Estimating an Executive Summary of a Time Series: The Tendency

TL;DR

The paper addresses extracting an executive summary T from a time series Y by decomposing it into T + r and selecting an ITD-based baseline via two criteria. It introduces the Stationarity Test Criterion (STC) and Maximum Extrema Prominence (MXEP) to choose the ITD baseline T = B^{j*}(i), and compares these tendencies with the Hodrick-Prescott filter, highlighting interpretability and multi-scale behavior. Through synthetic and real-world examples, the authors show that ITD-based tendencies capture structure not reducible to simple low-pass filtering, and that the best choice of tendency is context-dependent. The work provides robust, interpretable tools for generating time-series executive summaries applicable to diverse domains.

Abstract

In this paper we revisit the problem of decomposing a signal into a tendency and a residual. The tendency describes an executive summary of a signal that encapsulates its notable characteristics while disregarding seemingly random, less interesting aspects. Building upon the Intrinsic Time Decomposition (ITD) and information-theoretical analysis, we introduce two alternative procedures for selecting the tendency from the ITD baselines. The first is based on the maximum extrema prominence, namely the maximum difference between extrema within each baseline. Specifically this method selects the tendency as the baseline from which an ITD step would produce the largest decline of the maximum prominence. The second method uses the rotations from the ITD and selects the tendency as the last baseline for which the associated rotation is statistically stationary. We delve into a comparative analysis of the information content and interpretability of the tendencies obtained by our proposed methods and those obtained through conventional low-pass filtering schemes, particularly the Hodrik-Prescott (HP) filter. Our findings underscore a fundamental distinction in the nature and interpretability of these tendencies, highlighting their context-dependent utility with emphasis in multi-scale signals. Through a series of real-world applications, we demonstrate the computational robustness and practical utility of our proposed tendencies, emphasizing their adaptability and relevance in diverse time series contexts.

Figures (11)

• Figure 1: A times series generated by the SDE in \ref{['ex_SDE']}. The solution of the SDE was approximated by the Euler-Maruyama method. Here $t_i = i \Delta t$, $i=0:N=2000$ with $\Delta t = 0.05$.
• Figure 2: Successive (a) baselines, (b) and rotations associated with the signal appearing in Figure \ref{['f:SDE_time_series']}.
• Figure 3: The Hodrick-Prescott filter (see \ref{['eq:hp']}), applied to the SDE signal from Figure \ref{['f:SDE_time_series']}, with different values of the parameter $\lambda$, clearly suggesting that the filter is a windowed low-pass filter. The various outcomes would represent executive summaries of the signal. The choice of the appropriate $\lambda$ is relegated to the user or is inspired provenance of the data.
• Figure 4: SDE signal (grey) and executive summaries. In blue we depict the outcome of applying the HP filter ($\lambda = 1600$). In red, the ITD-generated tendencies, according to $\mathop{\mathrm{\mathsf{MXEP}}}\nolimits$ and $\mathop{\mathrm{\mathsf{STC}}}\nolimits$ criteria. Both of these chose the same baseline, in this case, the $j^*=3$. Inset: the 2 criteria, as a function of $j$, on the horizontal axis. The crossing in both cases is at $j=3$, see baseline decomposition of signal on the left row of Figure \ref{['f:sde_itd']}.
• Figure 5: Modulus of the discrete Fourier transform of: the SDE signal, the remainder $r(i)$ obtained by subtracting the HP filtered signal from $Y(i)$ with $\lambda = 1600$; and the remainder $r(i)$ using the ITD tendency for selected $j^*=3$ baseline. Only frequencies with moduli less than 250 are shown.