Prediction of the Number of Language Users Ngapak (Penginyongan) Using Linier and Logistic Model

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Introduction
The local language is one of the riches of the Indonesian nation as well as a characteristic of a person or a community in interacting. This means that regional languages play a role in differentiating one ethnic group from another, and one culture from another, as well as one geographical area from another. Thus, the concept of language contains ideas about ethnicity, culture, and geographic area. The destruction of a regional language means the destruction of ethnicity, culture, and the loss of that ethnic group from its original geographical area (Alamsyah, 2018;Gusnawaty and Nurwati, 2019).
Even though the concepts contained in language are very superior, nowadays many people don't care about the existence of regional languages, so that the sense of belonging to their regional languages also begins to decrease. This is indicated by the increasing number of people interacting using their local languages, so there is a concern that regional languages will become extinct.
One of the regional languages in Central Java Province is Ngapak or Ngapak Banyumasan. The term ngapak, although already very popular, contains a pejorative meaning so that experts and scholars are trying to find another, the more positive name for the name of the language that has been known as the Ngapak language (Huda, 2017;Pawestri, 2019).
The Ngapak Banyumasan language is not only used by people who inhabit the geographical area of Banyumas Regency but is also used by people in areas called barlingmascakebayu (Banjarnegara, Purbalingga, Banyumas, Cilacap, Kebumen, and Bumiayu). Geographically, the area includes six districts in the southern part of Central Java, which are directly adjacent to the speakers of Sundanese and Cirebon languages. Therefore, the idea emerged to name the language with the name Penginyongan language. Communities throughout the region use the term inyong to refer to me so that the term penginyongan contains messages and a spirit of selfishness, ego, self-existence, and pride in the identity of their language. The First Congress of Penginyongan Languages was held in 2017.
The trend is observed, the number of users of the Language of Penginyongan decreases in the sense that these users are no longer consistent in using the language of Penginyongan but mixed with Indonesian. In addition to combining the two languages in daily conversation, the ability to speak Penginyongan is also decreasing. Ownership of Penginyongan language vocabulary individually is also declining so that communication with that language requires substitution from other languages such as Indonesian. Therefore, we need an estimate of the development of the language of Penginyongan to find out whether in the long term the language of Penginyongan will experience extinction due to the decrease or exhaustion of individuals who can use the language. To estimate the number of users (speakers) of the Penginyongan language, the calculations are carried out using linear and logistical models.

Discussion
Penginyongan language is one of the regional languages which has quite a large number of speakers among the regional languages found in Indonesia. However, recently some of its functions have been taken over by the Indonesian language which has been designated as the state language. On the other hand, the life of the Javanese people who live in geographically barlingmascakebayu has also experienced quite basic social changes in their lives. As a consequence, the use of the Penginyongan language is decreasing (Ramadhan and Masykur., 2018). To predict this, it is necessary to calculate the prediction (projection) of the population of Banyumas who are still able to use the Penginyongan language in 2017-2021 using linear and logistical models.
In this modeling, it is assumed that the extinction of a language is due to the extinction (reduction) of users or speakers of the language. A dictionary of the local language may be available, but there are no more speakers. Another assumption is that in this modeling every baby born to a mother with a Penginyongan language will automatically be in the Penginyongan language. This birth will increase the number of speakers. Furthermore, the number of people who die and leave the geographic area barlingmascakebayu (withdrawal) will cause a decrease in the number of speakers. Meanwhile, it is assumed that immigrants from outside the region after domiciling in the Barlingmascakeb area will not use the Penginyongan language, including their descendants. However, for simplicity of the model, it is assumed that the number of people who withdraw (emigration) and the number of people who enter the barlingmascakeb area (immigration) are the same so that in this model the growth or decay of the number of speakers of the Penginyongan language is only influenced by two factors, namely birth (characterized by birth rate). and death (characterized by the rate of death).

Linear Model
Suppose (0) or This method is used if it is assumed that the increase in population occurs linearly. Besides, equations (1) and (2) can also be used for the linear extrapolation method, i.e. prediction/projection for > . But for values that are large enough (eg > 2 ), this method should not be used to perform linear extrapolation (Brown, 1997), unless the researcher assumes that the increase in population numbers will continue to occur linearly (Solomons, 2019;Ziesemer, 2020).

Logistic Model
This model was first introduced by a mathematician and also a Dutch biologist, namely Pierre Verhulst in 1838. The discovery of this model was because the natural growth model was not sufficiently precise for a large enough population and limited space so that obstacles arose due to the density of the population that would reduce the population itself. As a result, population growth will take place stationary/constant after a certain time (Mujib et al., 2019;Budiono et al., 2020;Panigoro and Rahmi, 2020).
This logistic population growth model is a refinement of the exponential growth model. In this model, the population size is influenced by the size of the carrying capacity of the environment such as language skills and the consistency of using a language. The logistic model assumes that at a certain time the population will approach the equilibrium point. At this point, the numbers of births and deaths are assumed to be equal so that the graph is approximately constant. The simplest form of relative growth rate that accommodates this assumption is where is maximum carrying capacity and is growth rate. Multiply equation (3) by , then we get a model for population growth known as the logistic differential equation If is small compared to , then / approaches 0 and / ≈ . However, if → (population is close to its carrying capacity), then / → 1, so / → 1. If the population is between 0 and , then the right-hand side of the equation above is positive, so / → 1 and the population goes up. But if the population exceeds its carrying capacity ( > ), then 1 − ( / ) is negative, so / < 0 and the population decreases. Logistic equation solutions can be obtained through the following steps dP From equation (5), if we give the initial value = 0 and (0) = 0 , then the value = ln( 0 /( − 0 )) will be obtained, then the value is substituted back into equation (5), so that obtained a specific solution from the logistics model as follows where is the total population at time , 0 is the initial population size at = 0, is the capacity of an area for the population, is the per capita population growth rate, and is time. Equation (6) is a simple form of a specific solution for the logistics model that will be used in carrying out population projections. The function graph for the logistics method can be seen in Figure 1.

Figure 1: Graph of functions for logistic methods
To find out the level of accuracy of the model, the following error measures can be used. By using MAPE whose value is at intervals of 0% -100% percent, the smaller the MAPE value, the better the model obtained. The formula for Mean Absolute Percentage Error (MAPE) is where is the actual -th data value, is the -th predicted value, and is the amount of data used.
In determining the projection of the number of Penginyongan speakers in 2017-2021, data on the population in Banyumas Regency from 2010-2015 were used and data on the number of Banyumas residents who used the Penginyongan language in 2010-2015 (assuming the number of Banyumas residents who used the Penginyongan language was 90%. of the total population of Banyumas).
Modeling with data from the Banyumas Regency area is due to the unavailability of population data in the entire Barlingmascakeb geographic area. Likewise, there is no data on the number of residents in the Banyumas district who speak Penginyongan, so it is assumed that 90% of the total population in Banyumas Regency speaks Penginyongan.
Population data according to BPS Banyumas Regency in 2010-2015 along with the number of Banyumas residents who use the Penginyongan language in 2010-2015 are given in Table 1.
From the data available in Table 1, it can be made a model for Penginyongan language speakers which is then used to predict the number of speakers in the following years.

Linear Growth Model
For the linear model, the derivation of the formula is carried out as follows: (0)  By using a linear growth model, the estimated population of Banyumas using the Penginyongan language and the error value in 2010-2015 is obtained as shown in Table 2. The error rate in using the linear model is Based on the table, the graph of the estimation of the development of the Ngapak Banyumasan language (Penginyongan) is obtained as follows.

Logistics Growth Model
To determine the logistic model from the data on the population of Banyumas, it was previously assumed that time ( ) was measured in years, and if = 0 in 2010, then the initial requirement was (0) = 1,401,900. Since the total population of Banyumas since 2010-2015 is still below 1,500,000, it is assumed that the capacity for the capacity is = 1,500,000, so that if the (0) value and the value are substituted into the equation, the logistic model solution will be obtained as follows Furthermore, equation (7) will look for a logistic model that can represent the growth rate of the Ngapak Banyumasan language users. For = 5 in 2015, (5) = 1,472,318, if substituted for equation (3) it is obtained 1,472,318 = 1,500,000 0.06998e −5k + 1 0.06998e −5k + 1 = 1,500,000 1,472,318 0.06998e −5k = 0.01880 e −5k = 0.26865 −5k = (0.26865) = 0.2629. The value obtained is substituted back in equation (3), yielding = 1,500,000 0.06998 −(0.2629) + 1 .
By using the logistical growth model, the estimated number of Banyumas residents who use the Ngapak Banyumasan language and the error values in 2010-2015 are obtained as shown in Table 3. The error rate in using the logistic model is Based on the table, the graph of the estimated development of the Penginyongan language is given in Figure 3.

Determining the Best Model
To determine the best model, the error rate of each model can be seen. The best models are those with the smallest MAPE. Based on the calculation results, the logistic growth model is preferred.

Estimated Population of Banyumas who uses Penginyongan Language
The projection of the number of Banyumas residents who use the Ngapak Banyumasan language in 2016-2021 based on the logistical growth model is given in Table 4. The results obtained show that the number of Penginyongan language users tends to increase towards the carrying capacity of = 1,500,000. This study determined the carrying capacity to be = 1,500,000 and the results showed that in an 11-year time interval (2010-2021) the carrying capacity figure could be said to be achieved. This means that for each year there is an increase in the population who use the Penginyongan language, on average, around 10,000 people per year.
To conclude whether the Ngapak Banyumasan language (Penginyongan) is headed for extinction is to make comparisons with the average population growth of Banyumas every year from 2010 to 2021. If the number is less than 10,000, then the Ngapak Banyumasan language will still exist.

Conclussion
Of the two models produced, the logistical growth model is a better model for estimating the number of Banyumas residents who use the Penginyongan language in 2016-2021. The logistic model that estimates the number of Banyumas residents who use the Penginyongan language is based on an equation Based on this model, it can be interpreted that after 2010 the number of Banyumas residents who use the Penginyongan language tends to increase every year.