LMFD: Latent Monotonic Feature Discovery

TL;DR

LMFD addresses the challenge of inferring a latent, monotonically increasing proxy for a system's age from multivariate time series by searching a grammar-defined space of equations and maximizing the absolute Spearman's rank correlation $| ho|$ with time. It combines a context-free grammar to generate candidate two-series expressions with monotonic transformations and a black-box optimizer (SMAC3) to fit constants, augmented by a causal EWMA convolution to capture history effects while penalising excessive smoothing. Across artificial and real datasets (Climate, InfraWatch), LMFD yields latent features with substantially higher $| ho|$ than raw sensors, demonstrating interpretable, two-sensor proxies for degradation or aging (e.g., a nearly equal-weight subtraction of two strain sensors achieving $| ho|=0.9545$). The approach provides a practical, explainable tool for monitoring aging-like processes in complex systems and informs sensor design and data fusion strategies for health monitoring.

Abstract

Many systems in our world age, degrade or otherwise move slowly but steadily in a certain direction. When monitoring such systems by means of sensors, one often assumes that some form of `age' is latently present in the data, but perhaps the available sensors do not readily provide this useful information. The task that we study in this paper is to extract potential proxies for this `age' from the available multi-variate time series without having clear data on what `age' actually is. We argue that when we find a sensor, or more likely some discovered function of the available sensors, that is sufficiently monotonic, that function can act as the proxy we are searching for. Using a carefully defined grammar and optimising the resulting equations in terms of monotonicity, defined as the absolute Spearman's Rank Correlation between time and the candidate formula, the proposed approach generates a set of candidate features which are then fitted and assessed on monotonicity. The proposed system is evaluated against an artificially generated dataset and two real-world datasets. In all experiments, we show that the system is able to combine sensors with low individual monotonicity into latent features with high monotonicity. For the real-world dataset of InfraWatch, a structural health monitoring project, we show that two features with individual absolute Spearman's $ρ$ values of $0.13$ and $0.09$ can be combined into a proxy with an absolute Spearman's $ρ$ of $0.95$. This demonstrates that our proposed method can find interpretable equations which can serve as a proxy for the `age' of the system.

Figures (6)

• Figure 1: Example of two parse trees derived from grammar $G_m$. The produced equations are $s1 * exp(c3 \cdot s2)$ and $s1/ewma(s2,\ c5)$.
• Figure 2: Convolution using EWMA kernel of $\tau = 2402$ hours (about 100 days), applied to two years of data (sensor 316). The derived signal (orange) is rather monotonic, but is not defined for $t < 4805 ( = 2\cdot 2402 + 1)$.
• Figure 3: Plot demonstrating the initial rise of $|\rho|$ as the kernel size increases and non-monotonic fluctuations in the data are being smoothed out, to an optimum of $|\rho| = 0.848$, after which the monotonicity declines due to ever smaller parts of the series being available to compute $|\rho|$.
• Figure 4: Graph of the latent monotonic feature found for the artificial dataset (red), for $s1$ (orange) and $s2$ (green). The y-axis shows the normalised values reported by the sensors and the proxy. Note that since the proxy combines the sensors, the unit can be undefined if sensors with different units are involved.
• Figure 5: Graph of the latent monotonic feature found for the climate change dataset (red), $s_{90S-24S}$ (green) and $s_{64N-90N}$ (orange). The x-axis shows the date of the recorded value. The y-axis shows the normalised values reported by the sensors and the proxy. Note that since the proxy combines the sensors, the unit can be undefined if sensors with different units are involved.