# A Visualization for Comparative Analysis of Regression Models

Nassime Mountasir, Baptiste Lafabregue, Bruno Albert, Nicolas Lachiche

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Nassime Mountasir, Baptiste Lafabregue, Bruno Albert, Nicolas Lachiche. A Visualization for Comparative Analysis of Regression Models. 2025. hal-04974691

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# A Visualization for Comparative Analysis of Regression Models

Nassime Mountasir*†, Baptiste Lafabregue*, Bruno Albert† and Nicolas Lachiche*

*ICube

Université de Strasbourg, 67400 Illkirch-Graffenstaden, France

Email: nassime.mountasir@etu.unistra.fr

†ENGLAB

Technology & Strategy, 67300 Schiltigheim, France

**Abstract**—As regression is a widely studied problem, many methods have been proposed to solve it, each of them often requiring setting different hyper-parameters. Therefore, selecting the proper method for a given application may be very difficult and relies on comparing their performances. Performance is usually measured using various metrics such as Mean Absolute Error (MAE), Root Mean Squared Error (RMSE), or R-squared (R²). These metrics provide a numerical summary of predictive accuracy by quantifying the difference between predicted and actual values. However, while these metrics are widely used in the literature for summarizing model performance and useful to distinguish between models performing poorly and well, they often aggregate too much information.

This article addresses these limitations by introducing a novel visualization approach that highlights key aspects of regression model performance. The proposed method builds upon three main contributions: (1) considering the residuals in a 2D space, which allows for simultaneous evaluation of errors from two models, (2) leveraging the Mahalanobis distance to account for correlations and differences in scale within the data, and (3) employing a colormap to visualize the percentile-based distribution of errors, making it easier to identify dense regions and outliers. By graphically representing the distribution of errors and their correlations, this approach provides a more detailed and comprehensive view of model performance, enabling users to uncover patterns that traditional aggregate metrics may obscure.

The proposed visualization method facilitates a deeper understanding of regression model performance differences and error distributions, enhancing the evaluation and comparison process.

**Index Terms**—visualization, regression, comparison, metrics, analysis

## I. INTRODUCTION

Regression is a widely studied problem in machine learning, with numerous methods proposed to tackle it, each demanding the tuning of different hyper-parameters. Therefore, selecting the proper method for a given application may be highly difficult, as it relies on effectively comparing their performances. Performance, in this context, refers to the ability of a model to make accurate predictions on new, unseen data. Traditionally, this assessment is conducted using a comparison of a variety of metrics such as Mean Squared Error (MSE), Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), and R-squared (R²) [1]. These metrics provide a numerical summary of the model predictive accuracy by quantifying the difference

between the predicted values and the actual values to decide which model is performing better compared to another.

The error (or residual) for each model is calculated as the difference $r$ between the predicted value $\hat{y}$ and the actual value $y$: $r = \hat{y} - y$, where $\hat{y}$ is the predicted value and $y$ is the true value. This error can be positive, indicating an overestimation when $\hat{y} > y$, or negative, indicating an underestimation when $\hat{y} < y$.

The focus of this article is on evaluating the predictions and qualifying the errors of one or more models for a regression problem, irrespective of the nature of the problem or the input data. The primary concern is the comparison between the model predictions and the actual observed values. To illustrate this, we use several datasets [2] [3] trained on multiple models, each providing actual values along with predictions. The datasets are described in Appendix.

Let us consider the performance of 10 regression models as measured by MAE and RMSE in Table I on Dataset A. Some models exhibit excellent results, with both low MAE and RMSE, i.e. models 1 or 10, while other perform poorly with both metrics significantly higher, such as models 5 or 7. As models 1, 2 and 10 have metrics that are way lesser than the other models, we could say that those models tend to overperform compared to the others, leaving only three models to compare instead of ten.

<table><caption>TABLE I
MAE AND RMSE VALUES FOR DATASET A</caption><thead><tr><td>Model</td><td>MAE</td><td>RMSE</td></tr></thead><tbody><tr><td>Model 1</td><td>12.91</td><td>16.72</td></tr><tr><td>Model 2</td><td>13.50</td><td>17.45</td></tr><tr><td>Model 3</td><td>32.37</td><td>37.22</td></tr><tr><td>Model 4</td><td>21.28</td><td>25.71</td></tr><tr><td>Model 5</td><td>35.56</td><td>40.74</td></tr><tr><td>Model 6</td><td>21.78</td><td>26.33</td></tr><tr><td>Model 7</td><td>39.19</td><td>44.18</td></tr><tr><td>Model 8</td><td>18.09</td><td>22.65</td></tr><tr><td>Model 9</td><td>19.25</td><td>24.75</td></tr><tr><td>Model 10</td><td>10.93</td><td>14.65</td></tr></tbody></table>

However, while these metrics are relevant for summarizing model performance and for discriminating between very good and very bad models, it becomes difficult to use them when
their values are close. In particular, they aggregate too much information, potentially obscuring important details about the nature and distribution of prediction errors. For example, in critical applications such as medical diagnosis or autonomous driving, it is crucial to avoid extreme values to prevent potentially life-threatening decisions whereas for financial forecasting or marketing strategies, more stable models might be preferred even if they occasionally produce outliers, as long as their overall predictive performance remains high. We identify three limitations in the use of metrics to compare two models.

First, consider Dataset B with ground truth values $y_{true}$ with a uniform distribution from 1 to 1000 as $y_{true} = [1, 2, 3, 4, ..., 999, 1000]$, for which a first model, model 1, predicts perfectly except for one value, as $y_{pred_1} = [1, 2, 3, 4, ..., 999, 1500]$, and another model, model 2, does moderate but constant errors, oscillating from -30 to 30, with the corresponding distribution shown in Figure 1. The corresponding MAE and RMSE for both models are shown in Table II. In this scenario, the MAE of model 1 is relatively low but has a high RMSE. For model 2, the MAE is higher but the RMSE is lower. Since MAE doesn't penalize high errors, it will favor model 1. RMSE is more sensitive to outliers, so RMSE will be in favor of model 2. The interpretation of performance of the model can differ greatly depending on the consideration of outliers or other variables that come into account.

<figure><img src="image_2.png" /><figcaption>Fig. 1. Distribution of the errors of model 2 on Dataset B</figcaption></figure>

<table><caption>TABLE II
METRICS RESULTS FOR EACH MODEL ON DATASET B</caption><thead><tr><td>Metric</td><td>Model 1</td><td>Model 2</td></tr></thead><tbody><tr><td>MAE</td><td><strong>0.5</strong></td><td>7.86</td></tr><tr><td>RMSE</td><td>15.81</td><td><strong>9.88</strong></td></tr></tbody></table>

The second limit is the impossibility to account for underestimations and over-estimations. For this scenario, consider Dataset C for which two models make predictions. The first model is generally underestimating, and the second model is generally overestimating. The corresponding distributions of errors is shown in Figure 2.

<figure><img src="image_3.png" /><figcaption>Fig. 2. Error distributions of two models on Dataset C</figcaption></figure>

<table><caption>TABLE III
METRICS RESULTS FOR EACH MODEL ON DATASET C</caption><thead><tr><td>Metric</td><td>Model 1</td><td>Model 2</td></tr></thead><tbody><tr><td>MAE</td><td><strong>9.02</strong></td><td>9.34</td></tr><tr><td>RMSE</td><td><strong>10.57</strong></td><td>10.94</td></tr></tbody></table>

When calculating their MAE and RMSE, as outlined in Table III both models share similar values of MAE and RMSE, with the MAE of 9.02 for model 1 and 9.34 for model 2, and the RMSE of 10.57 for model 1 and 10.94 for model 2. As the metrics take the average of the square of the errors, no information is given about the distribution of the errors and whether the models tend to underestimate or overestimate. The third one is that such metrics can obscure areas where one model might outperform another in specific regions of the data. For instance, two models could exhibit similar MAE and RMSE values, yet one model may deliver superior performance within certain data ranges while underperforming in others.

Until now, we have considered the deviation of a model from the ground truth. Another way of estimating the proximity between two models can be by comparing their predictions. Consider the predictions $\hat{y}_1$ of a model $m_1$ and the predictions $\hat{y}_2$ of a model $m_2$. The prediction difference, or deviation, is defined by $\delta(m_1, m_2) = \hat{y}_1 - \hat{y}_2$. Figure 3 presents the prediction difference (or deviation) histogram for two models on Dataset C. The idea is that if two models were nearly identical, they should have a consistent difference of 0 with a peek in the histogram, as they should do the same predictions for each individual.

<table><caption>TABLE IV
METRICS RESULTS FOR EACH MODEL ON DATASET D</caption><thead><tr><td>Metric</td><td>Model 1</td><td>Model 2</td></tr></thead><tbody><tr><td>MAE</td><td><strong>3.18</strong></td><td>3.19</td></tr><tr><td>RMSE</td><td><strong>3.76</strong></td><td>4.39</td></tr></tbody></table>

Although the MAE and RMSE presented in Table IV for the two models are nearly identical — Model 1 with an MAE of 3.18 and an RMSE of 3.76, and Model 2 with an MAE of 3.19 and an RMSE of 4.39 — the histogram reveals distinct patterns in their error distributions. Specifically, it highlights whether
<figure><img src="image_4.png" /><figcaption>Fig. 3. Prediction difference histogram of two models on Dataset D</figcaption></figure>

model 1 consistently underestimates relative to model 2 and vice versa. In the histogram of Figure 3, when the prediction difference is negative, model 1 does greater predictions than model 2, and when the errors difference is positive, model 1 predicts lower values. As shown in the histogram, model 1 tends to do generally lower predictions, with a very frequent difference of -13 on individuals, as well as a frequent difference of -5 to 0. On the opposite, the situations in which model 2 underestimates compared to model 1 is more equally distributed and less frequent, as shown by the distribution of error difference from 0 to 10. Those observations point out that, despite their similar values for MAE and RMSE stating that the two models have similar errors in average, the models don't predict similar values at all, revealing different patterns in their errors.

It is also important to note that MAE and RMSE are always positive since they represent aggregated magnitudes of errors, yet individual errors themselves can be either positive or negative. To address these four limitations, visualization techniques can be employed to provide a more nuanced and comprehensive view of model performance. By graphically representing the distribution of errors and other aspects of model behavior, these visualizations can reveal patterns that are not apparent from aggregate metrics alone. This article proposes a new comparison visualization technique for regression models. We show that it complements traditional performance metrics to provide more extensive insights on each model performance.

The remainder of the article is structured as follows. Section 2 presents the common visualization techniques used in the literature and their strengths and weaknesses. Section 3 explores and demonstrates the practical application of our proposed visualization tools. Finally, Section 4 summarizes the insights gained from these visualization techniques, highlighting their importance in providing a more comprehensive understanding of regression model performance beyond traditional metrics.

## II. STATE OF THE ART

### A. Literature review

While metrics like RMSE [4] and MAE [5] remain the most commonly used indicators of model performance, the visualization of regression model performance has evolved significantly, incorporating a variety of techniques to enhance the understanding of predictive accuracy and error distributions. Traditional methods, such as scatter plots, have long been used to compare predicted and actual values. However, they tend to become difficult to interpret with larger datasets due to point overlap and density issues [6]. Histograms and density plots offer a way to visualize the distribution of prediction errors, making it easier to identify skewness and outliers [7]. Residual plots, which visualize residuals against predicted values, provide more detailed insights into model fit and performance [8]. These methods help identify patterns that are not apparent in aggregate metrics, but are limited by their inability to handle large, complex datasets effectively. Taylor diagrams summarize multiple aspects of model performance, such as correlation, standard deviation, and RMSE, providing a comprehensive overview in a single diagram [9]. However, they are better suited for the comparison of multiple models on an individual basis rather than on an entire dataset. Overall, while traditional visualization techniques provide a panel of tools, they often fall short in handling the complexity of modern machine learning models and datasets, and do not effectively compare the performance of two regression models.

### B. Visualization examples

<figure><img src="image_5.png" /><figcaption>Fig. 4. Comparison of the predictions and the real values of Model 1 on Dataset A</figcaption></figure>

As mentioned before, a common way to compare regression models graphically is by displaying predicted values alongside real values in a scatter plot. Figure 4 shows the comparison between predictions and real values for Model 1 trained with Dataset A. Aligned points on the blue line represent a perfect prediction, which means a prediction equal to the real value. The further is a point from the line, the greater the error
is. This visualization helps identify points that are far from the ground truth. However, it can be hard to interpret when there are many points, and visualizing their distances from the ground truth can be challenging. Histograms are often used to address this issue by visualizing the error distribution. They help determine if the errors are symmetrically distributed or biased and identify any outliers. Figure 5 shows the distribution associated to the first figure. The dashline is the median of the histogram, representing the bin where 50% of the data are divided. For this model, the errors are distributed from -75 to 45, with the median being -9.29 and most of the values being between -20 and 10. This representation effectively shows a model propensity to make errors, but does not enable the comparison of several models as the range of predictions can be different.

<figure><img src="image_6.png" /><figcaption>Fig. 5. Error distribution histogram of Model 1</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 6. Comparison of the predictions and the real values of Model 1 and Model 11</figcaption></figure>

One approach to compare two models would be to plot the predictions of two models as function of the ground truth, to see if their range differ, as done in Figure 6. In this figure, both distributions are plotted, with in orange the errors of the first model and in green the errors of the second one. The diagonal represents the same information as in Figure

4, namely the distance from the predictions of the models to the ground truth. However, using only predictions does not provide an accurate overview of the models' performance, as it is entirely possible to have similar models that make significant mistakes. Additionally, it shares the same drawback as the figure above, especially as we don't have any information about the number of points. To address these limitations and effectively visualize 2D graphical comparisons of two regression models, we propose a novel approach that combines multiple statistical and graphical techniques. This approach aims to provide a more comprehensive and intuitive understanding of model performance. By plotting predictions of one model against another and incorporating error distributions, we can gain deeper insights into comparative performance of the models. In the following section, we will show how these visualization techniques are designed to compare two models more effectively, revealing their strengths and weaknesses and facilitating more informed decisions for selecting and refining models.

### III. 2D ERROR SPACE: GRAPHICAL COMPARISON OF THE ERRORS OF TWO MODELS

#### A. Space Definition

To compare two models, we propose to visualize the errors of one model over the other in a plot, called the 2D Error Space.

Here, rather than calculating metrics over the entire set, we are visualizing the error for each prediction individually, which gives a clearer picture of how each model performs on specific cases.

<figure><img src="image_8.png" /><figcaption>Fig. 7. Representation of the 2D Error Space on two points</figcaption></figure>

Figure 7 shows the representation of this space on two points A and B. Each point corresponds to the error of prediction associated to the same individual, where the x-axis is the error of one model and the y-axis represents the error of the other model. The coordinates of a point correspond to the errors of the first model and of the second model. In this example, A represents an individual for which the first model has an error of -12 and the second one an error of 17, and B an individual for which model 1 has an error of -34 and model
2 an error of -8. For A, model 1 is slightly better than model 2 as the absolute error is less, and for B, model 2 is way better than the other. In this space, the diagonal $y = x$ represents the points where the error of the models is the same. For situations where one of the models is overestimating as much as the other is underestimating, we can also use the diagonal $y = -x$.

<figure><img src="image_9.png" /><figcaption>Fig. 8. Comparison of the errors of Model 1 and Model 11</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 9. Zones where models are better</figcaption></figure>

The difference between the models can be visualized by looking at the closest diagonal from the point A. Finally, the horizontal and vertical axes provide information about whether the first and second model tend to underestimate or overestimate. The points located in the left and the right of the vertical axis are the points where model 1 is respectively underestimating and overestimating, and the points located below and above the horizontal axis are the points where the model 2 is respectively underestimating and overestimating. Here, A is located on the top left corner of the plot, meaning that model 1 is underestimating and model 2 is overestimating. B is on the bottom left corner, meaning that model 1 is still underestimating, but B is also underestimating. Such visualization allows to directly assess the performance of the models relative to each other, as we have both the information on the predictions made and the difference from the ground truth. This provides understandability on error distribution

and whether the models tend to make extreme predictions or predictions close to reality. Figure 8 shows the distributions of errors of two models, one in function of the other. The diagonals represent equal absolute errors for the models, so we can define two regions: one where model 1 is better - i.e. the absolute error is smaller -, and another where model 2 is better, creating two hourglasses, as featured in Figure 9, with these regions respectively in orange and green. We will refer to these spaces as "comparison zones" throughout the rest of the article. These hourglasses allow to effectively identify regions where there are more points than another, in case one is containing way more points than another.

## B. Statistical indicators

In addition to visualizing the differences between the models, the analysis can be enhanced by incorporating statistical measures. By calculating summary statistics for each region, such as the mean, the median, or by showing the errors histograms, we can have further insights into the performance disparity. As a reminder, the mean error gives an average measure of the prediction errors, which helps in understanding the overall bias of the model. A lower mean error indicates that the predictions are generally close to the actual values. The median error, being less sensitive to outliers, provides a robust central tendency measure. It highlights the typical error magnitude, offering a clearer picture of model performance in datasets with extreme values. Those two measures can help us identify inconsistency in the predictions. If both the mean and the median are aligned and close to zero, the model predictions are consistently accurate. Moreover, significant disparity between mean and median indicate skewed error distributions [10]. For instance, a mean much larger (respectively smaller) than the median suggests the presence of large positive (respectively negative) errors. In Figure 10, the dashed lines in blue represent the mean error and their corresponding standard deviations, and the ones in red represent the median error. These values can be compared to the density. If the intersection of the mean and standard deviation is not aligned with the densest zone, it means that they have a different distribution. In this figure, the mean is around -10 for model 1 and around 18 for model 2 with the same standard deviations for both models, indicating that model 1 is doing lower absolute errors than model 2 in average. The mean and median for both models are very close, so none of them tends to do significantly large errors.

While these summary statistics provide a picture into the overall bias and consistency of the models, they are incomplete. To gain a deeper understanding of the error distributions, histograms help us to identify how errors are spread across different ranges, highlighting areas where errors are concentrated and revealing any skewness or anomalies in the distributions. For instance, by comparing histograms for different models, we can visually assess which model has more concentrated errors and which has a wider spread, indicating higher variability. Figure 11 shows the error distribution histograms for the two models. Here, model 1 has a distribution almost
<figure><img src="image_11.png" /><figcaption>Fig. 10. Errors of the models with mean and median</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 11. Error distribution histograms of the two models</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 12. Errors of the models with density based on KDE (left) and proximity (right)</figcaption></figure>

symmetrical and uniform around -7, which is close from the median of 10. The second model is less uniform, with the same number of frequent values from 0 to 40 and a median of -3, indicating a smaller disparity but more concentrated values.

### C. Density and proximity with median

Using a scatter plot to visualize error distribution in the comparison zones can make it difficult to clearly identify the areas where most points are concentrated. This difficulty is especially pronounced with large datasets, where points may overlap. To solve this issue, we can represent the density of the points using a colormap, as proposed in Figure 12. The colormap on the left illustrates the density of points, calculated by Kernel Density Estimation (KDE) [11] [12]. This technique allows to estimate the probability density function of a random variable, that can then map to a color range the points depending of the density estimation. To calculate KDE, a kernel function, typically Gaussian, is applied to each data

point and smoothed by a bandwidth parameter, often chosen by methods such as Scott's [13] or Silverman's rule [14]. The resulting KDE is given by:

$$KDE(x) = \frac{1}{nh} \sum_{i=1}^{n} K\left(\frac{x-x_i}{h}\right), \text{ where } n \text{ is the number of data points, } h \text{ is the bandwidth, } x_i \text{ are the data points, and } K \text{ is the kernel function. The region in yellow indicates a higher density (more points), and the regions in purple a lower density (less points). In this figure, the densest zone is around -8 for model 1 and 25 for model 2, meaning that a lot of errors are around -8 and 25 for respectively model 1 and 2. These regions can then give information about whether one model tends to be always doing more errors or not by adding the information given by the comparison zones: if most of the points are in one of the zone, it means that one model is overall less prone to error, thus generally better performing.}$$

To address this issue, we propose an alternative visualization method that represents points using a colormap based on their proximity to the center of the distribution, which is chosen as the median of the distribution. This approach does not only simplifies the interpretation but also provides additional insights into the distribution of data points.

<figure><img src="image_14.png" /><figcaption>Fig. 13. Visualization of errors without (left) and with (right) density on Dataset E</figcaption></figure>

In the visualization shown in the right plot of Figure 12, each point is colored based on its distance from the mean value. The colormap also reflects the percentile of points, indicating the density of data around the median. Points closer to the median are depicted in warmer colors (e.g., red, orange), whereas those further away appear in cooler colors (e.g., blue). Additionally, a white boundary marks the region where the number of points inside equals the number of points outside this crown, providing a visual cue to quickly identify the core
<figure><img src="image_15.png" /><figcaption>Fig. 14. Visualization with Euclidean (left) and Mahalanobis (right) distances for Model 1 and 11</figcaption></figure>

distribution of the data. This helps in understanding the spread and density of points relative to the median.

The visualization of density becomes particularly relevant when there are outliers in one of the distributions. Outliers are defined as points that deviate significantly from the central distribution, either due to unusually large or small values compared to the rest of the dataset. In such cases, traditional representations (as shown in previous plots) fail to effectively capture this aspect. In particular, for this visualization, outliers represent a situation in which one of the model or both have done much more or much less errors than usual. By incorporating both density and distance into the visualization, outliers can be identified more intuitively, highlighting their relative rarity and impact on the overall distribution.

Figure 13 illustrates this phenomenon on another set of points. On the left side of the figure, most of the points (those in warm colors) appear to be concentrated within a small white circle around 0, as the scale of the graph (and the errors) is extremely large, ranging between -5000 and 5000. Some outliers, represented by the dark blue points, are located between 10000 and 15000, occupying a large area but actually representing less than 10%, or even less than 5%, of the total points. When examining the density of the points on the right side of the figure, considering their proximity to the median, it becomes clear that the majority of the points are indeed very close to 0. These outliers correspond to significant errors made by both models, where the predictions appear correct at first glance. Without the use of color to encode proximity, these outliers would not have been as easily detected.

The notion of proximity can be calculated using various metrics, with the Euclidean distance being the most common. However, this metric primarily highlights differences in scale rather than capturing the true distribution of the data. Unlike Euclidean distance, the Mahalanobis distance accounts for correlations between variables and differences in their scales [15]. It measures the distance between a point and a distribution, providing insights into the structure of the data and making it particularly useful for identifying outliers. The Mahalanobis distance $D_M$ is defined as:

$$D_M(x) = \sqrt{(x - \mu)^T \Sigma^{-1} (x - \mu)} \quad (1)$$

where $x$ is the vector corresponding to the observation, $\mu$ is the mean vector of the distribution, and $\Sigma^{-1}$ is the inverse of the covariance matrix. In Figure 14, the plots illustrate a distribution concentrated near the right diagonal, indicating that the two models share common errors. However, these errors are sparse, ranging from -70 to 50. When using Euclidean distance to measure proximity to the median, most points appear evenly distributed around a circular region with a radius of approximately 10. In contrast, the Mahalanobis distance reveals a different pattern: the majority of points lie within an elongated ellipse with a narrow semi-minor axis and a wide semi-major axis. This emphasizes the disparity along the diagonal and highlights the variability and correlations in the data that Euclidean distance fails to capture. For instance, as shown in the right panel of Figure 13, the dark blue point near 0 on the ascending diagonal is atypical. While it corresponds to a good prediction by both models (low errors relative to the median), it is far from the central distribution of errors. This point's outlier status is not apparent when using Euclidean distance, as illustrated in the left panel of Figure 13, but is effectively detected with Mahalanobis distance. By accounting for both the scale and correlation of variables, Mahalanobis distance better reveals such outliers and provides a clearer understanding of the true distribution of the data. The choice of the distance metric significantly impacts the visualization and interpretation of errors. Euclidean distance is simple to calculate and suitable for uniform distributions, while Mahalanobis distance provides more information on outliers by accounting for data correlation and scales.

## IV. CONCLUSION

Effective techniques for visualizing regression models are essential for their comparative analysis. In this work, we proposed a visualization method that highlights error distributions and performance disparities through a detailed exploration of residuals. By leveraging a two-dimensional residual space and accounting for correlations between variables using a robust distance metric, our approach allows to understand the relationships within error distributions. Additionally, the use of elements, such as colormaps, enables a clear representation of percentiles, facilitating the interpretation of model perfor-
mances. These contributions enhance the understanding of the strengths and limitations of regression models, supporting more informed decision-making. However, it is important to acknowledge that performance evaluation extends beyond error distributions to include aspects like computational efficiency and interpretability. Furthermore, as these tools are inherently visual, they remain subjective and context-dependent. To complement the insights provided by these visualizations, future work will focus on integrating statistical hypothesis tests, such as the Wilcoxon signed-rank test [16], to quantify performance differences observed visually. In particular, we aim to adapt these methods to large datasets, enabling both robust statistical evaluation and effective visual representation.

## ACKNOWLEDGMENT

This research was conducted as part of a CIFRE PhD funded by Technology & Strategy. The writing process was supported by GPT-4o.

## APPENDIX
### DATASETS AND ARCHITECTURES

Table V describes the datasets used in our experiments, and Table VI describes the architectures used for training for each dataset. As Datasets C to E are synthetically generated, the process to create them and their implementation are described in the repository defined below. Equations 2a, 2b and 2c define the custom loss functions [17] used for C-MAPSS models. As a reminder, the main focus is not on the specific implementation of the models or their tuning, as they are only intended as examples to illustrate how our visualization approach works. However, the complete implementation used to train the models is available here: https://anonymous.4open.science/r/visualization-tools

<table><caption>TABLE V
DESCRIPTION OF THE DATASETS USED</caption><thead><tr><td>No</td><td>Dataset</td><td>Description</td><td>Target</td></tr></thead><tbody><tr><td>A</td><td>C-MAPSS [2]</td><td>Turbofan degradation</td><td>RUL</td></tr><tr><td>B</td><td>Synthetic 1</td><td>Moderate errors vs Extreme errors</td><td>Dummy Target</td></tr><tr><td>C</td><td>Synthetic 2</td><td>Under and over estimations</td><td>Dummy Target</td></tr><tr><td>D</td><td>Synthetic 3</td><td>Similar errors but on different individuals</td><td>Target</td></tr><tr><td>E</td><td>Apartment [3]</td><td>Evolution of the price of apartments for rent in USA</td><td>Price</td></tr></tbody></table>

$$ \text{LIN-SE}(a) = \frac{1}{N} \sum_{i=1}^{N} \begin{cases} -a(\hat{y}_i - y_i) & \text{if } r < 0, \\ (\hat{y}_i - y_i)^2 & \text{otherwise.} \end{cases} \qquad (2a) $$

$$ \text{LIN-LIN}(a) = \frac{1}{N} \sum_{i=1}^{N} \begin{cases} -a(\hat{y}_i - y_i) & \text{if } r < 0, \\ b(\hat{y}_i - y_i) & \text{otherwise.} \end{cases} \qquad (2b) $$

$$ \text{QUAD-QUAD}(a) = \frac{1}{N} \sum_{i=1}^{N} \begin{cases} 2a(\hat{y}_i - y_i)^2 & \text{if } r < 0, \\ 2(-a + 1)(\hat{y}_i - y_i) & \text{otherwise.} \end{cases} \qquad (2c) $$

<table><caption>TABLE VI
DESCRIPTION OF THE ARCHITECTURES USED</caption><thead><tr><td>No</td><td>Model</td><td>Criteria</td><td>Dataset</td></tr></thead><tbody><tr><td>1</td><td>LSTM</td><td>Squared Error</td><td>C-MAPSS</td></tr><tr><td>2</td><td>LSTM</td><td>Absolute Error</td><td>C-MAPSS</td></tr><tr><td>3</td><td>LSTM</td><td>LIN-SE 3</td><td>C-MAPSS</td></tr><tr><td>4</td><td>LSTM</td><td>LIN-SE 4</td><td>C-MAPSS</td></tr><tr><td>5</td><td>LSTM</td><td>LIN-SE 5</td><td>C-MAPSS</td></tr><tr><td>6</td><td>LSTM</td><td>LIN-SE 6</td><td>C-MAPSS</td></tr><tr><td>7</td><td>LSTM</td><td>LIN-LIN 0.01 - 1.0</td><td>C-MAPSS</td></tr><tr><td>8</td><td>LSTM</td><td>LIN-LIN 0.05 - 1.0</td><td>C-MAPSS</td></tr><tr><td>9</td><td>LSTM</td><td>LIN-LIN 0.2 - 1.0</td><td>C-MAPSS</td></tr><tr><td>10</td><td>LSTM</td><td>LIN-LIN 0.3 - 1.0</td><td>C-MAPSS</td></tr><tr><td>11</td><td>LSTM</td><td>QUAD-QUAD 0.01</td><td>C-MAPSS</td></tr><tr><td>12</td><td>LSTM</td><td>QUAD-QUAD 0.03</td><td>C-MAPSS</td></tr><tr><td>13</td><td>LSTM</td><td>QUAD-QUAD 0.05</td><td>C-MAPSS</td></tr><tr><td>14</td><td>LSTM</td><td>QUAD-QUAD 0.07</td><td>C-MAPSS</td></tr><tr><td>15</td><td>LSTM</td><td>QUAD-QUAD 0.1</td><td>C-MAPSS</td></tr><tr><td>16</td><td>Decision Tree</td><td>Squared Error</td><td>Apartment</td></tr><tr><td>17</td><td>XGBoost [18]</td><td>Squared Error</td><td>Apartment</td></tr></tbody></table>

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