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Knowledge and Information Systems

Open Access 13-02-2025 | Regular Paper

Domino drift effect approach for probability estimation of feature drift in high-dimensional data

Authors: Gábor Szűcs, Marcell Németh

Published in: Knowledge and Information Systems

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Abstract

Concept drift (and data drift) is a common phenomenon in machine learning models, where the statistical properties of the input data change over time, leading to a decrease in model performance. Detecting data drift is crucial for maintaining the accuracy and reliability of machine learning models in real-world applications. While previous data drift detector approaches can identify if a drift has occurred, these approaches cannot localize which specific features have caused the drift. Feature drift detectors solve this deficiency, but the required number of detectors is equal to the number of dimensions, which is a resource-intensive solution in high-dimensional data. In this paper, we propose a novel approach for feature drift analysis and drift detection based on a domino effect caused by the correlation of features. Our approach, the so-called Domino drift effect (DDE), is based on the empirically proven assumption that an initial reference correlation can be utilized as a proxy for detecting other drifting features. The method analyzes the correlating and drifting behavior, and by using only a subset of all features, it derives inference about the drifting of the remaining features, if co-drifting phenomena occur in the data stream. At co-drifting phenomena, the DDE method can estimate the probability of feature drift, which is particularly useful in high-dimensional datasets. To evaluate the effectiveness of our approach, we conducted experiments on four real-world datasets. The results show that our approach can effectively be used to predict feature drift in the whole dataset, and it has potential industrial applications.

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1 Introduction

In data stream analysis, researchers and developers build models to capture information hidden in the data, either for description or prediction [1]. Methods for data stream analysis can be used in lots of potential application areas including financial time series analysis [2], industrial process monitoring, social media activity, or customer behavior data capturing. Standard machine learning techniques are widely employed for these data streams [3], but they are inadequate for the evolving nature of continuous changes in the non-stationary environment [4], e.g., when customers’ behavior is changing in credit card fraud detection [5]. Another example is the change in mobile usage; there has been a surge in interest in mobile data stream mining, aiming to construct near real-time models for data stream mining applications that run on mobile devices [6].

This kind of problem is often tackled by discarding past knowledge, despite its potential relevance in the case of recurrent concepts. The best solutions contain a concept drift detection method [7], and based on the result of the detector, they use only the new concept after the drift and they use the past knowledge until the previous concept.

Finding changes in multidimensional unlabeled data is still another kind of challenge. For a large number of dimensions, applying the PCA (principal component analysis) is a natural method for feature extraction prior to the change detection [8]. The authors investigated not only the data drift but feature drift as well, where their detector localized which features have drifted via conditional distribution tests [9]. In the literature, there are lots of drift detectors, like K-L distance and Hotelling’s t2 tests that are able to detect the changes in univariate or multivariate data streams [10]. The complex drift detectors contain an algorithm instead of a simple hypothesis test; these algorithms use a supervised or unsupervised approach. For example, margin density drift detection (MD3) algorithm [11] belongs to an unsupervised one; this tracks the number of samples in the uncertainty region of a classifier, as a metric to detect drift.

This paper is concerned with data drift, where multidimensional data consist of lots of features (the number of features is equal to the number of dimensions in the multidimensional dataset). Our research focused on feature drifts and relationships among them. The definition of feature drift (between t and $t+1$) with $i{\text {th}}$ feature can be seen in the next equation, i.e., the data belong to two different distributions.

$$\begin{aligned} {P}_t\left( f_i\right) \ne \ P_{t+1}\left( f_i\right) \end{aligned}$$

(1)

An interpretable drift detection method should also locate the source of the drift and characterize its nature [12]. While previous papers investigated the univariate or multivariate data drifts, none of them examined the relationships among the drifting features. The aim of our research was to investigate these relationships, particularly the relationship between the probability of feature drift and its correlation with other features. The further aim was to explore the possibility of the usage of the co-drifting phenomenon, i.e., if there is a strong connection between the drifting probability and the correlation inside the feature set, then this phenomenon can be used in the future for the prediction task. In the high-dimensional dataset, the number of features is large; in this problem, drift detectors for all features require a lot of resources. Our research is concerned with this problem, i.e., how only a few detectors can solve the drift prediction task for all features.

The main contribution of this paper is the data drift analysis in high dimensions, and it is the first to investigate the co-drifting phenomenon. Our contributions are as follows:

Introduction of a new concept, the Domino drift effect (DDE), for the co-drifting phenomenon, which is based on the assumption that the strong correlation between features is proportional to the probability of occurrence of the double drift phenomenon.

A method using feature correlation as a drift proxy, which can be used to assign probability values to the expected drift of the remaining features by analyzing only a subset of all features.

Calculations and predictions about the drifting chances of one of the feature members of the pair without actually performing a drift detection on it, purely based on the drift behavior of the other feature.

Design of the DDE probability lift curve, which can show whether the drift affects the individual features independently or affects groups of features based on the nature of the curve’s trend. We created an algorithm for the DDE method, and the implementation is published on GitHub with the corresponding datasets as well: https://github.com/marcell-nemeth/DominoDriftEffect.

Experiments on four real-world datasets with low and high dimensions, containing both artificially induced and naturally occurring data drifts.

The rest of the paper is organized as follows. In Sect. 2, we present the proposed approach with the co-drifting phenomenon. Section 3 contains the literature review. Section 4 contains the application of our method with the evaluation plan. In Sect. 5, we present the experimental results with four datasets. Section 6 consists of the conclusion about the proposed method.

2 Proposed approach

2.1 Notations for Domino drift effect (DDE)

The probability that a feature’s drift will also happen together with another feature’s drift whose relationship with previous feature (measured in the reference window) is stronger/greater (or equal) than a threshold can be determined with a sufficiently high number of drifts. This relationship is measured by the absolute value of the Pearson linear correlation coefficient ($\rho _{i j}$), and this threshold is denoted by z. Let’s call this phenomenon the Domino drift effect (DDE), and its probability is expressed by the formula below $ \operatorname {DDE}(z):=\operatorname {Pr}(\textrm{DDE} \mid z)\nonumber $ where $0\le z\le 1$ and

$$\begin{aligned} \operatorname {Pr}(\textrm{DDE} \mid z)=\operatorname {Pr}\left( P_t\left( f_j\right) \ne P_{t+1}\left( f_j\right) \left| P_t\left( f_i\right) \ne P_{t+1}\left( f_i\right) ,\right| \rho _{ij} \mid \ge z\right) \end{aligned}$$

(2)

As we do not have y labels, we can only measure feature drift (data drift) in an unsupervised way. Let $S_{feat}$ be the set of all features and denote their number by m:

$$\begin{aligned} m=\left| S_{feat}\right| \end{aligned}$$

(3)

Furthermore, let’s take the set of features where any drift occurred, denote this by $S_{drift}$ and the size of this set by k:

$$\begin{aligned} S_{drift}&=\left\{ j\ \left| \ P_t\left( f_j\right) \ne P_{t+1}\left( f_j\right) \right. \ \right\} \end{aligned}$$

(4)

$$\begin{aligned} k&=\left| S_{drift}\right| . \end{aligned}$$

(5)

Let us denote the number of drifting j features by $n_{i}$ for which the absolute value of linear correlation coefficient with $f_{i}$ is greater (or equal) than a threshold.

$$\begin{aligned} n_i\left( z\right) =\left| \ \left\{ \ j \ | \ \ j\in S_{drift}, \left| \rho _{ij}\right| \ge z\ \right. \right\} |,\ \ 0\le z\le 1 \end{aligned}$$

(6)

Let us denote the number of all j features by $N_{i}$ for which the absolute value of the linear correlation coefficient with $f_{i}$ is greater (or equal) than a threshold.

$$\begin{aligned} N_{i}(z)=\left| \ \left\{ \ j \ \left| \ \left| \rho _{ij}\right| \ge z\ \right. \right\} \right| ,\ \ 0\le z\le 1 \end{aligned}$$

(7)

Having these notations introduced, the aim for the following is to measure the probability of DDE. The next subchapters present the theoretical design of the DDE model and the probability lift curve.

2.2 DDE probability lift curve

The value of $n_{i}$ can show high variance depending on the value of i; however, $\frac{n_{i}}{N_{i}}$ usually results in the same ratio for different i values. In the following, we assume that for a given data set this ratio is approximately equal for two arbitrarily chosen features.

$$\begin{aligned} \frac{n_i\left( z\right) }{N_i\left( z\right) }\approx \frac{n_j\left( z\right) }{N_j\left( z\right) }\ \ \ \ \forall z,\ \forall i,j\in S_{feat} \end{aligned}$$

(8)

Even though this assumption is not a strong limitation, it helps a lot in the upcoming calculations, which is why this simplification is utilized in our DDE model. The $n_{i}(z)$ gives the number of drifting j features, where $\left| \rho _{ij}\right| \ge z$, and we can calculate the average number of drifting j features from arbitrary drifting feature i; this $n_{i}(z)$ average is denoted by n(z).

$$\begin{aligned} n\left( z\right) =\frac{\sum _{i=1}^{k}{n_i\left( z\right) }}{k} \end{aligned}$$

(9)

In a similar way, the average number of all j features from arbitrary drifting feature i where $\left| \rho _{ij}\right| \ge z$ can be also calculated; this $N_{i}(z)$ average is denoted by N(z).

$$\begin{aligned} N\left( z\right) =\frac{\sum _{i=1}^{k}{N_i\left( z\right) }}{k} \end{aligned}$$

(10)

Since $\frac{n_{i}(z)}{N_{i}(z)}$ results in approximately the same ratio for different i values, they will not only be equal to each other but also to the ratio of the two averages, i.e., n(z) divided by N(z).

$$\begin{aligned} \frac{n_i\left( z\right) }{N_i\left( z\right) }=\frac{n\left( z\right) }{N\left( z\right) }\ \ \forall z,\ \forall i\in S_{feat} \end{aligned}$$

(11)

To calculate the probability of the Domino drift effect (DDE), let’s take each drifting feature (all of them have a probability of $\frac{1}{k}$) and, starting from this, calculate the probability that its neighbor is also drifting (by neighbor we mean those features for which $\left| \rho _{ij}\right| \ge z$), then these should be summarized.

$$\begin{aligned} \Pr {\left( DDE\left| z\right. \right) }=\sum _{i=1}^{k}{\frac{1}{k}\frac{n_i\left( z\right) }{N_i\left( z\right) }}=\frac{n\left( z\right) }{N\left( z\right) } \end{aligned}$$

(12)

This ratio will be denoted by $r_{n/N}$, giving the DDE probability for different z values.

$$\begin{aligned} r_{n/N}\left( z\right) =\frac{n\left( z\right) }{N\left( z\right) } \end{aligned}$$

(13)

Both n(z) and N(z) values are the results of an average calculation. Let’s find another option for calculating the probability of DDE without considering each feature individually and then average them. Taking the feature pairs where the correlation is stronger (or equal to) z and within that, those with both features drifting, we can also examine their ratio. The former is denoted by B(z), the latter by b(z), and then, their proportion ($r_{b/B}$) can be formed.

$$\begin{aligned} B\left( z\right)&=\left| \ \left\{ \left( i,j\right) \ \left| \ \left| \rho _{ij}\right| \ge z\ \right. \right\} \right| \end{aligned}$$

(14)

$$\begin{aligned} b\left( z\right)&=\left| \ \left\{ \left( i,j\right) \ \left| \ i,j\in S_{drift},\ \left| \rho _{ij}\right| \ge z\ \right. \right\} \right| \end{aligned}$$

(15)

$$\begin{aligned} r_{b/B}\left( z\right)&=\frac{b\left( z\right) }{B\left( z\right) }. \end{aligned}$$

(16)

Let’s examine the relationship between $r_{b/B}$ and $r_{n/N}$. It follows from the definitions that the below equalities stand as

$$\begin{aligned} \sum _{i=1}^{k}{n_i\left( z\right) }&=2b\left( z\right) \end{aligned}$$

(17)

$$\begin{aligned} \sum _{i=1}^{k}{N_i\left( z\right) +\sum _{i=k+1}^{m}{N_i\left( z\right) }}&=2B\left( z\right) , \end{aligned}$$

(18)

where the first term contains the summation of drifting features, while the second term contains the summation of non-drifting features (since every pair was considered twice, there is a multiplier of 2 on the right).

$$\begin{aligned} r_{b/B}\left( z\right) & =\frac{b\left( z\right) }{B\left( z\right) } \nonumber \\ & =\frac{\sum _{i=1}^{k}{n_i\left( z\right) }}{\sum _{i=1}^{k}{N_i\left( z\right) +\sum _{i=k+1}^{m}{N_i\left( z\right) }}}=\frac{k n\left( z\right) }{k N\left( z\right) +\left( m-k\right) N\left( z\right) }=\frac{k n\left( z\right) }{m N\left( z\right) }\nonumber \\ \end{aligned}$$

(19)

$$\begin{aligned} r_{b/B}\left( z\right) & =\frac{k}{m} r_{n/N}\left( z\right) \end{aligned}$$

(20)

As can be seen from the formula, $r_{b/B}(z)$ can be clearly determined from the value of $r_{n/N}(z)$ by multiplying the ratio $\frac{k}{m}$, which is the ratio of drifting features. The advantage of our model is that no matter how discrete the distribution of $n_{i}$ is at different features (it may be a small value for one or a large value for another), we still have the possibility to calculate the probability of DDE through the feature pairs; plotting this on a diagram as a function of z, we get the DDE probability curve.

2.3 Theoretical design of the DDE model

In most cases, the specific value of the DDE probability curve is not interesting but the trend of the curve (e.g., horizontal or increasing). It is worth normalizing the curve by dividing the value taken at point 0, to obtain the DDE probability lift curve (it starts from 1 at zero).

$$\begin{aligned} r_{n/N}^{\left( lift\right) }\left( z\right) =\frac{r_{n/N}\left( z\right) }{r_{n/N}\left( 0\right) } \end{aligned}$$

(21)

Since all pairs must be taken into account for the $r_{b/B}(0)$ value, the ratio of the drifting and all pairs was included in the denominator, therefore

$$\begin{aligned} r_{b/B}^{\left( lift\right) }\left( z\right)&=\frac{r_{b/B}\left( z\right) }{r_{b/B}\left( 0\right) } \end{aligned}$$

(22)

$$\begin{aligned} \frac{r_{b/B}\left( z\right) }{\frac{k\left( k-1\right) /2}{m\left( m-1\right) /2}}&=\frac{r_{b/B}\left( z\right) m\left( m-1\right) }{k\left( k-1\right) } \end{aligned}$$

(23)

Furthermore, the two probability curves ($r_{b/B}(z)$ and $r_{n/N}(z)$) only differ by a constant multiplication, after normalization the two lift curves will be the same:

$$\begin{aligned} r_{b/B}^{\left( lift\right) }\left( z\right) =\ r_{n/N}^{\left( lift\right) }\left( z\right) \end{aligned}$$

(24)

The DDE probability lift curve shows whether, in the case of a stronger correlation, the probability that the neighbor of a drifting feature also drifting is higher. If this curve follows a horizontal trend, therefore regardless of the correlation, we always experience the same probability. On the other hand, if it is an increasing curve, then the drift does not only affect the individual features independently but rather affects groups of features between which the relationship is stronger.

Change detection [13, 14] (or drift detection) refers to the methodology that helps determine and identify a time instant when a change arises in the properties of the target object. Concept drift detection methods can be divided into supervised and unsupervised approaches. Supervised methods utilize labeled data to monitor and detect changes by tracking prediction errors or other performance metrics. Unsupervised methods operate without labeled data, focusing instead on the inherent characteristics of the data distributions to detect changes over time.

The unsupervised approach is a more difficult task, thus some solutions try to form a bridge between the two approaches so that the unsupervised can be led back to the supervised one. An unsupervised task can be solved using different conversion techniques, where a dummy classifier is constructed to detect concept drift by monitoring classification outputs [15‐17]. For example, unsupervised concept drift detection can be solved using a student–teacher learning approach [18], where an auxiliary model (student) mimics the behavior of the main model (teacher). In controlled experiments, this approach detects concept drift effectively; it signals a small number of false alarms, but requires more time to detect changes. Another solution to this conversion is the Probabilistic Concept Drift Detection (PCDD) [19]. If labeling samples in real-time streaming is not feasible due to resource utilization and time constraints, PCDD relies on the data stream classification process and provides concept drift without labeled samples. In our article, we used only the unsupervised approach, and it was not converted back to the supervised one.

There is no single drift detector that works better than all the others in all scenarios [20]; therefore, we investigated and used four detectors in our work: Jensen–Shannon [21] and ADWIN [22], Wasserstein [23], and Kolmogorov–Smirnov [24, 25] detectors [26]. Many concept drift detectors rely on the Jensen–Shannon detector, e.g., CDJD [27] or CDDDE [28]; the latter one uses Jensen–Shannon divergence with extension of exponentially weighted moving average. Kolmogorov–Smirnov is used in lots of unsupervised concept drift detectors without modifications [29, 30], or with amendment to extend the basic functionality [31]. Wasserstein distance is also an appropriate basic for concept drift detection [32]; although there are some advanced versions (e.g., Wasserstein distance learned feature representations—WDLFR [33]), we used the basic method.

Only a small portion of the papers investigated the correlation as a possibility in data drift detection in machine learning models. These methods utilize the correlation between the features in the input data to identify changes in their statistical properties. The basic idea is that changes in the correlation structure of the features can indicate changes in the data distribution. Similarly, Huang et al. [34] proposed a method based on not only higher-order statistical dependencies between the input features but also utilizing historical drift trends to estimate the probability of expecting a drift at different points across the stream.

Similarly to correlation-based methods, the covariate shift can also be used in drift detection [35‐37]. These methods measure the difference between the joint distribution of the input data and the joint distribution of the target variable and the input data. If the difference is significant, concept drift is detected. However, this method requires knowledge of the target variable, which is not always available in real-world applications.

Dynamic temporal dependencies between labels can be exploited using a label influence ranking method, which leverages a data fusion algorithm and uses the produced ranking to detect concept drift. Label dependency drift detector (LD3) [38], an unsupervised concept drift detector (in the multi-label classification problem area), uses these label dependencies within the data for multi-label data streams.

There is a similar work [39] to our paper, in which concept drift correlations are investigated. That work used Granger causality to determine linear and nonlinear correlations between concept drifts; then, authors used different techniques to find concept drift correlations. This correlation is a causality relationship, where the authors were interested in whether there was a causal relationship between two drifts that were different in time. A concept drift is represented by a snippet of the time series within which the drift is found, and the concept drift correlation is retrieved by the Granger causality definition. With Granger causality, it is not possible to examine simultaneous events, and we had to investigate the relationship among the features at the same time, because our focus was on features inside a drift situation. Another difference is that other papers dealt with concept drift in the multidimensional level, but our analysis was in the feature level.

Table 1 presents the comparison of the papers that uses correlation analysis in their concept drift methodology. Some of the unsupervised methods deal with prediction for the next drift, and none of them contains feature level analysis in a drift, except our paper. Furthermore, our research was computational resource-oriented work, so not only the goodness indicators were important but the performance (running time) as well.

Table 1

Comparison of the papers that uses correlation analysis for concept drift

References	Unsupervised	Prediction for the	Feature level	Comp. Resource
		next drift	analysis in a drift	oriented
[34]	Yes	Yes	No	No
[35]	No	No	No	No
[36]	No	Yes	No	No
[37]	Yes	No	No	No
[38]	Yes	No	No	No
[39]	Yes	Yes	No	No
Our paper	Yes	Yes	Yes	Yes

All these papers investigated the correlation among the drift or the features, but not among the drifting features. The previous works focused on drift detection, so the correlation was input information for these detectors. In this paper, we explored the relationship between the probability of feature drift and its correlation with other features, which is a different aim from the previous papers, because we are interested in the co-drifting phenomenon (this phenomenon describes the feature structure of the drift).

4 Prediction by the DDE method

4.1 Used datasets in the experiments

In this chapter, the usage of the applied DDE method is presented, putting the focus on the two main, drift curve fitting and prediction phases. During the research, we examined two lower-dimensional (CICIDS with 19 features and Covertype with top 16 features) and two high-dimensional datasets with the below-mentioned parameters (Insects with 200 features and Heartbeats with 280 features) in Table 2.

Table 2

Used datasets with main attributes

Dataset	No. Features	No. Classes	Drift source	Drift type
Covertype [40]	16	2	Artificially induced	Abrupt/incremental (synthetic)
CICIDS [41]	19	2	Natural	Abrupt
Insects [42]	200	3	Natural	Abrupt/incremental
Heartbeats [43]	280	4	Natural	Abrupt

4.2 Detecting drift points

However, there are many benchmark datasets in the drift detection literature; it can generally be said that drift change points are typically not clearly defined and labeled. Except for Covertype, where the drifts were artificially induced by changing the values of the features, the other datasets examined in our paper naturally contain drifts.

For all datasets, drift detection was achieved by using univariate, statistical methods. Four of the most popular, Jensen–Shannon (JS) [21], ADWIN [26], Wasserstein [23], and Kolmogorov–Smirnov [24] detectors were utilized to monitor changes between the distributions of two windows (reference and analysis windows) for each feature. Our aim was to use datasets that have at least two drift points to be able to utilize the DDE approach: one to fit the DDE curve and one to predict.

Detected drift points on the CICIDS dataset (as an example) via JS distance are shown in Fig. 1, where the drift points are marked with red, dashed lines, and different lines in the figure correspond to different features in the dataset. The horizontal axis shows the time steps, and the JS distance can be seen on the vertical axis. JS distance values are calculated by comparing the empirical distributions of consecutive data windows. Peaks in JS values indicate greater differences between two windows, therefore indicating drifts. In the Experimental Results chapter, different signaled drift points are evaluated to show the behavior of the DDE method-based drift likelihood prediction.

4.3 Drift curve fitting and prediction algorithm by DDE

In the following, the algorithm utilizing the DDE method will be detailed to fit an initial drift curve and predict DDE scores for future streams.

The whole procedure with drift detection and the details of our Domino drift effect (DDE) prediction algorithm is given in Algorithm 1. Let $S_{check}\subseteq S_{feat}$ be the set of features on which direct drift detection can be performed, these are the monitored features. Let’s denote the subset of the dataset by $D_{S_{check}}$ containing all features in $S_{check}$ and a slice of $D_{S_{check}}$ at time t as $D_{S_{check},t}$. For drift detection, set the size of the reference window $W_{ref}$ to r and the analysis window $W_{an}$ to q. Using the before-mentioned notations, drift detection will be performed by comparing time windows $W_{ref}=\{D_{S_{check},t}\ldots D_{S_{check},t+r}\}$ and $W_{an}=\left\{ D_{S_{check},t+r+1}\ldots D_{S_{check},t+r+q+1}\right\} \ of D_{S_{check}}$, where t is a running index. As a rule of thumb, r and q should be equal to achieve a statistically fair comparison.

Next, drift detection should be performed on every feature $f_i\in S_{check},\ \ i=1,2,\ldots ,\left| \ S_{check}\right| $ and denote the set of drifting features as $S_{drift}\subseteq S_{check}$, where $\left| S_{drift}\right| =k$. As the next step (if at least one feature triggers a drift), all possible feature pairs $S_{pairs}$ should be created as a Cartesian product of $S_{drift}$ and $S_{rest}=S_{feat}\ \setminus \ S_{check}$. For all feature pairs, the Pearson linear correlation coefficient and the double drift occurrences should be calculated. Based on these values, the DDE probability curve can be constructed (curve fitting phase) using the equations in Chapter 2.2, which could be utilized to predict future drift probabilities in the unmonitored features ($S_{rest}$).

This calculation must be performed at least once to obtain a DDE probability curve and is recommended to repeat regularly to refine every time a drift is detected. Once a DDE probability curve is constructed, the approximation (prediction phase) of the drift score of an unmonitored feature $f_{i}$ can be performed via aggregating DDE probability values from all pairs of which the specific feature is a member:

$$\begin{aligned} DDE_{score}\left( f_i\right) =\frac{\sum _{j=1}^{L_i}DDE\left( \ \rho _{ij}\right) }{L_i} \end{aligned}$$

(25)

where $L_i$ scaling factor is equal to the number of pairs in $S_{pairs}$ where $f_{i}$ can be found and $\rho _{ij}$ is the Pearson linear correlation coefficient between $f_i$ and $f_j$ features.

The correlation between features is calculated by the algorithm not only once at the beginning and not at each step, but after each drift detection. This allows the method to track the change in correlations to some extent on the one hand, and on the other hand, it is not as wasteful of resources as calculating these values at each step.

Summarizing the process detailed above, our method necessitates an initialization phase during which each feature is monitored by a drift detector until the first drift is detected. The resulting data from this initial drift are used to construct a curve, which then facilitates the prediction of subsequent drifts. Additionally, averaging the results from multiple drifts can enhance the accuracy of the curve. A limitation of this approach is its reliance on detecting at least one initial drift; without this initialization phase, it is not possible to construct the curve or make subsequent predictions, as all machine learning and statistics-based methods depend to some extent on historical information. Our assumption is the same that is the internal structure of across multiple features (the correlation of them) is similar with the past.

4.4 Further use cases

We can examine our DDE model for synthetic drift generation. Let’s assume that we have a real data set with m features, where the drift is (synthetically) generated by randomly taking k features and then manipulating them until drifting. Since we did not consider the correlation between the features, the probability of starting from a drifting $f_{i}$, another $f_{j}$ also drift will be $\frac{k-1}{m-1}$ independent of z. This will result in a horizontal DDE probability curve and a horizontal DDE probability lift curve.

Furthermore, correlation is a proxy for mutually drifting features; therefore, our method allows for the efficient use of resources in detecting and correcting drifting features, resulting in increased accuracy and efficiency in data analysis. This approach may find applications in various fields where checking each feature would be expensive. By identifying the correlation between features, this method can detect when one feature is likely to drift based on changes in another correlated feature, which may not be detected by univariate methods. Therefore, the method based on correlation can complement univariate drift detection methods and provide additional insight into the behavior of multiple features in a dataset.

In the following, a potential industrial application will be detailed through an easy-to-interpret example. Let us suppose that the goal is to monitor the terrain sensitivity of a self-driving vehicle. A digitally enhanced car typically has thousands of sensors, each of those contributing to describing the current state of the vehicle. For example, multiple vibration sensors are physically placed close to each other, therefore showing similar responses to terrain changes. Let us suppose that the driving assist model loses accuracy after leaving the highway to a countryside road and would like to explore the reason. We suspect the underlying phenomena should be related to the quality of the roads, therefore starting to analyze the corresponding data coming from the vibration sensors.

Having finished analyzing the time series of the first sensor, we detect significant changes in the vibration amplitudes due to the terrain quality changes and record it as a data/concept drift contributing to the decreased model accuracy. However, so far, there is only finished running univariate drift detection method on the first sensor feature data; the DDE method allows determining whether running the same tests on the other sensory features would be truly necessary due to the high initial correlation or would be unnecessary because of the drift probability is highly unlikely (because of the uncorrelated nature).

5 Experimental results

This chapter details the quantitative measurements of the DDE method on four datasets containing different types of drifts. In the first subsection, we present a lift curve example in a high-dimensional dataset as a results of the initialization phase.

5.1 Lift curve example as a results of the initialization phase

To obtain DDE probability and lift values, the number of overall and double drifting feature pairs should be calculated on different correlation threshold levels. We selected only a high-dimensional dataset for demonstrating a typical result of theoretical 3.2 and 3.3 subsections. The results are based on feature drifts in the initialization, i.e., on the first drift. Table 3 shows the before-mentioned pair numbers and the corresponding probability and lift curves (Fig. 2) on the first drift point of the Insects dataset. The lift values in the last column of Table 3 (and the curve in Fig. 2) come from Eq 20 and 21 (it can be seen at the end of our derivation in Eq 24 that they are the same). These lift curves were also constructed for the other three datasets, from which we could derive insights into drift characteristics based on the curve trends detailed in the next sections.

Table 3

Number of feature pairs on different correlation thresholds with the corresponding DDE probability and lift values on the $1{\text {st}}$ drift of the Insects dataset

Threshold	No. Feature pairs	No. Double drifting feature pairs	DDE probability	DDE lift
0.0	19,900	1176	0.24	1.00
0.1	13,755	1078	0.32	1.33
0.2	8401	722	0.35	1.45
0.3	5532	581	0.43	1.78
0.4	3332	446	0.55	2.27
0.5	2158	354	0.67	2.78
0.6	1501	295	0.80	3.33
0.7	1105	240	0.89	3.68
0.8	799	184	0.94	3.90
0.9	491	118	0.98	4.07

5.2 Aggregate results with multiple drift points

As previously mentioned in Chapter 2, the DDE lift curve trend characteristic should be considered as an indicator of the connection between the drifting probability of features and feature-wise correlation. An upward trending curve is a proxy for a strong connection, while a flat one is a sign of the lack of the co-drifting phenomena (i.e., lack of the DDE effect).

Figures 3 and 4 present the results of DDE probability curves with more drift points, where the average results are the dark lines in the middle, and the light areas show the variance of all curves belonging to different drift points. Results of two high-dimensional datasets and two low-dimensional datasets with natural and artificially induced drift points based on the Jensen–Shannon (JS) detector are shown in Fig. 3. While Fig. 3 contains the comparison between the curves on four different datasets by the same detector (Jensen–Shannon), Fig. 4 shows the comparison between the curves of different detectors (Jensen–Shannon, Wasserstein, Kolmogorov–Smirnov, ADWIN) on the same dataset (Insects). As pictured in these figures, all three datasets containing real drifts present the DDE phenomena via upward trending curves, while the Covertype shows a flattening lift. As mentioned earlier, Covertype contains artificially induced drifts by exchanging feature values; therefore, as expected a flat curve indicates a non-existent DDE effect (with the high variance of DDE likelihood values between drift points).

As a counterexample, we introduced an artificial drift point into the Insects dataset by selecting a specific part of the series, where no natural drift occurrence was detected. As a next step, we exchanged the values of 60 randomly chosen features in the analysis window (and we do not modify the reference window). Modifying the value range and distribution of the features resulted in an artificially induced drift point (between the two windows) without respect to any real correlation. The consequence of this artificially induced drift point was a reverting and then flattening lift curve (as a clear sign of no real correlation drift connection) as can be seen in Fig. 5.

5.3 Evaluation of the prediction of new feature drifts

The DDE method—as described in Chapter 4.3—can be applied to predict drift behavior for each feature in future data streams by monitoring only a subset of a dataset. The evaluation of the method can be interpreted as the evaluation of a binary classification task, during which we want to determine whether each feature will drift or not. The application of the method is real importance in the case of high-dimension datasets, where significant resource savings are possible by ignoring the monitoring of lots of features. For this reason, the prediction was performed for only the two high-dimensional, Insects and Heartbeats, datasets. Two figures in Appendix present two ROC curve examples of predictions for the $2{\text {nd}}$ drift point of these datasets, where the size of $S_{check}$ was 50.

During the evaluations, the size of $S_{check}$ sets varied between $1\ldots m$ (number of all features). The runs were performed several times (10 folds); in each fold, our algorithm randomly selects the monitored features and averages the results. To evaluate the binary classification, the F1-score and AUC metrics were measured, the values of which were calculated from the average values of the 10 unique folds (Figs. 6, 7, 8, 9 show the results).

Table 4

Performance measures as a function of monitored features—Heartbeats dataset

	Resources (%)	No. Monitored features
	Resources (%)	280	140	70	35	17	8
		100%	50%	25%	13%	6%	3%
DDE-ADWIN	Recall	1.0000	0.9992	0.9930	0.9672	0.9175	0.8436
	Precision	0.7871	0.5297	0.3964	0.3367	0.3079	0.2951
	F1	0.8808	0.6924	0.5666	0.4996	0.4610	0.4373
DDE-KS	Recall	0.9878	0.8950	0.8150	0.8004	0.7055	0.6924
	Precision	0.7623	0.6000	0.5147	0.4609	0.4170	0.3704
	F1	0.8605	0.7184	0.6309	0.5850	0.5242	0.4826
DDE-Wass	Recall	0.9911	0.9839	0.9759	0.9785	0.9491	0.8907
	Precision	0.7787	0.6674	0.5916	0.5614	0.5354	0.5318
	F1	0.8721	0.7953	0.7366	0.7134	0.6846	0.6660
DDE-JS	Recall	1.0000	1.0000	1.0000	1.0000	1.0000	0.7000
	Precision	0.7845	0.8061	0.7774	0.7975	0.7644	0.6750
	F1	0.8792	0.8922	0.8724	0.8760	0.8548	0.6833

Table 5

Performance measures as a function of monitored features—Insects dataset

	Resources (%)	No. Monitored features
	Resources (%)	200	100	50	25	12	6
		100%	50%	25%	13%	6%	3%
DDE-ADWIN	Recall	0.9800	0.9963	0.9642	0.8608	0.7254	0.5586
	Precision	0.8680	0.5712	0.4261	0.3634	0.3232	0.2952
	F1	0.9206	0.7261	0.5910	0.5110	0.4471	0.3863
DDE-KS	Recall	1.0000	1.0000	1.0000	0.9985	0.9455	0.7769
	Precision	0.7576	0.5955	0.4645	0.4007	0.3571	0.3429
	F1	0.8621	0.7465	0.6343	0.5719	0.5184	0.4758
DDE-Wass	Recall	1.0000	1.0000	1.0000	0.9984	0.8957	0.8654
	Precision	0.8730	0.5945	0.4580	0.3854	0.3348	0.3374
	F1	0.9322	0.7457	0.6283	0.5561	0.4874	0.4855
DDE-JS	Recall	0.9091	0.9229	0.9091	0.9213	0.8060	0.9000
	Precision	0.7576	0.7634	0.7867	0.7506	0.6615	0.5333
	F1	0.8264	0.8346	0.8407	0.8189	0.7013	0.5567

As can be seen in Figs. 6, 7, 8, 9, fitting a DDE function on the $1{\text {st}}$ reference drift to predict the drifting features during the $2{\text {nd}}$ occurrence performs accurately for both datasets. Furthermore, the increasing number of features to be added to $S_{check}$ not only increases the performance of the classifier but reduces the standard deviation of estimations also, as expected.

5.4 Experiments on the effects of performance as a function of the number of monitored features

We conducted experiments to analyze the effects of performance as a function of the number of monitored features. This analysis is interpreted as the resources needed for drift detection, considering that the drift detection of a detector for each feature is of equal length. Our main hypothesis that utilizing the Domino drift effect significantly reduces the number of features needed to be monitored to detect the same number of drifts.

Tables 4 and 5 show the results of drift detection performance on the Heartbeats (also visualized in Fig. 10) and Insects high-dimensional datasets as a function of the computational resources (number of features monitored with four different detectors). For both datasets, the resource requirements can be significantly decreased to achieve well-performing drift detection across the entire feature space (the larger difference can be seen in the right direction of the diagram in Fig. 10), purely based on the co-drifting phenomenon of the features described by the DDE method.

In Table 6, the results for the two lower-dimensional datasets are presented. In these cases, the reduction in the number of monitored features was limited by the lower number of dimensions themselves.

In Fig. 11, the overall performance measured by the F1-score for all datasets is shown. Here, the horizontal axis shows not the absolute numbers of monitored features but their relative proportions. The performance curves were similar in three natural drift datasets, but in the artificially induced Covertype dataset it decreased by a much larger amount.

Table 6

Drift detection performance on lower-dimensional datasets

Dataset	No. Monitored features	Resource (%)	Recall	Precision	F1
CICIDS	19	100.00	1.0000	0.9000	0.9400
	9	47.37	0.7758	0.7148	0.7122
COVERTYPE	16	100.00	1.0000	1.0000	1.0000
	7	43.75	0.6177	0.5215	0.5492

Table 7

Comparison of DDE method with three other methods on the Heartbeats dataset

	Resources (%)	No. Monitored features
	Resources (%)	280	140	70	35	17	8
		100%	50%	25%	13%	6%	3%
DDE	Recall	–	0.9992	0.9930	0.9672	0.9175	0.8436
	Precision	–	0.5297	0.3964	0.3367	0.3079	0.2951
	F1	–	0.6924	0.5666	0.4996	0.4610	0.4373
Reduced monitoring	Recall	–	0.7000	0.3500	0.1750	0.0850	0.0400
	Precision	–	0.8000	0.7800	0.8000	0.8100	0.7800
	F1	–	0.7467	0.4832	0.2872	0.1539	0.0761
Reduced random	Recall	–	0.4119	0.2594	0.2000	0.1630	0.1393
	Precision	–	0.6179	0.4548	0.3750	0.3151	0.2747
	F1	–	0.4943	0.3304	0.2609	0.2148	0.1849
Full monitoring	Recall	1.0000	–	–	–	–	–
	Precision	0.7940	–	–	–	–	–
	F1	0.8852	–	–	–	–	–

The best (largest) recall, precision, and F1 values are denoted by bolds

Table 8

Comparison of DDE method with three other methods on the Insects dataset

	Resources (%)	No. Monitored features
		200	100	50	25	12	6
		100%	50%	25%	13%	6%	3%
DDE	Recall	–	0.9963	0.9642	0.8608	0.7254	0.5586
	Precision	–	0.5712	0.4261	0.3634	0.3232	0.2952
	F1	–	0.7261	0.5910	0.5110	0.4471	0.3863
Reduced monitoring	Recall	–	0.5000	0.2500	0.1250	0.0570	0.0285
	Precision	–	0.8700	0.8700	0.9100	0.8000	0.8900
	F1	–	0.6350	0.3884	0.2198	0.1064	0.0552
Reduced random	Recall	–	0.4333	0.2626	0.1928	0.1646	0.1548
	Precision	–	0.6500	0.4583	0.3602	0.3204	0.3041
	F1	–	0.5200	0.3339	0.2512	0.2175	0.2052
Full monitoring	Recall	0.9800	–	–	–	–	–
	Precision	0.8680	–	–	–	–	–
	F1	0.9206	–	–	–	–	–

The best (largest) recall, precision, and F1 values are denoted by bolds

Table 9

Runtimes of the methods on the Heartbeats dataset (in secundum)

		No. Monitored features
		280	140	70	35	17	8
ADWIN	Full	714	–	–	–	–	–
	DDE	–	357.3	178.9	89.7	43.8	20.9
	Reduced monitoring	–	357.0	178.5	89.3	43.4	20.4
	Reduced random	–	357.1	178.7	89.5	43.6	20.7
KS	Full	238	–	–	–	–	–
	DDE	–	119.3	59.9	30.2	14.9	7.3
	Reduced monitoring	–	119.0	59.5	29.8	14.5	6.8
	Reduced random	–	119.1	59.7	30.0	14.7	7.1
Wass	Full	166.6	–	–	–	–	–
	DDE	–	83.6	42.1	21.3	10.6	5.2
	Reduced monitoring	–	83.3	41.7	20.8	10.1	4.8
	Reduced random	–	83.4	41.9	21.1	10.4	5.0
JS	Full	126.14	–	–	–	–	–
	DDE	–	63.4	31.9	16.2	8.1	4.1
	Reduced monitoring	–	63.1	31.5	15.8	7.7	3.6
	Reduced random	–	63.2	31.7	16.0	7.9	3.9

Table 10

Runtimes of the methods on the Insects dataset (in secundum)

		No. Monitored features
		200	100	50	25	12	6
ADWIN	Full	510	–	–	–	–	–
	DDE	–	255.3	127.9	64.1	31.0	15.7
	Reduced monitoring	–	255.0	127.5	63.8	30.6	15.3
	Reduced random	–	255.1	127.7	63.9	30.8	15.5
KS	Full	170	–	–	–	–	–
	DDE	–	85.3	42.9	21.6	10.6	5.5
	Reduced monitoring	–	85.0	42.5	21.3	10.2	5.1
	Reduced random	–	85.1	127.7	63.9	30.8	15.5
Wass	Full	119	–	–	–	–	–
	DDE	–	59.8	30.1	15.3	7.5	4.0
	Reduced monitoring	–	59.5	29.8	14.9	7.1	3.6
	Reduced random	–	59.6	29.9	15.1	7.3	3.8
JS	Full	90.1	–	–	–	–	–
	DDE	–	45.4	22.9	11.6	5.8	3.1
	Reduced monitoring	–	45.1	22.5	11.3	5.4	2.7
	Reduced random	–	45.2	22.7	11.4	5.6	2.9

We compared our solution with three other methods, two baseline methods (Full and Reduced monitoring methods) and an additional simple (so-called reduced random) method; they are described below.

Full monitoring method: it monitors all features during the whole time in every time step; obviously, this is a resource wasting solution, because the drift detectors are called far too often. This baseline method is only basic from the resource point of view, but from the goodness viewpoint it is a very good solution that cannot be expected to be better.

Reduced monitoring method: this solution investigates only a subset of all features during the whole time. The method postpones the possibility of finding a drift among the remaining features. Thus, this baseline represents a missed opportunity to find additional drifts besides the drifts among the monitored features (so this does not use DDE); thus, this baseline method is basic from the goodness point of view.

Reduced random method: this is similar to our DDE method but instead of drift prediction to rest features, this basic method predicts randomly. The solution investigates only a subset of all features during the whole time, and if a drift occurs among the monitored features, then it predicts additional drifts among the unmonitored features. This prediction is based on the global probability of a drift, where this probability is estimated by the ratio of the number of drifted features and the number of unmonitored features. Besides the global probability, there is no other information in this method, so the features are randomly predicted for drift.

The results of the comparison are shown in Tables 7 and 8 for Heartbeats and Insects datasets, respectively. The best recall and F1-scores are achieved by our DDE method (in almost all cases), and the highest precision values belong to the reduced monitoring method, but this was only due to the fact that the set of unmonitored features was not included in the decisions, so the number of errors that could be made here was in principle zero.

Tables 9 and 10 show the running times of the DDE and other three methods for Heartbeats and Insects datasets, respectively. In the full method, there is no initial value, but in the others the end users can choose a different number of monitored features according to their own available resources. This flexibility is a big advantage in our method. Despite the fact that two other methods have this advantage, the DDE method outperforms the others.

6 Conclusion

Data drift detection is a challenge in itself during the development and operation of machine learning models, but if the detection takes place even in high-dimensional data sets, it causes extra difficulties for researchers. In this research, we have developed a method using feature correlation as a drift proxy, which can be used to assign probability values to the expected drift of the remaining features by analyzing a low-dimensional subset.

We introduced a new concept, the Domino drift effect (DDE) for the co-drifting phenomenon, which is based on the assumption that the strong correlation between features is proportional to the probability of occurrence of the double drift phenomenon (both members of the feature pair drift). Our theoretical model takes advantage of this drift effect. Then we can make predictions about the drifting chances of one of the feature members of the pair without actually performing a drift detection on it, purely based on the drift behavior of the other feature.

We also proved the correctness and showed the practical application of our theory on four real-world datasets with low and high dimensions, containing artificially induced and naturally occurring data drifts also. The measurements were divided into two parts: we showed that in the case of natural drifts (Insects, Heartbeats, and CICIDS datasets), the DDE curve shows an increasing trend between the magnitude of the correlation and the DDE likelihood, but at the same time in the case of synthetically induced drifts (Covertype), this relationship does not hold above, a flat trend can be observed.

Although the necessity of detecting at least one initial drift is a limitation of our approach, the curve and subsequent predictions cannot be obtained without this initialization phase. The advantage of our solution is that it does not need to use drift detectors on all features after the initialization phase, but only on a subset. Even if our solution cannot reach the same goodness as the drift detectors on the unmonitored subset, it can achieve relatively good predictions using the DDE based on the co-drifting phenomenon.

Finally, we illustrated the practical importance of the method with drift prediction tasks performed on two high-dimensional data sets: we used models fitted to previous drifts to tell features whether they would drift or not during subsequent streams. The practical application of our theory performs well even with a small number of monitored features, but we also presented that by expanding the monitored feature set, the variance of the performance decreases.

Acknowledgements

Project No. KDP-IKT-2023-900-I1-00000957/0000003 has been implemented with the support provided by the Ministry of Culture and Innovation of Hungary from the National Research, Development and Innovation Fund, financed under the C2299763 funding scheme.

Declarations

Conflict of interest

The authors, Gábor Szűcs and Marcell Németh declare that they have no conflict of interest.

Ethical approval

Not applicable.

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Title: Domino drift effect approach for probability estimation of feature drift in high-dimensional data
Authors: Gábor Szűcs
Marcell Németh
Publication date: 13-02-2025
Publisher: Springer London
Published in: Knowledge and Information Systems
Print ISSN: 0219-1377
Electronic ISSN: 0219-3116
DOI: https://doi.org/10.1007/s10115-025-02362-0

Springer Professional

Domino drift effect approach for probability estimation of feature drift in high-dimensional data

Abstract

Publisher's Note

1 Introduction

2 Proposed approach

2.1 Notations for Domino drift effect (DDE)

2.2 DDE probability lift curve

2.3 Theoretical design of the DDE model

4 Prediction by the DDE method

4.1 Used datasets in the experiments

4.2 Detecting drift points

4.3 Drift curve fitting and prediction algorithm by DDE

4.4 Further use cases

5 Experimental results

5.1 Lift curve example as a results of the initialization phase

5.2 Aggregate results with multiple drift points

5.3 Evaluation of the prediction of new feature drifts

5.4 Experiments on the effects of performance as a function of the number of monitored features

6 Conclusion

Acknowledgements

Declarations

Conflict of interest

Ethical approval

Publisher's Note

Appendix

Premium Partner

Springer Professional

Abstract

Publisher's Note

1 Introduction

2 Proposed approach

2.1 Notations for Domino drift effect (DDE)

2.2 DDE probability lift curve

2.3 Theoretical design of the DDE model

3 Related works

4 Prediction by the DDE method

4.1 Used datasets in the experiments

4.2 Detecting drift points

4.3 Drift curve fitting and prediction algorithm by DDE

4.4 Further use cases

5 Experimental results

5.1 Lift curve example as a results of the initialization phase

5.2 Aggregate results with multiple drift points

5.3 Evaluation of the prediction of new feature drifts

5.4 Experiments on the effects of performance as a function of the number of monitored features

6 Conclusion

Acknowledgements

Declarations

Conflict of interest

Ethical approval

Publisher's Note

Appendix