反應曲面法在產品設計與開發階段是一門重要的技術。經由不同型式的方程式擬合量測值與估計值，以建立替代曲面，用以重現資料的原始背景與物理意義。一般反應曲面的使用是為了取代昂貴的實際測試與耗時的電腦模擬，但由於估計產生與實際值的誤差是不可避免的。早期研究中通常使用迴歸分析建立多項式曲面以觀察資料的趨勢，當多項式的非線性程度越高，越有可能得到擬合程度較高的曲面。而在反應曲面的發展與演進下，類神經網路與克利金法也逐漸被廣泛使用在工程領域。使用以上不同的反應曲面主要的目的在於預測一個從未測試或未開發的設計，以期能夠減少時間與金錢成本。一個較好的反應曲面能夠準確的預測資料，但用不同方式評估準確度間接影響了反應曲面在擬合該資料點時的優劣程度。早期的研究中，均方根誤差是最為廣泛使用於評估反應曲面優劣的基準。藉由減少均方根誤差，在相同背景下的反應曲面可以找到擬合程度較好的反應曲面參數。然而並非所有取得的資料點都符合實際的物理意義，其中可能含有些許經由重要或非重要資訊影響的量測誤差，或甚至量測到一組偏差較多的資料。在使用均方根誤差的評估中，此資料的不確定性是被忽略的，換句話說，量測到的資料點是被假設取自於沒有顯著偏差的物理意義方程式中。若考慮資料點取自於含有偏差的物理意義方程式，則通常使用貝式信賴度分析為評斷反應曲面優劣的基準，此分析基於考慮資料含有不確定性，在現有資訊下尋找一組最有可能重現資料物理意義的模型參數，以機率的形式評估反應曲面的優劣。在本論文中，我們組合了均方根誤差與貝式信賴度分析以尋找最佳的反應曲面，均方根誤差是做為資料確定性的評估基準，貝式信賴度分析是做為資料不確定性的評估基準。組合式評估基準是利用形成一個反應曲面估計結果的信賴區間，同時比較在資料含不含不確定性下反應曲面的平均表現。在一個最佳反應曲面中，必須考慮不同曲面可能帶來的含義，因此本論文使用複合式反應曲面，其中包含迴歸分析、類神經網路與克利金法。利用組合式評估基準，我們可以從多種組合中找到一個最佳的複合式反應曲面。本論文利用一些數學範例應證所提出的方法，並使用懸臂樑的可靠度分析作為工程範例加以應證。

Surrogate modeling, also known as response surface or meta modeling, is an important methodology in product design and development process. By fitting measurement data and testing outcomes with some types of basis functions, surrogate models can be created as a substitute of the original underlying physical phenomena of the data. Common practice of surrogate modeling is to replace expensive in-field tests and time-consuming computer simulations. Data fitting will inevitably associated with errors regardless of the basis being used. Researchers in the past have used polynomial functions as surrogate models to observe the trend of a given data set. With the increase of polynomial terms, one is able to provide a better fit. Among other nonlinear basis, neural network and Kriging are two of the most commonly applied surrogate models in engineering practice. The main advantage of using these surrogate models lies in the reduction of time and cost in `predicting' a design that has not been tested or even before its production. A better surrogate model fits the data `well'. However, different views on the data might affect the types of method used to quantify wellness of a surrogate model. In the past, the root mean square errors (RMSE) between measurements and predictions have been used as the most important index in selecting a good surrogate model. By reducing the RMSE, one is able to find the optimal parameters of a surrogate model given its basis forms. However, not all data at hands are perfect. They might contain important information as well as useless measuring errors or even wrongful biased outcomes. Unfortunately these uncertainties have primarily been omitted in surrogate modeling with RMSE. In other words, the data used in fitting are assumed to be in accordance with the main function trends, no significant variations exist. Another important index in quantifying the goodness-of-fit of a surrogate model is the evidence based on Bayesian theory. Evidence considers data with some form of underlying uncertainty in nature. Each set of model parameters is obtained with the most likely set to reproduce the data given existing prior information at hand. In this thesis we combine the RMSE and the Bayesian evidence in selecting the best surrogate model. We denote the RMSE as the deterministic metric while the evidence the probabilistic metric. The best surrogate model in this research is considered to have a form from all possible sources, therefore it is a composite surrogate model. An RMSE-evidence contour is proposed to form the region of trust. Comparing different RMSE-evidence contours enables us to balance between deterministic and probabilistic metrics of a surrogate. Optimal composition of all available surrogate models, in this work polynomial, neural network, and Kriging, is obtained by systematically tuning the parameters until no compositions outperforms the other one in both aspects. Several mathematical cases are used to demonstrate the process and usefulness of the proposed method. A cantilever beam with time-variant aging data is used to demonstrate the proposed method in predicting product reliability.

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