近年來台灣地區由於婦女受高等教育比例、就業率的增加等因素，使得育齡婦女的生育時間延後、生育總胎次減少，因此總生育率也逐年降低，在2001年時降至1.4的歷史新低，加速台灣地區人口老化的腳步。台灣地區婦女生育率的下降，在各年齡層有不同的趨勢，這種各年齡層生育率變化不同步的現象，使得許多常見的生育率模型(如Gamma及Lee-Carter模型)套用於台灣地區的生育率時產生不小的誤差。為尋求更精確的台灣地區生育率模型，除了考量直接預測年齡別生育率的各種模型外，本文也嘗試以世代(Cohort)的觀點預測未來的生育率，包含世代生育率模型，以及研究胎次別生育率與年齡別生育率之間的關係，再套用國外近年提出的擴散(Diffusion)模型。雖然世界各先進國家近幾年的總生育率皆是下降的，但唯一例外的是美國，在2002年有緩慢回升到2.07的現象；而處於歐洲中的瑞典，由於開發較早，總生育率一直處於極低的狀態。本文針對亞洲、美洲及歐洲三個總生育率變化狀況不同的國家分別找出適合的生育率模型，藉由交叉驗證評估模型的優劣。本文的實證研究台灣以西元1949至1996年(或1975至1996年，對胎次別資料而言)的資料為基礎，1997至2001年資料為檢測樣本；美國是以西元1917至1986年的資料為基礎，1987至1991年的資料為檢測樣本；而瑞典則是以西元1970至1994年的資料為基礎，1995至1999年的資料為檢測樣本。研究發現三個國家總生育率的變化趨勢並不相同，台灣的生育率持續下降，美國則是呈現波浪狀的變化，而瑞典的生育率一直以來維持較平穩的狀態，直到最近幾年有稍微下降的趨勢。另外台灣不同於美國及瑞典特別之處，在於台灣民眾由於中國習俗因而對十二生肖的好惡不同，因此台灣生育率還需將龍虎年的效應列入預測方法的修正，才能得到明顯的改善；而台灣、美國及瑞典相同之處在於世代模型可能因資料分析及估計的程序較為複雜，預測誤差反而高於單一年齡組個別估計法這類模型較為簡單的方法。

在無母數的迴歸當中，因為原始的函數類型未知，所以常用已知特定類型的函數來近似未知的函數，而spline函數也可以用來近似未知的函數，但是要估計spline函數就需要設定節點(knots)，越多的節點越能準確近似原始函數的內容，可是如果節點太多有較多的參數要估計, 就會變得比較不準確，所以選擇適合節點個數就變得很重要。 在本研究中，用交叉驗證的方式來尋找適合的節點個數， 考慮了幾種不同切割資料方式來決定訓練資料和測試資料, 並比較不同切割資料的方式下選擇節點的結果與函數估計的效果。 / In this thesis, I consider the problem of estimating an unknown regression function using spline approximation. Splines are piecewise polynomials jointed at knots. When using splines to approximate unknown functions, it is crucial to determine the number of knots and the knot locations. In this thesis, I determine the knot locations using least squares for given a given number of knots, and use cross-validation to find appropriate number of knots. I consider three methods to split the data into training data and testing data, and compare the estimation results.

閱讀是吸收知識的途徑，不同的閱讀模式所帶來的閱讀成效也會不同。如何透過機器學習的方式，從凝視點找出閱讀行為的關聯性，將是本研究的目標。實驗選擇低成本眼動儀紀錄讀者閱讀過程中的眼動資料，採用dispersion-based演算法找出凝視點，以計算凝視點特徵，包含凝視時間、凝視距離、凝視位置以及凝視方向。 本研究將閱讀模式分成五種類別，包含快讀、慢讀、精讀、跳讀與關鍵字識別，透過不同文章的呈現，引導30位測試者遵循其內容進行閱讀，藉此收集不同行為模式的眼動資料。實驗流程中所有的眼動資料會隨機被分成為兩份，依序建立不同維度的訓練資料，由交叉驗證的分類結果找出理想之特徵與維度。以每次挑選6位測試者的眼動數據為測試資料進行5次分類驗證，其平均正確率為78.24%、74.19%、93.75%、87.96%以及96.20%，均達到不錯的分類結果。 / Reading is one of the paths to acquire knowledge. The efficiency is different when different reading patterns are involved. It is the objective of this research to classify reading patterns from fixation data using machine learning techniques. In our experiment, a low-cost eye tracker is employed to record the eye movements during the reading process. A dispersion-based algorithm is implemented to identify fixation from the recorded data. Features pertaining to fixation including duration, path length, landing position and fixation direction are extracted for classification purposes. Five categories of reading pattern are defined and investigated in this study, namely, speed reading, slow reading, in-depth reading, skim-and-skip, and keyword spotting. We have recruited thirty subjects to participate in our experiment. The participants are instructed to read different articles using specific styles designated by the experimenter in order to assign label to the collected data. Feature selection is achieved by analyzing the predictive results of cross-validation from the training data obtained from all subjects. The average classification accuracies in five-fold cross-validation are 78.24%, 74.19%, 93.75%, 87.96% and 96.20% using the eye movements of the six randomly selected subjects as test data.

本研究目的，在利用PISA 2003資料庫為例，分辨數學自我概念、自我效能與數學成就關係之模式的建構。本研究選香港為研究對象，以參加PISA 2003的4402名香港的15歲學生為樣本來進行本研究。本研究運用探索性因素分析(EFA)檢視自我概念與自我效能之測量指標的信效度。分析結果顯示，「自我概念」與「自我效能」的測量模式的建構達良好的信效度。另一研究結果顯示，學生數學自我概念對數學成就之間沒有直接的影響效果，但會透過數學自我效能此中介變項，而產生對數學成就的間接影響效果。此外，在雙交叉驗證方面，顯示研究二組樣本具有交叉效度，研究模式之接受性均相當高。 / The purpose of this study was to examine the relationship among self-concept, self-efficacy, and performance in mathematics. The PISA 2003 Hong Kong data was used as an example. There were 4402 15-year-old participants in this survey. Explore factor analysis was used to identify the good measurement models of self-concept and self-efficacy in PISA 2003. The results showed that the measurement models had high reliability and validity. The other result showed self concept had no direct effects on the mathematics achievement. But under the mediation of the mediator, such as self-efficacy, there was indirect effect on the mathematics achievement. Analysis also showed that the two sets of samples have presented cross validity, the research model is highly acceptable.

在統計學上，高維度常造成許多分析上的問題，如進行多變量迴歸的假設檢定時，當樣本個數小於樣本維度時，其樣本共變異數矩陣之反矩陣不存在，使得檢定無法進行，本文研究動機即為在進行兩群多維常態母體的平均數檢定時，所遇到的高維度問題，並引發在分類上的研究，試圖尋找解決方法。本文研究目的為在兩種不同的正則化方法中，比較何者在檢定與分類上表現較佳。本文研究方法為以 Warton 與 Friedman 的正則化方法來分別進行檢定與分類上的分析，根據其檢定力與分類錯誤的表現來判斷何者較佳。由分析結果可知，兩種正則化方法並沒有絕對的優劣，須視母體各項假設而定。 / High dimensionality causes many problems in statistical analysis. For instance, consider the testing of hypotheses about multivariate regression models. Suppose that the dimension of the multivariate response is larger than the number of observations, then the sample covariance matrix is not invertible. Since the inverse of the sample covariance matrix is often needed when computing the usual likelihood ratio test statistic (under normality), the matrix singularity makes it difficult to implement the test . The singularity of the sample covariance matrix is also a problem in classification when the linear discriminant analysis (LDA) or the quadratic discriminant analysis (QDA) is used. Different regularization methods have been proposed to deal with the singularity of the sample covariance matrix for different purposes. Warton (2008) proposed a regularization procedure for testing, and Friedman (1989) proposed a regularization procedure for classification. Is it true that Warton's regularization works better for testing and Friedman's regularization works better for classification? To answer this question, some simulation studies are conducted and the results are presented in this thesis. It is found that neither regularization method is superior to the other.

傳統上國人寄望養兒防老，但面臨少子化及壽命延長，家庭已無法獨力負擔照顧老年人的責任，必須仰賴個人（老年人自己）、國家及政府分擔人口老化造成的需求，這也是政府在過去二十年來積極投入更多資源，制訂與老年人有關的社會保險、福利及政策的原因。像是1995年開辦的全民健康保險提升了全民健康，其中老年人受惠尤多；2005年的勞工退休金條例、2008年的國民年金保險等，則是因應我國國民壽命延長的社會保險制度。對於未來費用的需求估算，需要依賴可靠的死亡率預測，但大多數預測並沒有將死亡率改善列入考量，勢必低估長壽風險的衝擊，影響個人的財務規劃、增加國家負債。 有鑑於此，本文研究常用的死亡率模型，評估哪些適合用於描述高齡死亡率的變化，且能用於計算年金商品的定價。本文考量的模型大致分成兩類：關係模型(Relational Models)及隨機模型(Stochastic Models)，第一類包括常用於高齡的Gompertz、Coale-Kisker模型，以及Discount Sequence模型，第二類則有Lee-Carter及CBD等模型。模型比較的方式以長期預測和短期預測，選用交叉驗證的方式驗證死亡率模型的預測結果與觀察值之間的差異。研究結果顯示Discount Sequence、Lee-Carter、CBD隨機模型較能準確描述台灣、日本與美國等三個國家的死亡率特性；但這三個模型在年金險保費並沒有很明顯的訂價差異。另外，若用於短期預測、長期預測比較，又以Discount Sequence的預測結果優於Lee-Carter模型的預測。 / Traditionally in Asia, families played the main role in caring their own elderly (i.e., parents and grand-parents), but the declining fertility rates and longer life expectancy make it difficult for the families to take care of the elderly alone. The elderly themselves and the government need to share the burden caused by the aging population. In fact, most Taiwan’s major social policies in the past 20 years are targeting the elderly, such as National Health Insurance, Labor Pension Act and National Pension Insurance. Their planning and financial solvency rely on reliable mortality models and their projections for the elderly population. However, many mortality models do not take into account the mortality improvements and thus underestimate the cost. In this study, we look for elderly mortality models which can reflect the mortality improvements in recent years and use them to price the annuity products. Two types of mortality models are of interest: relational models and stochastic models. The first group includes the Gompertz model, Coale-Kisker model and Discount Sequence; the other group includes the Lee-Carter and CBD models. We utilize these mortality models to project future mortality rates in Taiwan, Japan and U.S., along with the block bootstrap and ARIMA for projection. The model comparison is based on cross-validation, and both short-term and long-term projections are considered. The results show that the Discount Sequence, Lee-Carter model and CBD model have the best model fits for mortality rates and, for the short-term and long-term forecasts, the Discount Sequence is better than the Lee-Carter model.

由於台灣地區的生育率變化較大，之前的研究發現其他各國的生育率模型不見得適用，亟需建立可反映我國國情的生育率模型。本文引用台灣、日本、荷蘭、美國（亞洲、歐洲、美洲）等經濟發達國家的出生資料，配適包括Gamma、Lee-Carter、主成份分析、年齡組個別估計法、擴散模型等較為常用的模型，比較這些國家配適結果的異同。分析發現，如果要預測總生育率，台灣、日本、美國都是以年齡組生育率個別配適擴散模型的結果最佳，荷蘭則是年齡組個別估計法；在年齡組生育率的預測方面，台灣、日本、荷蘭都是以年齡組生育率個別配適擴散模型最好，美國則是以年齡組個別估計最好。此外再從相對穩定性或相對效率的角度來評判，一樣是以年齡組生育率個別配適擴散模型或年齡組個別估計的總生育率預測結果最佳。最後還觀察到台灣地區和各縣市的有偶婦女比例和生育率呈正向關係，平均生育年齡和生育率呈反向關係，各縣市在有偶婦女比例、生育率、平均生育年齡的變化並不一致，各年齡組有偶婦女比例和生育率的改變也不盡相同。

台灣地區的人口隨著生育率及死亡率的雙重下降而呈現快速老化，其中生育率的降低影響尤為顯著。民國50年時，台灣平均每位婦女生育5.58個小孩，到了民國70年卻只生育1.67個小孩，去年（民國90年）生育率更創歷年新低，只有1.4。死亡率的下降可由平均壽命的延長看出，民國75年時男性為70.97歲，女性為75.88歲；到了民國90年，男性延長到72.75歲，女性延長到78.49歲。由於生育率的變化幅度高於死亡率，對人口結構的影響較大，因此本文分成兩個部份，主要在研究台灣地區15至49歲婦女生育率的變化趨勢，再將研究結果用於台灣地區未來人口總數及其結構的預測。 　　本研究第一部分是生育率的研究，引進Gamma函數、Gompertz函數、Lee-Carter法三種模型及單一年齡組個別估計法，以民國40年至84年（西元1951年至1995年）的資料為基礎，民國85年至89年（西元1996年至2000年）資料為檢測樣本，比較模型的優劣，尋求較適合台灣地區生育率的模型，再以最合適的模型預測民國91年至140年（西元2002年至2051年）的生育率。第二部分是人口推估，採用人口變動要素合成方法（Cohort Component Projection Method）推估台灣地區未來50年的人口總數及其結構，其中生育率採用上述最適合台灣地區的模型、死亡率則引進國外知名的Lee-Carter法及SOA法（Society of Actuaries），探討人口結構，並與人力規劃處的結果比較之。 / Both the fertility rate and mortality rate have been experiencing dramatic decreases in recent years. As a result, the population aging has become one of the major concerns in Taiwan area, and the proportion of the elderly (age 65 and over) increases promptly from 2.6% in 1965 to 8.8% in 2001. The decrease of fertility rate is especially significant. For example, the total fertility rate was 5.58 in 1961, and then decreases dramatically to 1.67 in 1981 (1.4 in 2001), a reduction of almost 70% within 20 years. 　　The goal of this paper is to study the population aging in Taiwan area, in particular, the fertility pattern. The first part of this paper is to explore the fertility models and decide which model is the most suitable based on age-fertility fertility rates in Taiwan. The models considered are Gamma function, Gompertz function, Lee-Carter method and individual group estimation. We use the data from 1951 to 1995 as pilot data and 1996 to 2000 as test data to judge which model fit well. The second part of this study is to project the Taiwan population for the next 50 years, i.e. 2002-2051. The projection method used is Cohort Component Projection method, assuming the population in Taiwan area is closed. We also compare our projection result to that by Council for Economic Planning and Development, the Executive Yuan of the Republic of China.

在二十世紀九十年代，學者提出地理加權迴歸（Geographically Weighted Regression;簡稱GWR)。GWR是一個企圖解決空間非穩定性的方法。此方法最大的特性，是模型中的迴歸係數可以依空間的不同而改變，這也意味著不同的地理位置可以有不同的迴歸係數。在係數的估計上，每個觀察值都擁有一個固定環寬，而估計值可以由環寬範圍內的觀察值取得。然而，若變數之間的特性不同，固定環寬的設定可能會產生不可靠的估計值。 為了解決這個問題，本文章提出CGWR(Conditional-based GWR)的方法嘗試修正估計值，允許各迴歸變數有不同的環寬。在估計的程序中，CGWR運用疊代法與交叉驗證法得出最終的估計值。本文驗證了CGWR的收斂性，也同時透過電腦模擬比較GWR, CGWR與local linear法(Wang and Mei, 2008)的表現。研究發現，當迴歸係數之間存有正相關時，CGWR比其他兩個方法來的優異。最後，本文使用CGWR分析台灣高齡老人失能資料，驗證CGWR的效果。 / Geographically weighted regression (GWR), first proposed in the 1990s, is a modelling technique used to deal with spatial non-stationarity. The main characteristic of GWR is that it allows regression coefficients to vary across space, and so the values of the parameters can vary depending on locations. The parameters for each location can be estimated by observations within a fixed range (or bandwidth). However, if the parameters differ considerably, the fixed bandwidth may produce unreliable or even unstable estimates. To deal with the estimation of greatly varying parameter values, we propose Conditional-based GWR (CGWR), where a different bandwidth is selected for each independent variable. The bandwidths for the independent variables are derived via an iteration algorithm using cross-validation. In addition to showing the convergence of the algorithm, we also use computer simulation to compare the proposed method with the basic GWR and a local linear method (Wang and Mei, 2008). We found that the CGWR outperforms the other two methods if the parameters are positively correlated. In addition, we use elderly disability data from Taiwan to demonstrate the proposed method.