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| Title: | 時間序列之相似度評估：振幅、時間與雜訊影響之探討
Similarity Evaluation of Time Series: An Investigation on the Effects of Amplitude, Time, and Noise |
| Authors: | 黃瑞華
Huang, Jui-Hua |
| Contributors: | 蔡炎龍
Tsai, Yen-Lung
黃瑞華
Huang, Jui-Hua |
| Keywords: | 時間序列相似度
模糊局部離群因子
時間軸交叉比
值域均分
模糊相似度
Time series similarity
Fuzzy Local Outlier Factor
Time-Axis Cross Ratio
Range Equally Divided
Fuzzy similarity |
| Date: | 2025 |
| Issue Date: | 2025-08-04 13:10:03 (UTC+8) |
| Abstract: | 時間序列相似度評估在金融、醫療與工業監控等領域中至關重要，但常因振幅與時間尺度變異，以及雜訊干擾，傳統相似度衡量方法常面臨準確性與穩健性不足之挑戰。特別是振幅離群點、時間軸錯位(如時間偏移與均勻伸縮)，以及振幅變動(如振幅偏移、均勻伸縮與加性雜訊) 等因素，皆可能嚴重扭曲時間序列間的相似關係。
本研究針對上述問題，提出一套三階段整合之時間序列相似度評估方法，涵蓋三項核心演算法：模糊局部離群因子(Fuzzy Local Outlier Factor, FLOF)、時間軸交叉比(Time-Axis Cross Ratio, TACR)與值域均分(Range Equally Divided, RED)，分別對應於振幅離群值檢測、時間軸校正與振幅變異處理。此方法兼顧精準性與運算效率，顯著提升相似度計算對異常值、時間錯位與振幅失真的抵抗能力。
首先，FLOF將傳統的局部離群因子概念結合模糊邏輯與局部密度估計，為每個資料點賦予離群程度的模糊隸屬度(而非二元的離群標記)。並透過分析滑動視窗大小與離群強度指標對局部密度偏差的影響，藉以量化離群點的嚴重程度。演算法進一步以非離群點模糊隸屬度的補數作為權重對離群點進行插值替補，從而在消除異常值的同時保留序列原有的趨勢。此方法有效去除異常峰值並維持時間序列的構造完整性。透過預先清除振幅離群點，FLOF 提升了相似度評估的穩健性。其可調參數機制使之可適應高頻波動、季節性週期、高雜訊等各類常見時間序列數據環境。實驗結果顯示，FLOF 能準確找出真實離群點且不對正常資料過度校正，是一種可靠的時間序列資料預處理方法。
其次，TACR演算法專為校正時間偏移(TS)和時間均勻伸縮(TUS)所導致的序列錯位問題。該方法採用三個步驟處理：首先對序列進行平滑處理以降低雜訊；再擷取序列中的局部極值點以構成代表形態的特徵序列；最終透過二序列極值點位置的交叉比(Cross Ratio)推估，並校正時間軸之伸縮比例與偏移量。由極值間距算得的交叉比是一種射影不變量，可直接推斷觀察序列相對查詢序列所需的時間伸縮因子和偏移量。依據這些參數調整觀察序列的時間軸，TACR得以精確對齊兩序列中對應事件的時間位置。相較於動態時間校正(DTW)等需要高計算量的傳統對齊方法，TACR僅利用局部極值等少量特徵點進行計算，大幅降低了時間複雜度。因而TACR在處理大規模資料時具備極高的效率。同時，由於對齊是以顯著特徵為基礎，而非對整個序列任意拉伸變形，序列整體的形態特徵得以保留。實驗分析顯示，TACR能以極小誤差復原各種時間差異(例如20% 的時間壓縮或拉伸，以及固定的時間延遲)，確保結構相似的序列得到正確對齊。透過消除非同步或時間尺度不一致造成的偏差，此方法顯著提升了後續相似度計算的可靠性。
最後，RED演算法以抗振幅變異為核心，處理振幅偏移(AS)、振幅均勻伸縮(AUS)和振幅加性雜訊(AAN)等問題。該方法透過將時間序列的數值範圍等間距地劃分為多個區間，並將資料點映射為符號表示，RED將序列轉換為對線性振幅變化不敏感的形式。在經FLOF與TACR預處理(離群值已移除、時間對齊且長度一致)後，將查詢序列與觀察序列的數值依其所屬的區間轉換為符號序列。進一步引入模糊邏輯，為對應時間點之離散值配對定義相似度的模糊隸屬函數，經由此模糊的符號轉換，即使兩序列絕對振幅不同，但只要形狀趨勢相同，其符號化後的模式也會高度相似。因此，峰谷等特徵將以不受絕對值影響的方式表達，確保即使存在固定偏移或均勻伸縮也不改變其形態表示。接著，RED計算兩個符號序列之間的模糊相似度，亦即比較它們的形狀特徵而忽略振幅差異。此方法在各種振幅條件下均能有效維持真正的模式相似性：對於形狀相同但量值不同，或含有中等雜訊的序列，使用RED所得到的模糊相似度仍能維持高準確性，有效維持序列形狀特徵的穩定表徵。測試顯示，即便序列經歷顯著的振幅平移、縮放及雜訊擾動，RED仍可將處理後的序列與原序列之相似度維持在0.97以上，展現出優異的抗雜訊能力與振幅不變性。透過著重比較振幅的相對分佈與排序，而非依賴平均值等整體統計量，RED降低了振幅與雜訊的影響，在消除振幅失真偏差的同時提供了穩健的最終相似度量。
綜合三項演算法，本研究建構一套完整且高效的FLOF-TACR-RED相似度評估框架，並透過系統化實驗加以驗證。測試資料涵蓋了具有代表性的合成時間序列模式(如「頭肩頂」形態及其他股價圖形型態)，模擬各類變異情境。首先，在振幅離群干擾測試中，針對原始序列加入不同比例(0%、5%、10%)的離群點，FLOF演算法均成功偵測並校正了離群點：在無離群值的情況下(0%)完全保留了序列結構，在離群點佔10%的高度污染情境下，仍有效清除離群而無錯判，且未破壞原有結構。其次，在時間軸變化測試中，對序列施加了設定的偏移與伸縮(例如偏移5%~20% 的長度、時間軸壓縮至0.8倍或拉伸至1.2倍)，TACR能準確估計出目標的偏移量與伸縮比例，使經校正後的序列之主要峰谷與原序列在時間上精確對齊。再次，在振幅變動與雜訊測試中，對序列加入固定基線偏移、最高±20%的整體振幅伸縮，以及均值為0、標準差達0.1的高斯雜訊；RED演算法處理後的序列與原始序列之模糊相似度皆維持在 0.97 以上，顯示儘管存在這些振幅與雜訊擾動，序列的相似性幾乎未受到影響。
此外，本研究亦將上述三階段方法與現有技術如局部極值動態時間扭曲(Locally Elastic Dynamic Time Warping, LE-DTW)技術進行對照。結果顯示，在同時存在離群點、時間錯位、振幅伸縮與雜訊的情境下，整合的FLOF-TACR-RED方法在所有測試案例中均顯著優於LE-DTW。LE-DTW所輸出之距離度量在這些變異因素的影響下易受到嚴重扭曲(例如在某測試模式中，僅引入輕微變異便導致LE-DTW計算的距離值飆升四倍以上)；相對之下，對於相同受擾序列，所提方法依然維持極高的相似度(模糊相似度≥ 0.98)。在四種不同圖形型態及多種組合干擾條件下，整合方法皆取得接近1.0的相似度分數，幾乎完全還原了序列之原始結構相似性。這些結果證實，所提出的各演算法(特別是結合運用時)大幅增強了時間序列相似度評估的穩健性。透過濾除振幅離群、校正時間錯位、抵消振幅差異，FLOF-TACR-RED 方法得以保留傳統方法在嚴苛條件下難以捕捉的序列真實相似關係。
總而言之，本研究提出了一套全面的解決方案，以改善在各種振幅與時間扭曲及雜訊干擾下的時間序列相似度評估。所提出的三種新穎演算法各自針對特定的變異類型，並集成形成一個高效的預處理與評估框架。透過系統性的理論推導與實驗驗證，本研究證明此整合方法相較傳統技術能夠提供更可靠且精確的相似度評估，大幅提升了對離群點、錯位與雜訊的抵抗力。研究成果不僅為時間序列分析提供了新的方法論見解，也對現實應用具有實際意義。無論是金融趨勢分析、生醫訊號比對，或工業感測監控，穩健的相似度量都是決策分析的關鍵，有賴於本研究所提出的方法來加以強化。
Similarity evaluation for time series data is crucial in finance, healthcare, and industrial monitoring domains. However, traditional similarity measurement methods often struggle with accuracy and robustness due to variations in amplitude and time scale and noise interference. Specifically, factors such as amplitude outliers, time-axis distortions (e.g., time shifts and uniform scaling), and amplitude transformations (e.g., shifts, scaling, and additive noise) can severely distort the perceived similarity between time series.
This study proposes an integrated three-stage similarity evaluation framework for time series data to address these challenges. It incorporates three core algorithms: Fuzzy Local Outlier Factor (FLOF), Time-Axis Cross Ratio (TACR), and Range Equally Divided (RED), which respectively handle amplitude outlier detection, temporal alignment, and amplitude variation correction. This framework balances precision and computational efficiency, significantly improving the resilience of similarity evaluation against outliers, time misalignments, and amplitude distortions.
FLOF extends the classical Local Outlier Factor (LOF) by incorporating fuzzy logic and local density estimation, assigning each data point a fuzzy degree of outlierness rather than a binary label. The algorithm quantifies the severity of anomalies by analyzing the impact of sliding window size and outlier strength indicators on local density deviations. Non-outlier points are then used to interpolate and replace outliers, weighted by the complement of their fuzzy membership values, effectively removing outliers while preserving the trend of the original sequence. This approach eliminates extreme peaks and maintains structural integrity, enhancing the robustness of subsequent similarity assessments. FLOF's adjustable parameters allow it to adapt to high-frequency fluctuations, seasonality, and high-noise environments. Experimental results confirm its precision in detecting true outliers without over-correcting clean data, making it a reliable preprocessing method for time series.
TACR, on the other hand, targets the correction of time shifts (TS) and uniform time scaling (TUS), which often cause misalignments. It operates in three steps: smoothing the sequence to reduce noise, extracting local extrema as representative features, and estimating scale and shift parameters based on the cross-ratio of extrema positions. Cross-ratios of interval lengths between extrema ”projective invariants” are used to calculate the required time-axis transformation. TACR aligns sequences based on key shape-defining features rather than warping the entire sequence, preserving the global shape and reducing computation. Compared to traditional approaches like Dynamic Time Warping (DTW), TACR is significantly more efficient and accurate in estimating 20% time compressions/stretching and fixed offsets. By correcting time-scale inconsistencies and misalignments, TACR significantly improves the reliability of similarity evaluations.
The RED algorithm addresses amplitude-related issues, including amplitude shift (AS), uniform scaling (AUS), and additive noise (AAN). It divides the value range into equal intervals and converts data points into symbolic representations, making the sequence less sensitive to linear amplitude variations. After preprocessing with FLOF and TACR (removing outliers, aligning time axes, and unifying sequence length), the query and reference sequences are transformed into symbolic sequences based on interval mapping. Fuzzy logic is applied to define membership-based similarity functions between corresponding symbols. Thus, even if the absolute amplitudes differ, similar trends and shapes yield highly comparable symbolic representations. Peaks and valleys are encoded in a distortion-invariant manner. The final fuzzy similarity metric focuses on shape characteristics rather than raw amplitudes. Experimental results show that RED maintains similarity scores above 0.97 despite amplitude shifts, scaling up to ±20%, and Gaussian noise with σ = 0.1. This demonstrates RED's effectiveness in preserving shape similarity under amplitude and noise disturbances. RED offers robustness and reliable similarity metrics by focusing on relative amplitude distribution and rank instead of global statistics like mean.
Combining these three algorithms, this study develops a comprehensive and efficient FLOF-TACR-RED similarity evaluation framework validated through systematic experiments. The experiments simulate various distortion scenarios using representative synthetic time series patterns (e.g., Head & Shoulders and other stock chart motifs). Under amplitude outlier conditions, sequences with 0%, 5%, and 10% outliers were tested. FLOF accurately identified and corrected outliers, preserving structure in clean sequences and eliminating up to 10% contamination without false positives. In time-axis variation tests (5%–20% shifts, scaling from 0.8x to 1.2x), TACR correctly estimated offsets and scaling factors, aligning key features in time. In amplitude and noise tests (fixed AS, AUS of ±20%, and Gaussian noise with σ = 0.1), RED maintained fuzzy similarity scores above 0.97, showing minimal impact from amplitude disturbances.
The integrated FLOF-TACR-RED approach was benchmarked against existing methods, such as Locally Elastic Dynamic Time Warping (LE-DTW). The results show that under combined distortions: outliers, time misalignment, amplitude changes, and noise, FLOF-TACR-RED consistently outperforms LE-DTW. Sometimes, minor distortions caused LE-DTW distances to quadruple, while the proposed method maintained fuzzy similarity scores ≥ 0.98. Across four shape types and multiple perturbation conditions, the proposed method consistently yielded similarity scores near 1.0, effectively recovering structural similarity. These findings confirm that the proposed algorithms (especially when integrated) significantly enhance the robustness of time series similarity evaluation. By removing outliers, aligning time, and normalizing amplitude, FLOF-TACR-RED reveals true underlying similarities that traditional methods often miss under noisy conditions.
In conclusion, this study offers a comprehensive solution to evaluating time series similarity under amplitude and time distortions and noise. The three novel algorithms, each targeting a specific type of variation, form a unified preprocessing and evaluation framework. Through rigorous theoretical formulation and experimental validation, the proposed method outperforms traditional techniques regarding reliability and precision. These results have practical significance for real-world applications, from financial trend analysis and biomedical signal matching to industrial monitoring, where robust similarity measures are critical for informed decision-making. |
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| Description: | 博士
國立政治大學
應用數學系
105751503 |
| Source URI: | http://thesis.lib.nccu.edu.tw/record/#G0105751503 |
| Data Type: | thesis |
| Appears in Collections: | [應用數學系] 學位論文
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