A Dictionary-based approach to Time Series Ordinal Classification

1. Problem Definition

The paper addresses a specific sub-domain of Time Series Classification (TSC) known as Time Series Ordinal Classification (TSOC).

• Context: In standard TSC, class labels are nominal (unordered). However, in many real-world applications (e.g., medical severity levels, stock market trends, energy fluctuation levels), labels possess a natural order (ordinality).
• The Challenge: Standard TSC algorithms treat misclassifications between distant classes (e.g., predicting "Low" instead of "High") the same as misclassifications between adjacent classes (e.g., predicting "Low" instead of "Medium"). This ignores the semantic distance between labels.
• Goal: To develop a dictionary-based classification method that explicitly exploits the ordinal nature of the target variable to minimize the "distance" of errors, rather than just minimizing the count of errors.

2. Methodology: Ordinal Temporal Dictionary Ensemble (O-TDE)

The authors propose O-TDE, an ordinal adaptation of the state-of-the-art Temporal Dictionary Ensemble (TDE) algorithm. The methodology modifies the TDE pipeline in three critical areas to accommodate ordinal data:

A. Ensemble Member Selection (Gaussian Process Optimization)

• Original TDE: Uses a Gaussian Process to select ensemble members based on Accuracy.
• O-TDE Adaptation: The selection criterion is changed to Mean Absolute Error (MAE). Since MAE quantifies the magnitude of the error on the ordinal scale, the Gaussian Process guides the selection of ensemble members that minimize the distance between predicted and actual ordinal labels, rather than just maximizing correct hits.

B. Symbolic Representation (Discretization)

The core of dictionary-based methods involves converting time series windows into symbolic "words."

• Feature Extraction: The algorithm applies a sliding window to the time series and uses Discrete Fourier Transform (DFT) to extract coefficients.
• Binning Strategy (IGB): Instead of standard binning, O-TDE uses Information Gain Binning (IGB) to determine thresholds for discretizing Fourier coefficients into symbols.
• Impurity Criterion: The Decision Tree Regressor (DTR) used to find optimal binning thresholds utilizes the Friedman-MSE (Mean Squared Error) criterion instead of standard classification impurity measures. This ensures the discretization process respects the continuous nature of the ordinal scale during the transformation from real values to symbols.

C. Classification Mechanism

• The transformed time series are represented as histograms of word counts.
• Classification is performed using a k-Nearest Neighbors (kNN) approach based on histogram distances.
• The final prediction returns the label of the nearest training instance, with the underlying logic that the distance metric inherently accounts for ordinal proximity.

3. Key Contributions

1. First Ordinal Dictionary-Based Approach: The paper introduces the first dictionary-based technique specifically designed for TSOC, filling a gap in the literature where ordinal time series were previously only addressed via shapelet-based methods (e.g., O-STC).
2. Algorithmic Adaptation: It successfully adapts the TDE architecture by integrating ordinal-specific metrics (MAE) into the ensemble selection and the symbolic transformation (Friedman-MSE) phases.
3. Expanded Benchmark: The authors significantly expanded the TSOC benchmark dataset. They curated 18 ordinal datasets (up from the previous 7), including:
   • 9 from the UEA/UCR archive.
   • 2 from the Monash/UEA/TSER archive (discretized).
   • 5 stock market return datasets (discretized).
   • 2 buoy sensor datasets (energy flux and wave height).
4. Comprehensive Evaluation: The study evaluates performance using both nominal metrics (Accuracy/CCR) and ordinal-specific metrics (MAE, Quadratic Weighted Kappa, 1-OFF Accuracy).

4. Experimental Results

The O-TDE was compared against four state-of-the-art nominal dictionary-based techniques: BOSS, cBOSS, WEASEL, and TDE.

• Performance Metrics:
  • MAE (Mean Absolute Error): O-TDE achieved the best results in 10 out of 18 datasets and was the second-best in 4. It significantly outperformed all nominal methods.
  • QWK (Quadratic Weighted Kappa) & 1-OFF: O-TDE consistently ranked highest, demonstrating superior ability to predict classes close to the true label.
  • CCR (Correct Classification Rate/Accuracy): Surprisingly, O-TDE also outperformed nominal methods in pure accuracy, suggesting that exploiting ordinality improves general classification robustness.
• Statistical Significance:
  • Using Critical Difference Diagrams (CDD) and the Nemenyi test (with Holm correction), the results showed that O-TDE is statistically significantly better than the nominal baselines, particularly for MAE and 1-OFF metrics.
  • The average rank of O-TDE (2.00) was superior to WEASEL (2.67), TDE (2.78), cBOSS (3.33), and BOSS (4.17).

5. Significance and Conclusion

• Paradigm Shift: The paper demonstrates that treating ordinal time series as nominal data leads to suboptimal performance. Explicitly modeling the order of labels yields significant gains.
• Practical Impact: The proposed O-TDE method provides a robust tool for domains where the degree of error matters (e.g., medical diagnosis, financial risk assessment, industrial monitoring).
• Future Work: The authors plan to expand the TSOC archive further and explore applying ordinal adaptations to other TSC paradigms, such as kernel-based, interval-based, and deep learning methods.

In summary, this work establishes O-TDE as the new state-of-the-art for ordinal time series classification, proving that dictionary-based approaches can be effectively adapted to leverage ordinal information for superior predictive performance.