以克利金法內插溫度之模型選擇與參數檢定研究

Abstract

本研究應用去一內插的效率係數，評估使用迴歸克利金(Regrssion Kriging, RK)法、簡單克利金(Simple Kriging, SK)法和通用克利金(Universal Kriging, UK)法時，選擇不同統計結構函數模型、參數檢定方法的優劣點，根據分析結果，建議各階段選擇模型、估計參數和內插面化使用的客觀評估方法與最佳方案。 檢定資料一、二階統計結構參數的方法，包括：(1)採用迴歸克利金(RK)法程序，先迴歸估計一階趨勢，再用迴歸殘餘值，依循建立半變異圖模型的傳統程序，計算二階統計模型參數。(2)使用內在假設概念估計半變異圖參數，並不去除原始半變異圖中的溫度趨勢值，但使用迴歸得到的一階趨勢函數，僅使用兩測站以經度、緯度、高程計算的溫度趨勢差值平方項總和低於選擇門檻值的原始半變異圖數據，來檢定半變異圖模式參數。(3)選擇趨勢和半變異圖模型，使用最大概似法(MLE)，同時檢定一、二階統計模型參數。(4)選擇趨勢和半變異圖模型，使用克利金法去一估計的交叉驗證法評量指標為目標函數，擾動迭代(趨勢和)半變異圖模型參數，使目標函數達到最佳值的方式，決定資料統計結構參數。這4種參數檢定方法，除了(2)只能應用於通用克利金(UK)法外，其他三種方法都能配合UK法應用，也能配合『先去除觀測值趨勢，再用簡單克利金(SK)法內插殘差值，最後加上估計點趨勢值』的程序應用。最後，利用克利金法去一估計誤差值與克利金法理論估計誤差標準偏差比值的統計顯著程度，以及迴歸模式的Cook’s D指標值大小，判斷某測站溫度是否合理或一致。 本研究採用2008至2012年間擁有最多有效觀測的小時溫度資料，計算日勳溫和月均溫等不同時間長度的資料，經資料測試各參數檢定作法後決定方法論，再利用2003至2009年間擁有最多觀測的小時溫度資料，選出多筆資料驗證方法論。經過測試2008至2012年擁有最多有效觀測的小時溫度資料、日均溫和月均溫資料，決定一階統計模型為經緯高一次項模式。二階統計模型則選用指數模型、球形模型、高斯模型和坑洞效應模型，經測試後以指數模型、球型模型最穩定且經常表現最佳。再經由2003至2009年中多筆小時觀測資料進行驗證後，利用最大概似(maximum likelihood estimation, MLE)法檢定參數與以效率係數作為目標函數的交叉驗證迭代最佳化(iteration for optimal cross validation ,IOCV)法程序皆可改進傳統法檢定的參數，然而以交叉驗證迭代最佳化(IOCV)法檢定參數時由於理論上仍存有疑慮，且利用最大概似(MLE)法與交叉驗證迭代最佳化法檢定的參數表現類似，因此建議使用最大概似(MLE)法作為克利金法的參數檢定方法。 在估計時，測試結果顯示利用最大概似(MLE)法檢定參數後進行去趨勢－簡單克利金(SK)法或以通用克利金(UK)法去一估計，其效率係數表現皆較傳統方法檢定的參數經去趨勢－簡單克利金(SK)法或通用克利金(UK)法去一估計之效率係數為佳，顯示該程序所檢定的參數應較傳統程序更有能力描述溫度樣本在空間中的相關性。而比較去趨勢－簡單克利金(SK)法與通用克利金(UK)法的去一估計效率係數顯示經最大概似(MLE)法檢定參數後以去趨勢－簡單克利金(SK)法去一估計的表現比經最大概似(MLE)法檢定參數後以通用克利金(UK)法的表現為佳。 本研究的結論建議使用克利金法時，可優先選擇指數、球型模型作為二階統計模型參數，經由最大概似(MLE)法程序檢定參數並再搜尋檢定參數後，再以最適用的半變異圖模型與最佳參數組合輸入去趨勢－簡單克利金(SK)法作空間面化估計。

This study applied coefficient of efficiency from leave-one-out (LOO) interpolation to evaluate the pros and cons of using three types of Kriging methods, regression Kriging (RK), simple Kriging (SK), and universal Kriging (UK), with different statistic structure models and parameter identification methods. According to the results of analysis, objective evaluation method has been proposed to select the appropriate methods in estimate procedure. This study compared four statistic structure parameter identification methods, which include first order and second order: 1.Applying RK procedure, which first estimates trend then follows the traditional procedure to establish variogram model with regression residual, compute statistical model parameters in second order; 2.Applying intrinsic hypothesis concept to estimate the parameter of variogram model. This method does not remove the temperature trend in the raw variogram, but identifies variogram parameters with raw variogram which is controlled by spatial trends squared difference term. Spatial trends squared difference term is the sum of the product of difference of latitude and longitude of any two stations and the first order trend function parameters; 3.Applying maximum-likelihood estimation (MLE) to identify variogram parameters; 4.Applying cross validation methods, uses assessment indicator, which is computed with Kriging error variance, as objective function. This method determines the statistical structure parameters with optimizing objective function by perturbing variogram parameters. Among these four parameter identification methods, only method #2 can only apply on UK. The remaining three methods can both apply on UK and the procedure which first removes trend, uses residual from SK, and ultimate add trend on. At last, this study used the ratio of Kriging error variance and Kriging expected error variance to judge the quality of temperature estimation. The five years (2008 to 2012) temperature hourly-data has been used to test in this study. This study only used the data from the station with high percentage of valid observations, and the data has been converted to hourly data, average daily data, and monthly average data to test. The test result shows the first order statistical model is a linear model with longitude, latitude, and height variables. After testing, exponential model and spherical model performed the most stable and best among four kinds of second order statistical models. This study used hourly data from 2003 to 2009 for validation. The validation results show that both MLE and iteration for optimal cross validation (IOCV) method performs better than traditional parameter identification methods. From a theoretical point of view, there are concerns about IOCV methods. Since there is only a little difference the performance between IOCV and MLE methods, the study recommends using MLE as parameter identification method for Kriging method. The results of test of estimation procedure show that applied MLE method in identified procedure produces better and more stable CoE compares with the other methods above-mentioned. In addition, among the procedure, which identify parameters by MLE, SK method produced CoE with better performance than UK. This study provides some recommendations of applying Kriging. First, exponential model and spherical model are preferences of second order model. Second, it is preferred to apply MLE method to identify parameters. Third, it is recommended to use SK method for interpolation.