單位伽瑪混合迴歸模型估計

Abstract

連續有界數據（範圍在 (0,1)）在醫學、金融、生物等領域有廣泛的應用。然而，傳統迴歸模型在處理異質性數據時往往存在局限，難以有效捕捉潛在的數據結構。因此，本研究用混合回歸模型（Mixture Regression Model）來建模這類包含多種不同成分的數據。本研究提出了一種基於單位伽瑪（Unit Gamma）分佈的混合迴歸模型，其中Unit Gamma 分佈是透過變數變換由 Gamma 分佈導出，能夠靈活描述對稱與偏態數據的特性。我們假設數據由多個潛在子群組成，每個子群均服從獨立的 Unit Gamma 分佈，並開發了一種基於期望最大化（EM）演算法和牛頓法（Newton's Method）的參數估計方法。EM 演算法用於最大化混合分佈的對數概似函數，其中 E-step計算各觀測值屬於不同子群的後驗機率，而 M-step 則更新各子群的參數，包括迴歸係數、分散參數及混合權重。最後，我們透過模擬實驗和真實資料來評估所提出模型的參數估計準確性與精確度，以驗證其在異質數據分析中的優勢。

Continuous bounded data (ranging from (0,1)) are widely applied in fields such asmedicine, finance, and biology. However, traditional regression models often havelimitations with heterogeneous data, making it difficult to capture underlying datastructures. Therefore, this study employs a Mixture Regression Model to model thistype of data, which comprises various distinct components. This research proposes a mixture regression model based on the Unit Gamma distribution. The Unit Gamma distribution is derived from the Gamma distribution through a variable transformation, enabling flexible description of both symmetric and skewed data characteristics. We hypothesize that the data consist of multiple latent subgroups, with each subgroup independently following a Unit Gamma distribution. We have developed a parameter estimation method based on the Expectation-Maximization (EM) algorithm and Newton's Method. The EM algorithm is used to maximize the log-likelihood function of the mixture distribution, where the E-step calculates the posterior probabilities of each observation belonging to different subgroups, and the M-step updates the parameters for each subgroup, including regression coefficient, shape parameter and mixing proportion. Finally, we evaluate the accuracy and precision of the proposed model's parameter estimation through simulation experiments, and real data to validate itsadvantages in heterogeneous data analysis.