Parameter Estimations of Normal Distribution via Genetic Algorithm

and Its Application to Carbonation Depth

SOMCHIT BOONTHIEM1, CHATCHAI SUTIKASANA2,

WATCHARIN KLONGDEE3, WEENAKORN IEOSANURAK3,∗

1Mathematics and Statistics Program, Sakon Nakhon Rajabhat University, Sakon Nakhon, THAILAND

2Logistics Department, Faculty of Business Administration and Information Technology,

Rajamangala University of Technology Isan Khonkaen Campus, THAILAND

3Department of Mathematics, Faculty of Science, Khon Kaen University, THAILAND

*Corresponding author

Abstract: - In this paper, we propose a method for estimating Normal distribution parameters using genetic al-

gorithm. The main purpose of this research is to identify the most efficient estimators among three estimators

for Normal distribution; Maximum likelihood method (ML), the least square method (LS), and genetic algorithm

(GA) via numerical simulation and three real data, carbonation depth of Concrete Girder Bridges data examples

which are based on performance measures such as The Root Mean Square Error (RMSE), Kolmogorov-Smirnov

test, and Chi squared test. The simulation studies are conducted to evaluate the performances of the proposed

estimators and provide statistical analysis of the real data set. The numerical results, χ2, show that the genetic

algorithm performs better than other methods for actual data and simulated data unless the sample size is small.

Key-Words: - Normal distribution, Parameter estimation, Maximum likelihood method, Genetic algorithm, The

Least square method, Carbonation depth

Received: July 11, 2022. Revised: January 12, 2023. Accepted: February 8, 2023. Published: March 2, 2023.

1 Introduction

The Normal distribution or Gaussian distribution

plays an important role in several fields of mathemat-

ics and its applications. The Normal distribution has

been widely used to describe the probability distri-

bution of carbonation depth, [1],[2],[3],[4]. The car-

bonation depth is a point deterioration factor to de-

termine the durability of the concrete structures. The

characterization of carbonation depth is essential for

the carbonation reliability analysis of Concrete Girder

Bridges. Carbonation depth is usually employed in

carbonation service life prediction of existing Con-

crete Girder Bridges as deterministic coefficients.

Several estimations, [5],[6],[7] for estimating pa-

rameters have been proposed. The authors in [7], us-

ing the Markov Chain Monte Carlo (MCMC) method

for estimating parameters. Li, Yan, Wang and Hou,

[6], proposed two parameter estimations for the Nor-

mal distribution: the least square method and the

Bayesian quantile method. They found that the least

square method is the best parameter estimation. Ge-

netic algorithm was first introduced by Holland, [8],

in 1992 and represented a population-based optimiza-

tion method. This algorithm is a method to find an ap-

proximate solution for optimization problems which

is used widely in several fields, [9],[10]. In parame-

ter estimation, some researchers, [11],[12],[13], stud-

ied a method to find parameters by using genetic al-

gorithms. The authors in [11], studied genetic algo-

rithms and proposed a new genetic algorithm. They

found that genetic algorithms are effective in perfor-

mance indicators improvement. The authors in [12],

used genetic algorithm (GA) to find estimators of

Skew Normal distribution. They found that the GA

has a high performance where traditional search tech-

niques fail. In this paper, we study the genetic al-

gorithm, a well-known search technique. The GA is

inspired by a metaphor of the evolution process ob-

served in nature.

The main purpose of this research is to identify the

most efficient estimators among three estimators for

the Normal distribution via actual data and simulated

data. The rest of this paper is organized as follows:

Section 2 discusses a short introduction of the Nor-

mal distribution, followed by the Normal distribution

parameter estimation in Section 3. Accuracy judg-

ment criteria are considered in Section 4. The per-

formances of all methods are compared via a detailed

simulation study in Section 5. Three parameter esti-

mations are applied to three real data sets of Carbon-

ation Depth of Concrete girder bridges, in Section 6.

Finally, the main conclusions of this study are sum-

marized in the last section.

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2 Normal Distribution

The Normal distribution is also called Gaussian dis-

tribution.

A random variable xhas a Normal distribution if

its probability density function is defined by Equation

(1).

The probability density function (pdf) of the Nor-

mal distribution can be written as

f(x) = 1

σ√2πe−1

2(x−µ

σ)2

,(1)

where µand σare the location parameter and the scale

parameter, respectively.

The cumulative distribution function (CDF) of the

Normal distribution is given by

F(x) = Zx

−∞

σ√2πe−1

2(t−µ

σ)2

dt, (2)

F(x) = Φ x−µ

σ,

where Φ(x) = 1

√2πZx

−∞

e−t2

2dt.

3 Estimation Methods

In this section, the three considered estimation meth-

ods were described to obtain the estimates of the pa-

rameters µand σof the Normal distribution.

3.1 Maximum Likelihood Method

Let x1, x2, . . . , xnbe observed values of

X1, X2, . . . , Xn,nindependent random variables

having the Normal distribution with parameters µ

and σ.

The maximum likelihood function of the sample,

denote by L(µ, σ|x1, x2, . . . , xn), is given by

L(µ, σ|x1, x2, . . . , xn) =

i=1

f(xi)

i=1

σ√2πe−1

2(xi−µ

σ)2

by taking ln, we get

ln L(µ, σ|x1, x2, . . . , xn)

=−n

2ln(2πσ2)−1

2σ2

i=1

(xi−µ)2.

(3)

To obtain the maximum likelihood estimators, we

have to maximize Equation (3), partial derivatives

of ln L(µ, σ|x1, x2, . . . , xn)functions with respect to

each parameter are taken, and we set them equal to 0

as follows:

∂

∂µ ln L(µ, σ|x1, x2, . . . , xn) = 0,

∂

∂σ ln L(µ, σ|x1, x2, . . . , xn) = 0.

The maximum likelihood estimators of µand σ, de-

noted by ˆµML and ˆσML, respectively, are obtained by

ˆµML =1

i=1

xi,

ˆσML =v

n−1

i=1

(xi−¯x)2,

where ¯xis the sample mean.

3.2 Least Square Method

Let X1, X2, . . . , Xnbe nindependent random vari-

ables having the Normal distribution with parameters

µand σ. Suppose that X(1) ≤X(2) ≤. . . ≤X(n)

are the order statistics. Let the empirical distribution

function of Xbe denoted by

Fn(x) =











0, x < X(1),

n, X(k)≤x < X(k+1), k = 1,2,...,n−1

1, x > X(n).

(4)

The following cumulative distribution function F(x)

is calculated by

F(x) = 1

σ√2πZx

−∞

e−1

2(t−µ

σ)2

dt. (5)

The CDF of Normal F(x)can be expanded in Taylor

series approximation as follow:

F(x) = Φ x−µ

σ,(6)

where

Φ(x)≈1

2+1

√2π

k=0

(−1)kx2k+1

2kk!(2k+ 1).

For the Normal distribution, the least square method

estimates ˆµand ˆσof the parameters µand σ, respec-

tively, are obtained by minimizing the function:

E(µ, σ) =

i=1

(F(xi)−Fn(xi))2,(7)

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where F(xi)and Fn(xi)are obtained by Equation (4)

and (5), respectively. By solving the following equa-

tions:

∂

∂µ E(µ, σ) = 0,

∂

∂σ E(µ, σ) = 0.

We denote by

A(xi, µ, σ) =

k=0

(−1)kxi−µ

σ2k

2kk!

and

B(xi, µ, σ) =

k=0

(−1)k+1 xi−µ

σ2k

2kk!(2k+ 1) ,

ˆµ, ˆσare the estimators of the parameters µand σ, re-

spectively. Using some algebraic manipulations, the

estimators satisfy the following equations:

ˆσ=Pn

i=1 Fn(xi)A(xi, µ, σ)

i=1 F(xi)1

σA(xi, µ, σ)

and

ˆµ=Pn

i=1(F(xi)−Fn(xi))xiA(xi, µ, σ)

i=1(F(xi)−Fn(xi))A(xi, µ, σ).

These equations are solved iteratively.

3.3 Genetic Algorithm

The main steps of the genetic algorithm (GA) are se-

lection, crossover, and mutation. In GA, each chro-

mosome (individual in the population, parameters)

represents a possible solution to a problem and is

composed of a string of genes. Kalra and Singh [14]

proposed a pseudo code of GA for optimization of

scheduling problems as follows:

Procedure GA

Determine the number of chromosomes generation

(Npop = 10000), and mutation rate (MR =

0.5).The number of chromosomes is 2 (µand σ).

1. Initialization: Generating initial population P

consisting of N= 100 chromosomes. Every

gene represents a parameter (variables) in the so-

lution. This collection of parameters that forms

the solution is the chromosome. Therefore, the

population is a collection of chromosomes. As-

sume that the initial population P(1) is denoted

P(1) = [w(1)

1, w(1)

2, . . . , w(1)

where w(1)

i= [µ(1), σ(1)]tis a vector of param-

eters for i= 1,2, . . . , N. Also, the vector of

w(m)

i, i = 1,2, . . . , N, m = 1,2, . . . , N pop

represents the values of the ith chromosome in the

population at mth iteration.

2. Fitness: Calculate the fitness value of each chro-

mosome using a fitness function. In this study,

the fitness values are defined by:

f(m)

i=1

i, m = 1,2, . . . , Npop

where f(m)

irepresents the fitness value of the ith

chromosome at mth iteration.

3. Selection: Select the chromosomes for produc-

ing the next generation using the selection op-

erator, the worst chromosomes are replaced by

new chromosomes generated randomly from the

search space. We used the roulette wheel selec-

tion. The probability of choosing chromosome i

is equal to

pi=f(m)

i=1 f(m)

where f(m)

iis the fitness value of the ith chromo-

some in the population at mth iteration.

4. Crossover: Perform the crossover operation on

the pair of chromosomes obtained in step 3.

5. Mutation: Perform the mutation operation on

the chromosomes. The chromosome kfor k=

1,2are mutated as follows:

random uk∈(0,1), if uk< M R, then we mu-

tated the chromosome i.

6. Replacement: Update the population P(m)for

m= 2,3, . . . by replacing bad solutions with bet-

ter chromosomes from offspring.

7. Repeat steps 3to 6until the stopping condition

is met. The stopping condition may be the max-

imum number of iterations or no change in the

fitness value of chromosomes for consecutive it-

erations.

8. Output the best chromosome (the best parame-

ter) as the final solution.

End Procedure

3.4 Goodness-of-Fit

To show how a theoretical probability function

matches with the observation data, three kinds of sta-

tistical errors are considered as the Goodness-of-fit.

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Generally, the smaller the errors, the better the fit is.

Let nbe the number of data and kbe the number of

classes, calculated by Sturges formula,

k=⌈1+3.322 log n⌉.

The First one is the root mean square error (RMSE)

defined as

RMSE =v

i=1

(Oi−Ei)2,(8)

where Oiis the actual value at time stage i, and Ei

is the value computed from correlation expression for

the same stage.

The second one is the Kolmogorov-Smirnov test

(KS), which is defined as the max error in CDFs

KS =max

x|Fn(x)−F(x)|,(9)

where Fn(x)is the empirical cumulative distribution

function not exceeding xand F(x)is the CDF of Nor-

mal distribution.

The third judgment criterion is the Chi-squared test

given as:

χ2=

i=1

(Oi−Ei)2

.(10)

4 A Simulation Study

In this section, a simulation study was performed to

compare the performance of the different methods

discussed in Section 3. 2000 random samples of sizes

n= 10,20,30,50,100,500,1000,and 2000 were

generated from the Normal distribution. Since any

Normal distribution data can be standardized to have

a location parameter of 0and scale parameter of 1,

only samples with parameters µ= 0 and σ= 1 were

generated. In order to compare the goodness-of-fit

of various pdfs to sample data, several statistics have

been used in studies related to data.

The RMSE is most useful when large errors are

particularly undesirable. The Kolmogorov–Smirnov

test has the advantage of considering the distribution

functions collectively. Advantages of the Chi-square

test include its robustness in data distribution, and

ease of calculation.

The most frequently used ones are the root mean

square error (RMSE), [15],[16], the Kolmogorov–

Smirnov test results (KS), [15],[17], and the Chi

squared test results (χ2), [15].

The results of the simulation study are presented in

Table 1-2. The following conclusions can be drawn:

1. All estimators of the parameters are unbiased, i.e.,

the estimators sometimes exceed the true value of the

nParameter Estimation

ML LS GA

10 µ0.5180 8.1768 0.6112

σ0.8437 0.3664 0.8263

RMSE 0.7920 1.0227 0.7729

KS test 0.9939 4.7753 1.2699

χ24.9517 16.5301 4.7405

20 µ0.2101 -7.6186 0.2315

σ0.9042 5.0261 0.9481

RMSE 0.7993 1.3470 0.7919

KS test 0.7447 8.5721 1.0788

χ23.7719 19.3985 3.7027

30 µ0.1314 0.1274 0.1330

σ0.8238 1.1688 0.9273

RMSE 1.4151 1.5568 1.4122

KS test 2.0833 3.7360 2.1236

χ213.7865 15.2259 12.9437

50 µ0.0932 4.7193 0.0890

σ0.9458 8.1408 1.0836

RMSE 1.2694 2.4223 1.4442

KS test 1.7180 15.7167 3.2420

χ213.6696 25.6064 9.7582

100 µ-0.0520 -2.8979 -0.0821

σ0.9920 3.3682 1.0383

RMSE 1.6756 3.7792 1.6742

KS test 2.6334 30.4071 1.9056

χ29.0434 46.5958 8.5079

500 µ0.0236 0.2656 0.0276

σ1.0694 1.2600 1.0831

RMSE 4.4704 7.1210 4.4631

KS test 5.0129 51.8562 5.9734

χ217.2298 48.2707 16.9679

Table 1: Comparison of the estimation methods for n=

10,20,30,50,100 and 500.

parameters. The biases of maximum likelihood esti-

mation and genetic algorithm of the parameters tend

to zero for large n

2. As the sample size increases, the estimates of µand

σgenerally approach their true values. An increase in

the sample size of the simulated The Normal distribu-

tion data generally results in the improvement of the

three methods. Overall, the RMSE, KS test, and χ2

values increase as the sample size increases.

3. The difference between the ML and GA is very lit-

tle for small sample sizes (n < 30), but it is slightly

more for larger sample sizes (n≥30).

4. The LS value is higher than the others.

Moreover, we found that:

1. The ML is a commonly used method for parameter

estimation because it is simple and fast.

2. The LS is the iterative method, so the best param-

eter is based on the initial parameter. The genetic al-

gorithm is the iterative method, but the genetic algo-

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nParameter Estimation

ML LS GA

1000 µ0.0119 8.6916 -0.0064

σ1.0588 7.3159 1.0875

RMSE 3.9538 28.6166 4.1776

KS test 6.9662 262.3597 5.2007

χ222.8076 383.9043 18.9865

2000 µ-0.0003 -0.2052 -0.0088

σ1.0471 1.5086 1.0568

RMSE 5.6892 32.7658 5.5773

KS test 14.8244 98.7142 6.0461

χ219.3856 358.0330 18.4112

Table 2: Comparison of the estimation methods for n=

1000 and 2000.

rithm performance is better than ML and LS as ML

performance is better than LS.

3. According to χ2, the genetic algorithm has a

smaller χ2value than other methods.

Therefore, ML and GA show identical perfor-

mance for estimating the µand σparameters of the

Normal distribution unless the sample size is larger.

However, the GA performs better for large sample

sizes than other methods considered here such as ML

and LS methods.

5 Application to Carbonation Depth

In this section, the parameter estimation methods

defined in Section 3 are applied to the real data-

carbonation depth. Three real data sets of carbonation

depth are analyzed to compare the considered three

estimation methods for the Normal distribution.

The first data set represents 12 measurements of the

carbonation depth of a reinforced concrete girder

bridge [6]: 12.5, 13.2, 13.9, 14.1, 14.3, 14.6, 14.9,

15, 15.3, 15.7, 16.4, 17.1 mm.

The second data set represents 18 measurements of

the carbonation depth of the Chorng-ching Viaduct

[2]: 8, 11, 15, 15, 17, 18, 20, 22, 22, 26, 28, 30, 30,

31, 33, 38, 38, 40 mm.

The third data set represents 27 measurements of the

carbonation depth of pier of a reinforced concrete

girder bridge [18]: 2, 2.1, 2.2, 2.3, 2.3, 2.3, 2.4, 2.5,

2.6, 2.7, 2.8, 2.9, 3.0, 3.2, 3.2, 3.3, 3.3, 3.3, 3.4, 3.4,

3.4, 3.5, 3.5, 3.6, 3.7, 3.8, 3.9 mm.

Method Estimated parameter RMSE KS test χ2

µ σ

ML 14.7500 1.2923 0.7851 0.7174 5.1521

LSM 14.5703 1.2197 0.7954 0.2285 5.9343

GA 14.8241 1.2976 0.7861 0.9394 5.0878

Table 3: Parameter estimations, RMSE, KS test, and Chi

squared test for the first data set.

Method Estimated parameter RMSE KS test χ2

µ σ

ML 24.5556 9.5808 1.1323 1.4957 11.7405

LSM 23.5642 10.6848 1.1626 1.1159 12.7121

GA 14.8356 1.2942 0.7865 0.9664 5.0868

Table 4: Parameter estimations, RMSE, KS test, and Chi

squared test for the second data set.

Method Estimated parameter RMSE KS test χ2

µ σ

ML 2.9852 0.5702 1.5836 2.7463 15.3773

LSM 2.9697 0.6770 1.5492 2.2868 15.0384

GA .0193 0.6512 1.5134 2.3795 14.6748

Table 5: Parameter estimations, RMSE, KS test, and Chi

squared test for the third data set.

Table 3, 4, and 5 show the estimators of µand

σparameters of the Normal distribution with val-

ues of RMSE, KS test, and χ2on carbonation depth

real data. According to χ2, the genetic algorithm

yields a smaller χ2value than other methods. Ac-

cording to the KS test, the least square method yields

a smaller KS value than other methods. According

to the RMSE, the genetic algorithm yields a smaller

χ2value than other methods. The genetic algorithm

yields a smaller χ2value than other methods.

The results indicate that the genetic algorithm is

better than other methods in terms of RMSE and χ2

values. Hence, for the real given data sets of carbon-

ation depth, we concluded that the genetic algorithm

method is the best among the three considered esti-

mation methods.

6 Conclusions

We proposed a parameter estimation to estimate pa-

rameters for the Normal distribution based on the ge-

netic algorithm. The proposed estimation and the

most common estimation were applied to real data

sets. We compare the performance of three methods

for the Normal distribution through a simulation study

and three real data sets of carbonation depth. There-

fore, it is concluded from both simulated and real data

sets that all the methods show identical performance

for estimating the parameters of the Normal distribu-

tion. However, the genetic algorithm performs better

than other methods, such as the maximum Likelihood

method and least square method. In future work, we

will adjust the genetic algorithm for estimating pa-

rameters.

Acknowledgements

This research is supported by Department of Math-

ematics, Faculty of Science, Khon Kaen University,

Fiscal Year 2022.

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Contribution of individual authors to

the creation of a scientific article

(ghostwriting policy)

Somchit Boonthiem: Investigation, Visualization and

Writing - original draft.

Chatchai Sutikasana: Writing - review & editing.

Watcharin Klongdee: Validation and Writing - review

& editing.

Weenakorn Ieosanurak: Project administration,

Methodology, Conceptualization, Visualization and

Writing - review & editing.

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Volume 22, 2023

This research is supported by Department of Math-

ematics, Faculty of Science, Khon Kaen University,

Fiscal Year 2022.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

Conflict of Interest

The authors have no conflicts of interest to declare

that are relevant to the content of this article.

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