comparison between markov chain techniques for future

© 2021, IJSRMSS All Rights Reserved 56

International Journal of Scientific Research in ___________________________ Research Paper Mathematical and Statistical Sciences

Volume-8, Issue-1, pp.56-71, February (2021) E-ISSN: 2348-4519

Comparison between Markov chain Techniques for Future

Forecasting Using Production and Consumption of Electric Energy

Mohammed Alqatqat

1*, Ma Tie Feng

1

1,2

School of Statistics, Southwestern University of Finance and Economics, Chengdu, China

*Corresponding Author: [email protected], Tel.: +8619960392796

Available online at: www.isroset.org

Received: 01/Jan/2021, Accepted: 03/Feb/2021, Online: 28/Feb/2021

Abstract—Forecasting of electric energy production and consumption is a fundamental phenomenon for a country’s

planning. This study aims to make a comparison between four forecasting methods namely Markov developed by Fiering

and Thomas , Markov chains Transformation , Fuzzy Time Series Markov Chain , and proposed method By combining

two methods Markov developed by Fiering and Thomas and Markov chains Transformation of predict future production

and consumption of electricity in 2016, 2017, 2018,2019 this study examined the accuracy of the prediction of production

and consumption of electric energy based on monthly data for production and consumption of electricity during 2016 to

2019 in China. By comparing the four models using the Mean Absolute Percentage Error (MAPE), proposed method was

found as the most appropriate model for predicting and analysing the data of interest. Based on this model, production and

consumption of electricity can be forecasted. The predictive values were consistent with the original values of the series,

which indicated the efficiency of the proposed method in production electric with lowest MAPE but the predictive of

consumption electric get low MAPE by Markov chains Transformation and we note that the random number in Markov

developed by Fiering and Thomas has impact in our proposed method.

Keywords— Time series analysis, predictive modelling, Markov chains, Fuzzy time series.

I. INTRODUCTION

Multiple uses of electric energy have been integrated into

all areas of life and it has become one of the foundations of

the infrastructure which is the basis for economic and

social progress of any country. Nowadays, the rate of

consumption and production of electric energy has become

an indicator of the standard of living and the degree of

urbanization. Electricity is a special kind of commodity

which is non-storable. Thus, to maintain stability in power

supply, a constant balance between electric energy

production and consumption is always a necessity [1].

Increasing economic activity has triggered the demand of

electricity but other factors like natural calamities, day to

day activities and business intensity (peak and off-peak

hours, holidays and non-holidays) [2] have immense

influence on the demand of electricity. All these together

made consumption forecasting (also known as demand

forecasting) a difficult job.

People’s increased dependency on electricity supply has

put the management under huge pressure to predict and

generate electricity in an efficient way. An accurate

forecasting ensures the electricity supply in a proactive

manner [3] whereas an in accurate forecasting can lead to a

total collapse in the economy of a country. A failed

forecasting system can damage a country’s economy in

Both ways, an overestimation leads to the wastage of

resources while an underestimation leads to the shortage of

electricity supply [4]. As a result, electricity consumption

forecasting has become one of the most important factors

in today’s world.

To cope with the challenges of forecasting of electricity

demand researchers developed and used a wide range of

forecasting models such as Exponential Smoothing

models [5], Double Seasonal Exponential Smoothing

models [6], Holter-Winters models [7], Markov model [8],

Grey-based models [9], fuzzy regression algorithm [10]

etc. In this paper we mainly focused on Exponential

Smoothing models and Markov model to forecast the

electrical production and consumption.

The main idea of this study is to focuses on the comparison

of statistical models for predicting production and

consumption of electric energy. The statistical methods

represented in the Markov models were chosen because

they can be applied to such studies and helps in adopting

the optimal method in the study analysis. Among the most

common models, Markov model have been used

successfully in predictive phenomena under the models of

linear time series analysis, a method Jenkins and Box

found. As for the Markov model, it is considered as one of

the mathematical models that has its applications in

various different fields. It also helps us in describing,

monitoring and forecasting, which depends on its

construction. This is the method, which increases the

accuracy of its description of the studied phenomenon, by

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Int. J. Sci. Res. in Mathematical and Statistical Sciences Vol. 8, Issue.1, Feb 2021


adopting an analysis study data on statistical programs

Microsoft Excel & R software. Fuzzy time series Markov

Models for time series forecasting is an important

statistical and inferential procedure that addresses

confusion or random errors. It has the process of fine-

tuning, and it is a type of assessment that has proven

successful by studying cases that depend or change with

time. The scientist The American-Azerbaijani "Lotfi Zada"

was the first to develop this method.

The aim of the study is to analyse the previous data of

electric energy production and consumption with several

statistical models to find out a model suitable for future

prediction. Then use of that suitable model to develop a

forecast for the future years so that the management system

can take accurate steps to fulfil the future demand of

electric energy.

II. RELATED WORK

Markov model was selected to be used in this research for

long-term forecasting. Traditionally, Markov models are

excellent to be used for short-term forecasting.

Additionally, the short-term and medium-term forecasts

are developed by identifying, modelling and extrapolating

the trends that are obtained in the historical data.

Most of the time, the historical data have inertia and

uniform properties, making it an excellent variable for

short-term and medium-term forecasting (Montgomery et

al., 2008). From the forecast that was obtained from the

available observation time, it can give a prognosis for (Box

et al., 1976):

Economic and market planning

Planning of production

Inventory and production control

Control and improvement of the industrial processes.

Brown (1962) explained that for each problem that

occurred will have a different lead time for the forecast.

Naturally, the forecast can be obtained at time t by using

the current month, Yt and the months before Y1, Y2,…,Yt-1

and the time from the future Ft+1, Ft+2,…,Ft+ m where it can

be retrieved from Y value forward.

To ensure the best forecast can be obtained, it is essential

to specify the forecast's accuracy. A convenient set of

probability limits on one side of the forecast such as 50%

and 95% can be used to define the forecasts' accuracy. By

this, the set-out probability limits will include the realized

value of the time series when it actually happens. In order

to make it clear, the author describes the previous research

works in the form of title, problem statement, objectives,

not repeat the information discussed in Introduction

Explores the integration of online energy demand forecasts

from various approaches with weights varying depending

on the specific time of the week and the forecast origin

Smooth transition approaches can be used to ensure a

smooth transition between various demand forecasting

methods [11]. However, rather than proposing a new

method to all others, using a variety of different

approaches would be a better approached to this study

[12]. Conventionally, the existing statistical method

contemplates with models in which results are considered

to be independent. Nevertheless, there is a massive set of

data in business, economics, engineering and natural

sciences in the form of time series in which observations

are dependent. The systematic methods available to

address the mathematical and statistical questions

presented by these set of dependent observations are called

time series analysis. The Markov chain is a special case

with the chance process in which the state space is

intermittent and carries the Markov property when the

receiver depends on the present only and does not depend

on the past and according to the formula below:

1 0 1( 0,......, ) ( )t t t t ijp x j x x k p x j x i p

…. (1)

Also, the set of possible values for the chance process is

called Space State, and that state space It is of two types,

limited or unlimited, depending on the number of its

elements. It can also be continuous if it takes continuous

values, or discontinued if it takes separate values. From

here it is clear that the stochastic process is determined by

two spaces, the status space and the parameter space [7],

[8].

The chance process is built on the basis of the

phenomenon's transition from one state to another based

on certain probability laws called transitional possibilities

that represent the probability of transition during a specific

time period we symbolize them, and the probability

distribution of transitions between cases can be

represented by an array called a transitional matrix [9]

which is a coincidence matrix It has the following

advantages:

1- All its components are negative (because they are

probabilities).

2- The sum of each of its classes equals one (because the

sum of the total odds is equal to one).

3- It should be a square array of degree as shown below

1 11 1

2

1

n

n m mn

s p p

p s

s p p

………………………. (2)

Markov chain cases are classified into

1- The Markov chain is irreducible, and it is the chain that,

if possible, can be transferred from any of its states to

other states and vice versa at any time.

2- The positive recurrent Markov chain is said to be the

case in the Markov chain as a positive return case if and

only if it is certain that the process will return to the same

condition that was previously left, i.e. if the final return to

it is confirmed.

3- The periodic state, and the case is said to be periodic if

the greatest common divisor of the number of cycles in

which the state appears is greater than the correct one and

vice versa.



4- The ergodic case is said to be the case if it is an evident

case if it is indivisible and has a positive and non-periodic

return

5- The stationary distribution is the probability distribution

of the cases of the Markov chain after a long time, and it is

a single distribution if the transitional matrix represents a

validation chain and is symbolized by the symbol B, and

this distribution can be found by solving the following

system [10].

1

1n

i

i

p

……………………………………………….………. ………..(3)

In this section the Markov models will be discussed in

detail. In order to predict the upcoming electrical

Production and Consumption from 2016 to 2019, the

monthly data of PC of electrical energy that has been

recorded in China will be used as the basis of this study.

[14] Explores the integration of online energy demand

forecasts from various approaches with weights varying

depending on the specific time of the week and the

forecast origin. Smooth transition approaches can be used

to ensure a smooth transition between various demand

forecasting methods [13]. However, rather than proposing

a new method to all others, using a variety of different

approaches would be a better approached to this study

[23]. Conventionally, the existing statistical method

contemplates with models in which results are considered

to be independent. Nevertheless, there is a massive set of

data in business, economics, engineering and natural

sciences in the form of time series in which observations

are dependent. The systematic methods available to

address the mathematical and statistical questions

presented by these set of dependent observations are called

time series analysis.

The aim of the time series analysis is generally to

understand and define the stochastic mechanism that

generated the observed series and then to estimate future

values of the series on the basis of past values on its own

[24]. The time series dissection in the time domain is

executed by a parameter known as the serial correlation

coefficient or also known as autocorrelation coefficient.

This parameter shows the dependency of the time series in

successive values. This coefficient is calculated in order to

find the successive values (elements) and also for elements

that are separate time intervals apart from those known as

the lag period. The correlogram is the graph of the

autocorrelation coefficient for the lag time.

The process is said to be absolutely random if the

correlogram shows zero or almost zero values for all lag

times. A value close to 1 indicates a dominant

deterministic mechanism [14]. The study of the time series

in the frequency domain is performed by a spectral density

that identifies the cyclic existence or periodicity of the

series. The density will be used as the period of the

deterministic data. It oscillates arbitrarily in a strictly

random operation. However, the object of streamflow

synthesis is not to evaluate a time series but to produce

data based on a series. This term does not include the

decomposition of the time series by the previously

mentioned analysis, rather an interpretation of its statistical

properties to replicate a series of related statistical features

[14].

There are other forecasting issues involving the application

of the time-series data. According to (Montgomery et al.

2008), forecasting issues can be divided into three

categories. The three categories are short-term, medium-

term and long-term. The following are the elaborations on

the forecasting issues:

• Short-term forecasting problem: Involve the

anticipation of events by only a few days, weeks, and

months.

• Medium-term forecasting problem: May reach out up

until one or two years into the future.

• Long-term forecasting problem: May extend for a very

long period and take years.

Most of the management on the operational level and

project development will utilize the short-term and

medium-term forecast. In contrast, for the long-term

forecast, strategic planning cases will be more suitable for

application [25].

The exponential smoothing methods possess the basic, yet

stable, prediction approaches quality. The distinctive

aspect of these methods is that time series are supposed to

be developed from unmeasured parameters such as level,

development, and seasonal impacts. Simultaneously, these

parameters need to be modified over time because the

demand series reveals the effects of systemic shifts in

commodity markets.

Markov model was selected to be used in this research for

long-term forecasting. Traditionally, Markov models are

excellent to be used for short-term forecasting.

Additionally, the short-term and medium-term forecasts

are developed by identifying, modelling and extrapolating

the trends that are obtained in the historical data.

Most of the time, the historical data have inertia and

uniform properties, making it an excellent variable for

short-term and medium-term forecasting [25]. From the

forecast that was obtained from the available observation

time, it can give a prognosis for [26]:

• Economic and market planning

• Planning of production

• Inventory and production control

• Control and improvement of the industrial processes.

[27] Explained that for each problem that occurred will

have a different lead time for the forecast. Naturally, the

forecast can be obtained at time t by using the current

month, Yt and the months before Y1, Y2,…,Yt-1 and the



time from the future Ft+1, Ft+2,…,Ft+ m where it can be

retrieved from Y value forward .

III. METHODOLOGY

Relevant details should be given including experimental

design and the technique (s) used along with appropriate

statistical methods used clearly along with the year of

experimentation (field and laboratory).

III.1 Markov Model

The general Markov procedure of data synthesis comprises:

(1) Determining statistical parameters from the analysis of

the historical record.

(2) Identifying the frequency distribution of the historical

data.

(3) Generating random numbers of the same distribution

and statistical characteristics.

(4) Constituting the deterministic part considering the

persistence and combining with the random part. The

various combinations of deterministic and random

components are recognized as different models [13]. The

details procedure is described below:

A. Define descriptive statistical measures and parameters

of the Markov model

There are four important milestones that were used during

the analysis of the phenomenon under study. These were

mean, standard deviation, torsional coefficient, and

correlation coefficient, which can be found by the following

formulas, respectively [14].

To calculate the mean:

1

1( )

n

i

i

X xn

………………..…..…………….. (4)

Where, X̅= mean observed (historical), n= total numbers

(values), iX = thi number of observations.

To calculate the standard deviation:

1

22

1

1

1

n

iiS X X

n

…………….. (5)

Where, S= Standard Deviation, ix = mean of the thi

observation, X̅= mean observed, n= total numbers to

calculate the coefficient Skewness:

The sample coefficient of skewness, g, which is a

measure of the lack of symmetry, is given by

3

1

3

( )

1 2

n

iin x x

gn n S

………………………….. (6)

Calculation of correlation coefficient:

The serial correlation coefficient is a measure of the extent

of production at any time to the production at another time.

The K- lag coefficient, in which the effect extends by K

time units, is given by

1

2

n k

i i kik

n x x x xr

n k S

……………. (7)

B. Learn about data distribution

Before starting the steps of building a random number

generation model, it is necessary to ensure that the time

series model used is subject to the normal distribution. If

this is not achieved, the method of transfers is used. The

Cox-Box method [15] shown in the equation

1 i

i

xq

……………………………………… (8)

One of the approved methods for converting the time series

to a normal distribution by imposing several values of λ is

between (-1, 1) for each month alone, to obtain new values,

the convolution coefficient is close to zero.

In general, the distribution used for data under study is the

natural distribution given in statistical applications. To

know the nature of the data and to reveal whether or not

they follow the normal distribution, they must lie on or near

the straight line.

C. Generate random numbers

Random numbers are generated from the Microsoft Excel

statistical program or via a table random numbers, and

convert them to follow the standard normal distribution.

Random numbers are generated using Microsoft Excel this

is done by RAND () to obtain the random variable t with an

arithmetic mean of zero and indicate the standard deviation

1, we use the function error inverse function 1erf

according to the following formula:

2 3 4

1 3 5 7 91 7 127 4369(

2 12 480 40320 5806080erf Z Z Z Z Z Z

……. (9)

As for finding the value of Z, it can be found through the

cumulative distribution function CDF (function

distribution) for the natural logarithm distribution as

follows:

2

1 1 ln()

2 2 2

xCDF erf RAND

..……. (10)

2

lnerf erf ( () 0.05)

2

xz RAND

..… (11)

Because the natural logarithm of random numbers has the

property that its mean is equal to the curvature of its waves

equal to one Correct, meaning that: 1, 𝜇 = 1 = σ.

ln 1

2

xerf Z

1 ln 1( )

2

xerf z

1ln 1 2 ( )x erf z

1ln 1 1 2 ( )t x erf z …………………….. (12)



D. Building a Markov model for prediction

Formulation of the Markov Model for annual flow (Gupta,

1989):

2

1 1 1( ) (1 )i i ix x r x x s r t ……….….. (13)

Where, iX Value in (ith) time, x̅: The mean of the data,

ir : Linear correlation coefficient, S: data standard

deviation,

it : Random number resulting from data generation and

follows the normal distribution.

As for the monthly Markov model developed by Fiering

and Thomas [16], it takes the following form

, 1, 1 ,1 (1 )i j j j i j i j j jjq q b q q t s r

……... (14)

Whereas

ix : Value in (ith) time.

x : The mean of the data.

ir : Linear correlation coefficient.

S: data standard deviation



III.2 Markov chains Transformation matrix Method

The specific Markov chain [17, 18, and 19] under

consideration often determines the natural notation for the

state space. In the general case where no specific Markov

chain is singled out, we often use N or Z+ as the state

space. We set , 1

1, i ii jp j ip x x

……………….. (15)

For fixed the (possibly infinite) matrix

P= p

ji

1,

,

is called the matrix of transition

probabilities (at time ). In our discussion of Markov

chains, the emphasis is on the case where the matrix P

is independent of l which means that the law of the

evolution of the system is time independent. For this

reason one refers to such Markov chains as time

homogeneous or having stationary transition probabilities.

The use of Markov chains in predicting oil prices

productivity goes through the following steps:

Step one: After preparing the data for the phenomenon that

we are about to predict its future path ٬ we start at the

beginning by dividing them into certain levels, after we

subtract the smallest value of the phenomenon from its

largest value ٬ then we divide the result of the subtraction

process by the number of previously determined levels.

Step two: We represent these levels in a horizontal,

horizontally cantered graph that expresses the time

(months). Its vertical axis expresses the levels specified in

Step one ٬ then we place each one at the productivity

values at the level at which they are located.

Step three: From the step two, we define the

transformation matrix ٬ as each element in this matrix

crosses on the possibility of the phenomenon moving from

one level to another.

Step four: After determining the various elements of the

transition matrix, we take the average of the values that fall

within the levels Specific to Step 1.

Step five : We form a line of its elements by the number of

levels defined by Step 1 ٬, all of which are equal to zero,

except An element that is equal to one of its location on the

line, corresponding to the level at which the last value of

the phenomenon is located.

Step sex: We multiply this line’s ray by a transition matrix

so we get a new line ray ٬ we multiply this last by the

transition matrix so we get a new line ray.

Step seven: We multiply the new line ray by the calculated

averages in step 4, and we get the expected value for the

phenomenon in the coming months.

Step eight: .and by repeating Step (6) and Step (7) on the

last line ray, we get the values of the phenomenon in the

coming months.

III.3 Fuzzy Time Series-Markov Chain Method

According to [20] the Markov Chain Fuzzy Time Series

forecasting procedure is defined in the following steps.

1) Step one: Gather historical data ( tY ).

2) Step two: Define the universe U set of data, where 1D

and 2D are the corresponding positive numbers.

min 1 max 2,U D D D D

3) Step three: Determine the number of fuzzy intervals, in

this research to calculate the number of fuzzy intervals

formed using the average based length method [18] with

steps as

Following:

a) Calculate the difference in absolute value from data

1iA and iA (i = 1, 2... n-1), then average the results.

b) Divide the two values generated in step a.

c) From the value obtained in step b, determine the base

value for the interval length based on (Table 1).

Table 1. Base mapping table

Range Basis 0, 1-1, 0 0, 1

1, 1-10 1 11-100 10

101-1000 100 1001-10000 1000



d) The number of fuzzy intervals can be calculated with,

1 min 1 min 1

2 min 1 min 1

2 min 1 min 1

,

, 2*

...

( 1)* , *

u D D D D base

u D D base D D base

u D D n base D D n base

Step four: Define the fuzzy set in universe of discourse U,

the fuzzy set iA expresses the linguistic variable of the

stock price with 1 ≤ i ≤ n.

Step five: Fuzzyfication of historical data. If a time series

data is included in the iu interval then the data is fuzzyfied

into iA .

Step sex: Determine the fuzzy logical relationship and

Fuzzy Logical Relationships Group (FLRG).

If the fuzzy set now is iA , and the fuzzy iA logic relation

group is unknown, for example iA , then ≠ will refer

to the fuzzy set iA

Step seven: Calculate forecasting results for time series

data, using FLRG, we can obtain the probability of a

statement to go to the next state. So that the Markov

probability transition matrix is used in calculating the

forecast value, the dimension of the transition matrix is n x

n. If the state iA makes a transition Go to state jA and

pass state kA , i, j = 1, 2... n, then we can get FLRG. The

transition probability formula is as follows [19].

, , 1, 2,...,ij

ij

i

MP i j n

M ………………………… (16)

With:

ijP =probability of transition from state iA to state jA one

step.

ijM =number of transitions from state iA to state jA one

step.

iM =the amount of data included in the iA state.

The probability matrix R of all states can be written as

follows:

11 1

1

n

n nn

P P

R

P P

………………………..……. (17)

Matrix R reflects the transition of the entire system [20]. If

F (t-1) = iA , then the process will be defined in the state

iA at the time (t-1), then the forecasting result F (t) will be

calculated using the lines 1 2, ,i i inP P P .

In matrix R The result of forecasting F (t) is the weighted

average value of 1 2, ,......, nm m m (midpoint

Of ( 1 2, ,......, nu u u ). The value of the output forecasting

result in F (t) can be determined using the following rules.

a) Rule 1: if the fuzzy logic relationship group Ai is one to

one (for example i kA A where

1ikP And 0,ijP j k then the forecast value F (t)

is the middle value of km , ku ( ) k ik kF t m P m

Rule 2: if the fuzzy logic relationship group Ai is one to

many (example 1 2, ,......, . 1,2,.....,j nA A A A j n ,

when Y (t-1) at time (t -1) included in the status of the

forecast F (t), are:

1 1 2 2 1 ( 1) 1 ( 1)( ) ... ( 1) ...j i j j j jj j j j n nF t m P m P m P Y t P m P m P

Where:

1 2, ,......, nm m m Is the middle

value 1 2, ,......, nu u u , ( 1)tY

Is the state value jA at time t-1.

Step Eight. Calculate the adjustment value ( )tD on the

forecast value. Here are the principles for calculating

Adjustment values.

a) If the state iA is related to iA , starting from the state

iA at time t-1 is stated as F (t-1) = iA , and experiencing

an increasing transition to state jA at time t where (i < j)

then the value the adjustments are: 1 ( )2

tD

Where is

the base interval.

b) If the state iA is related to iA , starting from the state

iA at time t-1 is stated as F (t-1) = iA , and experiencing a

decreasing transition to state jA at time t where (i > j) then

the value the adjustments are: 1 ( )2

tD

Where is the

base interval.

c) If the transition starts from state iA at time t-1

expressed as F (t-1) = iA , and experiences a jump forward

transition to state i sA at time t where (1 ≤ s ≤ n-i) then the

adjustment value is

2 ( ) ,12

tD v v i

…………..…………..……. (18)

Where , (s) is the number of leaps forward.



d) If the transition starts from state iA at time t-1

expressed as F (t-1) = iA , and experiences a jump

backward transition to state i vA at time t where (1 ≤ v ≤ -

i) then the adjustment value is

2 ( ) ,12

tD v v i

……………………………. (19)

Where, (v) is the number of leaps backward.

Step Nine. Calculate the adjusted forecast value

a) If FLRG iA is one to many and the 1iA state can be

accessed from the iA state where the iA state is related

to iA , the forecast results will be

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

t tF t F t D D F t

...… (20)

b) If FLRG iA is one to many and the 1iA state can be

accessed from the iA state where the iA state is not

related to iA , the forecast results will be

2( ) ( ) ( ) ( )2

tF t F t D F t

…………..….... (21)

c) If FLRG iA is one to many and the 1iA state can be

accessed from the iA state where the iA state is

communicate to iA , the forecast results will be

2( ) ( ) ( ) ( ) 2 ( )2

tF t F t D F t x F t

.... (22)

d) When v is the jump step, the general form of the result

of the forecast is:

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

t tF t F t D D F t v

… (23)

III.4 the Proposed Method

Step 1: Apply the Markov model developed by Fiering and

Thomas [15].

Step 2: take the calculation second part , (1 )i j j jt s r

Random Component





Step 3: The resulting predicted values from Markov chains

Transformation matrix we will add to Random Component.

Now we have this equation to get the result from combing

two methods

( ) ,ˆ ˆ (1 )Markov chains Transformati xt t io jn matri j jy y t s r

…… (24)

III.5 DATA COLLECTING AND PROCESSING

The data used in this research was gathered by the

https://www.ceicdata.com/en from 2003-01 to 2019-12. It

contained the data of production and consumption

electricity in china.

III.6 MODEL EVALUATION

The process of evaluating models is intended to evaluate

the field suitability of the model for the pattern in which

the series data is running or the accuracy of the model in

predicting the values of the current and future series, and

there are many measures of the suitability of the model all

depend on the degree of error, which is the difference

between the actual value of the series at a specific time

And the string value that the model expected at that time),

McGee and Yaffee [22]. In this study, we will rely on the

following methods to compare the two models used in this

paper to find out which one is more accurate in prediction.

A. Mean Absolute Percentage Error (MAPE)

1100 [| | / ] /

n

i i iiMAPE Y F Y n

……………. (25)

The scale that eliminates the problem that negative error

values cancel out positive values, and does not amplify the

error by squaring, as happens in the sum of squares of

errors. It is possible to compare models across different

series. It is called "Average absolute values of error ratios".

IV. RESULTS AND DISCUSSION

IV.1 Result Markov Chain Method and Discussion

First step : Define descriptive statistical measures and

parameters of the Markov model , Based on the theoretical

aspect that dealt with the developed Markov model by

Fiering-Thomas in equation number () , To find the

Parameters of the Markov model for monthly data from

January 2003 to December 2019, which are defined by 204

views, this is illustrated by the following (Table 2):

Table 2. Parameters of the Markov model

I ir

Jan 297.99346 13024.01 114.12279 0.4042 0.407773 0

Feb 554.69154 39457.759 198.63977 0.9878587 0.992857 297.99346

March 865.31292 96110.946 310.01765 0.9991141 1.002347 554.69154

April 1162.5062 170220.32 412.57765 0.9998027 1.002232 865.31292

May 1471.4507 270338.38 519.94075 0.9999641 1.001891 1162.5062

Jun 1791.6607 397144.71 630.19418 0.9998014 1.00139 1471.4507

July 2141.9333 563146.08 750.4306 0.9999422 1.001276 1791.6607



Aug 2496.1963 759324.87 871.39249 0.9998685 1.001017 2141.9333

Sep 2815.0096 950564.46 974.96895 0.9999476 1.000974 2496.1963

Oct 3126.4181 1157578.9 1075.9084 0.9999427 1.000873 2815.0096

Nov 3446.7341 1395118.8 1181.1515 0.9999306 1.000778 3126.4181

Dec 3860.3007 1913958.1 1383.4587 0.9984837 0.999206 3446.7341

Second step: Learn about data distribution and to know

that the data follow the normal distribution or not, so we

use the Klemgrove-Simernov test whereas:

1- Nondimposition: The sample data follows the normal

distribution.

2- Alternative hypothesis: Sample data do not follow the

normal distribution.

Table 3. Kolmogorov-Smirnov test results

K-S N Sig

0.097 204 0.0002

It is clear from (Table 3) of the results of the Klomgrove-

Simnerov test that the value-P value is greater than 0.0002

and is greater than the level of significance 0.05, where the

results indicate that the test is not significant.

This supports the validity of the assumption that the data

follow the normal distribution.

It is also clear from the following (Figure 1) that most of

the data is on the straight line and very close to it, and this

in turn is consistent with the results of the previous test,

which confirms the nature of the data.

probability plot of yt normal

Lognormal

yt

10.90.80.70.60.50.40.30.20.10

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Fig 1 .Series of Normal Probability Plot

Third step: Generate random numbers we explain it in

section 2.1.3 for example, the procedures for generating

random numbers for 2016 It is shown in Table (Table 4)

below:

Fourth step: The construction phase of the Markov model

for prediction The Markov model consists of two

Parts, the first deterministic part, which takes into account

the effect of the previous value on the model, and the

second part, which is random part which represents the

random part of the model.

By merging and combining these two parts, the monthly

Markov model is constructed for forecasting according to

the following formula:

, 1, 1 ,1 (1 )i j j j i j i j j jjq q b q q t s r …. (26)

The following (Table 5) shows the prediction of the

electric energy production for the year 2016, any year can

be predicted and constructed provided you know its

previous values.

The comparison between forecasted and actual production

and consumption of electricity in China from 2016 to 2019

by Markov developed by Fiering and Thomas method was

summarized in Appendix 1 and Appendix 2.

Table 4. Generate random numbers 2016

i RAND0 z crf t

Jan-16 0.1759883 -0.6480234 -0.6581163 0.4834966

Feb-16 0.8282821 0.6565642 0.669906 2.3616037

Mar-16 0.7749747 0.5499494 0.5340884 2.1695286

Apr-16 0.3092822 -0.3814356 -0.3520229 0.9163781

May-16 0.6376827 0.2753654 0.2490885 1.7664779

Jun-16 0.6412258 0.2824516 0.2557853 1.7759487

Jul-16 0.3048914 -0.3902172 -0.3608783 0.9038546

Aug-16 0.3576688 -0.2846624 -0.2578206 1.0496001

Sep-16 0.4387239 -0.1225522 -0.1089929 1.2600744

Oct-16 0.7562954 0.5125909 0.4910391 2.1086477

Nov-16 0.7381297 0.4762595 0.450834 2.0517891

Dec-16 0.5385768 0.0771535 0.0684793 1.5110579



Table 5. Prediction of the electric energy production for the year 2016 Deterministic Component Random Component Model Flow

i

, (1 )i j j jt s r

Jan-16 490.701 498.08813 0.483496608 50.46967779 548.5578115

Feb-16 856.086 1108.7976 2.361603727 72.87827534 1181.675901

Mar-16 1310.272 1622.667 2.169528565 28.3048526 1650.971813

Apr-16 1759.315 2058.5036 0.916378074 7.509361006 2066.012939

May-16 2218.6566 2529.5984 1.76647794 7.784230521 2537.382625

Jun-16 2709.0545 3030.9853 1.775948659 22.30349416 3053.288756

Jul-16 3220.8444 3572.9413 0.903854604 7.29380841 3580.235138

Aug-16 3738.0434 4093.9301 1.049600145 14.83305898 4108.763114

Sep-16 4198.9523 4519.4245 1.260074382 12.57787878 4532.002335

Oct-16 4651.12 4964.1313 2.108647689 24.29546178 4988.426791

Nov-16 5125.6727 5447.5439 2.051789134 28.5589158 5476.102814

Dec-16 5814.5733 6226.2597 1.511057917 115.0759745 6341.335722

IV.2 Result Markov Chain Method and Discussion

First step: prepare the data for 2 years 2015 and 2016 and

make difference between two years month by month, and

calculate the rate for every month by division the value of

the month on sum months multiplied 100% for 2016.

Second step: In this step we will form a matrix by rate multiplied difference on every month until we have matrix 12 12. (Table 6) shows the matrix which build in step two.

Table 6. Form a matrix by rate

Table 7. Transfer matrix Q (2016)

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Jan 0.9744819 -0.011384 0.0012132 0.001062 0.0013237 0.0013638 0.0024663 0.0037618 0.0047126 0.0057751 0.0066089 0.0086148

Feb -0.001504 0.9646017 0.0012132 0.001062 0.0013237 0.0013638 0.0024663 0.0037618 0.0047126 0.0057751 0.0066089 0.0086148

March -0.001504 -0.011384 0.977199 0.001062 0.0013237 0.0013638 0.0024663 0.0037618 0.0047126 0.0057751 0.0066089 0.0086148

April -0.001504 -0.011384 0.0012132 0.9770478 0.0013237 0.0013638 0.0024663 0.0037618 0.0047126 0.0057751 0.0066089 0.0086148

May -0.001504 -0.011384 0.0012132 0.001062 0.9773095 0.0013638 0.0024663 0.0037618 0.0047126 0.0057751 0.0066089 0.0086148


Jan

-

4.9530669 0.527835 0.4620711 0.5759376 0.5933742 1.0730352 1.636709934 2.0503805 2.5126547 2.8754563 3.7481797

Feb -

0.6543054 0.527835 0.4620711 0.5759376 0.5933742 1.0730352 1.636709934 2.0503805 2.5126547 2.8754563 3.7481797

March -2.037924

-

15.427008 1.439184 1.7938367 1.8481454 3.3421155 5.097758038 6.3861918 7.8260083 8.9560039 11.674221

April -

2.7048116

-

20.475322 2.1819999 2.3808495 2.4529301 4.4357851 6.765941833 8.476001 10.386981 11.886755 15.494478

May -3.410479

-

25.814691 2.751002 2.40825 3.0925829 5.5925089 8.530302826 10.686296 13.095604 14.986475 19.534987

Jun -

4.1498622 -31.4143 3.33477373 2.930637 3.6528228 6.8056114 10.38065864 13.00432 15.936245 18.237276 23.772431

July -

4.9808301

-

37.704696 4.0180879 3.5174674 4.3842636 4.5169978 12.45928069 15.608304 19.127317 21.889106 28.53262

Aug -

5.8306715

-

44.137963 4.7036638 4.1176262 5.1323173 5.2876989 9.5620727 18.27143 22.390866 25.623879 33.400925

Sep -

6.5766869

-

49.785271 5.3054823 4.6444631 5.7889806 5.9642427 10.785509 16.45123113 25.255705 28.90237 37.674465

Oct -

7.3157696

-

55.380099 5.901708 5.1664041 6.4395415 6.6344994 11.997575 18.30000699 22.925246 32.150395 41.90829

Nov -

8.0758009

-

61.133508 6.5148332 5.703139 7.108542 7.323754 13.243997 20.20118477 25.306937 31.012582 46.262119

Dec -

9.2233597 -

69.820484 7.440581 6.5135462 8.1186548 8.3644481 15.125949 23.07174185 28.903013 35.419422 40.533625



Jun -0.001504 -0.011384 0.0012132 0.001062 0.0013237 0.9773496 0.0024663 0.0037618 0.0047126 0.0057751 0.0066089 0.0086148

July -0.001504 -0.011384 0.0012132 0.001062 0.0013237 0.0013638 0.978452 0.0037618 0.0047126 0.0057751 0.0066089 0.0086148

Aug -0.001504 -0.011384 0.0012132 0.001062 0.0013237 0.0013638 0.0024663 0.9797476 0.0047126 0.0057751 0.0066089 0.0086148

Sep -0.001504 -0.011384 0.0012132 0.001062 0.0013237 0.0013638 0.0024663 0.0037618 0.9806984 0.0057751 0.0066089 0.0086148

Oct -0.001504 -0.011384 0.0012132 0.001062 0.0013237 0.0013638 0.0024663 0.0037618 0.0047126 0.9817609 0.0066089 0.0086148

Nov -0.001504 -0.011384 0.0012132 0.001062 0.0013237 0.0013638 0.0024663 0.0037618 0.0047126 0.0057751 0.9825947 0.0086148

Dec -0.001504 -0.011384 0.0012132 0.001062 0.0013237 0.0013638 0.0024663 0.0037618 0.0047126 0.0057751 0.0066089 0.9846006

Third Step: we will find the value of decreases by sum

every month from step 2, after that we find transfer matrix

in the (Table 7) by subtract the actual value for each month

from value of decreases the sum of these rows must be

equal to one let this matrix be Q (2016).

Fourth Step: In this step we will find P (2016) shown in

(Table 8) and it is matrix 1 12 by division the value of

the month on sum months for 2016.

Fifth step: Now multiply Q (2016) by P (2016) to get new

matrix 1 12 and let’s call it N (2016) show in (Table 9)

Table 8. Matrix P (2016)


0.011765 0.011765 0.03664377 0.04863503 0.0613176 0.0746184 0.08956 0.1048409 0.1182549 0.1315443 0.1452104 0.1658446

Table 9. Matrix N (2016)



by Markov Chain Transformation Matrix Method was


IV.3 Result Fuzzy Time Series-Markov Chain Method

and Discussion

First Step: Define universe of discourse U and partition it

into several equal-lengths intervals. The collected data is

shown in Table 1; we have the production electric data of

from Dec 2015 to Dec 2019 with = 13055 and = 19337.

We choose = 135.09 and = 146.57. Thus, U = [300,

7650]. U is divided into 7 intervals with

U1 = [300, 1350],

U2 = [1350, 2400],

U3 = [2400, 3450],

U4 = [3450, 4500]

U5 = [4500, 5550].

U6 = [5550, 6600],

U7 = [6600, 7650]

Second step: Define fuzzy sets on the universe U. The step

has the same defined fuzzy sets.

Third step: Fuzzify the historical data. The equivalent

fuzzy sets to each month are shown in (Table 11) and each

fuzzy set has 7 elements.

Fourth Step: Determine the fuzzy logical relationship

group.

The fuzzy logical relationship group is obtained as shown

in (Table 10).

Thus, using the fuzzy logical relationship group in Table 2,

the transition probability matrix R can obtained.

Fifth Step: Calculate the forecasted outputs. According to

the proposed rules in Step 7 the forecasting values are

obtained as in the third column of Table 5.3.3 The

forecasting value of Mar 2017 is F (Mar 2017) = (1/2) Y

(Feb 2017) + (1/2) (m2) = (1/2) (465.77) + (1/2) (1875) =

1170.3825.

Table 10. Fuzzy relationship groups

1 1 2 1 2 1 2 1 2, , , , , , ,A A A A A A A A A

2 2 3 2 3 2 2 3 2 3, , , , , , , ,A A A A A A A A A A

3 3 4 3 4 3 4 3 4, , , , , , ,A A A A A A A A A

4 4 5 4 5 5 5 5, , , , , ,A A A A A A A A

5 5 6 5 6 5 6 5 6, , , , , , ,A A A A A A A A A

6 1 6 1 6 7 6 7, , , , , ,A A A A A A A A

7 1A A

Sixth Step: Adjust the tendency of the forecasting values.

The relationships between the states, thus an adjusted value

should be considered, or vice versa.


0.011765 0.011765 0.03664377 0.04863503 0.0613176 0.0746184 0.08956 0.1048409 0.1182549 0.1315443 0.1452104 0.1658446



Table 11. Fuzzify the historical data

According to the proposed rules in Step 8, 9, the adjusted

values are obtained as in the fourth column of (Table 12)

Seventh step : Obtain adjusted forecasting values ,The

adjusted forecasting values are obtained in the last

column of (Table 12) The adjusted forecasting value for

Mar 2016 is F (Mar 2017) = F (Mar 2017) + 1050 =

2455.3.

Table 12. Adjusted values

Month/Year

Production

electric Data

Forecasting

value

Adjusted

value

Dec-2015 5802.00 4757.999143

Jan-2016 495.07 4757.999143 -3150

Feb-2016 876.23 1185.036559 0

Mar-2016 1352.36 1375.612755 1050

Apr-2016 1809.30 2051.309845 0

May-2016 2282.40 2305.164506 0

Jun-2016 2775.93 2568 1050

July-2016 3328.94 3375.465547 0

Aug-2016 3892.00 3651.968289 1050

Sep-2016 4388.50 4701.285714 0

Oct-2016 4877.55 4843.142857 1050

Nov-2016 5384.70 5550 0

Dec-2016 6129.71 5550 1050

Jan-2017 486.80 4898.446714 -3150

Feb-2017 935.60 1180.9 0

Mar-2017 1446.10 1405.3 1050

Apr-2017 1930.90 2103.388889 0

May-2017 2426.30 2372.722222 1050

Jun-2017 2950.80 3200.65 0

July-2017 3557.80 3462.9 1050

Aug-2017 4157.10 4605.8 0

Sep-2017 4688.80 4777.028571 1050

Oct-2017 5201.80 5550 0

Nov-2017 5733.10 5550 1050

Dec-2017 6363.60 4728.471429 0

Jan-2018 599.50 4998.685714 -3150

Feb-2018 1055.20 1237.25 0

Mar-2018 1587.80 1465.1 0

Apr-2018 2109.40 2182.111111 1050

May-2018 2662.80 2471.888889 0

Jun-2018 3229.10 3318.9 1050

July-2018 3877.50 3602.05 0

Aug-2018 4529.60 4697.142857 1050

Sep-2018 5106.10 5550 1050

Oct-2018 5655.20 5550 0

Nov-2018 6219.90 4695.085714 1050

Dec-2018 6900.20 4937.1 0

Jan-2019 617.20 825 1050

Feb-2019 1106.30 1246.1 0

Mar-2019 1679.50 1490.65 0

Apr-2019 2232.90 2233.055556 1050

May-2019 2799.30 2540.5 0

Jun-2019 3398.10 3387.15 1050

July-2019 4065.20 3686.55 0

Aug-2019 4742.20 4750.771429 1050

Sep-2019 5344.20 5550 1050

Oct-2019 5923.20 5550 0

Nov-2019 6514.40 4809.942857 1050

Dec-2019 7225.50 5063.314286 0



by Fuzzy Time Series-Markov Chain Method was


IV.4 Result proposed method and Discussion First step: we already apply the Markov model developed by Fiering and Thomas [13] in section 4.1.

Second step: from the results of first step we will take the calculation second part from equation Random Component.

For year 2017 of production electric , (1 )i j j jt s r



Table 13. Random component for the year 2017 Random Component

, (1 )i j j jt s r

0.52129353 54.415110047

1.97570877 60.9696903

2.42001728 31.57286497

1.39152664 11.4030182

-0.09360879 -0.41250014

1.13723947 14.28217742

2.39050512 19.29058751

3.45713718 48.85662399

2.37499913 23.70689509

0.49139161 5.661726306

2.13441658 29.7090098

2.29802132 175.0078802

Table 14. Predicted values from Markov chains Transformation matrix


369.0347881 3.63878814 1367.459 1794.673 2262.109 2743.658 3323.719 3923.17 4442.488 4961.438 5485.529 6304.46381

Table 15. Prediction for the year 2017


369.0237881 3.638788 1367.459 1794.673 2262.109 2743.658 3323.719 3923.17 4442.488 4961.438 5485.529 6304.463811

54.41510047 60.96969 31.57286 11.40302 -0.4125 14.28218 19.29059 48.85662 23.7069 5.661726 29.70901 175.0078802

314.6086877 -57.3309 1335.886 1783.27 2262.521 2729.376 3304.429 3874.313 4418.781 4955.777 5455.82 6129.455931





The following (Table 13) shows the random component for

year 2017.

Third Step: The resulting predicted values from Markov

chains Transformation matrix we will add to Random

Component. The following (Table 14) shows predicted

values from Markov chains Transformation matrix for year

2017.

Now we have this equation to get the result from combing

two methods

( ) ,ˆ ˆ (1 )Markov chains Transformati xt t io jn matri j jy y t s r

The prediction year 2017 Show in (Table 15) above. The

comparison between forecasted and actual production and

consumption of electricity in China from 2016 to 2019 by

proposed method was summarized in Appendix 1 and

Appendix 2.

V. CONCLUSION

The forecasts obtained utilizing Markov developed by

Fiering and Thomas, Markov chains Transformation, Fuzzy

Time Series Markov Chain, and proposed method by

combining two methods Markov developed by Fiering and

Thomas and Markov chains Transformation are discussed

in the present section. The aforementioned methods require

only the historical data series of electricity consumption to

build the forecast. This can be considered as an important

advantage, because the effort and cost linked to the data

mining are very limited [2]. These historical time series

data are analyzed to understand the past and predict the

future.

The results of predictive metrics of electricity production

and consumption indicated that Mean Absolute Percentage

Error Markov developed by Fiering and Thomas, Markov

chains Transformation, Fuzzy Time Series Markov Chain,

and proposed method (Table 6). So, this criterion clearly

indicated the superiority of Markov chains Transformation

Matrix in forecasting the

Consumption of electricity during 2016-2019 Similarly, in

case the MAPE of proposed method gave the lowest in the

production electricity during 2016-2019.

Similar to the prediction of electric energy production,

proposed method performed best when the consumption of

electric energy was predicted Markov chains

Transformation Matrix. But it is quite interesting that in

case of forecasting the electric energy consumption,

Markov chains Transformation Matrix performed better

than that of Fuzzy Time Series-Markov.



Statistical model forecast the future demand and production

by a mathematical combination of previous demand and

production as well as incorporate other exogenous factors

like weather conditions, seasonality etc. The forecasting

accuracy depends not only on the numerical efficiency of

the algorithms employed, but also on the quality of data

analyzed and the ability to incorporate important

fundamental factors, such as historical demand, demand

and consumption forecasts, weather forecasts of fuel prices

[1]. In our study, while all these factors were considered,

there is no doubt that Markov chain model performed the

best forecasting. So, this model was used to create a balance

sheet between future forecasting for electric energy

production and consumption.

The evaluation of the models for the consumption of

electricity showed that Markov chains Transformation

Matrix performed best with the lowest MAPE (1.28%) and

Fuzzy Time Series-Markov the worst performance with the

highest MAPE (4.21%) , (Table 6). The evaluation for the

production of electricity showed that Proposed Method

performed best with MAPE (2.84 %) while Fuzzy Time

Series-Markov had the worst performance with a MAPE

(9.14 %) (Table 16)

The reason for Proposed Method get the best MAPE in

production electric but Markov chains Transformation

Matrix get the best in consumption electric because when

we combine Markov chain developed by Fiering and

Thomas and Markov chains Transformation Matrix we note

that the random numbers in Markov chain developed by

Fiering and Thomas for production electric the most it is

less than 0.5 so we get the best MAPE but in the

consumption electric we note that the random numbers the

most more than 0.5 so we do not reach the best MPAE from

our Proposed Method .

Table 1 6.The values of evaluation metrics per Model

This paper compared several models to forecast production

and consumption electric energy for the period of 2016 to

2019. This study has fulfilled the objectives of the study to

propose the production and consumption of future

electricity by different forecasting methods like Markov

chain and Exponential Smoothing models, and then inspect

the accuracy of both models in forecasting ability.

After examining several models, it was found that that

Markov chains Transformation Matrix was the best for

consumption electric and our and Proposed Method the

best in production electric the both most appropriate to

apply to study data series of production and consumption

electricity. By analyzing the forecasted value using the

performance evaluation procedure, it is found that use of

Markov chains Transformation Matrix and Proposed

Method for forecasting production and consumption

electricity is better than Markov developed by Fiering and

Thomas and Fuzzy Time Series-Markov .

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85, 1962.



Appendix 1. Actual and forecasted values of production of electricity (GWh) during 2016-2019

Model

Month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Y

ear

Act

ual

Fo

reca

st

Act

ual

Fo

reca

st

Act

ual

Fo

reca

st

Act

ual

Fo

reca

st

Act

ual

Fo

reca

st

Act

ual

Fo

reca

st

Act

ual

Fo

reca

st

Act

ual

Fo

reca

st

Act

ual

Fo

reca

st

Act

ual

Fo

reca

st

Act

ual

Fo

reca

st

Act

ual

Fo

reca

st

Ma

rkov

dev

elo

pe

d b

y

Fie

rin

g

an

d

Th

om

as 2016 435 549 435 1182 1355 1651 1799 2066 2268 2537 2759 3053 3312 3580 3877 4109 4373 4532 4865 4988 5370 5476 6133 6341

2107 466 530 466 752 1459 1699 1938 2109 2437 2578 2960 3096 3570 3684 4166 4282 4689 4718 5194 5184 5712 5722 6604 6720

2018 523 605 523 730 1576 1781 2088 2237 2636 2760 3195 3294 3837 3944 4480 4542 5036 5013 5582 5505 6163 6050 7112 7052

2019 545 647 545 797 1675 1936 2220 2403 2781 2955 3367 3530 4030 4200 4703 4837 5297 5371 5874 5912 6480 6518 7503 7577

Ma

rkov

cha

ins

Tra

nsf

orm

ati

on

Ma

trix

2016 435 586 435 945 1355 1341 1799 1775 2268 2235 2759 2750 3312 3236 3877 3768 4373 4220 4865 4675 5370 5165 6133 5790

2107 466 423 466 65 1459 1399 1938 1806 2437 2262 2960 2758 3570 3343 4166 3972 4689 4466 5194 4967 5712 5515 6604 6479

2018 523 582 523 474 1576 1473 2088 1946 2636 2453 3195 2972 3837 3608 4480 4193 5036 4691 5582 5170 6163 5683 7112 6666

2019 545 678 545 561 1675 1627 2220 2102 2781 2652 3367 3211 4030 3838 4703 4470 5297 5033 5874 5582 6480 6201 7503 7160

Fu

zzy

Tim

e

Ser

ies-

Ma

rkov

2016 435 882 435 1085 1355 1803 1799 2432 2268 3019 2759 3100 3312 3567 3877 4383 4373 4894 4865 5466 5370 5596 6133 4692

2107 466 392 466 851 1459 1814 1938 2438 2437 3027 2960 3669 3570 3669 4166 4281 4689 4894 5194 5501 5712 6374 6604 4692

2018 523 358 523 868 1576 1891 2088 2359 2636 3055 3195 3669 3837 4383 4480 4894 5036 5472 5582 5607 6163 6323 7112 3975

2019 545 900 545 900 1675 2452 2220 2486 2781 3088 3367 3669 4030 4281 4703 4894 5297 5581 5874 6323 6480 4692 7503 606

Pro

po

sed

Met

ho

d 2016 435 485 435 800 1355 1284 1799 1760 2268 2220 2759 2705 3312 3221 3877 3739 4373 4194 4865 4627 5370 5107 6133 5560

2107 466 315 1185 -57 1459 1336 1938 1783 2437 2263 2960 2729 3570 3304 4166 3874 4689 4419 5194 4956 5712 5456 6604 6129

2018 523 348 523 456 1576 1454 2088 1948 2636 2429 3195 2949 3837 3564 4480 4154 5036 4686 5582 5176 6163 5652 7112 6593

2019 545 406 545 523 1675 1533 2220 2071 2781 2638 3367 3186 4030 3818 4703 4471 5297 5006 5874 5549 6480 6135 7503 7051



Appendix 2. Actual and forecasted values of consumption of electricity (GWh) during 2016-2019

Model

Month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Y

ear

Act

ual

Fo

reca

st

Act

ual

Fo

reca

st

Act

ual

Fo

reca

st

Act

ual

Fo

reca

st

Act

ual

Fo

reca

st

Act

ual

Fo

reca

st

Act

ual

Fo

reca

st

Act

ual

Fo

reca

st

Act

ual

Fo

reca

st

Act

ual

Fo

reca

st

Act

ual

Fo

reca

st

Act

ual

Fo

reca

st

Ma

rkov

dev

elo

pe

d b

y

Fie

rin

g

an

d

Th

om

as 2016 486 632 845 1085 1290 1598 1732 2055 2189 2513 2662 2996 3167 3523 3678 4053 4134 4480 4584 4922 5049 5417 5802 6213

2107 487 933 876 1115 1352 1662 1809 2116 2282 2604 2776 3107 3329 3693 3892 4279 4389 4733 4878 5205 5385 5716 6130 6629

2018 487 820 936 1130 1446 1760 1931 2245 2426 2729 2951 3287 3558 3928 4157 4540 4689 5019 5202 5548 5733 6073 6364 6827

2019 600 690 1055 1263 1588 1930 2109 2430 2663 2979 3229 3566 3878 4233 4530 4916 5106 5444 5655 5994 6220 6545 6900 7350

Ma

rkov

cha

ins

Tra

nsf

orm

ati

on

Ma

trix

2016 486 436 845 834 1290 1284 1732 1707 2189 2174 2662 2651 3167 3151 3678 3649 4134 4094 4584 4536 5049 4973 5802 5868

2107 487 446 876 840 1352 1343 1809 1791 2282 2249 2776 2737 3329 3308 3892 3874 4389 4394 4878 4907 5385 5429 6130 6030

2018 487 175 936 965 1446 1452 1931 1934 2426 2443 2951 2946 3558 3569 4157 4161 4689 4717 5202 5202 5733 5737 6364 6163

2019 600 562 1055 1095 1588 1563 2109 2096 2663 2675 3229 3225 3878 3872 4530 4496 5106 5084 5655 5616 6220 6184 6900 6802

Fu

zzy

Tim

e

Ser

ies-

Ma

rkov

2016 486 537 845 1083 1290 1847 1732 2350 2189 3019 2662 2998 3167 3771 3678 3425 4134 4806 4584 5386 5049 5481 5802 5124

2107 487 414 876 925 1352 1854 1809 2362 2282 3027 2776 3771 3329 3669 3892 3476 4389 4841 4878 5378 5385 4792 6130 5204

2018 487 426 936 1038 1446 1769 1931 2401 2426 3055 2951 3771 3558 3425 4157 4799 4689 5394 5202 5495 5733 4741 6364 657

2019 600 958 1055 1060 1588 2298 2109 2342 2663 3088 3229 3771 3878 3425 4530 4811 5106 5463 5655 4741 6220 5262 6900 51

Pro

po

sed

Met

ho

d 2016 486 436 845 834 1290 1284 1732 1707 2189 2174 2662 2651 3167 3151 3678 3649 4134 4094 4584 4536 5049 4973 5802 5868

2107 487 446 876 840 1352 1343 1809 1791 2282 2249 2776 2737 3329 3308 3892 3874 4389 4394 4878 4907 5385 5429 6130 6030

2018 487 175 936 965 1446 1452 1931 1934 2426 2443 2951 2946 3558 3569 4157 4161 4689 4717 5202 5202 5733 5737 6364 6163

2019 600 562 1055 1095 1588 1563 2109 2096 2663 2675 3229 3225 3878 3872 4530 4496 5106 5084 5655 5616 6220 6184 6900 6802

comparison between markov chain techniques for future

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