Average Based-FTS Markov Chain With  Modifications to the  Frequency Density Partition to Predict COVID-19 in Central Java


CAUCHY –Jurnal Matematika Murni dan Aplikasi 
Volume 7(2) (2022), Pages 231-239 
p-ISSN: 2086-0382; e-ISSN: 2477-3344 

Submitted: September 18, 2021 Reviewed: December 08, 2021 Accepted: December 21, 2021 
DOI: http://dx.doi.org/10.18860/ca.v7i1.13371     

Average Based-FTS Markov Chain With  Modifications to the  
Frequency Density Partition to Predict COVID-19 in Central Java  

Susilo Hariyanto*, Zaenurrohman, Titi Udjiani SRRM  

Department of Mathematics, Faculty Science and Mathematics, Diponegoro University, 
Indonesia 

 

*Corresponding Author 
Email: susilohariyanto@lecturer.undip.ac.id*,    

zaenurrohman8.zr@gmail.com, 
udjianititi@yahoo.com  

ABSTRACT 

Every day, new Covid-19 positive cases are discovered in Central Java. Many research has used 
various methodologies to try to forecast new positive instances. The fuzzy time series (FTS) 
approach is one of them. Many FTS are now in development, including the FTS Markov Chain. 
The duration of the gap in the FTS must be determined carefully because it will affect the FLR, 
which will be used to estimate the forecast value. The average-based method can be used to 
determine the optimum interval length; however, other research use frequency density 
partitioning to determine the optimal interval length in order to produce superior predicting 
values. The goal of this research is to improve the accuracy of forecasting values by modifying 
the frequency density partition on the average based-FTS Markov Chain. The approach utilized is 
average-based, with the length of the interval determined by the average, the forecast value 
determined by the FTS Markov Chain, and the frequency density partition modified to provide 
the ideal interval. The average-based FTS Markov Chain approach with adjustments to the 
frequency density partition achieves an accuracy rate of 89.3 percent, according to the findings 
of this study. Because changes to the frequency density partition can produce a good level of 
accuracy in forecasting new positive cases of Covid-19 in Central Java, it is hoped that this 
modification of the frequency density partition on the average-based FTS Markov chain can be 
used as a model for forecasting in fields other than new positive cases. Covid-19. 

Keywords: Average Based; FTS Markov Chain; Modified frequency density partitioning; 
COVID19; MAPE  

INTRODUCTION 

COVID-19 first appeared in the city of Wuhan, Hubei Province, China, which 
spread almost all over the world, including Indonesia. At the beginning of 2020, 
Indonesia experienced a COVID-19 pandemic, which, to this day, new positive cases are 
still being found[1]. The government is still thinking about how to make Indonesia free 
from COVID-19. 

Many experts have estimated the amount of new Covid-19 positive cases in 
Indonesia. Forecasting is the process of predicting what will happen in the future over a 
lengthy period of time[2]. However, no approach for accurately forecasting anything, 

http://dx.doi.org/10.18860/ca.v7i1.13371
mailto:susilohariyanto@lecturer.undip.ac.id*
mailto:zaenurrohman8.zr@gmail.com
mailto:udjianititi@yahoo.com


Average Based-FTS Markov Chain With  Modifications to the  Frequency Density Partition to 
Predict COVID-19 in Central Java 

Susilo Hariyanto 232 

including new Covid-19 positive cases, has been developed to yet. The fuzzy time series 
(FTS) is one of the forecasting approaches for determining the number of new positive 
cases in Indonesia. The concept of fuzzy logic is used in the forecasting of FTS.  Song and 
Chissom introduced the FTS in 1993[3], and it has since been widely developed, 
including the Markov method[4], Chen’s method[5], Chen and Hsu’s method[6], the 
weighted method[7], the multiple-attribute fuzzy time series method[8], the percentage 
change method[9], and Markov chain method[10]. 

Ruey Chyn Tsaur, developing fuzzy time series by merging the fuzzy time series 
method with the Markov chain concept [10].  Markov chain is a stochastic process in 
which future events only depend on today's events. Markov chain is used in the 
defuzzification process [11]. 

The determination of the length of the interval in the fuzzy time series does not 
have a definite formula, but the determination of the length of the interval in the fuzzy 
time series is based on the researcher. As a result, even if each researcher is utilizing the 
same data, the length of the interval will vary[3]. Even though the determination of the 
length of the interval is a very influential part in the formation of a fuzzy logical 
relationship (FLR).[12].  One method that can be used to determine the length of the 
interval is the average based. This average based uses an average-based method in 
determining the length of the interval [12]. 

Chen and Hsu in 2004 also developed a fuzzy time series. Chen and Hsu 
developed fuzzy time series by repartitioning based on frequency density. Chen and 
Hsu's frequency density repartitioning algorithm divides the interval with the highest 
frequency density into four sub-intervals, the interval with the second highest density 
into three sub-intervals, the interval with the third highest density into two sub-
intervals, and the interval with the lowest density into one sub-interval.[6][13]. 

Based on the description above, the researcher is interested in modifying the 
frequency density partitioning algorithm used by Chen and Hsu, namely by exchanging 
the partition between the interval with the first densest frequency with the third 
densest interval, which was originally the first densest interval partitioned into 4 sub-
intervals, the researcher changed the first densest was partitioned into 2 sub-intervals, 
and for the interval with the third densest frequency initially partitioned into 2 sub-
intervals the researcher changed it to 4 sub-intervals. 

Furthermore, the researcher will use the average-based method to determine the 
interval length in the fuzzy time series type Markov chain, and apply it to forecasting 
new positive cases of COVID-19 in Central Java.  

 

METHODS 

Fuzzy Time Series 

The definition of fuzzy time series was first introduced by Song dan Chisom 
(1993). Let 𝑈 universe of discourse, with 𝑈 = {𝑢1, 𝑢2, … , 𝑢𝑛} on a fuzzy set 𝐴𝑖 , defined 
as[3]: 

𝐴𝑖 =
𝑓𝐴(𝑢1)

𝑢1
+

𝑓𝐴(𝑢2)

𝑢2
+ ⋯ +  

𝑓𝐴(𝑢𝑛)

𝑢𝑛
 

(1) 

where 𝑓𝐴 is theemembership ofkthe fuzzypset 𝐴𝑖  and 𝑢𝑘  is anxelementuof the fuzzy set 
𝐴𝑖  and 𝑓𝐴(𝑢𝑘 ) shows the degreebof membership of 𝑢𝑘  in 𝐴𝑖  where 𝑘 = 1,2,3, … , 𝑛. 
Definition. If 𝐹(𝑡) is causeddby 𝐹(𝑡 −  1), then thecrelation in the first orderrmodel 
𝐹(𝑡)  cantbe stated assfollows: [5]. 



Average Based-FTS Markov Chain With  Modifications to the  Frequency Density Partition to 
Predict COVID-19 in Central Java 

Susilo Hariyanto 233 

𝐹(𝑡)  =  𝐹(𝑡 − 1) ○ 𝑅(𝑡, 𝑡 − 1) (2) 

where “○”  is Max-Min composition operator, and  𝑅(𝑡, 𝑡 − 1) is a relation matrix to 
describe the fuzzy relationship between 𝐹(𝑡 − 1) dan 𝐹(𝑡). 

Average-Based 

Average based is an algorithm that can be used to set the interval length that is 
determined at the initial stage of forecasting when using fuzzy time series. The steps of 
the average based algorithm are as follows [12], [14]: 

a. Determine the absolute difference (lag) between data 𝑛 + 1 and data 𝑛 with the 
formula: 

𝑙𝑎𝑔𝐷𝑛 = |(𝐷𝑎𝑡𝑎 𝑛 + 1) − (𝐷𝑎𝑡𝑎 𝑛)| (3) 
b. Determine the length of the interval 

𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 =  (
𝑡𝑜𝑡𝑎𝑙 𝑙𝑎𝑔

𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑜𝑓 𝑑𝑎𝑡𝑎
) : 2 

(4) 

c. Determine the basis value of the interval length according to Table 1. Following: 
Table 1. Basis Mapping Table 

Range Basis 
0,1 – 1,0 0,1 
1,1 -10 1 
11-100 10 

101-1000 100 

d. The length of the interval is then rounded up according to the interval basis table. 

Modification Frequency Density Partition 

In this study, modifications to the frequency density partition were used, the 
algorithm used is as follows: 

a. The interval with the first densest frequency is divided into 2 subintervals. 
b. The interval with the second densest frequency is divided into 3 subintervals. 
c. The interval with the third densest frequency is divided into 4 subintervals 
d. Eliminates intervals that have no frequency. 

Fuzzy Time Series Markov Chain 

Markov Chain's Fuzzy Time Series forecasting procedure is as follows[10]: 
Step 1. Collecting historical data (𝑌𝑡). 
Step 2. Defines the U universe set of data, with D1 and D2 being the corresponding 

positive numbers. 
𝑈 =  [𝐷𝑚𝑖𝑛 − D1, 𝐷𝑚𝑎𝑥 + D2] (5) 

Step 3. Specify the number of fuzzy intervals. 
Step 4. Defining the fuzzy set in the universe of discourse U, the Fuzzy Ai set declares 

the linguistic variable of the share price by 1 ≤  𝑖 ≤  𝑛. 
Step 5. Fuzzification of historical data. If a time series data is included in the 𝑢𝑖  

interval, then that data is fuzzification into 𝐴𝑖 . 
Step 6. Specifies fuzzy logical relationship (FLR) and Fuzzy Logical Relationships 

Group (FLRG). 
Step 7. Calculate forecasting results 

For time series data, using FLRG, a probability can be obtained from a 
state heading to the next state. In order to calculate the predicting value, a 
Markov probability transition matrix with a dimension of 𝑛 𝑥 𝑛 was used. If 



Average Based-FTS Markov Chain With  Modifications to the  Frequency Density Partition to 
Predict COVID-19 in Central Java 

Susilo Hariyanto 234 

state 𝐴𝑖  transition to a state 𝐴𝑗  and pass the state 𝐴𝑘 , 𝑖, 𝑗 =  1, 2, . . . , 𝑛, then we 

can obtain FLRG. The transition probability formula is as follows: 

𝑃𝑖𝑗  =  
𝑀𝑖𝑗

𝑀𝑖
, 𝑖, 𝑗 =  1, 2, … , 𝑛 

(6) 

 
with:  
𝑃𝑖𝑗  = probability of transition from state 𝐴𝑖  to state 𝐴𝑗  one step 

𝑀𝑖𝑗 = number of transitions from state 𝐴𝑖  to state 𝐴𝑗  one step 

𝑀𝑖  = the amount of data included in the 𝐴𝑖  
The probability matrix R of all states can be written as follows: 
  

𝑅 =  [
𝑃11 ⋯ 𝑃1𝑛

⋮ ⋱ ⋮
𝑃𝑛1 ⋯ 𝑃𝑛𝑛

] 
(7) 

Matrix R reflects the transition of the entire system. If 𝐹(𝑡 − 1) = 𝐴𝑖 , 
then the process will be defined in the 𝐴𝑖  at the time of (𝑡 − 1), then the 
forecasting results 𝐹(𝑡) will be calculated using the [𝑃𝑖1, 𝑃𝑖2, … , 𝑃𝑖𝑛 ] on the 
matrix R. Forecasting results 𝐹(𝑡) is the weighted average value of the 𝑚1, 
𝑚2, ..., 𝑚𝑛 (midpoint of 𝑢1, 𝑢2, ..., 𝑢𝑛 ). The forecasting output result value on 
𝐹(𝑡) can be determined by using the following rules:  
Rule 1: if fuzzy logical relationship group 𝐴𝑖  is one-to-one (suppose 𝐴𝑖  →  𝐴𝑘  
where 𝑃𝑖𝑘  =  1 and 𝑃𝑖𝑗  =  0, 𝑗 ≠  𝑘) then the forecasting value of 𝐹(𝑡) is 𝑚𝑘  

the middle value of the 𝑢𝑘 . 
𝐹(𝑡)  =  𝑚𝑘 𝑃𝑖𝑘  =  𝑚𝑘  (8) 

Rulet2: if the FLRG 𝐴𝑖  is one-to-many (e.g. 𝐴𝑗  →  𝐴1, 𝐴2, . . . , 𝐴𝑛 .  𝑗 =

 1, 2, . . . , 𝑛), when 𝑌(𝑡 − 1) at time (𝑡 − 1) is included in state 𝐴𝑗  then the 

forecasting 𝐹(𝑡), is: 
𝐹(𝑡) =  𝑚1𝑃𝑗1  +  𝑚2𝑃𝑗2  +  …  +  𝑚𝑗−1𝑃𝑗(𝑗−1)  +  𝑌(𝑡

− 1)𝑃𝑗𝑗  +  𝑚𝑗+1𝑃𝑗(𝑗+1) +  …  +  𝑚𝑛𝑃𝑗𝑛  
(9) 

where 𝑚1, 𝑚2, . . . , 𝑚𝑗−1, 𝑚𝑗+1, … , 𝑚𝑛 is the middle value 

𝑢1, 𝑢2, . . . , 𝑢𝑗−1, 𝑢𝑗+1, . . . , 𝑢𝑛 , and  𝑌(𝑡 − 1) are state values 𝐴𝑗  at time 𝑡 − 1. 

Rule 3: if the FLRG 𝐴𝑖  is empty (𝐴𝑖 → ∅) forecast value 𝐹(𝑡) is 𝑚𝑖  which is the 
middle value of 𝑢𝑖  with the following equation: 

𝐹(𝑡) = 𝑚𝑖  
(10) 

Step 8. Adjusting the trend of forecasting values with the following rules: 
 If state 𝐴𝑖  communicates with 𝐴𝑖 , startingffrom state 𝐴𝑖  at time 𝑡 − 1 

expressed as 𝐹(𝑡 − 1)   =  𝐴𝑖 , and undergoing an increasingatransition to 
state 𝐴𝑗  at the time 𝑡 where (𝑖 < 𝑗), thendthe adjustment value is: 

𝐷𝑡1 =  (
𝑙

2
) 

(11) 

where 𝑙 is the basis interval.  
 Ifestate 𝐴𝑖  communicatesswith 𝐴𝑖 , startinggfrom state 𝐴𝑖  atpthestime 𝑡 − 1 

expressed as 𝐹(𝑡 − 1)   =  𝐴𝑖 , anddexperiencing a 
decreasingmtransitionxto state 𝐴𝑗  atzthevtime 𝑡 where (𝑖 >  𝑗) the 

adjustment valueeis: 

𝐷𝑡1  =  − (
𝑙

2
) 

(12) 



Average Based-FTS Markov Chain With  Modifications to the  Frequency Density Partition to 
Predict COVID-19 in Central Java 

Susilo Hariyanto 235 

 Ifzstate 𝐴𝑖  at the time 𝑡 − 1 is expressed 𝐹(𝑡 − 1)  =  𝐴𝑖 , and undergoes a 
jumpbforward transitionrtoqstate 𝐴𝑖+𝑠 atythegtime 𝑡 where (1 ≤  𝑠 ≤
 𝑛 − 𝑖) then the adjustment value is:  

𝐷𝑡2 =  (
𝑙

2
)𝑠 

(13) 

where 𝑠 is the number of forward jumps. 
 If state 𝐴𝑖  atpthemtime 𝑡 − 1 is as 𝐹(𝑡 − 1)  =  𝐴𝑖 , andhundergoes anjump-

backwardxtransition topstate 𝐴𝑖−𝑣  at thertime 𝑡 where (1 ≤  𝑣 ≤ 𝑖) 
thenlthe adjustment value is: 

𝐷𝑡2  =  − (
𝑙

2
)𝑣 

(14) 

where 𝑣 is the number of jumps backward. 
Step 9. Determine the final forecast value based on the adjustment of the trend of 

the forecasting value 
If FLRG 𝐴𝑖  is one-to-many and state 𝐴𝑖+1 can be accessed from state 

𝐴𝑖  where state 𝐴𝑖  is related to 𝐴𝑖  then the forecasting result becomes ’(𝑡) =

 𝐹(𝑡) + 𝐷𝑡1 + 𝐷𝑡2 =  𝐹(𝑡)  +  (
𝑙

2
)  +  (

𝑙

2
) . IfQFLRG 𝐴𝑖  isoone-too-many 

andastate 𝐴𝑖+1 can be accessed from 𝐴𝑖  wherewstate 𝐴𝑖  is not related to 𝐴𝑖  

then the forecasting values becomes 𝐹’(𝑡) =  𝐹(𝑡) + 𝐷𝑡2  =  𝐹(𝑡) +  (
𝑙

2
).  If 

FLRG 𝐴𝑖  is one to many and state 𝐴𝑖−2 can be accessed from state 𝐴𝑖  where 𝐴𝑖  
is not related to 𝐴𝑖  then the forecasting result is 𝐹’(𝑡) =  𝐹(𝑡) −  𝐷𝑡2 =

𝐹(𝑡)  −  (
𝑙

2
) 𝑥 2   =  𝐹(𝑡)–  𝑙.  If 𝑣 is jumpxstep, theggeneralmform ofpthe 

forecast is:    

𝐹’(𝑡)  =  𝐹(𝑡)  ±  𝐷𝑡1  ±  𝐷𝑡2  =  𝐹(𝑡)  ±  (
𝑙

2
)  ±  (

𝑙

2
) 𝑣. 

(15) 

Forecasting Error Measurement 
The reliability of a forecast can be determined by looking at mean average 

percentage error (MAPE), this MAPE formulas[15]: 

𝑀𝐴𝑃𝐸 =
1

𝑛
∑

|𝑌(𝑡) − 𝐹′(𝑡)|

𝑌(𝑡)
𝑥 100%

𝑛

𝑡=1

 (16) 

with 𝑌𝑡  : actual data period 𝑡, 𝐹′𝑡 : 𝑡 period forecasting value, and 𝑛  : the 
predictable amount of data. 

 

RESULTS AND DISCUSSION 

Forecasting with an average based-FTS Markov Chain with modified frequency 
density partitioning, the first step is to collect COVID-19 in the Central Java period June 
25, 2021 until August 20, 2021 as a universe discourse (U). Next, determine the greatest 
value (𝐷𝑚𝑎𝑥 = 5655) and smallest values (𝐷𝑚𝑖𝑛 = 1428), and the value of D1 = 8 and D2 
= 5, so it can be defined 𝑈 = [1428 − 8, 5655 + 5] = [1420, 5660]. Then, calculate the 
absolute difference from historical data, the average absolute difference from 57 data 
points is 658,554, which is then divided by 2 to yield 329,277. The value of 329,277 is 
then determined using Table 1. The basis of the length of the interval is 100, so that U 
can be partitioned into the same interval length, namely 
𝑢1, 𝑢2, 𝑢3, 𝑢4, 𝑢5, … , 𝑢39, 𝑢40, 𝑢41, 𝑢42, 𝑢43 successively the value for each interval is  
  



Average Based-FTS Markov Chain With  Modifications to the  Frequency Density Partition to 
Predict COVID-19 in Central Java 

Susilo Hariyanto 236 

Table 2. Universe Discourse of New Positive Cases of Covid-19 
𝑢1 = [1420, 1520] ⋮ 
𝑢2 = [1520, 1620] 𝑢41 = [5420, 5520] 
𝑢3 = [1620, 1720] 𝑢42 = [5520,5620] 
𝑢4 = [1720, 1820] 𝑢43 = [5620, 5720] 

 
The next step is to distribute the data to each interval and determine the 

frequency density, resulting in the densest interval, which is then repartitioned using a 
modified method. The following outcomes were achieved: 

Table 3. Frequency Density and Repartition Interval 
Interval Frequency Redivided interval 

𝑢29 = [4220, 4320] 5 𝑢29,1 = [4220, 4270], 

𝑢29,2 = [4270, 4320]  
𝑢28 = [4120, 4220] 4 𝑢28,1 = [4120, 4153.33]  

 𝑢28,2 = [4153.33, 4186.67] 

𝑢28,3 = [4186.67, 4220]  
𝑢16 = [2920, 3020] 3 𝑢16,1 = [2920, 2945], 

𝑢16,2 = [2945, 2970],  

𝑢16,3 = [2970, 2995]  

𝑢16,3 = [2995, 3020]  
𝑢17 = [3020, 3120] 3 𝑢17,1 = [3020, 3045], 

𝑢17,2 = [3045, 3070],  

𝑢17,3 = [3070, 3095], 

𝑢17,4 = [3095, 3120]  
𝑢19 = [3220, 3320] 3 𝑢19,1 = [3220, 3245], 

𝑢19,2 = [3245, 3270],  

𝑢19,3 = [3270, 3295], 

𝑢19,4 = [3295, 3320]  
𝑢27 = [4020, 4120] 3 𝑢27,1 = [4020, 4045], 

𝑢27,2 = [4045, 4070],  

𝑢27,3 = [4070, 4095], 

𝑢27,4 = [4095, 4120]  
𝑢32 = [4520, 4620] 3 𝑢32,1 = [4520, 4545], 

𝑢32,2 = [4545, 4570],  

𝑢32,3 = [4570, 4590], 

𝑢32,4 = [4590, 4620]  
𝑢33 = [4620, 4720] 3 𝑢33,1 = [4620, 4645], 

𝑢33,2 = [4645, 4670],  

𝑢33,3 = [4670, 4695], 

𝑢33,4 = [4695, 4720]  

𝑢2, 𝑢3, 𝑢4, 𝑢5, 𝑢6, 𝑢8, 𝑢11, 

𝑢14, 𝑢20, 𝑢23, 𝑢26, 𝑢34, 𝑢42 

0 Removed 

 
Next, look for the middle value (𝑚1) for each interval, we get: 
 

Table 4. Middle Value 
𝒖𝒊 𝒎𝒊 𝒖𝒊 𝒎𝒊 
𝑢1 1470 ⋮ ⋮ 

𝑢7 2070 𝑢39 5270 

𝑢9 2270 𝑢40 5370 
𝑢10 2370 𝑢41 5470 
𝑢12 2570 𝑢43 5670 

 



Average Based-FTS Markov Chain With  Modifications to the  Frequency Density Partition to 
Predict COVID-19 in Central Java 

Susilo Hariyanto 237 

Furthermore, defining fuzzy sets, fuzzy sets that can be formed from the universe 
conversation are 44 fuzzy sets. The fuzzy sets formed is as follows: 

𝐴1 = {
1

𝑢1⁄ +
0,5

𝑢7
⁄ + 0 𝑢9⁄ +

0
𝑢10⁄ + ⋯ +

0
𝑢40⁄ +

0
𝑢41⁄ +

0
𝑢43⁄ } 

𝐴2 = {
0,5

𝑢1
⁄ + 1 𝑢7⁄ +

0,5
𝑢9

⁄ + 0 𝑢10⁄ + ⋯ +
0

𝑢40⁄ +
0

𝑢41⁄ +
0

𝑢43⁄ } 

𝐴3 = {
0

𝑢1⁄ +
0,5

𝑢7
⁄ + 1 𝑢9⁄ +

0,5
𝑢10

⁄ + ⋯ + 0 𝑢40⁄ +
0

𝑢41⁄ +
0

𝑢43⁄ } 

⋮ 

𝐴49 = {
0

𝑢1⁄ +
0

𝑢7⁄ +
0

𝑢9⁄ +
0

𝑢10⁄ + ⋯ +
1

𝑢40⁄ +
0,5

𝑢41
⁄ + 0 𝑢43⁄ } 

𝐴50 = {
0

𝑢1⁄ +
0

𝑢7⁄ +
0

𝑢9⁄ +
0

𝑢10⁄ + ⋯ +
0,5

𝑢40
⁄ + 1 𝑢41⁄ +

0,5
𝑢43

⁄ } 

𝐴51 = {
0

𝑢1⁄ +
0

𝑢7⁄ +
0

𝑢9⁄ +
0

𝑢10⁄ + ⋯ +
0

𝑢40⁄ +
0,5

𝑢41
⁄ + 1 𝑢43⁄ } 

The next step is to perform fuzzification, the data from the fuzzification results 
are presented in the following table: 

Table 5. Fuzzification Results 

t 
Actual 
Data 

fuzzy 
Data 

t 
Actual 
Data 

fuzzy 
Data 

1 2311 𝐴3 ⋮ ⋮ ⋮ 
2 2064 𝐴2 54 3263 𝐴18 
3 3079 𝐴14 55 3078 𝐴14 
4 2702 𝐴6 56 1428 𝐴1 
5 2932 𝐴8 57 1432 𝐴1 

The next step, determine the FLR and FLRG, as shown in Table 6. And Table 7: 
 

Table 6. FLR 

Data 
Order 

FLR Data Order FLR 

1-2 𝐴3 → 𝐴2 ⋮ ⋮ 
2-3 𝐴2 → 𝐴14 54-55 𝐴18 → 𝐴14 
3-4 𝐴14 → 𝐴6 55-56 𝐴14 → 𝐴1 
4-5 𝐴6 → 𝐴8 56-57 𝐴1 → 𝐴1 

 

Table 7. FLRG 

Current 
State 

Next State Current 
State 

Next State 

𝐴1 (1)𝐴1 ⋮ ⋮ 
𝐴2 (1)𝐴14 𝐴49 (1)𝐴35, (1)𝐴42 
𝐴3 (1)𝐴2 𝐴50 (1)𝐴48 
𝐴4 (1)𝐴6 𝐴51 (1)𝐴43 

 
The initial forecast will be calculated next. For example, for 𝑡 = 2, June 26, 2021, 

the forecast computation based on formulas (8), (9), and (10) is 𝐹(2) =  𝑚𝑘 𝑃𝑖𝑘  =  𝑚𝑘 =
2070. The summary of the initial forecasting results is as follows: 

 
Table 8. Initial Forecasting Results (𝐹(𝑡)) 

Period Actual Data 𝑭(𝒕) Period 
Actual 
Data 

𝑭(𝒕) 

6/25/21 2311 Na ⋮ ⋮ ⋮ 
6/26/21 2064 2070 8/19/21 1428 2070 
6/27/21 3078 3082,5 8/20/21 1432 1428 



Average Based-FTS Markov Chain With  Modifications to the  Frequency Density Partition to 
Predict COVID-19 in Central Java 

Susilo Hariyanto 238 

After we get the initial forecasting, the next step is to adjust the forecasting trend. 
For example, adjustment value for June 26, 2021, the next step is 𝐴2 and the current 
state is 𝐴3 then the adjustment calculation uses the forecast adjustment rule (14) we get 

𝐷𝑡2 = − (
𝑙

2
) 𝑣 = − (

100

2
) 1 = −(50). Dor the calculation of other forecasting value 

adjustment using equations (11), (12), (13), and (14). The following is a forecast 
adjustment Table. 

Table 8. Forecasting Trend Adjustment Value 
Period FLR 𝐷𝑡𝑛  Period FLR 𝐷𝑡2 

6/25/21 𝐴3 → 𝐴2 Na ⋮ ⋮ ⋮ 
6/26/21 𝐴2 → 𝐴14 -50 8/19/21 𝐴14 → 𝐴1 -650 
6/27/21 𝐴14 → 𝐴9 600 8/20/21 𝐴1 → 𝐴1 0 

 
Calculate the final forecast value. The final forecast is the sum of the initial 

forecast value with the forecast adjustment value by following the equation (15). For 
example, the final forecast value for June 26, 2021 data is 𝐹′2 = 𝐹2 ± 𝐷𝑡2 = 2070 +
(−50) = 2020. By doing the same way, the summary of the final forecasting result is as 
follows: 

 
Table. 9. Final Forecast Value 

Period Y(t) 𝐹′𝑡  Period Y(t) 𝐹′𝑡  
6/25/21 2311 Na ⋮ ⋮ ⋮ 
6/26/21 2064 2020 8/19/21 1428 1907.5 
6/27/21 3078 3682,5 8/20/21 1432 1428 

    
Furthermore, the average based-FTS Markov chain is based on the modified 

frequency density partition to forecast data for new positive cases on August 21, 2021, 
the current state is 𝐴1, from FLRG it is known that the next state of 𝐴1 is 𝐴1, then based 
on equations (8), (9), and (10) the forecasting result is 1 times the data of the previous 
new positive case 𝑌(𝑡 − 1) is 1432. 

The last step is to calculate the forecast accuracy value using MAPE. The MAPE values 
of average based-FTS based on a modified frequency density partitioning is 10,7%. For 
forecasting results using average based-FTS based on a modified frequency density 
partitioning, are presented in the following figure: 

 

 
 

 
Figure 1. Graph of Forecasting Results Using Average Based-FTS Markov Chain Based on Modified 

Frequency Density Partitioning 

0

1000

2000

3000

4000

5000

6000

7000

Actual Data

Forecasting Value



Average Based-FTS Markov Chain With  Modifications to the  Frequency Density Partition to 
Predict COVID-19 in Central Java 

Susilo Hariyanto 239 

 

CONCLUSIONS  

Forecasting new positive cases of COVID-19 in Central Java using Average Based-FTS 
Markov Chain Based on Modified Frequency Density Partitioning has a good level of 
accuracy, this can be seen from the MAPE value obtained which is 10.7%. And for the 
predicted value of new positive cases on August 21, 2021, it is 1432 new positive cases 
of COVID-19 in Central Java. 

 

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