皮爾森相關係數是計算兩變數$X,Y$之相關性常用的指標，而本論文主要想推廣出兩變數$X,Y$在另一變數$T=t_0$附近的變化局部相關函數(varying local correlation function)，來觀察在不同$T$之下兩變數的相關係數是如何變化，在文獻上已有探討更簡單的情形，即為兩變數$X,Y$在$X=x_0$附近的局部相關函數(local correlation function)，Bjerve and Doksum (1993)提出相關曲線(correlation curve)，可以適用於$X,Y$間為非線性關係。首先我們從皮爾森相關係數與線性模型的性質及關係延伸到加權皮爾森相關係數與加權簡單線性迴歸的關係，發現可以將加權皮爾森相關係數的權重設為核函數來當作本論文的估計式，並且從Jones (1996)提出的local dependence function 調整核函數成為理論值，並推廣出兩變數$X,Y$在另一變數$T=t_0$附近的變化局部相關函數。我們會探討局部相關函數及變化局部相關函數的基本性質以及推導近似性質，最後利用模擬及實際的資料來驗證理論結果。

This thesis aims to develop the varying local correlation function of two variables $X$ and $Y$ given another variable $T$ in a neighborhood of $t_0$, which can capture the relationships between $X$ and $Y$ among defferent $t_0$'s. In the literature, Bjerve and Doksum (1993) introduced the correlation curve of two variables $X$ and $Y$ given $X$ in a neighborhood of $x_0$, which applies to two variables with a nonliner relationship. First, we reveiw the relationships between the Pearson correlation and the simple linear regression, and extend it to the relationships between the weighted Pearson correlation and the weighted simple linear regression. We replace the weights of the weighted Pearson correlation by the kernel function to form an estiamtor of the local correlation between $X$ and $Y$ given $X$ in a neighborhood of $x_0$. Then we show that the local correlation enjoys some properties similar to the weighted Pearson correlation, and it has a connection with the correlation curve when estimating by local linear regression. Similarly, we replace the bivariate kernel function of the local dependence function (Jones (1996)) by the univariate kernel function to serve as the theoretic definition of the local correlation. For the varying local correlation function, it is based on the bivariate local dependence function (Jones (1996)) but with kernel weights assigned in a neighborhood of $T=t_0$. We discuss the properties of the local correlation function and the varing local correlation function and derive some asymptotic properties. Finaliy, we verify the theoretical results through simulations and a real-data example.

Chatfield, C. (2004). The Analysis of Time Series. CRC Press.
Chuang, C. L., Jen, C. H., Chen, C. M., & Shieh, G. S. (2008). A pattern recognition approach to infer time-lagged genetic interactions. Bioinformatics, 24(9), 1183-1190.
Doksum, K., Blyth, S., Bradlow, E., Meng, X. L., & Zhao, H. (1994). Correlation curves as local measures of variance explained by regression. Journal of the American Statistical Association, 89(426), 571-582.
Doksum, K. A., & Froda, S. M. (2000). Neighborhood correlation. Journal of Statistical Planning and Inference, 91(2), 267-294.
Fan, J., & Gijbels, I. (1996). Local Polynomial Modelling and Its Applications. CRC Press.
Golub, G. H., Heath, M., & Wahba, G. (1979). Generalized cross-validation as a method for choosing a good ridge parameter. Technometrics, 21(2), 215-223.
He, G., Müller, H. G., & Wang, J. L. (2004). Methods of canonical analysis for functional data. Journal of Statistical Planning and Inference, 122(1), 141-159.
Holland, P. W., & Wang, Y. J. (1987). Dependence function for continuous bivariate densities. Communications in Statistics-Theory and Methods, 16(3), 863-876.
Huang, L. S., & Chen, J. (2008). Analysis of variance, coefficient of determination and F-test for local polynomial regression. The Annals of Statistics, 36, 2085-2109.
Huang, L. S., & Su, H. (2009). Nonparametric F-tests for nested global and local polynomial models. Journal of Statistical Planning and Inference, 139(4), 1372-1380.
Jones, M. C. (1996). The local dependence function. Biometrika, 83(4), 899-904.
Minas, C., Waddell, S. J., & Montana, G. (2011). Distance-based differential analysis of gene curves. Bioinformatics, 27(22), 3135-3141.
Papadimitriou, S., Sun, J., & Yu, P. S. (2006, December). Local correlation tracking in time series. In Data Mining, 2006. ICDM'06. Sixth International Conference on (pp. 456-465). IEEE.
Simonoff, J. S. (1996). Smoothing Methods in Statistics. Springer.
\bibitem{Shreve}
Weisberg, S. (2005). Applied Linear Regression. John Wiley and Sons.
Yule, G. U. (1926). Why do we sometimes get nonsense-correlations between Time-Series?--a study in sampling and the nature of time-series. Journal of the Royal Statistical Society, 89(1), 1-63.