在本研究中，我們考慮了兩組不同的多變量變數，分別記為 Y 和 X，目標是建立一個迴歸模型來描述 Y 和 X 之間的關係。雖然多變量迴歸模型來處理這種問題是一種蠻直覺的方法，而且過去也有很多人研究過，但資料仍存在一些重要問題尚未被充分解決，包括 Y 和 X 中的測量誤差、資訊變數 X 的選擇，以及 Y 中網絡結構。此外，現有的方法大多限制是只考慮線性模型和連續型的 Y。為了應對這些挑戰並提供可靠的估計方法，我們首先提出了一種一般性的非參數方法來建模 Y 和 X，其中 Y 被允許是混合分布的隨機向量。我們將隨機森林方法結合迴歸校正（regression calibration），處理測量誤差並偵測 X 中的資訊變數及 Y 的網絡結構。在某些情況下，Y 和 X 被多變量線性模型描述的特例下，我們開發了兩種不同的誤差修正策略，能同時解決測量誤差問題以及變數選擇和網絡偵測問題。最後，透過模擬研究和資料分析，數值結果驗證了我們的新方法是有效。

In this study, we consider two different multivariate variables, denoted Y and X, respectively, and aim to build up the regression model to characterize Y and X. While the multivariate regression model is an intuitive strategy and has been studied in past years, there still exist some crucial issues in the dataset and have not been fully addressed, including measurement error in both Y and X, selection of informative X, and detection of network structure in Y. Moreover, the other critical limitation in the existing methods is the consideration of linear models and continuous variable Y. To tackle these challenges and provide reliable estimator, we first consider a general nonparametric approach to model Y and X, and Y is allowed to be a mixture distributed random vector. We extend the random forest method with regression calibration approach to handle measurement error and detect informative X and network structure in Y. Moreover, under a special case that Y and X are characterized by multivariate linear models, we develop two different error-corrected strategies to address measurement error and handle variable selection and network detection simultaneously. Throughout simulation studies and data analysis, numerical results verify the validity of our novel methods.

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