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題名 混合變數之互信息的假設檢定
Hypotheses testing of the mutual information of two mixed-type variables
作者 王俞惠
Wang, Yu-Hui
貢獻者 薛慧敏
Hsueh, Huey-Ming
王俞惠
Wang, Yu-Hui
關鍵詞 互信息
相關性檢定
漸近原理
排列檢定
檢定力
型一錯誤率
Mutual information
Test of dependence
Asymptotic theory
Permutation test
Power
Type one error rate
日期 2022
上傳時間 5-十月-2022 09:01:50 (UTC+8)
摘要 在大數據時代，很多實務資料同時涵蓋連續型與離散型變數（稱為混合型變數），在資料分析過程中常牽涉測量這些變數間的相關性。文獻上有許多方法以檢定連續型變數之間或離散型變數之間的獨立性，本研究則針對混合型變數之間相關性的檢定問題，我們將資訊理論中的互信息量延伸至此問題用以測量混合型變數間的相關性。Beknazaryan et al.(2018)定義混合型變數之互信息並且提出無母數估計方法，作者們並推導出該估計量的漸近常態性質。本研究延續該研究成果發展相關的檢定方法，包括常態漸近檢定和排列檢定。透過模擬實驗觀察到漸近檢定表現並不好，其型一誤差率遠低於顯著水準。我們進一步發現即便增加樣本數至3000時，該無母數估計量的抽樣分配與常態分配仍有相當歧異。相反地排列檢定的表現則過度激進，其型一誤差率無法有效控制在顯著水準下。為參考起見，模擬中也發現兩個檢定方法在特定對立假設下都有良好檢定力以偵測混合變數的相關性。本研究將所發展的統計檢定應用至一組羽球實證資料。本研究的結論是這兩個方法都有待改進。
In the generation of big data, many practical data including both continuous and discrete variables (also known as mixed variables), and an analytical procedure often involves measuring the dependence between mixed variables. In literatures, many methods are developed to measure and test the dependence among continuous variables or among discrete variables. In this study, we focus on the test of the dependence between mixed variables. We extend the mutual information criterion in information theory to describe dependence between mixed variables. In the study of Beknazaryan et al. (2018), the authors defined the mutual information of mixed variables and proposed a nonparametric estimation method which employed kernel estimation. Further, the authors showed the asymptotic normality of the estimator. In this study, we follow their research and develop two correspondent hypothesis tests: an asymptotic test and permutation test. Through simulation experiments, it is seen that the two testing procedures do not have satisfactory performance in controlling the type I error rate, the asymptotic test is too conservative and the permutation test is too liberal. Further, we find that even when the sample size is increased to 3000, the sampling distribution of the nonparametric estimator is still deviated from a normal distribution. However, we find that both tests have good power to detect two dependent mixed variables under certain alternative hypothetical scenarios in the simulation. The two tests are applied to a real badminton data set for demonstration. We conclude that both methods have room for improvement.
參考文獻 劉宇軒，蔡子傑與薛慧敏(2022)。穿戴式裝置資料在羽球訓練上可行性分析。
K. J. Begum & A. Ahmed (2015). The Importance of Statistical Tools in Research Work. International Journal of Scientific and Innovative Mathematical Research, Vol. 3, 50-58.
A. Beknazaryan, X. Dang & H. L. Sang (2018). On Mutual Information Estimation for Mixed-Pair Random Variables. Statistics & Probability Letters, Vol. 148, 9-16.
N. R. Cox (1974). Estimation of the Correlation between a Continuous and a Discrete Variable. Biometrics, Vol. 30, 171-178.
T. M. Cover & Joy A. Thomas (2006). Elements of Information Yheory. John Wiley & Sons, Inc.
Evie McCrum-Gardner (2007). Which is the Correct Statistical Test to Use? British Journal of Oral and Maxillofacial Surgery, Vol. 46, 38–41.
S. Geisser (1975). The Predictive Sample Reuse Method with Applications. Journal of the American Statistical Association, Vol. 70, 320-328.
W. H. Gao, S. Kannan, S. W. Oh & P. Viswanath (2017). Estimating Mutual Information for Discrete-Continuous Mixtures. Paper presented at the 31st Conference on Neural Information Processing Systems, Long Beach, CA, USA.
J. Scott Long (1997). Regression Models for Categorical and Limited Dependent Variables. Sage Publications, Inc.
P. Laarne, M. A. Zaidan & T. Nieminen (2020). Ennemi: Non-Linear Correlation Detection With Mutual Information. SoftwareX, Vol. 14, 100686.
K. Pearson (1909). On a New Method of Determining Correlation Between a Measured Character A, and a Character B, of which Only the Percentage of Cases Wherein B Exceeds (or Falls Short of) a Given Intensity is Recorded for Each Grade of A. Biometrika, Vol. 7, 96-105.
M. Stone (1974). Cross-Validatory Choice and Assessment of Statistical Predictions. Journal of the Royal Statistical Society Series B (Methodological), Vol. 36, 111-147.
B. W. Silverman (1986). Density Estimation for Statistics and Data Analysis. Chapman & Hall, Inc.
R. F. Tate (1954). Correlation Between a Discrete and a Continuous Variable. Point-Biserial Correlation. The Annals of Mathematical Statistics, Vol. 25, 603-607.
R. F. Tate (1955). The Theory of Correlation Between Two Continuous Variables when One is Dichotomized. Biometrika, Vol. 42, 205-216.
A. Wehrl (1978). General Properties of Entropy. Reviews of Modern Physics, Vol. 50, 221-260.
描述 碩士
國立政治大學
統計學系
109354002
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0109354002
資料類型 thesis
dc.contributor.advisor 薛慧敏zh_TW
dc.contributor.advisor Hsueh, Huey-Mingen_US
dc.contributor.author (作者) 王俞惠zh_TW
dc.contributor.author (作者) Wang, Yu-Huien_US
dc.creator (作者) 王俞惠zh_TW
dc.creator (作者) Wang, Yu-Huien_US
dc.date (日期) 2022en_US
dc.date.accessioned 5-十月-2022 09:01:50 (UTC+8)-
dc.date.available 5-十月-2022 09:01:50 (UTC+8)-
dc.date.issued (上傳時間) 5-十月-2022 09:01:50 (UTC+8)-
dc.identifier (其他 識別碼) G0109354002en_US
dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/142069-
dc.description (描述) 碩士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 統計學系zh_TW
dc.description (描述) 109354002zh_TW
dc.description.abstract (摘要) 在大數據時代，很多實務資料同時涵蓋連續型與離散型變數（稱為混合型變數），在資料分析過程中常牽涉測量這些變數間的相關性。文獻上有許多方法以檢定連續型變數之間或離散型變數之間的獨立性，本研究則針對混合型變數之間相關性的檢定問題，我們將資訊理論中的互信息量延伸至此問題用以測量混合型變數間的相關性。Beknazaryan et al.(2018)定義混合型變數之互信息並且提出無母數估計方法，作者們並推導出該估計量的漸近常態性質。本研究延續該研究成果發展相關的檢定方法，包括常態漸近檢定和排列檢定。透過模擬實驗觀察到漸近檢定表現並不好，其型一誤差率遠低於顯著水準。我們進一步發現即便增加樣本數至3000時，該無母數估計量的抽樣分配與常態分配仍有相當歧異。相反地排列檢定的表現則過度激進，其型一誤差率無法有效控制在顯著水準下。為參考起見，模擬中也發現兩個檢定方法在特定對立假設下都有良好檢定力以偵測混合變數的相關性。本研究將所發展的統計檢定應用至一組羽球實證資料。本研究的結論是這兩個方法都有待改進。zh_TW
dc.description.abstract (摘要) In the generation of big data, many practical data including both continuous and discrete variables (also known as mixed variables), and an analytical procedure often involves measuring the dependence between mixed variables. In literatures, many methods are developed to measure and test the dependence among continuous variables or among discrete variables. In this study, we focus on the test of the dependence between mixed variables. We extend the mutual information criterion in information theory to describe dependence between mixed variables. In the study of Beknazaryan et al. (2018), the authors defined the mutual information of mixed variables and proposed a nonparametric estimation method which employed kernel estimation. Further, the authors showed the asymptotic normality of the estimator. In this study, we follow their research and develop two correspondent hypothesis tests: an asymptotic test and permutation test. Through simulation experiments, it is seen that the two testing procedures do not have satisfactory performance in controlling the type I error rate, the asymptotic test is too conservative and the permutation test is too liberal. Further, we find that even when the sample size is increased to 3000, the sampling distribution of the nonparametric estimator is still deviated from a normal distribution. However, we find that both tests have good power to detect two dependent mixed variables under certain alternative hypothetical scenarios in the simulation. The two tests are applied to a real badminton data set for demonstration. We conclude that both methods have room for improvement.en_US
dc.description.tableofcontents 第一章 介紹……………………………………………………………………………………………………………………1
第二章 研究方法…………………………………………………………………………………………………………3
2.1 統計假設…………………………………………………………………………………………………………………3
2.2 點估計………………………………………………………………………………………………………………………4
2.3 假設檢定方法………………………………………………………………………………………………………7
2.3.1 漸近檢定(Asymptotic Test)………………………………………………………………7
2.3.2 排列檢定(Permutation Test)……………………………………………………………8
第三章 模擬驗證…………………………………………………………………………………………………………9
3.1 模擬驗證…………………………………………………………………………………………………………………9
3.2模擬結果…………………………………………………………………………………………………………………15
第四章 實證…………………………………………………………………………………………………………………25
4.1 資料介紹………………………………………………………………………………………………………………25
4.2 檢定結果………………………………………………………………………………………………………………25
第五章 結論…………………………………………………………………………………………………………………27
參考文獻…………………………………………………………………………………………………………………………29		zh_TW
dc.format.extent 1359076 bytes-
dc.format.mimetype application/pdf-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0109354002en_US
dc.subject (關鍵詞) 互信息zh_TW
dc.subject (關鍵詞) 相關性檢定zh_TW
dc.subject (關鍵詞) 漸近原理zh_TW
dc.subject (關鍵詞) 排列檢定zh_TW
dc.subject (關鍵詞) 檢定力zh_TW
dc.subject (關鍵詞) 型一錯誤率zh_TW
dc.subject (關鍵詞) Mutual informationen_US
dc.subject (關鍵詞) Test of dependenceen_US
dc.subject (關鍵詞) Asymptotic theoryen_US
dc.subject (關鍵詞) Permutation testen_US
dc.subject (關鍵詞) Poweren_US
dc.subject (關鍵詞) Type one error rateen_US
dc.title (題名) 混合變數之互信息的假設檢定zh_TW
dc.title (題名) Hypotheses testing of the mutual information of two mixed-type variablesen_US
dc.type (資料類型) thesisen_US
dc.relation.reference (參考文獻) 劉宇軒，蔡子傑與薛慧敏(2022)。穿戴式裝置資料在羽球訓練上可行性分析。
K. J. Begum & A. Ahmed (2015). The Importance of Statistical Tools in Research Work. International Journal of Scientific and Innovative Mathematical Research, Vol. 3, 50-58.
A. Beknazaryan, X. Dang & H. L. Sang (2018). On Mutual Information Estimation for Mixed-Pair Random Variables. Statistics & Probability Letters, Vol. 148, 9-16.
N. R. Cox (1974). Estimation of the Correlation between a Continuous and a Discrete Variable. Biometrics, Vol. 30, 171-178.
T. M. Cover & Joy A. Thomas (2006). Elements of Information Yheory. John Wiley & Sons, Inc.
Evie McCrum-Gardner (2007). Which is the Correct Statistical Test to Use? British Journal of Oral and Maxillofacial Surgery, Vol. 46, 38–41.
S. Geisser (1975). The Predictive Sample Reuse Method with Applications. Journal of the American Statistical Association, Vol. 70, 320-328.
W. H. Gao, S. Kannan, S. W. Oh & P. Viswanath (2017). Estimating Mutual Information for Discrete-Continuous Mixtures. Paper presented at the 31st Conference on Neural Information Processing Systems, Long Beach, CA, USA.
J. Scott Long (1997). Regression Models for Categorical and Limited Dependent Variables. Sage Publications, Inc.
P. Laarne, M. A. Zaidan & T. Nieminen (2020). Ennemi: Non-Linear Correlation Detection With Mutual Information. SoftwareX, Vol. 14, 100686.
K. Pearson (1909). On a New Method of Determining Correlation Between a Measured Character A, and a Character B, of which Only the Percentage of Cases Wherein B Exceeds (or Falls Short of) a Given Intensity is Recorded for Each Grade of A. Biometrika, Vol. 7, 96-105.
M. Stone (1974). Cross-Validatory Choice and Assessment of Statistical Predictions. Journal of the Royal Statistical Society Series B (Methodological), Vol. 36, 111-147.
B. W. Silverman (1986). Density Estimation for Statistics and Data Analysis. Chapman & Hall, Inc.
R. F. Tate (1954). Correlation Between a Discrete and a Continuous Variable. Point-Biserial Correlation. The Annals of Mathematical Statistics, Vol. 25, 603-607.
R. F. Tate (1955). The Theory of Correlation Between Two Continuous Variables when One is Dichotomized. Biometrika, Vol. 42, 205-216.
A. Wehrl (1978). General Properties of Entropy. Reviews of Modern Physics, Vol. 50, 221-260.		zh_TW
dc.identifier.doi (DOI) 10.6814/NCCU202201498en_US