Ec x As Es

xE x r x r x

T

x xEc r x k Es r x k

T T

/ /(2) (2)

2 2( ) cos cos 2 , ( ) cos sin 2 .x xx T x T

k k

x x x x

x x x xEc r x k Es r x k

T T T T

(30)

Here the known functions r1,2(x) should be associated with the reduced values of the smoothed roots

1,2(t). The functions (2) (2)

0( ), ( ), ( )k kE x Ec x Es xtake into account the fact that the root r2(x) is

negative. The function F(x; K, Tx) contains only two nonlinear fitting parameters that can be found

from the minimization of the relative error surface

( ) ( ; ,

min 100%( )

xstdev y x F x K TRelError

mean y x

, (31)

that are given by (K, Tx). Usually, the mean period Tx is not known and lies in the interval (0.5Tin < T

< 2Tin), Tin = (x1 – x0)length(x). The minimal value of the final mode K is found from the condition

that the level of the relative error should be located in the acceptable interval (1% – 10%). After

minimization of the value (31), the desired amplitudes (1,2) (1,2)

0 , ( ), ( )k kA Ac x As x are found by the

LLSM. The result of the fit of expression (30) to the reduced function y(x) is shown in Fig. 8. The

quality of the fitting curve (30) is rather high because the total number of the amplitudes 4K = 72 is

comparable with number of the reduced points R=100. The total distribution of amplitudes is shown

by the figure 9(a). Actually, this distribution together with other fitting parameters (shown in the

Table 1) represents itself the desired IM expressed in terms of the GPS. We should stress also the

importance of the BLC (Fig. 9(b)) that serves as a useful tool for analysis of spectrograms containing


~ 26 ~

large number of the discrete amplitudes (>100). The separated distributions of the amplitudes

(1,2) (1,2)( ), ( )k kAc x As x are shown in Fig. 10.

-10 0 10 20 30 40 50 60 70 80

-400

-300

-200

-100

0

100

200

300

400A

mpl

itude

s -

72

1< k < 72

Amp_tot

Odr_Amp

Figure 9 (a). This plot shows the total distribution of the amplitudes figuring in the fitting function

(30). This presentation is convenient when the number of amplitudes is sufficiently large (in our case

K = 72). Actually, these amplitudes form a specific "piano" and each mode shows the intensity of each

"key".

-10 0 10 20 30 40 50 60 70 80

0

1000

2000

3000

4000

Total distribution of amplitudes presented in the form of BLC

The fit of the BLC to beta-distribution function

BLC

and

its

fit to

bet

a-di

strib

utio

n

0 < k < 71

A=10.653, =0.861

=1.656, B=5.521

RelErr=0.08%

Figure 9(b). For large number of amplitudes one can suggest another convenient presentation

expressed in the form of the BLC. This curve can be fitted by expression (21) and the parameters of

this distribution can characterize the GPS at whole. For the case considered (person 00s20are) these

fitting parameters are placed inside the figure.


~ 27 ~

0

-400

-300

-200

-100

0

100

200

300D

istr

ibu

tion

of a

mp

litu

de

s A

c 1,2(k

) a

nd

As 1

,2(k

)

Ac1(k)

As1(k)

Ac2(k)

As2(k)

Distribution of amplitudes for the person(1) s0020are

K=18

Figure 10. This is conventional presentation of the amplitudes figuring in the fitting function (30).

The analysis of these amplitudes can be realized with traditional methods accepted in any

spectroscopy.

In the same manner, we can treat other HBs that were recorded for other persons and collected in

the same site. Three stages of the proposed algorithm are remained the same and so we show only the

final results: (1) the statistical variability of the HBs (as it was presented by the bell-like curve in Fig.

3(b)) then (2) the fitting of the reduced HBs (similar to Fig. 8) and, finally, (3) the distributions of

amplitudes that figure in expression (30). Other additional parameters associated with the GPS are

collected in Table 1 shown above. Figures 11(a,b,c) show the desired BLC and their fit. This new

source of information signifies about the stability and statistics of the treated HBs forming

approximately by the samplings equaled M=100. The heights of these distributions indicate to the

character of deviations from the slope equaled one. The larger heights of the BLCs correspond to the

stronger deviations of the slopes from the unit value. Figures 12(a,b) demonstrate the fit of the

functions defined above as y(x). Combined together they can give a new source of information

signifying about the detailed behavior of the individual HBs for each tested person. The figures

13(a,b,c,d) demonstrate the final result of the whole procedure: the distribution of the amplitudes

forming each mean HB. The analysis of the GPS of these signals can contain new source of

information in cardiology if one starts to compare these new parameters with conventional approach

associated with diagnosis of different cardio-diseases.


~ 28 ~

-10 0 10 20 30 40 50 60 70 80 90 100 110

0.0

0.5

1.0

1.5

BLC_s0022lre

Fit to BLC

Line

BLC

and

its

fit to

bet

a-di

strib

utio

n

0 < m <103

Nup=30Nmn=47

Ndn=26

Rt=46% - "acceptable" HBs

Figure 11(a). The distribution of 104 HBs for the person (2) s0022lre. This distribution can be

characterized as "acceptable". Only 46% of beats are located in the mean interval (1-dn, 1+up). The

blue line corresponds to the fitting curve (21). All fitting parameters are collected in Table 2 given

above.

-10 0 10 20 30 40 50 60 70 80 90 100 110

0

2

4

BLC_s0026lre

Fit_BLC

Line

BLC_s0039lre

Fit_BLC

Line

Be

ll-lik

e c

urv

es

an

d th

eir

fit

0 < m< M=104

Nup=42 Nmn=39 Ndn=25

Rt=37.14% "acceptable" HBs

Nup=23

Nmn=62Ndn=20

Rt=60.2% "good" HBs

Figure 11(b). The comparison of distributions corresponding to the persons s0026lre and s0039lre.

As one can notice from this figure the range of the BLC will be more higher for "bad" and

"acceptable" HBs. If the deviations from the unit slope will be small then the height of the

beta-distribution will be less.


~ 29 ~

-10 0 10 20 30 40 50 60 70 80 90 100 110

0.0

0.5

1.0

BLC_s0043lre

Fit to Beta-distr

Line

Bel

l-lik

e cu

rve

and

its fi

t for

per

son

s004

3lre

0 < m < 100

Ndn=4

Nup=31

Nmn=65

Rt=65, "good" HBs

Figure 11(c). These three Figs.11 (a,b,c) contains new source of information related to the

distribution of the HBs. We chose approximately 100 beats in order to describe them quantitatively in

terms of the fitting parameters of expression (21). The heights of these distributions correlate with

stability of the HB sequence. The strong deviations from the unit value correspond to unstable beats,

while the small heights of these BLCs correspond to the sequence of stable heartbeats.

0 100 200 300 400 500 600

-1000

-500

0

500

1000

1500

y(x)_s0022lre

Fit_y(x)_s0022lre

y(x)_s0026lre

Fit_y(x)_s0026lre

Fit

the

re

du

ced

HB

s: y

(x)

x=tmn

Figure 12(a). Here we compare the fit of the single HBs (the reduced and mean curve y(x)) for two

persons s0022lre and s0026lre, correspondingly. So, in the case of application of this theory the

modern cardiology will receive a chance to compare the single HBs in terms of the fitting parameters

expressed by the GPS.


~ 30 ~

0 100 200 300 400 500 600

-800

-600

-400

-200

0

200

400

600

800

1000

1200

1400

y(x)_s0039lre

Fit y(x)_s0039lre

y(x)_s0043lre

Fit y(x)_s0043lreF

it th

e r

ed

uce

d H

B-

the

fu

nct

ion

s y(

x)

x=tmn

Figure 12(b). This figure demonstrates the comparison of two other persons s0039lre and s0043lre,

accordingly. The quantitative comparison and further analysis of the single HBs opens new source of

information in cardiology.

0

-500

-400

-300

-200

-100

0

100

200

300

Dis

trib

utio

n o

f a

mp

litu

de

s A

c1(k

), A

s 1(k

)

Ac 2

(k)

an

d A

s 2(k

) fo

r s0

02

2lr

e

Ac1(k)

As1(k)

Ac2(k)

As2(k)

Distribution of amplitudes for the person (2) s0022lre

K=18

Figure 13(a). Here we demonstrate the distribution of the amplitudes for the HBs recorded from the

person s0022lre. The fit of the single HB is shown by cyan line in Fig. 12(a).


~ 31 ~

0

-100

-50

0

50

100

150

200

250

Dis

trib

utio

n o

f a

mp

litu

de

s A

c1

,2(k

)

an

d A

s 1,2(k

)

Ac1(k)

As1(k)

Ac2(k)

As2(k)

K=18

Distribution of amplitudes for the person (3) s0026lre

Figure 13(b). Here we demonstrate the distribution of the amplitudes for the HBs recorded from the

person s0026lre. The fit of the single HB is shown by red line in Fig. 12(a).

0

-1400

-1200

-1000

-800

-600

-400

-200

0

200

400

600

800

1000

1200

Dis

trib

utio

n o

f th

e a

mp

litu

de

s A

c1

,2(k

) a

nd

As

1,2(k

)

Ac1(k)

As1(k)

Ac2(k)

AS2(k)

Distribution of the amplitudes for the person s0039lre

K=18

Figure 13(c). The distribution of the amplitudes for the HBs recorded for the person s0039lre. The fit

of the single HB corresponding to this distribution is shown by blue line in Fig. 12(b).


~ 32 ~

0

-1000

-800

-600

-400

-200

0

200

400

600

800

1000

Dis

trib

utio

n o

f th

e a

mp

litu

de

s A

c1

,2(k

)

an

d A

s 1,2(k

)

Ac1(k)

As1(k)

Ac2(k)

As2(k)

K=18

Distribution of the amplitudes describing the single HB

for the person s0043lre

Figure 13(d). The distribution of the amplitudes for the HBs recorded from the person s0043lre. The

fit of the single HB corresponding to this distribution is shown by the rose line in Fig. 12(b). The

comparisons of these three similar figures give us a specific distribution of "sounds" that form a

universal "piano" and each key reflects the contribution of the mode forming the HB signal. This new

source of information reflecting the behavior of the HBs is needed in further analysis. This type

analysis is out of the scope of this paper because it needs some specific cardiologic knowledge in

order to fit this information for diagnostic purposes.

4. The results and discussion

In this paper we tried to collect the justified arguments associated with existence of the IM. The

proposed self-consistent theory based on only one assumption related to l la x T a x

enables to suggest the IM associated quantitatively with the fitting parameters of the GPS. From

mathematical point of view it implies the following transformation

1,2 1,2( ) ( ), Pr ( )S t GPS r t t . (32)

The reduced functions r1,2(t) reflect the influence of the uncontrollable factors, while the periodic

functions Pr1,2(t) can be associated with an "ideal" experiment. The possible elimination of these

uncontrollable factors for the case of short memory (L=2) are shown by expressions (26) and (28).

We should stress also that this IM is "universal" and a lot of experiments can be expressed in terms of

the quantitative parameters belonging to the GPS. This is only the beginning of a new direction in the

conventional theory of measurements and definitely new findings are waiting of a potential

researcher moving by this way. The selected example associated with the fit of the single HBs

confirms the effectiveness of this new unification scheme.

We want to stress here new possible applications that can be extracted from the proposed theory.


~ 33 ~

1. Calibration of different equipment based on the pattern device and creation of the universal

metrological standard based on quantitative parameters associated with the GPS.

2. The creation of the fully computerized mini-laboratories when the final measurement results are

expressed in the form of the functions related to the universal IM. Any results obtained in the frame of

this mini-laboratory can be accessible for any interested researcher through the Internet.

3. The proposed theory enables to transform any quality to a quantity by means of the specially tuned

measurements. Then these measurements can be presented by a set of the fitting parameters

characterizing the segment of the GPS. The comparison of the tested product with the pattern one

(associated with the quality of the produced product) can be realized with the help of the

"traffic-light" principle: all acceptable products will pass on the "green" light, while the abandoned

product marked by "red light" can be a subject of analysis of the qualified personnel.

Finishing this final section we should stress the principal difference between the proposed theory and

general approach accepted as the basic trend in the fitting of different random signals. In the

conventional approach additional information which is necessary for the fitting of the random signal

is contained in a priori verified hypothesis. All traditional methods are aimed for estimation of the

fitting parameters that are contained in the proposed hypothesis with the compared random signal. In

the proposed theory additional information is contained in the sampling of the random measurements

and the fitting function is derived from the given sampling without usage of an additional hypothesis.

From our point of view, it will give a unique possibility for more reliable comparison of the different

models/fitting functions with each other.

Acknowledgments: The authors(1,2,3)

want to express their thanks for the support of academic

exchanges from “High-end Experts Recruitment Program” of Guangdong province, China. This work

was supported in part by the NSF project of China (No. 61302131), project of International S&T

Cooperation Program of China (ISTCP) (No. 2015DFI12970), projects of Guangdong Science and

Technology Program (No.2015B010105012, 2014B050505011, 2013B010136002,

2015B020214004, 2014A050503046, 2015B020233010), and project of Guangzhou Science and

Technology Program(No. 201508020083).

References

[1]. Rabiner L.R. and Gold B., (1975), "Theory and application of digital signal processing".

Englewood Cliffs, NJ, Prentice-Hall, Inc.,.

[2]. Singleton Jr, Royce A.,, Straits B.C., and Straits M. M., (1993), "Approaches to social research".

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