利用希爾柏特－黃轉換和狀態空間模型之瞬時頻率分析

The Analysis of Instantaneous Frequency by Using Hilbert-Huang Transformation and State-Space Model.

劉倍齊(Liu, Bei-Chi)

指導教授 ： 林財川

國立臺北大學/商學院/統計學系/碩士(2011年)

摘要

本篇論文利用經驗模態分解 (Empirical Mode Decomposition, EMD) 將非平穩 (nonstationary) 時間序列分解成數條平穩序列的總和加上一殘餘項，用標準狀態空間模型 (state-space model, SSM) 以及移動平均 (running window) 的方法估計時間序列模型的係數矩陣，進一步建立隨時間變動的損害指標 (Damage Index,DI)，判斷資料在何時間點有太大的能量變化。接著定義瞬時特徵頻率以及對每條本質模態函數 (Intrinsic Mode Function, IMF) 用三次樣條 (cubic spline) 函數計算出的上包絡線 (upper envelopes)，經過希爾柏特－黃轉換法來做出希爾柏特頻譜圖(Hilbert spectra)，能改善原本在能量急遽變化時對於希爾柏特頻譜圖IMFs交雜相混的狀況。最後用隨機產生的時間序列資料、心電圖 (ECG)、和地震資料的模擬結果與過去文獻之結果相比較。本研究所建構的損害指標與希爾柏特頻譜圖更為精確，並可把此方法應用在建築物結構損害偵測或是工業生產線品質管理上面，提供快速找到問題發生所在的時間與地點的依據。

關鍵字

希爾柏特黃轉換 ； 狀態空間模型

並列摘要

This paper uses empirical mode decomposition to decompose non-stationary time series into a summation of several non-stationary series and a residual item. Next, it uses the standard state space representation of dynamic autoregressive moving average model to estimate the coefficient matrix and establish instantaneous frequency and the damage index. Finally, it defines the instantaneous eigen-frequency and derives the upper envelopes through all the maxima of intrinsic mode functions by using the cubic smoothing spline. With the upper envelopes and the instantaneous frequencies, one can derive the Hilbert spectrum. Through real data applications, we find that he proposed method can improve the performance of Hilbert spectrum at the high-frequency part change.

並列關鍵字

hilbert-huang transformation ； state-space model

參考文獻

1.Anderson P., and Kirkegaard P. H. (1998). Statistical damage detection of Civil Engineering Structures using ARMAV Models. Aalborg University.

5.Dong Y. F., Li Y. M., Xiao M. K., and Lai M. (2007). Analysis of earthquake ground motions using an improved Hilbert-Huang transform. Soil Dynamics & Earthquake Engineering; 28(1):7-19.

6.Dong Y. F., Li Y. M., Xiao M. K., and Lai M. (2010). Structural damage detection using empirical-mode decomposition and vector autoregressive moving average model. Soil Dynamics & Earthquake Engineering; 30:133-145.

7.Huang N. E., Shen Z., Long S. R., Wu M.C., Shih H. H., and Zheng Q., et al. (1998). The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis. Proceedings of Royal Society of London Series A; 454:903-95.

9.Titchmarsh E. C. (1948). Introduction to the theory of Fourier integrals. Oxford Uni-versity Press.