Spline based Hermite quasi-interpolation for univariate time series

Antonella Falini; Francesca Mazzia; Cristiano Tamborrino

doi:10.3934/dcdss.2022039

Article Contents

2022, Volume 15, Issue 12: 3667-3688. Doi: 10.3934/dcdss.2022039

This issue Previous Article Stability and errors estimates of a second-order IMSP scheme Next Article A nonautonomous Cox-Ingersoll-Ross equation with growing initial conditions

Spline based Hermite quasi-interpolation for univariate time series

1.
Dipartimento di Informatica, Università degli Studi di Bari Aldo Moro, via Orabona, 4, Bari 70125, Italy
2.
Dipartimento di Matematica, Università degli Studi di Bari Aldo Moro, via Orabona, 4, Bari 70125, Italy

^* Corresponding author: Francesca Mazzia
^* Corresponding author: Francesca Mazzia

Received: August 2021

Revised: December 2021

Early access: February 28, 2022

Published online: February 28, 2022

The research of Antonella Falini is founded by PON Project AIM 1852414 CUP H95G18000120006 ATT1. The research of Francesca Mazzia is founded by the PON "Ricerca e Innovazione 2014-2020", project "MAIA: Monitoraggio attivo dell'infrastruttura", n. ARS01 00353. The research of Cristiano Tamborrino is funded by PON Project "Change Detection in Remote Sensing" CUP H94F18000260006.

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Abstract

In this article the authors introduce a spline Hermite quasi-interpolation technique for the preprocessing operations of imputation and smoothing of univariate time series. The constructed model is then applied for the forecast and for the anomaly detection. In particular, for the latter case, algorithms based on the combination of quasi-interpolation, dynamic copulas and clustering have been proposed. Some numerical results are included showing the effectiveness of the presented techniques.

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Mathematics Subject Classification: Primary: 65D07, 65D10, 62M10; Secondary: 62M20.

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Figure 1. Results on "tsAirgap": the dots in blue are the known data, the magenta diamonds the missing values and the yellow squares are the imputed values

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Figure 2. Three random walk patterns are generated. The quadratic continuous model $ s $ is computed by the smoothing technique QIH-LSQ$ (15, 15, 1, 1) $

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Figure 3. The same time series are now approximated with the quadratic continuous model obtained by using QIH-LSQ$ (10, 10, 2, 2) $

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Figure 4. Given the original three random walk patterns, the use of QIH-I-LSQ$ (15, 15, 1, 1) $ produces a cubic smooth continuous model $ s $

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Figure 5. We see a cubic, smooth continuous model $ s $ generated with QIH-I-LSQ$ (10, 10, 2, 2) $ approximating the random walk dataset

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Figure 6. Smoothing and forecast of livestock, sheep in Asia. Comparison of SES, DES, QIH-LSQ$ (0, 4, 1, 0) $ and QIH-I-LSQ$ (0, 1, 0, 0) $

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Figure 7. PS time series regularized with QIH-LSQ$ (5, 10, 2, 1) $ and QIH-I-LSQ$ (5, 10, 1, 1) $

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Figure 8. Zoom in of the forecasting task performed on the last $ 8 $ PS values. Results obtained with QIH-LSQ$ (5, 10, 2, 1) $, QIH-I-LSQ$ (5, 10, 1, 1) $ SES and DES

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Figure 9. A1Benchmark-real19 time series and Kendall's tau time-varying copula

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Figure 10. A1Benchmark-real25 time series and Kendall's tau time-varying copula

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Figure 11. A4Benchmark-TS10 time series and Kendall's tau time-varying copula

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Figure 12. A4Benchmark-TS11 time series and Kendall's tau time-varying copula

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Figure 13. Scatterplot of $ x(t)-s(t) $ vs $ s^\prime(t) $ for the QIH-I-LOF (a) and of $ x(t)-s(t) $ vs Kendall's tau (b) for QIH-I-DC-DBSCAN, both for problem A4Benchmark-TS10. Computed normal points: magenta bullet. Computed anomalies: green diamond. True normal behavior: blue +. True anomalies: yellow x

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Table 1. Statistics ran on the results for the livestock sheep dataset

	NRMSE	KS-Test		Theil's
		$ \texttt{statistic}$	$ \texttt{p-value}$	$ U_2 $
SES	$ 1.20 $	$ 0.71 $	$ 0.053 $	$ 1.47 $
DES	$ 0.63 $	$ 0.43 $	$ 0.58 $	$ 0.998 $
QIH-LSQ	$ 0.59 $	$ 0.43 $	$ 0.42 $	$ 0.69 $
QIH-I-LSQ	$ 0.60 $	$ 0.43 $	$ 0.42 $	$ 0.76 $

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Table 2. Statistics ran on the results for PS time series

	NRMSE	KS-Test		Theil's
		$\texttt{statistic}$	$\texttt{p-value}$	$ U_2 $
SES	$ 0.93 $	$ 0.625 $	$ 0.087 $	$ 1.129 $
DES	$ 0.99 $	$ 0.625 $	$ 0.087 $	$ 0.84 $
QIH-LSQ	$ 0.98 $	$ 0.625 $	$ 0.087 $	$ 0.86 $
QIH-I-LSQ	$ 1.10 $	$ 0.5 $	$ 0.282 $	$ 0.749 $

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Table 3. Mean values of the recall, overall accuracy and ROC-AUC for the A4Benchmark for the compared algorithms

	RECALL	OA	ROC-AUC
QIH-I-DBSCAN	0.917	0.942	0.929
QIH-I-DC-DBSCAN	0.939	0.980	0.959
DBSCAN	0.984	0.067	0.523
QIH-I-LOF	0.973	0.955	0.964
QIH-I-DC-LOF	0.897	0.984	0.940
LOF	0.208	0.991	0.601
QIH-I-IF	0.940	0.791	0.865
QIH-I-DC-IF	0.958	0.882	0.920
IF	0.620	0.728	0.674

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