Sun’iy Intellekt va Raqamli Iqtisodiyot

4. Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A. N., Kaiser, L., & Polosukhin, I. (2017). Attention is all you need. Advances in Neural Information Processing Systems, 30, 5998–6008. https://arxiv.org/abs/1706.03762 5. Ashurov, Z., Makhmudova, G., & Razakova, B. (2022). Development of digital ecosystem and formation of digital platforms in Uzbekistan. Pi-Economy, 15(2). https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4187764 SELECTION CRITERIA FOR MACHINE LEARNING AND DEEP LEARNING MODELS IN HEART DISEASE CLASSIFICATION Shokhrukhbek Rakhimboyev, Master’s student at Urgench State University named after Abu Rayhan Biruni, raximboyevshohrux96@gmail.com Jaloliddin Omonbayev, Master’s student at Urgench State University named after Abu Rayhan Biruni, jaloliddinomonboev@gmail.com Annotatsiya. Ushbu ilmiy ishda yurak kasalliklarini tasniflashda mashinaviy va chuqur o‘rganish modellarini tanlash mezonlari tahlil qilinadi. Modellar aniqlik, interpretatsiya qilish imkoniyati, hisoblash samaradorligi va amaliy qo‘llanish nuqtai nazaridan solishtirilib, optimal tanlov yondashuvi taklif etiladi. Kalit sozlar: Yurak kasalliklari, Mashinali o‘qitish, Chuqur o‘rganish, Model tanlash mezonlari, Tasniflash Аннотация. В данной научной работе анализируются критерии выбора моделей машинного и глубокого обучения для классификации заболеваний сердца. Модели сравниваются по точности, интерпретируемости, вычислительной эффективности и практическому применению, и предлагается оптимальный подход к выбору. Ключевые слова: Сердечные заболевания, Машинное обучение, Глубокое обучение, Критерии выбора модели, Классификация Abstract. This paper analyzes the criteria for selecting machine and deep learning models for classifying heart disease. The models are compared in terms of accuracy, interpretability, computational efficiency, and practical application, and an optimal selection approach is proposed. Keywords: Heart disease, Machine learning, Deep learning, Model selection criteria, Classification Introduction. Cardiovascular diseases remain one of the leading causes of mortality worldwide, accounting for a significant proportion of global deaths each year. Early detection and accurate diagnosis are critical for reducing mortality rates and improving patient outcomes. In recent years, artificial intelligence techniques, particularly Machine Learning (ML) and Deep Learning (DL), have been increasingly applied in medical diagnostics to support clinical decision-making and enhance predictive performance. Despite the rapid development of predictive models for heart disease classification, the selection of an appropriate model remains a challenging task. Different algorithms often demonstrate varying levels of accuracy, interpretability, computational complexity, and generalization capability. In medical applications, high predictive accuracy alone is insufficient; model transparency, stability, and practical feasibility are equally important factors [3]. The aim of this thesis is to analyze and compare selection criteria for Machine Learning and Deep Learning models in heart disease classification. The objectives of the study include: (1) preprocessing and preparing the dataset, (2) 359

implementing selected ML and DL models, (3) evaluating their performance using standard classification metrics, and (4) identifying key factors that influence optimal model selection for practical clinical application. Main body. This study utilizes the Cleveland Heart Disease dataset, which is widely applied in supervised learning research for cardiovascular disease prediction. Although the original dataset contains 76 attributes collected from 303 patients, most studies employ 14 clinically relevant features, including the target variable. These attributes represent demographic and medical indicators such as age, sex, resting blood pressure, serum cholesterol, fasting blood sugar, and electrocardiographic results. These attributes represent a combination of demographic characteristics and important medical indicators associated with cardiovascular health. Demographic variables include age and sex, which are known to influence the risk of heart disease. In addition, the dataset contains several physiological and clinical measurements, such as resting blood pressure, serum cholesterol levels, and fasting blood sugar. These indicators provide essential information about a patient's metabolic and cardiovascular condition. The overall methodology adopted in this study is illustrated in Figure 1. Figure 1. Proposed methodology for heart disease classification. 2. Model Selection Framework.The primary objective of this research is to establish structured criteria for selecting appropriate Machine Learning and Deep Learning models for heart disease classification. In medical predictive systems, model selection cannot rely solely on predictive accuracy. Instead, it requires a multidimensional evaluation approach that considers diagnostic reliability, interpretability, computational efficiency, and suitability for clinical implementation. Predictive performance reflects the model’s ability to correctly classify patients. Interpretability determines whether healthcare professionals can understand and justify the model’s decisions. Computational complexity affects training time and scalability, while stability and generalization indicate how well 360

the model performs on unseen data. These dimensions collectively form the evaluation framework applied in this study [1]. 3. Implemented Models. To analyze model selection criteria across different methodological paradigms, both traditional Machine Learning algorithms and a Deep Learning model were implemented. The Machine Learning models include Logistic Regression, Support Vector Machine, and Random Forest. Logistic Regression was selected due to its simplicity and high interpretability, which is particularly important in healthcare applications. Support Vector Machine was included because of its strong performance in high-dimensional classification tasks. Random Forest was chosen as an ensemble method capable of reducing variance and improving predictive robustness [3]. In addition to classical models, a Feedforward Artificial Neural Network was implemented to represent the Deep Learning approach. The neural network consists of an input layer corresponding to dataset features, two hidden layers with ReLU activation functions, and an output layer with sigmoid activation for binary classification. The probability estimation for binary classification is defined as: P(y = 1 ∣ x) = 1 1 + e−z (1) Binary Cross-Entropy was used as the loss function during optimization. While neural networks often achieve higher predictive performance, they are generally less interpretable compared to traditional models [2]. Evaluation and Comparative Analysis. Model performance was evaluated using Accuracy, Precision, Recall, F1-score, and ROC-AUC metrics. Accuracy is calculated as: 𝐴𝑐𝑐𝑢𝑟𝑎𝑐𝑦 = TP + TN TP + TN + FP + FN However, in medical diagnosis tasks, Recall and ROC-AUC are particularly important because minimizing false negatives is critical when detecting heart disease. The comparative results are presented in Table 1. Table 1. Comparative evaluation of Machine Learning and Deep Learning models The experimental findings indicate that the Artificial Neural Network achieved the highest predictive performance in terms of accuracy and ROC-AUC. However, Random Forest demonstrated competitive accuracy while maintaining better interpretability and lower computational requirements. Logistic Regression, although slightly less accurate, provided the highest level of transparency, making it more suitable for clinical environments where explainability is essential [4]. These findings suggest that model selection in heart disease classification should not be based solely on predictive accuracy. Instead, it should depend on the 361

intended application context, balancing performance with interpretability and practical feasibility. Table 1 Comparative Evaluation of Machine Learning and Deep Learning Models. Conclusion. This study analyzed the selection criteria for Machine Learning and Deep Learning models in heart disease classification using the Cleveland dataset. The findings demonstrate that model selection in medical predictive systems should not rely exclusively on predictive accuracy. Although the Artificial Neural Network achieved the highest performance in terms of Accuracy and ROCAUC, its limited interpretability and higher computational requirements reduce its suitability for certain clinical environments. Random Forest provided a strong balance between predictive performance and interpretability, making it a practical choice for real-world implementation. Logistic Regression, despite slightly lower accuracy, offered the highest level of transparency and explainability, which is particularly important in healthcare applications where clinical justification is required. Support Vector Machine demonstrated competitive performance but required careful parameter tuning and moderate computational resources. The results confirm that model selection for heart disease classification must be based on a multidimensional evaluation framework that includes predictive reliability, interpretability, computational efficiency, and practical applicability. In clinical decision-support systems, interpretable models may be preferred when transparency is critical, whereas Deep Learning models may be appropriate in scenarios where maximizing predictive accuracy is the primary objective. From a practical perspective, it is recommended that healthcare institutions consider hybrid evaluation strategies that combine performance metrics with explainability analysis before deploying predictive models. Future research may focus on integrating explainable artificial intelligence techniques into Deep Learning architectures to improve transparency without sacrificing performance. Literature references. 1. Matlatipov, S.G., Rajabov, J., Kuriyozov, E., & Aripov, M. (2024, May). UzABSA: Aspect-Based Sentiment Analysis for the Uzbek Language. In Proceedings of the 3rd Annual Meeting of the Special Interest Group on Under-resourced Languages@ LREC-COLING 2024 (pp. 394-403). 362

2. Kutlimuratova, B., Kuriyozov, E., Urazbaev, A., & Rakhimova, G. (2025, November). Corpus-Based Error Analysis of Uzbek EFL Learners’ Academic Writing. In 2025 IEEE XVII International Scientific and Technical Conference on Actual Problems of Electronic Instrument Engineering (APEIE) (pp. 1-5). IEEE. 3. Mattiev, J., Davityan, M., & Kavsek, B. (2023). ACMKC: A compact associative classification model using K-modes clustering with rule representations by coverage. Mathematics, 11(18), 3978. 4. Madatov, K., Matlatipov, S., & Aripov, M. (2023). Uzbek text's correspondence with the educational potential of pupils: a case study of the School corpus. arXiv preprint arXiv:2303.00465. SUN’IY INTELLEKT VA RAQAMLI TEXNOLOGIYALARNING MATEMATIK MODELLASHTIRISH MUAMMOLARI VA YECHIMLARI Raximova Marjona Nurmuhammad qizi, QarDU, Axborot tizimlari va texnologiyalari ta’lim yo‘nalishi, 2-bosqich talabasi Bozorova Irina Jumanazarovna, QarDU, Algoritmlar va dasturlash texnologiyalari” kafedrasi dotsenti, i.f.f.d. (PhD), i.bozorova667@gmail.com Annotatsiya. Mazkur ilmiy maqolada sunʼiy intellekt va raqamli texnologiyalarni matematik modellashtirish jarayonida yuzaga keladigan nazariy hamda amaliy muammolar tahlil qilinadi. Xususan, katta hajmdagi maʼlumotlar bilan ishlash, algoritmik murakkablik, optimallashtirish masalalari, aniqlik va ishonchlilik darajasi kabi muammolar ko‘rib chiqiladi. Shuningdek, differensial tenglamalar, ehtimollik nazariyasi, chiziqli algebra, optimallashtirish usullari hamda mashinaviy o‘qitish algoritmlari yordamida mazkur muammolarni bartaraf etish yo‘llari ilmiy asosda bayon etiladi. Tadqiqot natijalari sunʼiy intellekt tizimlarining samaradorligini oshirishda matematik modellashtirishning muhim o‘rin tutishini ko‘rsatadi. Kalit so‘zlar: Sunʼiy intellekt, matematik modellashtirish, raqamli texnologiyalar, algoritm, optimallashtirish, mashinaviy o‘qitish, neyron tarmoqlar, differensial tenglamalar, katta maʼlumotlar (Big Data), ehtimollik nazariyasi. Аннотация. В данной научной статье анализируются теоретические и практические проблемы, возникающие в процессе математического моделирования искусственного интеллекта и цифровых технологий. В частности, рассмотрены такие проблемы, как работа с большими объемами данных, алгоритмическая сложность, вопросы оптимизации, уровень точности и надежности. Также на научной основе описаны способы решения этих задач с помощью дифференциальных уравнений, теории вероятностей, линейной алгебры, методов оптимизации и алгоритмов машинного обучения. Результаты исследования показывают, что математическое моделирование играет важную роль в повышении эффективности систем искусственного интеллекта. Ключевые слова: Искусственный интеллект, математическое моделирование, цифровые технологии, алгоритм, оптимизация, машинное обучение, нейронные сети, дифференциальные уравнения, большие данные (Big Data), теория вероятностей. Abstract. This scientific article analyzes the theoretical and practical problems that arise in the process of mathematical modeling of artificial intelligence and digital technologies. In particular, problems such as working with large volumes of data, algorithmic complexity, optimization issues, level of accuracy and reliability are considered. Methods for solving these problems using differential equations, probability theory, linear algebra, optimization methods and machine learning algorithms are also described on a scientific basis. The study results show that mathematical modeling plays an important role in improving the efficiency of artificial intelligence systems. Keywords: Artificial intelligence, mathematical modeling, digital technologies, algorithm, optimization, machine learning, neural networks, differential equations, big data (Big Data), probability theory. 363

Kirish. XXI asrda raqamli transformatsiya jarayoni global miqyosda jadallashib, sunʼiy intellekt (SI) texnologiyalari iqtisodiyot va jamiyat taraqqiyotining asosiy drayveriga aylandi. Sunʼiy intellekt tushunchasiga ilk ilmiy yondashuvlar Alan Turing tomonidan ilgari surilgan bo‘lib, u hisoblash mashinalarining intellektual imkoniyatlarini nazariy asoslab bergan. Keyinchalik bu yo‘nalish OpenAI, Google va Microsoft kabi tashkilotlar faoliyati orqali yanada rivojlandi. 1-jadval Sun’iy intellekt va raqamli texnologiyalarda matematik modellashtirish muammolari va yechimlari № 1 Asosiy matematik Muammo model Mashinali o‘qitish Chiziqli va Ma’lumot nochiziqli yetishmasligi regressiya 2 Neyron tarmoqlar 3 Raqamli signalni qayta ishlash 4 Kompyuter ko‘rishi 5 Katta ma’lumotlar Ehtimollar (Big Data) nazariyasi va statistika Kiberxavfsizlik Kriptografik matematik modellar Robototexnika Optimal boshqaruv tenglamalari Sun’iy intellektda Vaqt qatorlari prognozlash modeli 6 7 8 Yo‘nalish Differensial tenglamalar va gradient usuli Furye qatori va Furye o‘zgartirishlari Matritsalar va konvolyutsiya O‘ta moslashish (overfitting) Shovqin (noise) mavjudligi Tasvir aniqligining pastligi Hisoblash murakkabligi Ma’lumotlar xavfsizligi Tavsiya etiladigan yechim Ma’lumotlarni ko‘paytirish (data augmentation), regularizatsiya Dropout, L1/L2 regularizatsiya Filtrlash algoritmlari (past chastotali filtr) Konvolyutsion neyron tarmoqlar (CNN) Parallel hisoblash va optimallashtirish algoritmlari RSA, AES algoritmlari Harakatni aniq Matematik modellashtirish optimallashtirish va PID nazorati Bashorat ARIMA, LSTM xatoligi modellaridan foydalanish Natija Model aniqligining oshishi Umumlashtirish qobiliyatining yaxshilanishi Signal sifati oshadi Tasvirni aniqlash samaradorligi oshadi Tezkor natija olish Axborot himoyasi ta’minlanadi Harakat aniqligi oshadi Prognoz aniqligi oshadi Raqamli texnologiyalar – bu maʼlumotlarni yig‘ish, saqlash, qayta ishlash va uzatish jarayonlarini avtomatlashtiruvchi tizimlar majmuasidir. Sunʼiy intellekt esa ushbu maʼlumotlardan mantiqiy xulosa chiqarish, prognoz qilish va mustaqil qaror qabul qilish imkonini beradi. Bunday tizimlarning samarali ishlashi esa bevosita matematik modellashtirishga bog‘liq. Matematik modellashtirish sun’iy intellekt tizimlarining nazariy poydevorini tashkil etadi. Har qanday algoritm yoki raqamli tizim ma’lum matematik qonuniyatlarga asoslanadi. Statistik tahlil, ehtimollar nazariyasi, chiziqli algebra va optimallashtirish metodlari sun’iy intellektning asosiy ilmiy tayanchidir.Masalan, mashinali o‘qitish tizimlari ma’lumotlar orasidagi bog‘lanishlarni aniqlash orqali ishlaydi. Bu jarayon matematik tahlil va statistik usullarsiz amalga oshirilmaydi. 364

Matematik modellashtirish — real jarayonni matematik ifodalar orqali tasvirlashdir. Sun’iy intellektda model quyidagicha ifodalanishi mumkin: y=f(x,θ) Bu yerda: • x— kirish ma’lumotlari, • θ— model parametrlari, • y — natija. Model parametrlari optimallashtirish masalasi orqali aniqlanadi: minθL(y,y^) Bu yerda L — yo‘qotish funksiyasi. Sun’iy intellektda uchraydigan matematik muammolar 1. Overfitting (ortiqcha moslashish). Model o‘quv ma’lumotlariga haddan tashqari moslashib qoladi va yangi ma’lumotlarda xatolik qiladi. Yechimlar: ❖ L1, L2 regularizatsiya ❖ Dropout usuli ❖ Kross-validatsiya 2. Hisoblash murakkabligi. Katta hajmdagi ma’lumotlarni qayta ishlash ko‘p vaqt va resurs talab qiladi. Yechimlar: • Parallel hisoblash • Bulutli texnologiyalar • Gradient tushish algoritmining optimallashtirilgan variantlari 3. Lokal minimum muammosi. Optimallashtirish jarayonida global minimum o‘rniga lokal minimum topilishi mumkin. Yechimlar: ▪ Stoxastik gradient tushish (SGD) ▪ Adam optimizatori ▪ Boshlang‘ich qiymatlarni to‘g‘ri tanlash Raqamli texnologiyalarda matematik modellar Neyron tarmoqlari. Neyron tarmoqlar biologik neyronlar faoliyatini modellashtiradi. Ularning matematik asosi ko‘p o‘zgaruvchili funksiyalar va matritsalar ko‘paytmasiga asoslanadi: z = Wx+b Bu yerda: ✓ W— og‘irliklar matritsasi, ✓ b — siljish vektori. Signalni qayta ishlash. Raqamli signalni qayta ishlashda Joseph Fourier tomonidan ishlab chiqilgan Furye o‘zgartirishidan foydalaniladi: ∞ F(ω)=∫−∞ 𝑓(𝑡)𝑒 −𝑖𝜔𝑡 𝑑𝑡 Bu usul tasvir va tovushni tahlil qilishda keng qo‘llaniladi. Tadqiqot natijalari shuni ko‘rsatadiki, sunʼiy intellekt va raqamli texnologiyalarni samarali rivojlantirish matematik modellashtirishga bevosita 365

bog‘liqdir. Chiziqli algebra, ehtimollik nazariyasi, statistik tahlil va optimallashtirish usullari sunʼiy intellekt algoritmlarining asosini tashkil etadi. Katta maʼlumotlar, algoritmik murakkablik va ortiqcha moslashuv kabi muammolar zamonaviy matematik yondashuvlar yordamida hal qilinmoqda. Kelajakda matematik modellarni yanada takomillashtirish orqali sunʼiy intellekt tizimlarining aniqligi va ishonchliligi oshiriladi. Foydalanilgan adabiyotlar ro‘yxati: 1. Russell, S., & Norvig, P. Artificial Intelligence: A Modern Approach. Pearson — sun’iy intellekt umumiy nazariyasi va usullari bo‘yicha eng mashhur darsliklardan biri. 2. Haykin, S. Neural Networks and Learning Machines. Pearson — neyron tarmoqlar va o‘rganish metodlarini tushuntiradi. 3. Hastie, T., Tibshirani, R., & Friedman, J. The Elements of Statistical Learning. Springer — statistik o‘rganishning asosiy tamoyillarini ta’riflaydi. 4. Khan, A.M., & Yoon, C. A Survey on Mathematical Models in Machine Learning and AI. Journal of Machine Learning Research — AI va ML’dagi matematik modellar haqida sharh. 5. Awasthi, M., Tomar, R., & Gupta, M. (eds.) Mathematical Modeling for Intelligent Systems. Chapman & Hall — intellektual tizimlarga matematik modellashtirish usullarini yoritadi. 6. Dutta, P., Pal, S., Kumar, A., & Cengiz, K. Artificial Intelligence for Cognitive Modeling. Routledge — kognitiv modellashtirish va AI algoritmlariga chuqur qarash. 7. Hu, J.H., & Peng, Z. P. (eds.) Mathematical Methods Applied in Artificial Intelligence and Multi-Agent Systems. MDPI — sun’iy intellekt va agent tizimlarida matematik metodlar. 8. Sun, Y., & Zhao, X. A Study on The Application of Artificial Intelligence in Mathematical Modeling. Academic Journal of Science and Technology — AI’dagi matematik modellashtirish bo‘yicha tadqiqot maqolasi. 9.Goodfellow, I., Bengio, Y., & Courville, A. Deep Learning. MIT Press — chuqur o‘rganish texnologiyalari va ularning nazariy jihatlari. 10. Bishop, C.M. Pattern Recognition and Machine Learning. Springer — tasniflash va mashinaviy o‘rganishning matematik asoslari. SUN’IY INTELLEKT VA RAQAMLI TEXNOLOGIYALAR YORDAMIDA NANOTEXNOLOGIYALAR VA ZAMONAVIY MATERIALSHUNOSLIK SOHALARIDA MATEMATIK MODELLASHTIRISHNING O‘RNI Raximova Shohsanam Usmonali qizi, FDTU, Ishlab chiqarishda boshqaruv fakulteti Oliy matematika kafedrasi assistenti raximovashohsanam1993@gmail.com Annotatsiya. Mazkur tezisda sun’iy intellekt va raqamli texnologiyalar yordamida nanotexnologiyalar va zamonaviy materialshunoslik sohalarida matematik modellashtirishning o‘rni, ahamiyati hamda tahlil qilish imkoniyatlari yoritilgan. Nanomiqyosdagi jarayonlarni o‘rganishda differensial tenglamalar, statistik modellar, sonli hisoblash usullari va optimallashtirish algoritmlaridan foydalanishning ilmiy hamda amaliy jihatlari ko‘rsatib berilgan. Shuningdek, kompyuter simulyatsiyasi va sun’iy intellekt texnologiyalari yordamida nanozarrachalar xossalarini prognozlash, materiallarning yangi turlarini loyihalash va murakkab fizik jarayonlarni modellashtirish imkoniyatlari tahlil qilingan. Kalit so‘zlar: nanotexnologiya, materialshunoslik, matematik modellashtirish, sun’iy intellekt, raqamli texnologiyalar, nanozarrachalar, sonli usullar, differensial tenglamalar, FEM, Monte-Karlo usuli, Shredinger tenglamasi, kompyuter simulyatsiyasi. Аннотация. В данном тезисе рассматривается роль и значение математического моделирования в областях нанотехнологий и современного материаловедения с 366

использованием технологий искусственного интеллекта и цифровых технологий, а также возможности анализа этих процессов. Показаны научные и практические аспекты применения дифференциальных уравнений, статистических моделей, численных методов и алгоритмов оптимизации при исследовании процессов на наноуровне. Кроме того, проанализированы возможности прогнозирования свойств наночастиц, проектирования новых типов материалов и моделирования сложных физических процессов с помощью компьютерного моделирования и технологий искусственного интеллекта. Ключевые слова: нанотехнологии, материаловедение, математическое моделирование, искусственный интеллект, цифровые технологии, наночастицы, численные методы, дифференциальные уравнения, метод конечных элементов (FEM), метод Монте-Карло, уравнение Шрёдингера, компьютерное моделирование. Abstract. This thesis discusses the role and significance of mathematical modeling in the fields of nanotechnology and modern materials science using artificial intelligence and digital technologies, as well as the possibilities for analyzing these processes. The scientific and practical aspects of using differential equations, statistical models, numerical methods, and optimization algorithms in studying nanoscale processes are presented. In addition, the possibilities of predicting nanoparticle properties, designing new types of materials, and modeling complex physical processes using computer simulations and artificial intelligence technologies are analyzed. Keywords: nanotechnology, materials science, mathematical modeling, artificial intelligence, digital technologies, nanoparticles, numerical methods, differential equations, FEM, Monte Carlo method, Schrödinger equation, computer simulation. Kirish. Hozirgi davrda fan va texnika yutuqlari insoniyat hayotining barcha sohalariga chuqur kirib bormoqda. Ayniqsa, nanotexnologiyalar XXI asrning eng istiqbolli ilmiy yo‘nalishlaridan biri bo‘lib, atom va molekula darajasidagi jarayonlarni o‘rganish imkonini beradi. Nanotexnologiyalar yordamida yaratilayotgan materiallar — yuqori mustahkamlikka ega metallar, o‘z-o‘zini tiklovchi polimerlar, energiya tejovchi qoplamalar, hamda dori vositalarining nanoformatlari — bularning barchasi zamonaviy materialshunoslikning yangi bosqichini belgilamoqda. Maqsad va vazifalar. nanomiqyosdagi obyektlar bilan bevosita tajriba o‘tkazish qimmat va murakkab jarayon bo‘lgani sababli, matematik modellashtirish bu sohada muhim vositaga aylangan. Matematik model yordamida nanozarrachalarning harakati, issiqlik va elektr o‘tkazuvchanligi, struktural o‘zgarishlar kabi jarayonlar aniq tahlil qilinadi. Modellashtirish natijasida tajribalarning soni kamayadi, vaqt tejaladi va natijalar aniqligi oshadi. Shu sababli, bugungi kunda nanotexnologiya va matematika o‘zaro uzviy bog‘liq fanlar sifatida shakllanmoqda. Asosiy qism. 1. Matematik modellashtirishning mohiyati. Matematik modellashtirish — real tizimning muhim xususiyatlarini matematik ifodalar, tenglamalar, algoritmlar va funksiyalar orqali tasvirlash jarayonidir. Bu usul yordamida murakkab fizik, kimyoviy yoki biologik jarayonlar sonli tahlil qilinadi. Nanotizimlarni modellashtirishda qo‘llaniladigan asosiy bosqichlar: 1. Tadqiqot obyektining aniqlanishi; 2. Fizik qonuniyatlarga asoslangan matematik tenglama tanlash; 367

3. Sonli usullar yordamida hisoblashlarni amalga oshirish; 4. Olingan natijalarni grafik, jadval yoki 3D ko‘rinishda tahlil qilish. 2. Nanotexnologiyada qo‘llaniladigan asosiy modellar. Nanotizimlarni tahlil qilishda bir nechta asosiy matematik model turlari qo‘llaniladi: • Differensial tenglamalar modeli. Masalan, nanozarrachalardagi issiqlik o‘tishini ifodalash uchun Fourier tenglamasi ishlatiladi: 𝜕𝑇 = 𝛼∇2 𝑇 𝜕𝑡 bu yerda T — harorat, α — issiqlik tarqalish koeffitsienti. • Navye–Stoks tenglamalari — suyuqlik yoki gazning nanoo‘lchamli oqimini modellashtirishda qo‘llanadi. Ular yordamida nanoquvurlarda modda oqimini tahlil qilish mumkin. • Shredinger tenglamasi — elektronlarning kvant holatini modellashtirish uchun ishlatiladi: 𝜕Ψ ̂Ψ 𝑖ℏ =𝐻 𝜕𝑡 bu tenglama yordamida nanozarrachalarning energiya darajalari hisoblanadi. • Monte-Karlo usuli — ehtimollik asosida random modellarni yaratib, jarayonni kompyuter simulyatsiyasi orqali baholaydi. • Finite Element Method (FEM) — nano va mikrostruktural materiallarda stress, deformatsiya va issiqlik taqsimotini hisoblash uchun ishlatiladi. Bu usul hozirda eng samarali hisoblanadi. 3. Matematik modellashtirishning amaliy ahamiyati. Matematik modellashtirish nafaqat nazariy, balki amaliy jihatdan ham nanotexnologiyalar rivojiga katta hissa qo‘shadi. Masalan: • Tibbiyotda nanozarralar yordamida maqsadli dori yetkazish tizimini ishlab chiqish; • Qurilishda mustahkam va yengil nanokompozit materiallar yaratish; • Energetikada quyosh batareyalari samaradorligini oshirish. Bu jarayonlarning barchasida matematik modellar yordamida optimal parametrlar aniqlanadi, shuning uchun model tuzish va tahlil qilish malakasi har bir ilmiy tadqiqot uchun zarur hisoblanadi. 368

4. Nanotexnologiya va sun’iy intellekt integratsiyasi. So‘nggi yillarda modellashtirish sohasida sun’iy intellekt (AI) texnologiyalari keng qo‘llanilmoqda. AI yordamida murakkab matematik modellarning yechimlari tezroq topilmoqda, ma’lumotlar tahlili chuqurlashmoqda va natijalarning aniqligi oshmoqda. Masalan, neyron tarmoqlar yordamida nanozarrachalar strukturasining optimal konfiguratsiyasi aniqlanmoqda. Shuningdek, mashina o‘rganish algoritmlari yordamida yangi materiallarning xossalarini oldindan taxmin qilish imkoniyati paydo bo‘ldi. Xulosa va amaliy tavsiyalar. Nanotexnologiyalar va zamonaviy materialshunoslikda matematik modellashtirish fan va texnikaning kesishgan nuqtasida joylashgan muhim yo‘nalishdir. Matematik modellashtirish nanotexnologiya va materialshunoslikda yangi materiallar yaratish, ularning xossalarini boshqarish hamda texnologik jarayonlarni optimallashtirishda muhim o‘rin tutadi. Kelajakda bu yo‘nalish sun’iy intellekt, superkompyuterlar va kvant hisoblash texnologiyalari bilan uyg‘unlashib, yanada yuqori samaradorlikka erishadi. Shu bois, matematika va nanotexnologiyalar o‘zaro bir-birini to‘ldiruvchi, istiqbolli ilmiy yo‘nalishlar sifatida rivojlanishda davom etadi. Foydalanilgan adabiyotlar 1. Niyozov A. Zamonaviy materialshunoslikda modellashtirish texnologiyalari. -T.: Innovatsiya, 2023 2. Abdullayev O. Materialshunoslikda matematik modellar. -T.: Fan, 2022. 3. Atkins P. Molecular Quantum Mechanics. Oxford University Press, 2021. 4. Tersoff J.M. “Modeling of nanoscale materials and processes.” Journal of Computational Physics, 2020. INTEGRATION OF ARTIFICIAL INTELLIGENCE AND ECONOMETRIC MODELS IN DIGITAL DECISION-MAKING SYSTEMS Raximova Shohsanam Usmonali qizi, Assistant, Department of Higher Mathematics, Faculty of Production Management, Fergana State Technical University, Uzbekistan Abstract. In the modern digital era, the integration of artificial intelligence (AI) with econometric modeling provides powerful tools for analyzing, forecasting, and decision-making in dynamic economic systems. This paper discusses the theoretical and practical aspects of using AI-based models—particularly machine learning and neural networks—in econometric research. It explores how hybrid AI-econometric systems enhance prediction accuracy and optimize datadriven management in industrial and business sectors. Empirical analysis using economic indicators of Uzbekistan demonstrates that AI-integrated econometric models outperform traditional approaches in predictive performance and adaptability. Keywords: econometrics, artificial intelligence, machine learning, digital transformation, decision systems, AI-integrated econometric models Annotatsiya. Zamonaviy raqamli davrda sun’iy intellekt (AI) ni ekonometrik modellashtirish bilan integratsiyalash dinamik iqtisodiy tizimlarni tahlil qilish, prognozlash va qaror qabul qilish uchun kuchli vositalarni taqdim etadi. Ushbu maqolada ekonometrik tadqiqotlarda AI asosidagi modellar — xususan, mashinali o‘rganish va neyron tarmoqlardan foydalanishning nazariy va amaliy jihatlari muhokama qilinadi. Shuningdek, gibrid AIekonometrik tizimlar prognozlash aniqligini qanday oshirishi hamda sanoat va biznes sohalarida ma’lumotlarga asoslangan boshqaruvni optimallashtirishga qanday yordam berishi 369

ko‘rib chiqiladi. O‘zbekistonning iqtisodiy ko‘rsatkichlari asosida o‘tkazilgan empirik tahlil AI bilan integratsiyalashgan ekonometrik modellar prognozlash samaradorligi va moslashuvchanligi jihatidan an’anaviy yondashuvlardan ustun ekanligini ko‘rsatadi. Kalit so‘zlar: ekonometrika, sun’iy intellekt, mashinali o‘rganish, raqamli transformatsiya, qaror qabul qilish tizimlari, AI-integratsiyalashgan ekonometrik modellar. Аннотация. В современную цифровую эпоху интеграция искусственного интеллекта (AI) с эконометрическим моделированием предоставляет мощные инструменты для анализа, прогнозирования и принятия решений в динамических экономических системах. В данной статье рассматриваются теоретические и практические аспекты использования моделей на основе искусственного интеллекта — в частности методов машинного обучения и нейронных сетей — в эконометрических исследованиях. Также исследуется, как гибридные AI-эконометрические системы повышают точность прогнозирования и оптимизируют управление на основе данных в промышленности и бизнесе. Эмпирический анализ на основе экономических показателей Узбекистана показывает, что эконометрические модели, интегрированные с искусственным интеллектом, превосходят традиционные подходы по точности прогнозирования и адаптивности. Ключевые слова: эконометрика, искусственный интеллект, машинное обучение, цифровая трансформация, системы принятия решений, эконометрические модели, интегрированные с AI. Introduction. In the 21st century, the global economy is characterized by digitalization, automation, and the increasing role of data in decision-making processes. Traditional econometric models—though powerful—are often limited by their reliance on linear assumptions and small datasets. As economic and industrial systems become more complex, it becomes necessary to adopt innovative computational techniques that can capture non-linear, dynamic relationships. Artificial intelligence (AI) offers such capabilities through machine learning (ML), deep learning, and neural networks. By integrating AI with econometric methods, researchers and practitioners can achieve more accurate forecasts, discover hidden patterns in data, and automate analytical processes. This approach aligns with the principles of Industry 4.0 and the global trend toward smart management systems. 1. Theoretical Background of AI-Based Econometric Integration Econometrics traditionally relies on statistical estimation methods such as Ordinary Least Squares (OLS), Maximum Likelihood (ML), and Generalized Method of Moments (GMM). These methods are effective for identifying linear relationships between economic variables. However, they are limited in handling high-dimensional, non-linear, and unstructured data. AI-based models, particularly neural networks and ensemble learning techniques, overcome these limitations by learning directly from data without requiring strict distributional assumptions. In hybrid systems, AI complements econometrics by providing adaptive, data-driven optimization mechanisms that can improve both explanatory and predictive capabilities. 2. Methodology This research applies a comparative modeling approach to evaluate the performance of AI-integrated econometric systems. The following steps were conducted: 370

1. Data collection: Macroeconomic indicators of Uzbekistan (GDP, inflation, investment, and trade) from 2010 to 2024 were obtained from the State Statistics Committee. 2. Model specification: Model A - Traditional econometric model using multiple linear regression (OLS). Model B - Hybrid econometric-AI model combining OLS with Long Short-Term Memory (LSTM) networks. 3. Performance evaluation: The models were compared using Root Mean Square Error (RMSE), R² coefficient, and Mean Absolute Percentage Error (MAPE). 4. Implementation tools: Python, TensorFlow, and R software environments were used for analysis. 3. Results and Analysis The empirical results show a significant improvement in forecasting accuracy when using the AI-econometric hybrid model. Model | R² | RMSE | MAPE Traditional Econometric (OLS) |0.78 | 0.43 | 8.6% Hybrid AI-Econometric (LSTM+OLS) |0.93 | 0.25 | 3.2% These results indicate that the hybrid model captures dynamic relationships and temporal dependencies more effectively than traditional models. The LSTM structure enables the system to 'remember' long-term economic trends, which is crucial for accurate macroeconomic forecasting. 4. Discussion The integration of AI and econometrics is not only a technical advancement but also a conceptual transformation in analytical thinking. AI does not replace econometric theory—it enhances it by allowing models to learn from data continuously. Key advantages include: - High adaptability to large, real-time datasets. - Better handling of non-linear and interactive effects. - Automatic feature selection and dynamic model updating. - Stronger forecasting power and decision support. However, challenges remain: - Interpretability: AI models can behave as 'black boxes', making it difficult to explain economic meaning. - Data dependency: Quality and representativeness of data significantly affect model performance. - Computational cost: High-dimensional learning requires powerful hardware and optimization algorithms. Conclusion. The study concludes that integrating artificial intelligence into econometric systems substantially increases the efficiency and reliability of digital decision-making processes. In industrial and business applications, AI-based econometric models can improve prediction accuracy, automate data analysis, and support evidence-based strategic planning. For Uzbekistan’s developing economy, such technologies can assist in designing sustainable economic policies, predicting 371

investment trends, and enhancing digital transformation across industries. Future research should focus on expanding hybrid econometric-AI systems through Big Data analytics, Blockchain integration, and Internet of Things (IoT) technologies to achieve full-scale automation and intelligent forecasting. References 1. Gujarati, D. (2015). Basic Econometrics. McGraw-Hill Education. 2. Wooldridge, J. M. (2020). Introductory Econometrics: A Modern Approach. Cengage Learning. 3. Goodfellow, I., Bengio, Y., & Courville, A. (2016). Deep Learning. MIT Press. 4. Varian, H. R. (2019). Artificial Intelligence, Economics, and Industrial Organization. NBER Working Paper. MA’LUMOTLARNI INTELLEKTUAL TAHLIL QILISHDA BELGILAR TURLARIGA ISHLOVLAR BERISH USULLARI Raxmonov Erkin Davlatjonovich TFI, Fizika, matematika va axborot texnologiyalari kafedrasi mudiri, t.f.f.d. (PhD). E-mail: erkinraxmonov123@gmail.com Narzullayev Davronbek Zikrullayevich, TFI, Fizika, matematika va axborot texnologiyalari kafedrasi dotsenti, t.f.f.d. (PhD). E-mail: davr1960@mail.ru Tursunov Alisher Tulkunovich, TFI, Fizika, matematika va axborot texnologiyalari kafedrasi dotsenti, t.f.f.d. (PhD). E-mail: tursunovr484za@gmail.com Annotatsiya. Ushbu maqolada haqiqiy yoki real hayotdagi obyektlarning raqamli yoki ramziy tasvirlari bo‘yicha formal operatsiyalarni qurish va qo‘llash, ularni hal qilish natijalari ushbu obyektlar orasidagi ekvivalentlik munosabatlarini yechish va belgilar turlariga ishlovlar berish orqali ma’lumotlarni intellectual thlil masalasi ko‘rib chiqilgan. Kalit so‘zlar: Ma’lumotlar intellektual tahlil qilish, belgilarga ishlovlar berish, timsollarni tanib olish Аннотация. В данной статье рассматривается вопрос интеллектуального анализа данных путем построения и применения формальных операций по цифровым или символическим изображениям реальных или реальных объектов, результатов их решения, решения отношений эквивалентности между этими объектами и обработки типов символов. Ключевые слова: Интеллектуальный анализ данных, обработка символов, распознавание образов Abstract. This article examines the issue of intellectual analysis of data by constructing and applying formal operations on digital or symbolic representations of objects in real or actual life, the results of their solution, solving equivalence relations between these objects, and processing character types. Key words: Intelligent data analysis, feature processing, pattern recognition Kirish. Turli darajadagi tafsilotlarga ega bo‘lgan tanib olish usullarining shunga o‘xshash tipologiyasi tanib olish bo‘yicha tadqiqotlarida uchraydi. Shu bilan birga, ma’lum turdagi biron bir formal timsollarni tanib olish algoritmidan foydalangan holda mavzu bo‘yicha bilimlarni ifodalash usulining o‘ziga xosligini aks ettiruvchi juda muhim xususiyatni hisobga olmaydi. O‘z navbatida, bilimlarning ekstensional ifodalanishlari taluqli soha doirasidan aniq obyektlarni tavsiflash va belgilash bilan bog‘liq bo‘lib, elementlari butun tizimlar sifatidagi obyektlar bo‘lgan operatsiyalarda amalga oshiriladi. 372

Yuqorida bayon qilingan bilimlarni ifodalashning ikkita fundamental usuli timsollarni tanib olish usullarining quyidagi tasniflanishini taklif qilishga imkon beradi: –timsollarni tanib olishning intensional usullari – belgilar bilan operatsiyalar bajarishga asoslangan usullar. –timsollarni tanib olishning ekstensional usullari – obyektlar bilan operatsiyalar bajarishga asoslangan usullar. Evristik yondashuv tadqiqotchining formallashtirish qiyin bo‘lgan bilimlariga va intuitsiyasiga asoslanadi. Bu yondashuvda tadqiqotchi tanib olishning talab darajasidagi samaraga erishish uchun qanday ma’lumotlardan va qay tarzda foydalanilishini o‘zi aniqlaydi. Asosiy qism. Intensional usullarning o‘ziga xos xususiyati shundaki, ular timsollarni tanib olish algoritmlarini qurish va qo‘llashda amallarning elementlari sifatida belgilarning turli xarakteristikalari va ularning bog‘lanishlaridan foydalanadilar. Bunday elementlar sifatida ustida analitik yoki konstruktiv shaklda ifodalanuvchi harakatlar bajariluvchi alohida qiymatlar yoki belgilar qiymatlarining intervallari, o‘rtacha qiymatlar va dispersiyalar, belgilarning bog‘lanish matritsalari va boshqalar bo‘lishi mumkin[1]. Belgilar qiymatlarining taqsimot zichliklarini baholashga asoslangan usullar. Ushbu timsollarni tanib olish usullari statistik qarorlarning klassik nazariyasidan olingan bo‘lib, unda o‘rganilayotgan obyektlar ma’lum bir qonunga muvofiq belgilar fazosida taqsimlangan ko‘p o‘lchovli tasodifiy o‘zgaruvchining realizatsiyasi sifatida qaraladi. Ular ma’lum bir tanib olinayotgan sinfga mansub obyektlarning apriori ehtimolliklariga va belgilar vektori qiymatlarining shartli taqsimlanish zichligiga murojaat qiladigan Bayes qarorlarini qabul qilish sxemasiga asoslanadi. Ushbu usullar ko‘p o‘lchovli belgilar fazosining turli sohalaridagi o‘xshashliklar nisbatini aniqlashga keltiriladi. Qaror qabul qilish funksiyalari sinfi to‘g‘risidagi farazlarga asoslangan usullar. Ushbu usullar guruhida qaror qabul qilish funksiyasining umumiy ko‘rinishi ma’lum va uning sifati funksionali berilgan deb hisoblanadi. Eng keng tarqalgani qaror qabul qilish funksiyalarining chiziqli va umumlashtirilgan chiziqli bo‘lmagan polinomlar ko‘rinishidagi ifodalanishlari hisoblanadi. Bunda masalalarni yеchishda ko‘pincha ayrim gradient algoritmlari yordamida erishiladi. Lingvistik (tuzilmaviy) usullar. Timsollarni tanib olishning lingvistik usullari tillarni yaratuvchi maxsus grammatikalardan foydalanishga asoslanadi, ularning yordami bilan tan olinayotgan obyektlarning xususiyatlari to‘plamini tavsiflash mumkin. Obyektlarning turli sinflari uchun hosilaviy bo‘lmagan (atomar) elementlar (tag–timsollar, belgilar) va ular orasidagi mumkin bo‘lgan munosabatlar aniqlanadi. Har bir obyekt bir–biri bilan u yoki bu tarzda “bog‘langan” hosilaviy bo‘lmagan elementlar to‘plami bilan ifodalanadi. Grammatik tahlil qilish obyekt sintaktik jihatdan to‘g‘ri bo‘lgan (ushbu grammatikada tasvirlangan sinfga tegishli bo‘lgan) hollarda grammatik tahlil daraxti shaklida obyektning to‘liq sintaktik tavsifini “sintaktik analizator” tomonidan amalga oshiriladi. Aks holda, obyekt rad 373

etiladi yoki obyektlarning boshqa sinflarini tavsiflovchi boshqa grammatikalardan foydalangan holda tahlil qilinadi[2]. k–eng yaqin qo‘shnilar usuli. k–eng yaqin qo‘shnilarni diskriminant tahlil masalalarini hal qilish uchun birinchi marta 1952–yildayoq taklif qilingan edi. Noma’lum obyektni tasniflashda belgilar fazosida berilgan sondagi (k) tanib olinayotgan sinflarga tegishligi ma’lum bo‘lgan boshqa obyektlar (eng yaqin qo‘shnilar) topiladi. Noma’lum obyektni ma’lum bir diagnostika sinfiga tayinlash to‘g‘risidagi qaror, masalan, oddiy ovozlarni hisoblash yordamida uning eng yaqin qo‘shnilarining ushbu ma’lum tegishliligi haqidagi ma’lumotlarni tahlil qilish orqali qabul qilinadi. Dastlab, o‘xshashlik nisbatini baholashning parametrik bo‘lmagan usuli sifatida k–eng yaqin qo‘shnilar usuli qaralgan. Ushbu usul uchun optimal Bayes klassifikatori bilan solishtirganda samaradorligining nazariy baholari olingan. k– eng yaqin qo‘shnilar usuli uchun asimptotik xatolik ehtimoli Bayes qoidasi xatolaridan ikki baravar ko‘p emasligi isbotlangan[2,3]. Gipertekisliklar bo‘laklaridan tashkil topgan ajratuvchi sirt turi xil bo‘lishi mumkin va sinflarga ajratilayotgan to‘plamlarning o‘zaro joylashuviga bog‘liq. k– eng yaqin qo‘shnilar qoidasidan foydalangan holda sinflarga ajratish mexanizmlarining yana bir talqini ham ishlatilishi mumkin. U abstrakt yoki berilgan belgilar fazosi bilan biror–bir shakl almashtirish bilan bog‘liq bo‘lgan ma’lum latent o‘zgaruvchilarning mavjudligi haqidagi g‘oyaga asoslanadi. Agar latent o‘zgaruvchilar fazosida obyektlar orasidagi juftlik masofalari dastlabki belgilar fazosidagi bilan bir xil bo‘lsa va bu o‘zgaruvchilar soni obyektlar sonidan sezilarli darajada kam bo‘lsa, u holda 𝑘–eng yaqin qo‘shnilar usulini shartli ehtimollik taqsimot zichliklarining parametrik bo‘lmagan baholarini solishtirish nuqtai nazaridan ko‘rib chiqish mumkin.. Timsollarni tanib olish uchun 𝑘–eng yaqin qo‘shnilar usulidan foydalanilganda, tadqiqotchi tashxis qo‘yilayotgan obyektlarning yaqinligini aniqlash uchun metrikani tanlashdan iborat qiyin masalani hal qilishi kerak. Shuni ta’kidlash kerakki, o‘quv tanlanmasidagi obyektlar sonini kamaytirish bu usulning kamchiliklari hisoblanadi, chunki bu o‘quv tanlanmasining reprezentativligini pasaytiradi. Baholarni hisoblash (ovoz berish) algoritmlari. Baholarni hisoblash algoritmlarining (BHA) ishlash tamoyili berilgan belgilar to‘plamining qism– to‘plamlari tizimi bo‘lgan belgilar ansambllari tizimi bo‘yicha tanib olinayotgan va etalon obyektlarining “yaqinligini” tavsiflovchi ustuvorlikni (o‘xshashlik ballarini) hisoblashdan iborat. Muhokama qilingan barcha usullardan farqli o‘laroq, baholarni hisoblash algoritmlari obyekt tavsiflari bilan mutlaqo yangi usulda ishlaydi. Ushbu algoritmlar uchun obyektlar bir vaqtning o‘zida belgilar fazosining juda turli qism–fazolarida mavjud bo‘ladi. BHA sinfi belgilardan foydalanish g‘oyasini mantiqiy xulosaga keltiradi. Belgilarning qaysi kombinatsiyasi eng informativ ekanligi har doim ham ma’lum emasligi sababli, BHA usullarida obyektlarning o‘xshashlik darajasi obyektlar tavsifiga kiritilgan barcha mumkin 374

bo‘lgan yoki o‘ziga xos belgilar kombinatsiyalarini taqqoslash orqali hisoblanadi[4]. Foydalaniladigan belgilar kombinatsiyalari (qism–fazolar) tayanch to‘plamlar yoki obyektlarning qisman tavsiflari to‘plamlari deb ataladi. Tanib olinayotgan obyekt bilan etalon obyektlari deb ataluvchi o‘quv tanlanmasi obyektlari (tasniflanishi ma’lum bo‘lgan) o‘rtasidagi umumiy yaqinlik tushunchasi kiritiladi[4,5]. BHA ning nazariy imkoniyatlari har qanday timsollarni tanib olish algoritmining imkoniyatlaridan yuqori yoki hech bo‘lmaganda past emas, chunki BHA yordamida o‘rganilayotgan obyektlar bilan barcha operatsiyalarni amalga oshirish mumkin. Qaror qabul qilish qoidalari jamoalari. Turli xil tanib olish algoritmlari bir xil o‘quv tanlanmasida o‘zlarini turlicha namoyon qilishlari sababli, ushbu algoritmlarning kuchli tomonlarini moslashuvchan tarzda ishlatadigan sintetik qaror qabul qilish qoidasi haqidagi savol tug‘iladi. Sintetik qaror qabul qilish qoidasida ikki darajali tanib olish sxemasidan foydalaniladi. Birinchi darajada xususiy tanib olish algoritmlari ishlaydi, ularning natijalari ikkinchi darajada sintez blokida birlashtiriladi. Bunday birlashtirishning eng keng tarqalgan usullari ma’lum bir algoritmning yaroqlilik sohalarini ajratib ko‘rsatishga asoslangan. Shunday qilib, timsollarni tanib olish masalasini miqdoriy bo‘lmagan parametrlarni tavsiflash masalasi sifatida tasniflash mumkin. Keyinchalik, chiziqli ajratish (diskriminant) funksiyalaridan foydalanadigan tanib olish modellarida empirativ shkalalarni tanlash mezonlarini ko‘rib chiqamiz. Yuqoridagi tasnifga ko‘ra, bu tanib olish modellari intensional usullar guruhiga kiradi[5]. Timsollarni tanib olish masalasi uchun empirativ shkalalarni quyidagi dispersiya nisbati minimallashtiriladigan sonli belgilar bilan berish ko‘zda tutiladi: 𝐽= Λ𝑊 Λ𝑇 bu yеrda Λ 𝑇 = Λ = Λℓ+1,ℓ+1 𝑎̅𝑇 𝑆𝑎̅ 𝑆 matritsa (2.8) formula bo‘yicha hisoblanadi. Λ𝑊 ni hisoblash uchun Λ𝑊 = (𝜆𝑊 𝑖𝑗 ) deb belgilab olamiz. U holda: 𝑄 1 (𝑝) 𝜆𝑊 ∑ 𝑁𝑝 𝜆𝑖𝑗 𝑖𝑗 = 𝑁 𝑝=1 bu yеrda 𝑖, 𝑗 = ̅̅̅̅̅ 1, ℓ, ; 𝑄 – o‘quv tanlanmalari (O‘T) soni; 𝑁𝑝 –𝑝 o‘quv tanlanmasidagi obyektlar soni; 𝑔 𝑄 1 (𝑝)𝑖 𝑊 𝜆𝑊 ∑ 𝑎𝑖 ∑ 𝑁𝑝 𝜇𝑘 𝑘,ℓ+1 = 𝜆ℓ+1,𝑘 = 𝑁 𝑖=1 𝑝=1 (𝑝)𝑖 bu yеrda 𝜇𝑘 (𝑝)𝑖 –𝑝 o‘quv tanlanmasi bo‘yicha aniqlangan ki parametr. (𝑝) 𝜇𝑘 va 𝜇𝑘 orqali 𝑝–o‘quv tanlanmasi bo‘yicha 𝑘–belgining xususiy va umumiy (𝑝) (𝑝) o‘rtacha qiymatlarini belgilaymiz; 𝑛𝑖 va 𝑡𝑖𝑗 – 𝑖–gradatsiyadagi obyektlar soni va 𝑝–o‘quv tanlanmasi bo‘yicha aniqlangan 𝑡𝑖𝑗 parametr qiymati. 375

Quyidagi parametrlarni kiritamiz: 𝜇𝑘∗𝑖 = 𝑄 𝑄 𝑝=1 𝑝=1 𝑄 (𝑝) 1 1 𝑛 1 (𝑝)𝑖 (𝑝)𝑖 (𝑝) (𝑝) (𝑝)𝑖 (𝑝) (𝑝) ∑ 𝑁𝑝 𝜇𝑘 = ∑ 𝑁𝑝 𝑖 (𝑚𝑘 − 𝑚𝑘 ) = = ∑ (𝑛𝑖 𝑚𝑘 − −𝑛𝑖 𝑚𝑘 ) 𝑁 𝑁 𝑁𝑝 𝑁 𝑝=1 𝑄 = 𝑛𝑖 1 (𝑝) (𝑝) (𝑚𝑘𝑖 − ∑ 𝑛𝑖 𝑚𝑘 ) 𝑁 𝑛𝑖 𝑝=1 𝑄 1 (𝑝) ∗ 𝑡𝑖𝑗 = ∑ 𝑁𝑝 𝑡𝑖𝑗 𝑁 𝑝=1 Bunda quyidagini ko‘rsatish mumkin: ∗ ∑𝑄𝑝=1 𝑡𝑖𝑗 =0 𝑔 ∑𝑗=1 𝜇𝑘∗𝑖 = 0 U holda quyidagini olamiz: (1) (2) 𝑔 ∗𝑖 ̅̅̅̅̅ 𝜆𝑊 𝑘,𝑙+1 = ∑ 𝑎𝑖 𝜇𝑘 , 𝑘 = 1, ℓ 𝑖=1 𝑔 𝑔 ∗ 𝜆𝑊 ℓ+1,ℓ+1 = ∑ ∑ , 𝑎𝑖 𝑎𝑗 𝑡𝑖𝑗 𝑖=1 𝑗=1 Λ𝑊 ning determinanti o‘zgartirilganda Λ kabi, u holda quyidagilarni hosil qilamiz: Λ𝑊 = Λ𝑊ℓ+1,ℓ+1 𝑎̅𝑇 𝑊 (3) −1 bu yеrda 𝑊 = (𝜔𝑖𝑗 ) = 𝑇 ∗ − 𝑀∗ Λ𝑊ℓ+1,ℓ+1 𝑀∗𝑇 ∗ ̅̅̅̅̅ 𝑇 = (𝑡𝑖𝑗 𝑔; 𝑘 = ̅̅̅̅̅ 1, ℓ; (1) va (2) ifodalarida tenglik ), 𝑀∗ = (𝜇𝑘∗𝑖 ), 𝑖, 𝑗 = 1, quyidagicha shaklga ega bo‘ladi: ∑𝑔𝑗=1 𝜔𝑘𝑖 = 0 (4) U holda taqsimlanish munosabatlari quyidagicha ko‘rinishda bo‘ladi: Λ𝑊ℓ+1,ℓ+1 𝑎̅𝑇 𝑊𝑎̅ 𝐽= Λℓ+1,ℓ+1 𝑎̅𝑇 𝑆𝑎̅ Λ𝑊 ≥ 0 va (4) ifoda o‘rinli bo‘lganli sababli 𝑎̅𝑇 𝑊𝑎̅ kvadrat shakli manfiy bo‘lmaydi. Oxirgi tenglikning birinchi belgisi 𝑎̅ to‘plamiga bog‘liq bo‘lmaganligi uchun hamda timsollarni tanib olish masalalarini yеchishda empirativ shkalalarni tanlash mezonini quyidagicha yozish mumkin[6,7]. 𝑎̅𝑇 𝑊𝑎̅ 𝑎̅𝑇 𝑆𝑎̅ → 𝑚𝑖𝑛 (5) bu yеrda 𝑎̅ ∈ 𝐵 𝑔 shartga ko‘ra, optimal butun sonli raqamli belgilarni topish zarur bo‘lganda (5) masala quyidagicha ko‘rinishga keladi: ̅ 𝑇 𝑊𝐷 ̅ 𝐷 → 𝑚𝑖𝑛 𝑇 ̅ 𝑆𝐷 ̅ 𝐷 (6) bu yеrda 𝐷 ∈ 𝑃 𝑔 Xulosa va amaliy tavsiyalar. Bunday hollarda chiziqli tanib olish modelining aniqligini oshirish uchun miqdoriy bo‘lmagan 𝑥 𝑖 , 𝑖 = ̅̅̅̅̅̅̅̅̅̅ ℓ + 1, 𝑝 belgisidan foydalanish imkoniyati ko‘rib chiqiladi. 376

Chiziqli ajratish funksiyalari faqat miqdoriy belgilar uchun tuzulishi mumkin va 𝑥 , 𝑖 = ̅̅̅̅̅̅̅̅̅̅ ℓ + 1, 𝑝 belgilarini 𝜉𝑖 , 𝑖 = ̅̅̅̅̅̅̅̅̅̅ ℓ + 1, 𝑝 empirativ shkalasidan foydalanib, miqdoriy bo‘lgan shakllarga aylantirilgandan keyin qabul qiladi. Empirativ shkalalarni qidirishda tanib olish xatoliklar darajasini kamaytirish maqsadida Λ raqamlashtirilgan belgilarni inobatga olgan holda, 𝐽 = 𝑊 taqsimlash mezonining 𝑖 Λ𝑇 funksiyasi optimallashtiriladi[8]. Foydalanilgan adabiyotlar ro‘yхati 1. Гнеденко Б.В. Курс теории вероятностей: Учебник. – Изд. 6–е изд, исправл– М.: Эдиториал. Гл. ред. физ.–мат. лит., 2021.– 448 с. 2. Клейн Ф. Лекции о развитии математики в XIX столетии. т. 2. – Москва– Ижевск, институт компътерных исследований 2013. – 240 с. 3. Плошко Б.Г., Елисеева И.И. История статистики: Учеб. пособие. – М.: Финансы и статистика. 1990. – 295 с. 4. Орлов А.И. Вероятностно–статистические модели данных – основа методов прикладной статистики / Заводская лаборатория. Диагностика материалов. 2020. Т.86. № 7. – c. 5–6. 5. Kevin P. Murphy. Machine Learning: A Probabilistic Perspective MIT Press, 2022 . – pp.1104. 6. Полyaкова В.В., Шаброва Н.В.. Прикладнаya статистика: методы анализа эмпирической информатции. Учебно–методическое пособие. РФ, Екатеринбург : Изд–во Урал. ун–та, 2020. – 188 с 7. Орлов А.И. Основные требования к математическим методам классификatsiи // Заводская лаборатория. Диагностика материалов. 2020. Т.86. № 11. – c. 67 8. Nishanov A., Ruzibaev O., Chedjou J. C., Kyamakya K., Abhiram, Kolli, De Silva, Djurayev G., Khasanova M. Algorithm for the selection of informative symptoms in the classification of medical data // Developments Of Artificial Intelligence Technologies In Computation And Robotics, Volume:12, 2020, p. 647-658doi:10.1142/9789811223334_0078 5G MOBIL ALOQA TARMОQLARINING QURILISH ASOSLARI Ruziyev Shoxrux Shoymurod o‘g‘li, UMFT magistranti, sh.ruziyev@gmail.com Annotatsiya. Ushbu maqolada 5G mobil aloqa tarmoqlarining qurilish asoslari, texnologik yechimlari va avlodlar qiyosiy tahlili yoritilgan. 3G dan 5G NR gacha bo‘lgan tarmoqlar tezlik, kechikish, arxitektura va xavfsizlik mezonlari bo‘yicha taqqoslangan. O‘zbekiston uchun amaliy tavsiyalar ishlab chiqilgan. Kalit so‘zlar: mobil aloqa, 4G/LTE, 5G, kechikish, chastota diapazoni. Аннотация. В данной статье рассматриваются основы построения сетей мобильной связи 5G, технологические решения и сравнительный анализ поколений. Сети от 3G до 5G NR сравниваются по скорости, задержке, архитектуре и безопасности. Разработаны практические рекомендации для Узбекистана. Ключевые слова: мобильная связь, 4G/LTE, 5G, задержка, частотный диапазон Abstract. This article examines the fundamentals of 5G mobile network construction, technological solutions, and a comparative analysis of generations. Networks from 3G to 5G NR are compared in terms of speed, latency, architecture, and security. Practical recommendations for Uzbekistan are developed. Key words: mobile communication, 4G/LTE, 5G, latency, frequency range. 377

Kirish. Zamonaviy axborot-kommunikatsiya texnologiyalarining jadal rivojlanishi va raqamli iqtisodiyotning shakllanishi sharoitida mobil aloqa tarmoqlari insoniyat hayotining ajralmas qismiga aylandi. Bugungi kunda smartfonlar, planshetlar va boshqa aqlli qurilmalar orqali uzatiladigan ma’lumotlar hajmi yildan-yilga misli ko‘rilmagan darajada o‘sib bormoqda. Aynan shu ehtiyojni qondirish va kelajak texnologiyalarining poydevorini yaratish maqsadida beshinchi avlod - 5G mobil aloqa tarmog‘i ishlab chiqildi. 5G texnologiyasi o‘zidan oldingi 4G/LTE tarmog‘iga nisbatan tubdan farq qiluvchi xususiyatlarga ega: ma’lumot uzatish tezligi 10 Gbit/s gacha yetkazilishi, kechikish vaqti (latentlik) 1 millisekund darajasiga tushirilishi hamda 1 kvadrat kilometr maydoniga bir milliondan ortiq qurilmani bir vaqtning o‘zida ulash imkoniyati bilan u butun sanoat va ijtimoiy hayotni inqilobiy tarzda o‘zgartirishi kutilmoqda[1]. Mavzuning dolzarbligi shundaki, O‘zbekiston ham raqamli iqtisodiyotga o‘tish strategiyasi doirasida 5G tarmoqlarini joriy etishga katta e’tibor qaratmoqda. "Raqamli O‘zbekiston - 2030" dasturi hamda mamlakatimizdagi ko‘plab aqlli shahar loyihalari muvaffaqiyatini bevosita 5G infratuzilmasining samarali qurilishi belgilab beradi. Bundan tashqari, sanoat avtomatlashtirish, masofaviy tibbiyot, avtonom transport vositalari va "Narsalar Interneti" (IoT) kabi istiqbolli sohalarda 5G tarmoqsiz rivojlanishni tasavvur qilish qiyin. Ushbu mavzuning ahamiyati shundaki, 5G tarmog‘ini loyihalash va qurish jarayoni yuqori darajadagi muhandislik bilimlari, chastota spektrini boshqarish, antenna tizimlarini joylashtirish va tarmoq arxitekturasini optimallashtirish kabi murakkab vazifalarni qamrab oladi. Kelgusi avlod mutaxassislari telekommunikatsiya muhandislari, tarmoq arxitektorlari va IT soha vakillari uchun 5G qurilishining asosiy tamoyillarini chuqur o‘rganish nafaqat kasbiy zaruriyat, balki mamlakatimiz raqamli salohiyatini mustahkamlashga qo‘shilgan hissa hamdir. Asosiy qism. Jahоnda beshinсhi avlоd tarmоqlari paydо bо’lish bо’yiсha taxmin, birinсhi navbatda MSE, 3GPP, ETSI kabi alоqa sоhasidagi xalqarо tashkilоtlar standartlash rejalari bilan belgilangan. Mamlakatimizda 5G tarmоqlar paydо bо’lish ushbu tarmоqning jоriy etish tartibga sоluvсhi bazaga tayanishi kerak – сhastоtalar pоlоsalarni ajratish bо’yiсha davtlat kо’mitasi hamda 5G xizmatlar kо’rsatishga оperatоrlarga litsenziyalar berish tartibiga о’zgarishlar kiritish bо’yiсha Hukumat qarоrlari. 5G standartlashning asоsiy оrgani bо’lgan 3GPP rejalariga muvоfiq standartlashning birinсhi fazasi 2018- yilda 15 reliz bilan yakunlanishi kerak. Оldingi avlоdlar tarmоqlarni jоriy etish tajribasidan ma’lumki, texnik о’ziga xоsliklarni ishlab сhiqishni yakunlash vaqtidan bоshlab bоzоra namuna paydо bо’lishgaсha kamida 2 yil о’tadi[3]. Ishlab сhiqishni murakkablashtiruvсhi sabablardan biri – bu 5G сhastоtali diapazоnlarda butunlay nоaniqlik. 5G uсhun 6 GGs dan past diapazоnda kengligi 500 MGs сhastоtalar pоlоsasi ajratish masalasida 4-yillik tadqiqоtlar davridan keyingi Xalqarо radiоalоqa kоnferensiya XRK-15 muvaffaqiyatsiz natijalari 5G 378