ABSTRACT
Adana dewlap pigeons are a special kind of pigeon bred in the Adana region of Turkey. These pigeons differ from other pigeon species due to their flight characteristics. In this study, the mathematical models that explain the growth dynamics of Adana pigeons were examined and the one capable of the best predictions was determined. The incubation period of pigeons happens two or three times per year. For this reason, only one or two pairs of pigeons can be yearly reared from an adult pair of pigeons. The hatchlings die before they reach adulthood for various reasons. In this study, 43-day body weight measurements of 88 pigeons reared over a 7-year period were used. The study was carried out by taking the average of the 43-day weight measurements of these 88 pigeons. Special Matlab codes were developed to estimate the parameters in the mathematical growth models used. The performance of the most commonly used growth models when explaining the growth dynamics of Adana pigeons is discussed in this study, an issue that to the best of our knowledge had not yet been studied. Among these growth models, the Richards growth function gave the best results. The new data set obtained from 43-day weight measurements of 88 pigeons was considered as a time series. For this purpose, a discrete-time stochastic Gompertz model (DTSGM) was used to explain the growth dynamics, and modeling was done to explain the growth of Adana pigeons. It is assumed that the unknown parameter in this model changes with time. The Kalman filter (KF) was used for the time-varying estimation of this parameter. The aim of the study was to investigate whether the model and estimation method considered in this way is suitable for explaining the growth dynamics of these birds. The results obtained by considering DTSGM and estimating its parameter with KF showed that MSE provides a good analysis tool for modeling daily weight in terms of mean absolute percent error (MAPE) and R 2 criteria. It is suggested that the use of DTSGM and KF is appropriate.
Keywords: Adana Pigeons; growth model; Kalman Filter
INTRODUCTION
Pigeons are among the species that played an important role in human history. Known as a religious symbol in ancient times, these birds were later used as a means of conveying messages in wars, and today it have become a symbol of peace. Domestic pigeon species originate from Columba livia (rock pigeon), a type of Columba species in the Columbinae subfamily of the Columbidae family (Secord, 1981). Genome analysis was performed by (Shapiro, 2013). The period of domestication of the pigeon cannot be precisely fixed (Yılmaz & Ertuğrul, 2012). Growth is one of the important characteristics of animals. It can be defined as any change in body size per unit of time and is affected by genotype and environment. In a more general sense, growth refers to a mass development due to the increase in the size and number of living things. Mathematical functions called ‘growth models’ are used to describe the growth dynamics of poultry species. In poultry science, Gompertz, Logistic, and Richards functions are among the most widely used to model and identify the growth processes of birds (Narinç et al., 2017; Gompertz, 1825; Bertalanffy, 1949; Richards, 1959). Adana dewlap pigeons are a special kind of pigeon bred in the Adana region of Turkey. These pigeons differ from other pigeon species due to their flight characteristics. They have developed unique color, form and flight characteristics through the addition of different features from breeding over many years in Turkey since the Ottoman period. They are currently also bred in Syria, being known in all regions under the name Adana. Adana pigeons, which are related to the wattled pigeons known as “dewlap” in English, are not a distinct variety from wattled pigeons, but are considered a separate regional group because of their different regional characteristics. Adana pigeons are thus a separate breed among wattled pigeons.
In this study, the mathematical models in the literature were examined to explain the growth dynamics of Adana pigeons and the mathematical/statistical model that best explained the growth of these pigeons was determined. The incubation period of pigeons happens two or three times a year. For this reason, only one or two pairs of pigeons can be reared from an adult pair of pigeons per year. The hatchlings die before they reach adulthood for various reasons. In this study, 43-day body weight measurements of 88 pigeons reared over a 7-year period were used. The study was carried out by taking the average of the 43-day weight measurements of these 88 pigeons. Special codes were developed in the Matlab program to estimate the parameters in the mathematical growth models used. The performance of the most commonly used growth models when explaining the growth dynamics of Adana pigeons is discussed in this study. Among these growth models, the Richards growth function gave the best results. The Gompertz model (GM) is well known and widely used in many sub-fields of biology (Zwietering et al., 1990; Gerlee, 2013). Numerous parametrizations and re-parametrizations of the GM may be found in the literature (Kathleen& Tjørve, 2017). The new data set obtained from 43-day weight measurements of 88 pigeons in this study was considered as a time series. For this purpose, discrete-time stochastic Gompertz model (DTSGM) was used to explain the growth dynamics and modeling was done to explain the growth of Adana pigeons. It is assumed that the unknown parameter in this model changes with time. The Kalman filter (KF) was used for the time-varying estimation of this parameter. The aim of the study was to investigate whether the model and estimation method considered in this way is suitable for explaining the growth dynamics of these birds. The results obtained by considering DTSGM and estimating its parameter with KF showed that MSE provides a good analysis tool for modeling daily weight in terms of mean absolute percent error (MAPE) and R 2 criteria. It is suggested that the use of DTSGM and KF is appropriate. The mathematical equations of the most commonly used growth models in the literature are given in Table 1.
88 Adana pigeons were used in the study. Weight data of pigeons from 1 to 43 days were analyzed. Parameter estimates were made using the average of the daily weights of these 88 pigeons. MSE, MAPE and R 2 statistics were used to compare which of the models performed the best. Table 2 shows the MSE, MAPE and R 2 values for the models discussed. The table shows that the most consistent model in terms of both MAPE and MSE is the Richards model. The actual measurements and the results of the Richard model are shown in Figure 1. As a result of the estimation obtained with the data, the best analysis for modeling the growth was made by the Richards model for both MAPE and MSE. It has been determined that the use of this model is appropriate. The results of the study by (Özbek, 2022) are consistent with the results of this study.
MATERIALS AND METHODS
The Continuous GM, Recursive Linear Model and Linear Regression Model have been used as broiler chickens growth models (Aerts et al., 2003). Zheng et al. (1998) have also used a recursive linear model to estimate the model parameters of a broiler chicken growth model. The authors stated that the recursive linear model gives more accurate estimates than the static linear growth model. They proposed a class of complex population dynamic models that combines new time-varying parameters to describe univariate ecological time series data. They used second-order stochastic Ricker and GM and also KF to estimate model parameters in ecological time-series data sets. Moreover, they also explained the advantage of using KF in ecological time series analysis. The structural time series modeling method using KF has many advantages over classical Box-Jenkins methods. In the structural time series model, it is easy to include and estimate unobserved components (Harvey, 1989). Knape and Valpine (Jonas et al., 2012) analyzed 627 data sets in the The Global Population Dynamics Database using Gompertz population models and accounting for uncertainty via the KF.
The 43-day weight measurements of the 9 pigeon data used in the research are shown in Figure 2. In order to model daily weight measurements using time series methods, the Dickey-Fuller test was applied to find out whether there is a unit root in the data. A pigeon with 99 days of weight measurements was used to perform this test. The time series regarding the weight measurements of this pigeon is given in Figure 3.
The ADF (Augmented Dickey-Fuller) test stated that the series does not contain a unit root at 0.1% significance level. On the other hand, stationarity means that the data moves around a certain average. However, looking in detail with specific intervals, the data set including the first 36 observations is not stationary. The stationarity begins after the 36th observation. The ADF Test result for the whole data set is given in Table 3. The ADF tests result for the first 36 observations is given in Table 4.
From Table 4, it is clear to say that the data for the first 36 observations is not stationary. In order to eliminate the unit root problem, we used KF analysis to study the growth of Adana pigeons. As far as we know, there has been no study analyzing the growth of Adana pigeons through this methodology.
This study emphasizes the growth model in order to estimate the growth of the Adana pigeons using DTSGM and AKF. The estimations presented here are based on the total data of the 88 pigeons. The Gompertz model that we used in our study is linear and simpler compared to those used in the studies discussed above. There is only one parameter in the discrete-time stochastic GM, autoregressive time-series AR(1) model we used, and that parameter is estimated on-line through AKF.
The weight of the pigeon observed until day t is given by the following DTSGM, where wt denotes the weight of the pigeon at time t:
a and b are constants, and et is defined by et~N(0, σ 1 2) . On the logarithmic scale, the GM is a linear, autoregressive time-series model AR(1)
where yt=lnwt and c=b+1 . The statistical properties of the DTSGM are well-known (Dennis et al., 2006).
KF is widely used in many fields. In this study, KF was used to estimate the time-varying parameter of DTSGM. KF is a recursive estimator used to estimate time-varying parameters. If in Equation 2, is obtained.
In the case where the c parameter in Equation (3) is time-varying and presumed as random walk process, state-space model is used. Here, vt is distributed as vt~N(0, σ 2 2) , and the state variable is an unobservable, time-varying ct parameter, which can be estimated through KF.
RESULTS
According to the estimation results obtained by using the daily weight measurements in the DTSGM, MSE, MAPE, and R 2 were calculated (see Table 5). Actual logarithmic daily weight values and estimated values using DTSGM and KF are given in Figure 4. Actual daily weight values and estimated values are given in Figure 5.
These calculated values show that the fit of the model with the real data is quite high. This tells us that estimating daily weight with DTSGM is a reliable method. KF is the most advantageous aspect of this method, as it only uses the observation at time t and the previous estimate.
CONCLUSION
In this study, which was the best non-linear growth function present in the literature at explaining the growth dynamics of Adana dewlap pigeons. Then, the time series consisting of the average of daily weight measurements was examined, and the characteristics of this time series were examined. This time series was modeled with the DTCM model, and it was seen that this model gave the best results. The time series modeled with DTCM is the AR(1) model. The parameter in this model was estimated sequentially using the KF. When the estimation results obtained from 88 Adana pigeons were examined, the most effective analysis for modeling daily weight were DTSGM and KF, according to MSE, MAPE and R 2 criteria. This proposed method is suitable for weight estimation. This proposed method is a very simple method for estimating growth, which can be conducted by modeling the daily weight time series using AR(1) time series and KF.
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DATA AVAILABILITY STATEMENT
Data will be available upon request.
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DISCLAIMER/PUBLISHER’S NOTE
The published papers’ statements, opinions, and data are those of the individual author(s) and contributor(s). The editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions, or products referred to in the content.
APPENDIX
State-space model and adaptive Kalman filter (AKF). Considering a general discrete-time stochastic system given by
where xt is an system, yt is an observation vector. The covariance matrices wt and wt are Wt~N(0, Ԛ t ) , vt~N(0, R t ) . The filtering problem is the problem of determining the best estimate of its xt condition, given its observations Yt=(y0, y1, ..., yt) . Let the initial state be assumed to have a normal distribution in the form of x0~N(x̅ 0 , P 0 ) . The optimum KF equations are,
In the above equations, xt|t-1 is the a priori estimation and xt is the a posteriori estimation of xt . Also, Pt|t-1 and Pt|t are the covariance of a priori and a posteriori estimations, respectively (Jazwinski, 1970; Chen, 1993; Özbek, 2016). In some cases, divergence problems may occur in the KF due to the incorrect installation of the model. In order to eliminate divergence in the KF, adaptive methods are used (Özbek, 1998; Özbek & Efe, 2004). One of these is the use of the forgetting factor. A forgetting factor is proposed by Özbek and Aliev (1998) as follows:
Data availability
Data will be available upon request.
Publication Dates
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Publication in this collection
25 Nov 2024 -
Date of issue
2024
History
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Received
03 Sept 2024 -
Accepted
08 Oct 2024