Analysis of Signals from Air Conditioner Compressors with Ordinal Patterns

Most machines have devices that monitor their operation. In particular, air conditioners are routinely monitored through several measurements. A desirable outcome of such monitoring is identifying when the device will likely require maintenance. We present the use of Ordinal Patterns, a symbolic transformation of time series, that enables the visual assessment of the type of operation. We juxtapose two machines in different operational conditions, from which six variables are measured. We analyze the expressiveness of these measurements and identify those that best separate the two machines. The technique is visually appealing because it outputs points in a plane whose position reveals hidden dynamics.


Introduction
The analysis of signals using ordinal patterns, first proposed by Bandt and Pompe [1], has raised significant interest due to its ability to reveal crucial information in various contexts.The literature covers a wide range of successful applications of this approach, demonstrating its versatility and relevance in diverse areas.Bandt and Pompe [1] introduce the transformation of time series into ordinal patterns, highlighting the importance of the relative order rather than the absolute values.This concept is the foundation for the subsequent analysis of ordinal patterns in different contexts.
Ordinal Patterns have been extensively employed in various machine diagnostics and monitoring applications, spanning a variety of equipment and domains.The approach offers valuable insights for the early detection of failures and understanding the intrinsic characteristics of the machines under study.
Predictive maintenance has benefited significantly from applying ordinal patterns in machine signal analysis.Wang and Li [2] explore the effectiveness of this approach in detecting bearing failures, especially in identifying anomalous patterns in vibration time series.They apply the permutation entropy to improve fault detection and isolation in jet engines.
Several studies, including those by Chen et al. [3], Wan et al. [4], Zheng et al. [5], Fu et al. [6], Yu et al. [7], Zhang et al. [8], Yan et al. [9], Liang and Zhang [10], Yin et al. [11], and Liu et al. [12], have in common the approach of combining several analysis techniques to identify abnormal patterns that may indicate incipient failures in wind turbines.Using ordinal pattern analysis, such as the Permutation Entropy, these studies can identify changes in bearing vibration patterns, signaling the presence of possible future problems.
Ren et al. [13] address fault detection in diesel engines The authors introduce an improvement in Variational Mode Decomposition and apply it to fault diagnosis.The technique reveals its usefulness in the analysis of complex and non-linear systems.Sharma [14] uses Variational Mode Decomposition and Entropy Permutation for gear failure detection.This approach combines instantaneous frequency estimation with ordinal pattern analysis techniques, making it a powerful tool for identifying anomalies.
Applying Signal Analysis and Ordinal Patterns is not just limited to turbines and engines.Shi et al. [15] investigate the diagnosis of faults in steam turbine rotors.They use Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN) and CBBO-SVM to identify patterns indicative of operational problems.
Furthermore, the ordinal patterns methodology has found application in the medical [16] and financial and economic domains [17].The evolution of methodologies has accompanied the growing application of ordinal patterns.A notable example is the work by Mao et al. [18], which extends Bandt and Pompe's approach to multidimensional time series.Other studies have used these techniques to assess the impact of attacks on complex networks [19], understand the behavior of vehicles [20], analyze textured images [21], and even identify undue interference in the financial market [22].This broadening of the scope of applications further solidifies the continued relevance of this pioneering methodology.However, these techniques' direct application in analyzing signals from industrial machines was not evidenced in the reviewed works.
Given this gap, it is essential to highlight the relevance of future research to explore the potential of Ordinal Patterns and techniques based on the "Entropy-Complexity Plane" (H ×C) in this specific context.This work aims to fill this gap, using these methodologies to distinguish time series from air conditioning compressors.The main objective is to detect and characterize points in H ×C that allow discriminating machines in good working order from those with defects or failures.By applying these techniques in this specific scenario, we seek to enhance diagnostic and predictive maintenance strategies in industrial machines, particularly those related to air conditioning systems.
We pose the following research questions: RQ1 [Techniques].How can Ordinal Patterns be effectively applied for the analysis of time series from air conditioning machines?
This question explores the feasibility and challenges of using Ordinal Patterns and derived features to analyze time series generated by air conditioning systems, including parameter selection, pre-processing, and assessing the sensitivity of the techniques to different operating conditions.
RQ2 [Standards].How can the features be correlated with the operating state of the machines, distinguishing between normal and faulty conditions?
This question seeks to understand the relationship between extracted features and air conditioning machines' operational status.We will investigate how these features manifest in different operating scenarios, aiming to distinguish between operating and faulty machines.
RQ3 [Efficiency].What is the effectiveness of the proposed techniques in the early detection and diagnosis of failures in air conditioning systems in industrial machines, compared to traditional approaches of time series analysis?
This question focuses on evaluating the effectiveness of the Ordinal Patterns and H × C techniques compared to traditional time series analysis methods.In future works, we will perform a comparative analysis to determine whether the proposed methodologies can accurately identify abnormal conditions, allowing early detection and reliable diagnosis of faults in air conditioning systems.
This work shows that the points in the H × C plane provide helpful information and discriminate two machines: one in working condition and another failing and needing maintenance.This diagnostic is shown graphically and is easy to assimilate by human operators.

The Entropy-Complexity Plane
t=1 be a real-valued time series of length T , without ties.Consider for X its time delay embedding representation, with embedding dimension D ≥ 2 (D ∈ N) and time delay τ ≥ 1 (τ ∈ N, also called "embedding time"): for t = 1, 2, . . ., N with N = T − (D − 1)τ .Then, the vector X (D,τ ) t can be mapped to one of the D! symbols in the set π (D) = {π 1 , π 2 , . . ., π D! }.The most frequent way to define the mapping . In the remainder of this article, we use τ = 1.After computing all the symbols, one obtains the histogram of proportions h = h(j) 1≤j≤D! .Such a histogram estimates the (unknown, in general) probability distribution function of these patterns.The next step in the characterization of the time series is computing descriptors from this histogram.
The first descriptor is a measure of the disorder of the system.The most frequently used feature for this is the Normalized Shannon entropy, defined as with the convention that terms in the summation for which h(j) = 0 are null.This quantity is bounded in the unit interval.It is zero when h(j) = 1 for some j (and, thus, all other bins are zero), and one when h(j) = 1/D!for every j (the uniform probability function).Eq. 2 is often called the "Permutation Entropy" (PE) of X .Although very expressive, the PE is not able to describe all possible underlying dynamics.In particular, for intermediate values of H, there is a wide variety of situations worth characterizing.The Jensen-Shannon distance between h and the probability function u = u(1), u(2), . . ., u(D!) stems as an adequate measure of such a distance: The quantity (3) is also called "Disequilibrium."The normalized disequilibrium is Q = Q ′ / max{Q ′ }, and we chose to work with the uniform probability distribution function as the equilibrium law.Under these conditions holds that With this, Lamberti et al. [23] proposed C = HQ as a measure of the Statistical Complexity of the underlying dynamics.Notice that H and Q are normalized quantities, as is C.Moreover, C can discern different degrees of periodicity, among other relevant properties.A time series can then be mapped into a point in the H × C plane.Fig. 1 shows deterministic functions (log(x + 0.1), sin(2x) cos(2x), and two versions of the Logistic map: x t = 3.6x t−1 (1 − x t−1 ) and x t = 4x t−1 (1 − x t−1 )), with stochastic signals of type 1/f k for k = 0, 1/2, 1, 3/2, 2, 5/2, 3.Each time series is identified by a color, and the point onto it is mapped in the H × C plane.The reader is referred to the work by Chagas et al. [24] for details and examples.
Several packages implement metrics derived from ordinal patterns.We have used the R package statcomp by Snippel et al. [25].

Nature of Time Series Data
We analyzed time series data collected from air conditioning compressors.Six characteristics were measured, as shown in Table 1.

Attribute Description
Att1 ADT: air discharge temperature Att2 Inlet Temperature: air intake temperature Att3 Internal pressure Att4 pN: system pressure Att5 pNloc: system pressure (whole network) Att6 Oil separator pressure differential These data are measurements over time in multiple dimensions, represented as "Att1" through "Att6."We have the same attributes from a machine in working condition ("Good", in teal in all plots) and a faulty machine ("Bad", in red).They are shown in Figure 2. The analysis aims to identify patterns of complexity and entropy that differentiate between these two machines.

Results
We explored the points in the H × C plane for different attributes and dimensions of the dataset after performing preprocessing to filter and handle null values.We present all features (columns), dimensions (rows), and machines (colors) involved in the data set in Figs. 3. The former shows the points obtained with the whole series, while the latter only uses values different from zero.
In the following, we analyze both types of series.The thin lines are the lower and upper bounds of the feasible region for points that define the H × C plane.

All Attributes, Machines, and Dimensions No−Zero
Figure 3: Visualization of all measures ("Att1" to "Att6", one per column), machines ("Good" in teal and "Bad" in orange), embedding dimensions (3, 4, 5, and 6, one per row), and feasible bounds in the H × C plane.We observe that "Att6" discriminates between the machines when using the complete series, suggesting that it has significant predictive power; cf.Fig. 4. Furthermore, dimension D = 3 provides the best separation among the available dimensions.The Euclidean distances between machine points in each scenario support these observations; cf.Table 2.The combination of "Att6" and D = 3 provides a clear and effective way of delineating the variations in the H × C plane.Interestingly, the distances between points from "Att3" are the only ones preserved by removing zeroes.Moreover, although these distances may seem very small, the results by Rey et al. [26,27] pave the way to perform statistical tests on such differences.
Another interesting property is the location of points.Both "Att3" and "Att6" from the complete series place the faulty machine closer to the white noise point (1, 0).The placement switches in "Att6" when considering the series without zeroes.We notice an interesting behavior when considering attributes "Att3" and "Att6" after removing the zero values.The Euclidean distances between these attributes are smaller in lower dimensions, suggesting that removing the zero values reduced the variability in the attributes, making them closer in lower dimensions.Therefore, if we want to identify distinct patterns in smaller dimensions after excluding zero values, "Att3 No-Zero" and "Att6 No-Zero" may be more effective in providing meaningful patterns.

Attribute
Choosing the best attribute to provide patterns depends on the specific goals of the analysis.If the goal is to find consistent patterns in higher dimensions, "Att3" and "Att6" are promising.However, if the priority is to identify distinct patterns in smaller dimensions after excluding zero values, "Att3 No-Zero" and "Att6 No-Zero" are more suitable.
Furthermore, the Euclidean distances calculated for "Att6" indicated a gradual increase in the dispersion of the points as the dimension increases; cf.Table 2.This suggests that, in higher dimensions, "Att6" can better distinguish machines, making it a robust choice for pattern discrimination.
On the other hand, "Att3" also showed discrimination ability, although less prominently than "Att6".Its Euclidean distance also increases with increasing dimensions, indicating a more evident dispersion of the points.However, the results suggest that "Att6" is more efficient in separating machine patterns.
Considering both the discrimination capacity and the Euclidean distances in different dimensions, we conclude that "Att6" is the most reliable choice for discrimination and machine selection.This provides a solid foundation for making informed decisions in machine selection, ensuring an accurate and effective approach to identifying distinct patterns between machines.

Conclusions
Comprehensive analysis of attributes, dimensions, Euclidean distances, and dispersion patterns allowed us to identify and select the most suitable attribute for pattern discrimination and machine selection based on the Entropy-Complexity plane."Att6" emerged as the soundest choice, exhibiting a greater ability to distinguish machines.Its effectiveness was reinforced by the analysis of Euclidean distances between points, which demonstrated its consistency and discrimination power in different dimensions.
These results provide insights into which attributes are most effective in differentiating patterns and have practical implications for optimizing industrial processes, where the correct selection of machines based on relevant metrics can lead to significant improvements in efficiency and quality of production.
At the broadest level, this review underscores the importance of rigorous analytical approaches to informed decision-making.Combining techniques such as data visualization, Euclidean distance calculation, and pattern interpretation allows us to draw reliable conclusions and guide strategies for improvements in several areas.

Figure 1 :
Figure 1: Several time series, and their corresponding points in the Entropy-Complexity plane (H × C) for D = 6.From Chagas et al. [24].

Figure 2 :
Figure2: Temporal Evolution of Attributes across machines: Time Series Comparison of "Good" and "Bad" Machines for Each Attribute "Att1" through "Att6".The best candidates for discrimination in the H × C plane are "Att3" and "Att6" (right column).

Figure 4 :
Figure 4: All measures and dimensions.

Table 1 :
Measurements extracted from the two machines under study.
At a glance, it is possible to see that the points that represent the machines separate better in some configurations of Attribute (columns) and Dimension (rows) than in others.Figures 4 show a closer look at these points, and allow us to identify the best candidates regarding separability. 6

Table 2 :
Euclidean distances between points in different configurations of Attribute, Dimension, and removal of zeroes.