國際匯率與經濟的關係密不可分，匯率的預測對於金融決策亦有重要價值。馬可夫鏈在各個領域都有廣泛的應用，也經常被用於金融與經濟中一系列現象的建模，藉由觀測過去一段時間的狀態變化，推測未來不同狀態的發生率。一階馬可夫鏈並不具有記憶性，其轉移矩陣僅能描述一步的狀態變化率。然而現實中的現象往往不僅與過去一步的狀態有關，因此我們引入更高階的馬可夫鏈。常規的高階馬可夫模型中，獨立參數的數量隨著階數的上升呈指數型成長，造成計算上的嚴重困難。為了避免此問題，我們使用調整後的模型，並將馬可夫矩陣轉換成一個非負張量，在求解其Z特徵值問題後，便得到平穩機率分布向量，此向量代表未來各個匯率值的可能機率，計算後的期望值即為預測的匯率結果。

Exchange rate forecasting is of great value to financial decisions. Markov chain is commonly used in many practical system including finance and economics. By observing the changing of states during the past time, it predicts the incidence possibility of each state. First-order Markov chain is memoryless whose transition probability matrix can only describe the change of one step. However, the phenomenon in reality is often not only related to the state of one step in the past. Therefore, we apply higher-order Markov chains. The number of independent parameters in a conventional higher-order Markov model increases exponentially with respect to the order, causing serious difficulties in computation. To avoid such problem, we used a modified model and turn transition matrix into an nonnegative tensor. After solving the Z-eigenvalue problem corresponding to this tensor, we find the stationary probability distribution which represents the possibility of any exchange rate in the future. The forecasting result is then computed by the expect value.

1 Introdution....................................1
1.1 Notation Proclaim...........................2
2 Background.....................................3
2.1 Markov chain................................3
2.2 Limiting properties of Markov chains........5
2.3 Tensor......................................6
3 Higher-order Markov chain......................8
3.1 Models......................................8
3.2 Parameter estimation.......................10
3.3 Conversion to Z-eigenvalue problem.........12
4 Z-eigenvalue problem for nonnegative tensor...14
4.1 Computing Z-eigenpair......................14
5 Numerical experiments.........................16
5.1 Database...................................16
5.1.1 Data preprocessing......................17
5.1.2 Data classification.....................17
5.2 Model construction.........................18
5.3 Measure of predictive power................19
5.4 Numerical results..........................19
6 Conclusions...................................25

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