Fast, Interpretable, and Deterministic Time Series Classification With a Bag-of-Receptive-Fields

Francesco Spinnato; Riccardo Guidotti; Anna Monreale; Mirco Nanni

cs.LGNov 29, 2023

Fast, Interpretable, and Deterministic Time Series Classification With a Bag-of-Receptive-Fields

Authors: Francesco Spinnato, Riccardo Guidotti, Anna Monreale, Mirco Nanni

Organizations: Scuola Normale Superiore, Pisa, Italy · University of Pisa, Pisa, Italy · ISTI-CNR, Pisa, Italy

Abstract

The current trend in the literature on Time Series Classification is to develop increasingly accurate algorithms by combining multiple models in ensemble hybrids, representing time series in complex and expressive feature spaces, and extracting features from different representations of the same time series. As a consequence of this focus on predictive performance, the best time series classifiers are black-box models, which are not understandable from a human standpoint. Even the approaches that are regarded as interpretable, such as shapelet-based ones, rely on randomization to maintain computational efficiency. This poses challenges for interpretability, as the explanation can change from run to run. Given these limitations, we propose the Bag-Of-Receptive-Field (BORF), a fast, interpretable, and deterministic time series transform. Building upon the classical Bag-Of-Patterns, we bridge the gap between convolutional operators and discretization, enhancing the Symbolic Aggregate Approximation (SAX) with dilation and stride, which can more effectively capture temporal patterns at multiple scales. We propose an algorithmic speedup that reduces the time complexity associated with SAX-based classifiers, allowing the extension of the Bag-Of-Patterns to the more flexible Bag-Of-Receptive-Fields, represented as a sparse multivariate tensor. The empirical results from testing our proposal on more than 150 univariate and multivariate classification datasets demonstrate good accuracy and great computational efficiency compared to traditional SAX-based methods and state-of-the-art time series classifiers, while providing easy-to-understand explanations.

Figures & tables

Fig. 1: Two receptive fields (blue and red) with a length of $w=9$ , divided into segments of length $q=3$ . Dilation is the number of steps between consecutive observations within the receptive field. $d\cdot q$ is the segment hop, i.e., the number of steps between consecutive segments. The parameter $s=5$ is the stride, i.e., the distance between consecutive receptive fields.

Fig. 2: A simplified schema of Algorithm 2 for the two receptive fields of Figure 1 , extracted from signal $\textbf{x}_{i,j}$ . First, the moving average $\bm{\mu}^{\mathit{seg}}$ is computed. Values in $\bm{\mu}^{\mathit{seg}}$ are used to fill $M$ as in Equation 11 . The two receptive fields are then normalized, binned into the SAX words $[0,1,0]$ and $[0,1,1]$ , and hashed into the integers and respective counts $10,1$ and $11,1$ . This allows the update of $\mathcal{{Z}}$ . E.g., for the red word, $z_{i,j,11}=1$ .

Panel of Figure 3 — Fig. 3: Example instances from the three classes of the “Cylinder-Bell-Funnel” dataset.

Fig. 4: Local explanation for a Cylinder instance of the CBF dataset. On the left, the original time series colored based on the saliency map, highlighting the most relevant observations for the classification. On the right, the medoid of the most important pattern that is not contained in the time series. The grey area represents all the possible alignments of the pattern ‘0,1,1,2’ found in the dataset.

Panel of Figure 5 — Fig. 5: Global explanation for three of the most important patterns for the CBF dataset. On the left, in dark gray, the medoid of all alignments of the pattern in the CBF dataset. On the right, the count of appearances of each pattern in each time series in the dataset, divided by label. The color represents the importance of the pattern count in terms of SHAP values.

		interpretable
Model		features	classifier	deterministic
Dictionary-Based
BORF	(ours)	✓	✓	✓
BOP	[ 14 ]	✓	✓	✓
HYDRA	[ 2 ]	✗	✓	✗
MRSEQL	[ 19 ]	✓*	✓	✓

TABLE I: List of competitor classifiers, divided by classifier family, with information about feature space and classifier interpretability, and stochasticity. Models that have an interpretable feature space and classifier, and are also deterministic are highlighted in gray.

Panel of Figure 7 — Fig. 6: CD Plots for all benchmarked methods. Best models to the right.

Panel of Figure 8 — Fig. 7: CD Plots for SAX-based, dictionary-based, and interpretable methods. Best models to the right.

	$\Delta$ accuracy
	median	mean	win	tie	loss	p-val
MR-HYDRA	$+0.022\,192$	$+0.037\,400$	109	22	27	$0.000\,100$
RDST	$+0.020\,242$	$+0.030\,600$	103	20	35	$0.000\,100$
ROCKET	$+0.016\,769$	$+0.022\,100$	100	17	41	$0.000\,100$
HYDRA	$+0.011\,855$	$+0.022\,700$	95	19	44	$0.000\,100$
MINIROCKET	$+0.010\,000$	$+0.020\,500$	96	17	45	$0.000\,100$

TABLE II: Comparison Matrix between BORF and competitor approaches. Models are sorted by median difference in accuracy with BORF . Wins, ties, and losses are to be read “against BORF ”. Best models on top. Grey rows are statistically tied with BORF using the Wilcoxon signed-rank test.

	$\Delta$ runtime (sec)
	median	mean	win	loss	p-val
BORF	$1.305\,673$	$298.748\,806$
HYDRA	$+0.240\,600$	$-294.475\,100$	55	103	$0.4150$
MRSQM	$+1.445\,971$	$+1717.956\,800$	16	142	$0.000\,100$
MR-HYDRA	$+1.842\,159$	$-286.187\,900$	25	133	$0.000\,100$
SAXVSM	$+2.037\,075$	$+8465.654\,600$	14	144	$0.000\,100$

TABLE III: Comparison Matrix between BORF and competitor approaches. Models are sorted by median difference in runtime with BORF . Wins, and losses are to be read “against BORF ”. Best models on top. Grey rows are statistically tied with BORF using the Wilcoxon signed-rank test.

Panel of Figure 11 — Fig. 8: Comparison of median accuracy performance against median runtime (seconds). Best models are on the top left.

Fig. 9: Runtime comparison between naive PAA and our proposal (wPAA). On the left: runtime when changing the window size, and keeping time series length fixed. On the right: runtime when changing the time series length and setting the window size to half the number of points.

Fig. 10: Space complexity. On the left: dataset size against the number of extracted patterns. Each blue point represents the empirical number of extracted features for each dataset, and each orange point is the upper bound on the number of features for that specific dataset. On the right, the dataset size against non-zero elements in $\mathcal{{Z}}$ and fitted log-log regression line.

	$\theta$	$d$	ash	$l$	$w$	Acc	Time (s)
baseline	✓	✓	✓	2,4,8	all	0.820175	$17.954\,614$
w/o $\theta$	✗	✓	✓	2,4,8	all	$-0.012\,570$	$+7.954\,003$
w/o ash	✓	✓	✗	2,4,8	all	$-0.019\,762$	$-0.053\,850$
w/o dilations	✓	✗	✓	2,4,8	all	$-0.009\,656$	$-10.144\,182$
$\max{l}=2$	✓	✓	✓	2	all	$-0.185\,247$	-17.196495
$\max{l}=4$	✓	✓	✓	2,4	all	$-0.050\,413$	-14.253021

TABLE IV: Performance delta in median Accuracy (higher is better) and median runtime (lower is better) for various alternative hyperparameter configurations with respect to the BORF baseline heuristic presented in Section V . The two best-performing models for each metric are in bold.

Fig. 11: Electrode locations based on the International 10-20 system for encephalography recording. The FingerMovements dataset contains recordings from the blue regions.

Fig. 12: Local explanation for a Left-Hand instance of the FingerMovements dataset. On the left, $12$ out of the $28$ signals of the original time series colored based on the saliency map, highlighting the most relevant observations for the classification. On the right, for each signal, the medoid of the most important pattern that is not contained in that signal of the time series.

Panel of Figure 17 — Fig. 13: Global explanation for four of the most important ‘2,0’ patterns for the FingerMovements dataset. On the left, in dark gray, the medoid of all alignments of the pattern in the FingerMovements dataset. On the right, the count of appearances of each pattern in each time series in the dataset, divided by label. The color represents the importance of the pattern count in terms of SHAP values.

Appendix figures & tables9 assets

Supplementary material from the paper’s appendix.

Appendix

Time Series Data
$\mathcal{X},X,\textbf{x},x$	time series dataset, instance, signal, entry
$\textbf{y},y$	labels vector, value
$n$	number of instances in a dataset
$c$	number of signals in a time series
$m$	number of observations in a signal
Convolution

Appendix

TABLE V: Summary of notation.

Panel of Figure 19 — Fig. 14: CD Plots for SAX-based methods. Best models to the right.

Panel of Figure 20 — Fig. 15: CD Plots for dictionary-based methods. Best models to the right.

Panel of Figure 21 — Fig. 16: CD Plots for interpretable methods. Best models to the right.

Fig. 17: CD Plots for all benchmarked methods. Best models to the right.

Fig. 18: Comparison of median F1 performance against median runtime (seconds). Best models are on the top left.

Fig. 19: Runtime boxplots (seconds), Lower is better. BORF median and interquartile range are highlighted in red.

	BORF	DR CIF	HYDRA	IT	MINI ROCKET	MR HYDRA	MR SEQL	MR SQM	RDST	ROCKET	RS TSF	ST	W+ MUSE
ACSF1	0.610	0.880	0.850	0.920	0.910	0.880	0.710	0.770	0.920	0.880	0.890	0.890	0.920
Adiac	0.652	0.829	0.818	0.813	0.816	0.844	0.739	0.509	0.742	0.785	0.803	0.801	0.824
AGW-X	0.656	0.647	0.686	0.789	0.703	0.736	0.601	0.590	0.693	0.699	0.643	0.590	0.487
AGW-Y	0.706	0.661	0.750	0.817	0.769	0.791	0.680	0.624	0.754	0.774	0.711	0.626	0.563
AGW-Z	0.633	0.661	0.671	0.784	0.679	0.724	0.579	0.546	0.653	0.716	0.674	0.589	0.516
ArrowHead	0.829	0.823	0.834	0.834	0.863	0.863	0.754	0.709	0.857	0.823	0.749	0.806	0.874

Appendix

TABLE VI: Accuracy of the top 13 best-performing models on all datasets. Missing values are due to exceeded runtime limits or out-of-memory errors.

	BORF	DR CIF	HYDRA	IT	MINI ROCKET	MR HYDRA	MR SEQL	MR SQM	RDST	ROCKET	RS TSF	ST	W+ MUSE
IEPG-ST	0.928	1.000	1.000	1.000	0.992	1.000	0.896	0.795	0.992	0.980	1.000	0.976	1.000
IWingImp	0.519	0.664	0.679	0.692	0.673	0.667	-	0.303	0.482	0.399	-	-	-
IWingSound	0.633	0.682	0.655	0.627	0.672	0.667	0.599	0.552	0.665	0.666	0.668	0.648	0.627
ItaPwrDem	0.954	0.968	0.966	0.971	0.963	0.968	0.907	0.926	0.856	0.970	0.972	0.962	0.948
JapVow	0.795	0.973	0.978	0.989	0.992	0.978	0.965	0.930	0.965	0.854	0.965	0.762	0.954
LSST	0.603	0.565	0.607	0.506	0.651	0.636	0.681	0.576	0.647	0.636	0.628	0.639	0.603

Appendix

Table 26

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