Çocuk gelişiminin kesirli kalkülüs ile modellenmesi ve bir uygulama
Modelling child growth with fractional calculus and an application
Yazar: İSMAİL DEMİRBİLEK
Danışman: PROF. DR. ERTUĞRUL KARAÇUHA
İstanbul Teknik Üniversitesi / Bilişim Enstitüsü / Bilişim Uygulamaları Ana Bilim Dalı
Derece: Yüksek Lisans, 2019
Konu: Bilgisayar Mühendisliği Bilimleri-Bilgisayar ve Kontrol = Computer Engineering and Computer Science and Control ; Matematik = Mathematics
Dizin: Bilgisayar yazılımları = Computer softwares ; Uygulama yazılımı = Application software ; Çocuk gelişimi = Child development

Özet:
Daha iyi çocuk bakımı için çocukların fiziksel gelişimine katkı sağlayan faktörleri belirleyen güvenilir ve faydalı sonuçlar gerekliliği açısından çocuk gelişiminin takibi her geçen gün daha fazla önem kazanmaktadır.
Bu çalışmada yaşa bağlı olarak çocukların boy uzunluğu, vücut ağırlığı ve vücut kitle endeksi verilerinin modellenmesini sağlayan matematiksel bir metot oluşturularak çocukların fiziksel gelişiminin takibine yönelik bir uygulama geliştirilmiştir. Çalışmada Türkiye'deki çocuklara ait boy uzunluğu, vücut ağırlığı ve vücut kitle endeksi değerlerini içeren 7 yüzdelik dilim grubundan oluşan veri seti kullanılmıştır. 0 - 18 yaş aralığı için verilen 7 gruptaki ayrık boy uzunluğu, vücut ağırlığı ve vücut kitle endeksi değerleri kullanılarak, en küçük kareler yöntemiyle sürekli bir fonksiyon, kesirli kalkülüs teorisi yardımıyla geliştirilmiştir.
Bu matematiksel model 0 - 18 yaş aralığı için verilen 7 gruptaki ayrık, boy uzunluğu, vücut ağırlığı ve vücut kitle endeksi değerlerini sürekli birer fonksiyona dönüştürmektedir. Örneğin veri setindeki 0 ay, 3 ay, 6 ay, 12 ay v.b. şeklinde aralıklı sürelerde yapılan ölçümlere dayalı olarak herhangi bir andaki normal şartlar altında beklenen gelişim değerinin hesaplanması mümkün hale gelmiştir.
Böylece, çocukların sadece 7 yüzdelik dilim grubunda sınırlandırmak yerine, 3-97 yüzdelik dilimleri arasında hangi dilime dahil olduklarını ve herhangi bir zamanda bu dilime dayalı olarak normal şartlarda ne kadar gelişme göstereceğini bulmak mümkün hale gelmiştir.
Geliştirilen model, Lineer ve Polinom modellerine nazaran oldukça başarılı sonuçlar üretmektedir. Bu model ile çocukların yüzdelik grubunu belirleyerek gelişiminin takip edilmesinin yanında çocukların geçmişte elde edilen ölçümleri ile 18 yaşına kadarki değerlerini tahmin etmeyi de sağlamaktadır.
Böylece, geliştirilen matematiksel modelden faydalanılarak hem sağlık personelleri hem de ebeveynler için kolay kullanılabilir ve modern bir web uygulaması ile; Android ve iOS platformları için mobil uygulamalar geliştirilmiştir. Bu uygulamalar ile çocukların periyodik gelişim takiplerinin daha kolay ve daha verimli yapılması amaçlanmaktadır.
Uygulamalar alternatiflerine kıyasla kullanıcı dostu arayüzü ve sağladığı kolay kullanım itibarıyla farklılık göstermektedir. Ayrıca alternatiflerinden farklı olarak ek özellikler ile hasta kaydına, gelişim geçmişinin her hasta için özel grafikler ile değerlendirilmesine imkan sağlamaktadır.

Abstract:
Development of a mathematical model out of observations regarding surrounding physical environment is always a need for scientists. Because, nature of science requires to be deterministic. In this study an answer is looked for the question: "How to model mathematically children's growth?".
It is getting more and more important to keep track of the children development, because good care is vital for children. Determining quality of the care is a must to take precautions in this regard. To do so, various growth charts are developed by taking samples from children raised under good conditions. These charts include weight, height and body mass index values with respect to age in 7 percentile groups. Percentile groups are seperated by birth weight, birth height and birth body mass index.
In this study, a mathematical approach developed for analysis of factors such as weight, body height and body mass index with respect to age. 7 groups for weight, body height and body mass index values from Percantage Chart of Turkey (Neyzi et al.) is used in the study. A continous curve is developed which is valid for any time interval by using  discrete values of weight, height and body mass index data of 0 - 18 years old children along with the least squares method. 
By doing so, it became possible to find the percentage of the children. This new mathematical method is advanced with help from fractional calculus theory. Formula (9) shows the curve function developed with fractional calculus. Function is divided to two parts in formula (17) and formula (19) for convenience.
Fractional calculus has been in the literature for more than 300 years. Fractional calculus is expanded version of calculus to include derivative and integrals of arbitrary orders. It is applied in the fields of science, engineering and mathematics. Fractional calculus allowed us to construct better curves for the growth charts of children's of Turkey. To calculate fractional derivative Laplace Transfrom and Inverse Laplace Transform are used.
Related works in this area use linear and polynomial regression. Fractional calculus allowed us to produce better curves that minimizes margin of errors based on the dataset of percentage chart. Coefficients in the mathematical model developed with fractional calculus, is found with help from least squares method. Regression is done along with inverse Laplace transform following the Laplace transform. With this way not only coefficients for each percentile is found, but also best alpha factors are determined too. Coefficients with alpha and f(0) values of height – percentile of boys is shown on Figure 2.1; height – percentile of boys is shown on Figure 2.2; body mass index – percentile of boys is shown on Figure 2.3. Coefficients with alpha and f(0) values of height – percentile of girls is shown on Figure 2.4; height – percentile of girls is shown on Figure 2.5; body mass index – percentile of girls is shown on Figure 2.6. f(0) values in these tables are taken from the dataset used (Neyzi et al.). 
 
The new method produces quite successful results compared to Linear and Polynomial Model analysis. The newly developed method makes it possible to predict expected development values of children for the future by using previous data obtained.
Results show that fractional calculus model performs better in regard of error margins settled with the Mean Absolute Percentage Error (MAPE). MATLAB program is used for comparison of the three model which are Linear, Polynomial and Fractional Model in these two error calculations. Table 3.1 shows MAPE scores in age – height distribution of the Linear, Polynomial and Fractional models. Table 3.2 shows MAPE scores in age – weight distribution of the Linear, Polynomial and Fractional models. Table 3.3 shows MAPE scores in age –body mass index distribution of the Linear, Polynomial and Fractional models. In Figure 3.1 continuous curves from linear, polynomial and fractional model are shown for each percentiles of boys' and girls' weight distribution. In Figure 3.2 continuous curves from three models are shown for each percentiles of boys' and girls' height distribution. In Figure 3.3 continuous curves from three models are shown for each percentiles of boys' and girls' body – mass index distribution.
A ready to be used modern web application along with the native mobile applications for Android and iOS platforms is also developed based on the newly introduced mathematical model. Source code is in three parts. One library for core functions to be shared with the native mobile application and the web application; one for Android / iOS application; one for the web application. Source code is written in JavaScript for interoperability between web and native platforms. JavaScript allowed us to operate in maximum code reuse with high cohesion and low coupling.
Web application is developed on top of open source, massively adopted defacto UI library React. Component architecture and clean API helped a lot during development.
For mobile applications it is inevitable to be cross platform for efficiency. So for code reuse and share work as much as possible, on JavaScript strong and trustworthy React Native is engaged. In this way we shared core library between these two projects and provided the same functionality in three platforms (web, Android, iOS). Applications provide useful functions especially for  pediatric endocrinologists. They provide not only trusted expected growth measures but also consolidate doctors with built in patient records, easily perceived results evaluation and custom history charts for each patient. Figure 3.4 shows new patient entry user interface. Figure 3.5 shows patient measures evaluation while follow up dignosis.
It is also very important for individual patient data privacy. So there is no central place for data transfer and service point. All of the produced data stored in the computer, mobile phone or tablet of the user. And there is no list of stored patients anywhere in the applications. Endocrinologists should enter patient identity number to access individual patient data. This way data privacy of patients is secured. Figure 3.6 shows patient search user interface.
Finally, a mathematical method based on fractional calculus is developed. Regression on the model is done with help from Laplace transforms and Least Squares method. Continuous curves developed with the model introduced are compared with Linear and Polynomial alternatives and results showed that newly introduced fractional model performs significantly better. With the continous curve function which is valid for any time interval and for any percentile, we can calculate the expected body weight, height and body mass index of an individual child. If child's measures taken during follow up diagnosis is easily evaluated. So future growth measures can also be calculated along with current expected measures with the model we introduce. On top of this mathematical model a up-to-date, modern computer and mobile applications are developed to make evaluation even easier. So pediatric endocrinologists can easily start using our method with a single click.