隨著科技日新月異，我們所能蒐集到的資料也越來越多樣化，從原本的向量到矩陣，甚至是多維度的陣列，例如：腦電圖（EEG）與磁振造影（MRI），都成了統計分析中常見的變數。張量迴歸正是將這種高維度陣列資料的視為解釋變數，進而建構迴歸模型並進行分析的方法。但在處理張量迴歸的問題時，秩（rank）的選擇卻是一道難題，選擇良好的秩可以簡化所需要的參數，選擇過少的秩卻又會失去過多的資訊。同時我們也可以透過秩觀察到張量結構，也因此秩的選擇在張量迴歸中有著舉足輕重的地位，正如同變數選擇問題在線性迴歸中所扮演的角色。George and McCulloch (1993)提出了 Stochastic Search Variable Selection (SSVS)，在線性迴歸模型中加入指標變數，並透過貝氏方法達到變數選擇的目的。在本論文中，不同於以往的貝氏張量迴歸，我們仿照SSVS的做法，在張量迴歸模型中加入了新的指標變數，透過吉布斯採樣建立演算法，再依據這個指標變數找到迴歸模型中最適合的秩。論文的最後，使用MNIST資料集進行手寫數字的辨識，我們的方法能達到與Lasso相近的準確度，同時透過適當的秩觀察數字辨別的結構。

With the rapid development of technology, the information we can collect is becoming more and more various. Not only a vector or a matrix, but also a multi-dimensional array. Electroencephalography (EEG) and Magnetic Resonance Imaging (MRI) are examples. Tensor regression treat the high-dimensional array data as explanatory variables. Instead of variable selection, rank selection is a major problem when we deal with tensor regression problem. Choosing a suitable rank reduces the parameters we need, whereas choosing too small a rank loses relevant information. We can also observe the structure of the tensor through the rank at the same time. Therefore, rank selection plays an important role in tensor regression analogous to variable selection in regression problem, rank selection can be done in a similar way. George and McCulloch (1993) proposed Stochastic Search Variable Selection (SSVS) approach for variable selection. They added indicator variables to linear regression model, and solve the variable selection problem by the Bayesian method. In this thesis, different from the previous Bayesian tensor regression, we follow the idea of SSVS, and add the new indicator variables to the tensor regression model. We find the suitable rank in the regression model based on these indicator variables by our Gibbs sampling algorithm. At the end of the thesis, we use MNIST database considering handwritten digits classification. Our method and Lasso are compatible in terms of prediction accuracy. Additionally, we learned the structure of images through the tensor with different ranks.

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