Prediction of rare events in the operation of household equipment using co-evolving time series

Throughout this paper, we consider a time series in coevolution $X_{t}=\vec{x_{1}},\vec{x_{2}},\ldots,\vec{x_{t}}$, where $\vec{x_{i}}$ represents the vector of observations recorded in time $t_{i}$ from different sensors describing the functioning of an equipment. Among the observations there are the ones relative to physical features such as the temperature, speed, noise, and other relatives to the operating state of the equipment such as the time since last failure and past failure frequency at each moment $t_{i}$. Discussing the redundancy of these measurements is crucial in our analysis. Some of these measurements may exhibit correlations, such as the temperature and noise levels in specific equipment, or the time since the last failure and the past failure frequency at each time point, as pointed out. Redundancy can provide further insights into the phenomena under study, but it can also impact the conception of our prediction model. For instance, a strong correlation between two variables can pose challenges when utilizing them in a regression model. Therefore, it is essential to assume that the time series data we are working with are not highly correlated. To validate this assumption, we perform experimental assessments by measuring the pairwise correlations between the time series. Each observation $\vec{x_{i}}$ is associated with a binary variable $y_{i}$, which takes the value $1$ when a failure occurs at time $t_{i}$, and $0$ otherwise. The binary variables also form a time series $Y_{t}=y_{1},y_{2},\ldots,y_{t}$. Our objective is to predict the occurrence of a failure at time $t+\Delta t$ using $X_{t}$ and $Y_{t}$; that is $y_{t+\Delta t}=1$.

Autologistic regression is particularly well suited to modeling the probability of a failure, as it is designed to specify the probability of failure by leveraging various quantitative variables, which can be binary, categorical, or real. This method has demonstrated considerable success in predicting failure. Notably, when compared to complex models like neural networks, our approach offers a balance between frugality and explainability. It’s frugal in the sense that it requires fewer data during the learning phase, making it a resource-efficient choice. In addition, it is explainable because it makes it possible to evaluate the role of each variable in estimating the probability of failure. In our specific case, we argue that incorporating $Y_{t}$ as a regressor introduces a temporal correlation in the predictions, effectively reducing randomness in the outcomes. This logistic regression model, enhanced with the $Y_{t}$ regressor, is referred to as autologistic. The approach offers a valuable perspective for improving prediction accuracy while maintaining model interpretability. To specify the prediction model, we define $y_{t+\Delta t}$ as a Bernoulli variable, $y_{t+\Delta t}=1$ if a failure occurs with the probability $p_{t+\Delta t}$ in the interval $\Delta t$ and $y_{t+\Delta t}=0$ otherwise. The conditional probability of occurrence of this failure is given by:

where $p_{t+\Delta t}$ is the following logistic function:

The vector $\theta$ is the set of the autologistic regression parameters $\theta=\left\{\vec{a_{0}},\cdots,\vec{a_{q}},b_{0},\cdots,b_{q},c)\right\}$. In addition, $\lambda_{t+\Delta t}$ takes the value 1 if a failure occurs between times t and t + $\Delta t$, and 0 otherwise. Furthermore, $q$ represents the start date of the memory for each time series. The same start date for all chronological series simplifies the model, as it is a compromise between the different coevolving chronological series. Selecting the value of $q$ involves identifying the time lags that leads to an acceptable prediction error. In this work, we utilized the Forward Feature Selection (FFFS) method to select this value.

The determination of a failure occurrence is established by contrasting the probability derived from Equation 7 with a predefined threshold. If the calculated probability surpasses this threshold, it signifies a potential indication of failure, prompting the decision. Multiple threshold values were experimentally examined to fine-tune the decision process in alignment with the model’s effectiveness. Upon surpassing the choosing threshold, signaling a heightened probability of failure, the verdict is made that a fault is indeed present.

There are several methods to estimate the parameter vector $\theta$ in autologistic regression such as Maximum Likelihood Estimation (MLE) [13] and Bayesian estimation [14]. In this paper, we employ MLE for robust parameter estimation by maximizing the likelihood, ensuring principled model fitting and capturing underlying relationships effectively. For that, we define the likelihood as the probability of $y_{t+\Delta t}=\lambda_{t+\Delta t}$ given $Y_{t}$, $X_{t}$ and $\theta$:

Applying conditional probabilities rule and since $y_{t+\Delta t}$ depends conditionally on the $q$ previous observations $\vec{x}_{t},\vec{x}_{t-1},\ldots,\vec{x}_{t-q}$ and $y_{t-1},y_{t-2},\ldots,y_{t-q}$, as well as on the parameter set $\theta$, we can therefore write:

For computational convenience, we use the log of the previous equation given by:

As mentioned in equation 1, we can deduce that the conditional probability of observing $y_{t+\Delta t}$ can be modeled using a Bernoulli distribution as follows:

where $p_{n+\Delta t}$ is the logistic function as mentioned in equation 2.

In this paper, we used the gradient descent algorithm [16] to minimize $-l(\theta)$ with respect to $\theta$. Note that the values of the probability in Equation 1 can vary abruptly in time. These fluctuations can cause temporal discontinuities in our predictions, leading to false alarms. To overcome this challenge, we added a smoothing probabilities constraint. Supposing that a failure lasts for a minimum duration L, we express this constraint by choosing to calculate the average of the last L probabilities using a moving average. This average probability $P_{t+\Delta t}$ is then used to make a final decision on the presence or absence of the failure, as mentioned in equation 7.

By averaging the L probabilities, we mitigate the impact of any outliers, i.e. abnormally high or low probabilities, often associated with noise. So, if a probability has been abnormally influenced by an outlier, this influence is reduced when averaged with L other values, helping to stabilize our predictions.

Recall that a binary variable $y_{t}$ is a label associated with $x_{t}$. The rarity of failure leads to having fewer observations $x_{t}$ with associated labels $y_{t}=1$. This imbalance leads to biased estimates and failure prediction errors [4]. To counter this, we incorporate class weighting into the likelihood. Assigning higher weight to the minority class and lower weight to the majority class rebalances their influence. In this case, equation 6 is rewritten as follows:

Two methods were often used for estimating weights $w_{0}$ and $w_{1}$: simple weighting and adaptive weighting [4]. The simple weighting method involves assigning weights to classes based on their frequencies in the dataset. For the majority class (class 0), $w_{0}$ is calculated as $w_{0}=\frac{n_{0}}{n}$, where $n_{0}$ is the number of minority class observations, and $n$ is the total number of observations. Conversely, for the minority class (class 1), $w_{1}$ is computed as $w_{1}=\frac{n_{1}}{n}$, with $n_{1}$ representing the number of majority class observations. This approach has limitations, notably its ineffectiveness in the presence of extreme class imbalance and outliers. This can give too much emphasis to the majority class, which hurts model performance for underrepresented classes. Our choice, the Adaptive Class Weights approach, addresses these limitations. Initially, we can use the weights calculated by the simple method instead of considering them equal as mentioned in [4]. Then, at each training iteration, weights are updated based on prediction errors as suggested in equation 9. The algorithm adaptively calculates the weights, giving priority to classes with higher prediction errors. This adaptability readjusts the model’s focus and enhances its predictive capabilities for underrepresented classes.

In our analysis, we concentrated on various home equipment failure data sources to evaluate our model’s performance. We began by generating synthetic data, creating a controlled environment to evaluate the model’s efficacy. This synthetic dataset, consisting solely of sensor readings, served as a benchmark for comparison. Subsequently, we introduced simulated data from HVAC (Heating, Ventilation, and Air Conditioning) systems, featuring 4 failures. Additionally, we tested our model’s proficiency using real-world data from a water pump. This pump logs its sensor readings and operational status every minute and recorded a total of 7 failures.

The table 1 below succinctly summarizes these datasets. It provides a clear overview of the duration, granularity of data collection, and, most importantly, the number of failures experienced by each equipment. It’s important to note that each sensor in these datasets provides a time series, which is integral to our modeling approach.

In this section, we aim to measure the performance of the proposed approach by evaluating our autologistic regression model across various types of equipment. For this evaluation, we have considered five metrics, specifically chosen to align with and facilitate direct comparison with state-of-the-art methods. The first is Accuracy, which indicates the proportion of correct predictions relative to all predictions. The second is Recall, representing the proportion of failures that the model correctly identified. Specificity measures the proportion of normal operation identified correctly, while the F-Score is the harmonic mean of precision and recall. Additionally, we evaluate the Number of false alarms, counting failure predictions while the equipment is operating normally. Beyond these metrics, we also delve into investigating the effects of varying time intervals, understanding the influence of weighting, and gauging the impact of other considered phenomena.

Table 2 summarizes the scores: For the ”pump” equipment type over a 10-day interval, the model exhibits an accuracy of 0.9997, a recall of 0.7255, and only 3 false alerts. When the interval is shortened to 5 days, the accuracy stands firm at 0.9750, but the recall dips to 0.5056, accompanied by 154 false alerts, while retaining the same decision threshold of 0.9. On a 1-day interval, the model’s recall drops significantly to 0.2185, although its accuracy remains at 0.896. Synthetic data demonstrates exceptional performance without any imbalance.

In this approach, the data division for training and testing the model is based on the proportion of failures. Specifically, 80% of the failures are used for training the model, and the remaining 20% are used for testing. For instance, with synthetic data that contains a total of 30 failures, the split is as follows: 24 failures (which is 80% of the total) are utilized for training the model, and the remaining 6 failures (20% of the total) are used for testing the model’s predictions at different time intervals. This method ensures a balanced approach, where the majority of data is used for learning, and a significant portion is reserved for evaluating the model’s predictive accuracy.