Review Tendency Signal (RTS) Analysis

• Review Tendency Signal (RTS) is an executive summary of a time series that preserves dominant, macroscopic features while filtering out noise through intrinsic time decomposition.
• It employs adaptive decomposition methods by selecting baselines based on maximum extrema prominence and statistical stationarity to capture the signal’s most meaningful structure.
• RTS distinguishes itself from conventional low-pass filters by combining geometric and statistical selection rules to yield a less complex yet interpretable representation of the original series.

Searching arXiv for the cited papers and closely related RTS/tendency work. Review Tendency Signal (RTS) (Editor’s term) denotes the tendency selected from a time series as an executive summary: a component that preserves the signal’s “notable characteristics” and ignores parts treated as residual or seemingly random variation. In the formulation based on Intrinsic Time Decomposition (ITD), the tendency is a generalization of a trend, written for an $N$-point series $\{Y(i)\}_{i=1}^N$ as $\{T(i)\}_{i=1}^N$, with residual $r(i)=Y(i)-T(i)$ and decomposition $Y(i)=T(i)+r(i)$. The cited work is explicit that this object is not just a conventional trend: it is a decomposition-dependent summary chosen from ITD baselines, intended to retain dominant or macroscopic structure while being less complex and more interpretable than the original series (Alves et al., 2024).

1. Definition and conceptual scope

The tendency is defined as an executive summary of a signal. Its intended role is to preserve the “dominant / macroscopic features” of $Y$, remain less complex than $Y$, and provide an interpretable representation of the data-generating structure. The residual is the complement of that summary: $r(i)=Y(i)-T(i), \qquad i=1,\dots,N,$ so that

$Y(i)=T(i)+r(i).$

Within this framework, the tendency is not equivalent to an arbitrary smoother. It is selected from the adaptive ITD hierarchy and is meant to keep what is described as physically / structurally meaningful. For the stationarity-based selection rule, the summary is said to contain “known physics” and possibly an “epistemic component” of $Y$, while the residual is intended to contain largely aleatoric, near-stationary, or noise-like content. For the maximum-extrema-prominence rule, the tendency retains larger-scale extrema structure and leaves out smaller-scale variability.

A common misconception is to identify the tendency with a classical low-frequency trend. The underlying paper explicitly rejects that equivalence. The tendency is framed as a summary selected from a multilevel nonlinear decomposition, not as the output of a fixed low-pass operator.

2. ITD formulation and candidate tendency construction

The ITD representation decomposes the signal into baselines and rotations: $\{Y(i)\}_{i=1}^N$0 Recursively,

$\{Y(i)\}_{i=1}^N$1

with

$\{Y(i)\}_{i=1}^N$2

The ITD step is written as a nonlinear operator $\{Y(i)\}_{i=1}^N$3,

$\{Y(i)\}_{i=1}^N$4

and the tendency is chosen as one of these baselines,

$\{Y(i)\}_{i=1}^N$5

Equivalently,

$\{Y(i)\}_{i=1}^N$6

The construction is organized around extrema. If $\{Y(i)\}_{i=1}^N$7 are the extrema times of $\{Y(i)\}_{i=1}^N$8, these extrema are called knots and denoted

$\{Y(i)\}_{i=1}^N$9

The next baseline is constructed piecewise linearly between successive knots: $\{T(i)\}_{i=1}^N$0 The associated rotation is

$\{T(i)\}_{i=1}^N$1

A key structural property is that the rotation is monotonic between adjacent extrema. Boundary handling is also specified: the method mentions “free” boundary conditions, where the end knots are handled by averaging neighboring extrema, and notes that periodic signals can use periodic boundary conditions (Alves et al., 2024).

3. Selection rules: maximum extrema prominence and stationary rotations

Two operational rules are proposed for selecting $\{T(i)\}_{i=1}^N$2, and therefore the tendency.

The first rule is Maximum Extrema Prominence, denoted $\{T(i)\}_{i=1}^N$3. Its motivation is morphological: ITD removes structure scale-by-scale, so the tendency should be chosen just before the most prominent large-scale extrema are erased. For an extremum $\{T(i)\}_{i=1}^N$4 with neighboring extrema $\{T(i)\}_{i=1}^N$5 and $\{T(i)\}_{i=1}^N$6, the local prominence is defined as

$\{T(i)\}_{i=1}^N$7

The maximum prominence over the full series is

$\{T(i)\}_{i=1}^N$8

The tendency is then selected as the last baseline before the maximum prominence falls the most: $\{T(i)\}_{i=1}^N$9

The second rule is Statistical Stationarity of Rotations, denoted $r(i)=Y(i)-T(i)$0. It uses the Augmented Dickey–Fuller (ADF) test in the autoregressive form

$r(i)=Y(i)-T(i)$1

where

$r(i)=Y(i)-T(i)$2

The test uses

• $r(i)=Y(i)-T(i)$3 — unit root present,
• $r(i)=Y(i)-T(i)$4 — stationary, no unit root.

Writing the level-$r(i)=Y(i)-T(i)$5 rotation as

$r(i)=Y(i)-T(i)$6

the selected baseline index is

$r(i)=Y(i)-T(i)$7

The reported threshold is $r(i)=Y(i)-T(i)$8, with the remark that in the examples the selected baseline was stable even if $r(i)=Y(i)-T(i)$9.

The two criteria encode different notions of importance. $Y(i)=T(i)+r(i)$0 is described as more statistical, because it removes structure until the associated rotation looks stationary. $Y(i)=T(i)+r(i)$1 is described as more geometric or “topological in nature,” because it preserves large extrema features. In some signals they agree; in others they do not. The work explicitly states that neither is universally “better” (Alves et al., 2024).

4. Residual structure and distinction from low-pass filtering

The tendency–residual split is contrasted directly with conventional low-pass filtering, especially the Hodrick–Prescott (HP) filter. For a series $Y(i)=T(i)+r(i)$2, the HP trend $Y(i)=T(i)+r(i)$3 is defined