Master of Science / Department of Statistics / Weixing Song / A robust estimation procedure for mixture linear regression models is proposed in this report by assuming the error terms follow a Laplace distribution. EM algorithm is imple- mented to conduct the estimation procedure of missing information based on the fact that the Laplace distribution is a scale mixture of normal and a latent distribution. Finite sample performance of the proposed algorithm is evaluated by some extensive simulation studies, together with the comparisons made with other existing procedures in this literature. A sensitivity study is also conducted based on a real data example to illustrate the application of the proposed method.

In the thesis, the Wilcoxon approach to linear regression problems is combined with the fractal image compression to form a novel Wilcoxon fractal image compression. When the original image is corrupted by noise, we argue that the fractal image compression scheme should be insensitive to those outliers present in the corrupted image. This leads to the new concept of robust fractal image compression. The proposed Wilcoxon fractal image compression is the first attempt toward the design of robust fractal image compression. Four different numerical methods, i.e., steepest decent, line minimization based on quadratic interpolation, line minimization based on cubic interpolation, and least absolute deviation, will be proposed to solve the associated linear Wilcoxon regression problem. From the simulation results, it will be seen that, compared with the traditional fractal image compression, Wilcoxon fractal image compression has very good robustness against outliers caused by salt-and-pepper noise. However, it does not show great improvement of the robustness against outliers caused by Gaussian noise.

Penalized methods are becoming more and more popular in statistical research. This dissertation research covers two major aspects of applications of penalized methods: variable selection and nonparametric function estimation. The following two paragraphs give brief introductions to each of the two topics. Infinite variance autoregressive models are important for modeling heavy-tailed time series. We use a penalty method to conduct model selection for autoregressive models with innovations in the domain of attraction of a stable law indexed by alpha is an element of (0, 2). We show that by combining the least absolute deviation loss function and the adaptive lasso penalty, we can consistently identify the true model. At the same time, the resulting coefficient estimator converges at a rate of n^(?1/alpha) . The proposed approach gives a unified variable selection procedure for both the finite and infinite variance autoregressive models. While automatic smoothing parameter selection for nonparametric function estimation has been extensively researched for independent data, it is much less so for clustered and longitudinal data. Although leave-subject-out cross-validation (CV) has been widely used, its theoretical property is unknown and its minimization is computationally expensive, especially when there are multiple smoothing parameters. By focusing on penalized modeling methods, we show that leave-subject-out CV is optimal in that its minimization is asymptotically equivalent to the minimization of the true loss function. We develop an efficient Newton-type algorithm to compute the smoothing parameters that minimize the CV criterion. Furthermore, we derive one simplification of the leave-subject-out CV, which leads to a more efficient algorithm for selecting the smoothing parameters. We show that the simplified version of CV criteria is asymptotically equivalent to the unsimplified one and thus enjoys the same optimality property. This CV criterion also provides a completely data driven approach to select working covariance structure using generalized estimating equations in longitudinal data analysis. Our results are applicable to additive, linear varying-coefficient, nonlinear models with data from exponential families.

The ordinary least squares (OLS) is one of the most widely used methods for modelling the functional relationship between variables. However, this estimation procedure counts on some assumptions and the violation of these assumptions may lead to nonrobust estimates. In this study, the simple linear regression model is investigated for conditions in which the distribution of the error terms is Generalised Logistic. Some robust and nonparametric methods such as modified maximum likelihood (MML), least absolute deviations (LAD), Winsorized least squares, least trimmed squares (LTS), Theil and weighted Theil are compared via computer simulation. In order to evaluate the estimator performance, mean, variance, bias, mean square error (MSE) and relative mean square error (RMSE) are computed.

隨著WiFi訊號在都市的涵蓋率逐漸普及，基於WiFi訊號強度值的定位方法逐漸發展。WiFi匹配定位(Matching Positioning)是透過參考點坐標與WiFi訊號強度(Received Signal Strength Indicator, RSSI)的蒐集，以最小二乘法(Least Squares, LS)計算RSSI模型參數；然後，利用模型參數與使用者位置的WiFi訊號強度，推估出使用者的位置。然而WiFi訊號強度容易受到環境因素影響，例如降雨、建物遮蔽、人群擾動等因素，皆會使訊號強度降低，若以受影響的訊號強度進行定位，將使定位成果與真實位置產生偏移。 為了降低訊號強度的錯誤造成定位結果的誤差，本研究嘗試透過具有穩健性的最小一乘法( Least Absolute Deviation, LAD)結合WiFi匹配定位，去克服WiFi訊號易受環境影響的特性，期以獲得較精確的WiFi定位成果。研究首先透過模擬資料的建立，測試不同粗差狀況最小一乘法WiFi匹配定位之表現，最後再以真實WiFi訊號進行匹配定位的演算，並比較最小一乘法WiFi匹配定位與最小二乘法WiFi匹配定位的成果差異，探討二種方法的特性。 根據本研究成果顯示，於模擬資料中，最小一乘法WiFi匹配定位相較於最小二乘法WiFi匹配定位，在面對參考點接收的AP訊號與檢核點接收的AP訊號強度含有粗差的情形皆能有較好的穩健性，且在參考點接收的AP訊號含有粗差的情況有良好的偵錯能力。而於真實環境之下，最小一乘法WiFi匹配定位之精度也較最小二乘法WiFi匹配定位具有穩健性；在室外資料的部份，最小一乘法WiFi匹配定位之精度為8.46公尺，最小二乘法WiFi匹配定位之精度為8.57公尺。在室內資料的部份，最小一乘法WiFi匹配定位之精度為2.20公尺，最小二乘法WiFi匹配定位之精度為2.41公尺。 / Because of the extensive coverage of WiFi signal, the positioning methods by the WiFi signal are proposed. WiFi Matching Positioning is a method of WiFi positioning. By collecting the WiFi signal strength and coordiates of reference points to calculate the signal strength transformation parameters, then, user’s location can be calculated with the LS (Least Squares). However, the WiFi signal strength is easily degraded by the environment. Using the degraded WiFi signal to positioning will produce wrong coordinates. Hence this research tries to use the robustness of LAD (Least Absolute Deviation) combining with WiFi Matching Positioning to overcome the sensibility of WiFi signal strength, expecting to make the result of WiFi positioning more reliable. At first, in order to test the ability of LAD, this research uses simulating data to add different kind of outliers in the database, and checks the performance of LAD WiFi Matching Positioning. Finally, this research uses real data to compare the difference between the results of LAD and LS WiFi Matching Positioning. In the simulating data, the test result shows that LAD WiFi Matching Positioning can not only have better robust ability to deal with the reference and check points AP signal strength error than LS WiFi Matching Positioning but also can detect the outlier in the reference points AP signal strength. In the real data, LAD WiFi Matching Positioning can also have better result. In the outdoor situation, the RMSE (Root Mean Square Error) of LAD WiFi Matching Positioning and LS (Least Squares) WiFi Matching Positioning are 8.46 meters and 8.57 meters respectively. In the indoor situation, the RMSE (Root Mean Square Error) of LAD WiFi Matching Positioning and LS (Least Squares) WiFi Matching Positioning are 2.20 meters and 2.41 meters respectively.

空載光達系統於建置國土測繪基本資料扮演關鍵角色，依國土測繪法，為確保測繪成果品質，應依測量計畫目的及作業精度需求辦理儀器校正。國土測繪中心已於102年度建置航遙測感應器系統校正作業中，提出矩形建物之平屋頂面做為空載光達系統校正之可行性，而其所稱之校正，是以點雲精度評估待校件空載光達系統所得最終成果品質，並不對儀器做任何參數改正，但其校正成果可能因不同人員操作而有差異，因此本研究嘗試建立一套空載光達點雲半自動化精度評估程序，此外探討以山形屋脊線執行點雲精度評估之可行性。 由於光達點雲為離散的三維資訊，不論是以山形屋脊線或矩形建物之平屋頂面作為標物執行點雲精度評估，均須先萃取屋頂面上之點，為避免萃取成果受雜訊影響，本研究引入粗差偵測理論，發展最小一乘法結合李德仁以後驗變方估計原理導出的選擇權迭代法(李德仁法)將非屋頂點視為粗差排除。研究中分別對矩形建物之平屋頂面及山形屋脊線進行模擬及真實資料實驗，其中山形屋脊線作為點雲精度評估之可行性實驗中發現不適合用於評估點雲精度，因此後續實驗僅以萃取矩形建物之平屋頂面點雲過程探討粗差比率對半自動化點雲精度評估程序之影響。模擬實驗成果顯示最小一乘法有助於提升李德仁法偵測粗差數量5%至10%；真實資料實驗，以含有牆面點雲的狀況為例，則有助提升5%的偵測粗差數量。本研究由逐步測試結果提出能夠適用於真實狀況的半自動化之點雲精度評估程序，即使由不同人員操作，仍能獲得一致的成果，顯示本研究半自動化精度評估程序之可信度。 / The airborne LiDAR system plays a crucial role in building land surveying data. Based on the Land Surveying and Mapping Act, to ensure the quality of surveying, instrument calibration is required. The approach proposed by National Land Surveying and Mapping Center (NLSC) in 2013 was confirmed the feasibility for airborne LiDAR system calibration using rectangular horizontal roof plane. The calibration mean to assess the final quality of airborne LiDAR system based on the assessment of the accuracy of the point cloud, and do not adjust the instrument. But the results may vary according to different operators. This study attempts to establish a semi-automatic procedure for the accuracy assessment of point clouds from airborne LiDAR system. In addition, the gable roof ridge lines is discussed for its feasibility for the accuracy assessment of point cloud. No matter that calibration is performed using rectangular horizontal roof plane or gable roof ridge line, point clouds located on roof planes need to be extracted at first. Therefore, Least Absolute Deviation (LAD) combined with the Iteration using Selected Weights (Deren Li method) is developed to exclude the non-roof points which regarded as gross errors and eliminate their influences. The simulated test and actual data test found that gable roof ridge lines are not suitable for accuracy assessment. As for the simulated test using horizontal roof planes, LAD combined with Deren Li method prompts the rate of gross error detection about 5% to 10% than that only by Deren Li method. In actual test, data contains wall points, LAD combined with Deren Li method can prompt about 5%. Meanwhile, a semi-automatic procedure for real operations is proposed by the step-by-step test. Even different operators employ this semi-automatic procedure, consistent results will be obtained and the reliability can achieve.