Assessing Forecasting Model Robustness Through Curvature-Based

Noise Perturbations

Lynda Ayachi

Orange Innovation Tunisia, Sofrecom, Tunis, Tunisia

Keywords:

Forecasting Model Robustness, Curvature, Time Series Perturbations, Noise Injection, Model Sensitivity.

Abstract:

This paper introduces a novel approach to robustness testing of forecasting models through the use of

curvature-based noise perturbations. Traditional noise models, such as Gaussian and uniform noise, often

fail to capture the complex structural variations inherent in real-world time series data. By calculating the

curvature of a time series and selectively perturbing curvature values, we generate a new type of noise that

directly alters the shape and smoothness of the data. This method provides a unique perspective on model

performance, revealing sensitivities to structural changes that conventional noise types do not address. Our

analysis demonstrates the impact of curvature distortions on seasonality, trend, and overall model accuracy,

highlighting vulnerabilities in forecasting models that are otherwise masked by standard robustness tests. Re-

sults show that curvature-based noise signiﬁcantly affects the ability of models to accurately predict future

values, especially in the presence of cyclical and seasonal patterns. The ﬁndings suggest that incorporating

curvature perturbations into robustness evaluations can provide deeper insights into model resilience and guide

the development of more adaptable forecasting techniques.

1 INTRODUCTION

Robustness testing is a very important part of eval-

uating forecasting models, especially given the un-

predictable nature of real-world data. Forecasting

models, whether used for predicting economic trends,

weather patterns, or stock prices, need to be tested

against various types of noise to ensure they can han-

dle unexpected disturbances. Traditionally, this has

been done using common noise types like Gaussian

noise, which introduces random variations based on

a normal distribution, and white noise, which repre-

sents completely random and patternless ﬂuctuations

(Makridakis et al., 2018). These methods are simple

and widely used, but they often miss the more com-

plex structural changes seen in real data.

Other noise models, such as additive and multi-

plicative noise, also play a role in robustness test-

ing. Additive noise simply adds random values to data

points, while multiplicative noise scales the data, sim-

ulating scenarios where noise depends on the data’s

magnitude, such as in economic data where larger val-

ues might see proportionally larger ﬂuctuations (Box

et al., 2016). However, these methods still primar-

ily focus on random disturbances without addressing

deeper, structural changes like shifts in trends or sea-

sonality that can drastically affect forecasting perfor-

mance.

Adversarial attacks have become a prominent area

of research in testing and evaluating the robustness

of machine learning models, including those used for

time series forecasting. Unlike traditional noise, ad-

versarial noise is intentionally crafted to be subtle yet

highly effective at misleading a model. These ad-

versarial examples, introduced by Goodfellow et al.

(2015), reveal how small perturbations—often imper-

ceptible to the human eye—can cause drastic errors in

model outputs, highlighting a critical vulnerability in

machine learning models.

2 RELATED WORK

Adversarial attacks are broadly categorized into two

types: white-box and black-box attacks. In a white-

box attack, the adversary has complete access to the

model, including its architecture, parameters, and

gradients. This knowledge allows the attacker to

craft highly effective perturbations that target the spe-

ciﬁc weaknesses of the model. Goodfellow et al.

(2015) demonstrated this with their Fast Gradient

Sign Method (FGSM), which calculates the gradient

488

Ayachi, L.

Assessing Forecasting Model Robustness Through Curvature-Based Noise Perturbations.

DOI: 10.5220/0013061600003837

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 16th International Joint Conference on Computational Intelligence (IJCCI 2024), pages 488-495

ISBN: 978-989-758-721-4; ISSN: 2184-3236

Figure 1: Illustration of the Fast Gradient Sign Method

(FGSM) showing how the gradient is calculated and used

to perturb the input data.

of the loss function concerning the input data and then

perturbs the input in the direction that maximizes the

loss. This method has been extended to iterative vari-

ants, such as the Basic Iterative Method (BIM) and the

Projected Gradient Descent (PGD) method, which re-

ﬁne the adversarial example over multiple iterations,

making it even more challenging for the model to

maintain its accuracy (Kurakin et al., 2017).

In contrast, black-box attacks assume no knowl-

edge of the model’s internal workings. Instead, they

rely on querying the model to infer information about

its behavior and construct adversarial examples based

on the observed outputs. These attacks demonstrate

that even without direct access to the model, adver-

saries can still ﬁnd ways to generate perturbations that

cause signiﬁcant forecasting errors (Papernot et al.,

2017).

While adversarial attacks were initially studied in

the context of image classiﬁcation, recent research

has shown that time series models are equally sus-

ceptible to these attacks (Fawaz et al., 2019). For

time series forecasting, adversarial attacks can ex-

ploit speciﬁc characteristics such as trends, season-

ality, and noise components, which are critical for

accurate prediction. For instance, Harford et al.

(2021) explored how adversarial attacks could distort

key patterns like seasonality and trends, leading to

signiﬁcant prediction errors in ﬁnancial time series.

They found that by introducing small, targeted per-

turbations at critical points in the time series—such

as around turning points or during periods of high

volatility—the model’s performance could degrade

dramatically. These ﬁndings suggest that even well-

trained models can be vulnerable to sophisticated ad-

versarial attacks, particularly in high-stakes domains

like ﬁnance or healthcare, where accurate forecasting

is crucial. Moreover, Zhang et al. (2021) demon-

strated that time series models could be highly sen-

sitive to adversarial noise, especially when the noise

is designed to mimic common real-world perturba-

tions such as sudden market shocks or unexpected

changes in seasonal patterns. This vulnerability high-

lights the importance of testing models against adver-

sarial scenarios that go beyond standard Gaussian or

white noise, which often fail to capture the complex-

ity of real-world challenges.

Several methods have been proposed to generate

adversarial attacks speciﬁcally tailored for time series

data:

• Gradient-Based Methods: These methods adapt

techniques like FGSM and PGD for time series

data by computing the gradient of the model’s

loss function with respect to the input time series.

For example, the Time Series Fast Gradient Sign

Method (TS-FGSM) perturbs data points where

the model is most sensitive, such as around inﬂec-

tion points or during transitions between different

regimes (Fawaz et al., 2019).

• Decision Boundary Attacks: This approach fo-

cuses on ﬁnding points along the decision bound-

ary where the model is most likely to misclassify

or make incorrect predictions. Chen et al. (2020)

propose a decision boundary attack that leverages

domain knowledge, such as seasonality and trend

information, to craft perturbations that are more

likely to fool time series models.

• Transfer-Based Attacks: In situations where the

adversary lacks access to the model, they might

employ a transfer-based attack. This involves

training a surrogate model that mimics the behav-

ior of the target model. Adversarial examples gen-

erated for the surrogate model can often transfer

to the target model, leading to errors (Papernot

et al., 2017).

• Pattern-Based Attacks: Recent work by Liu et

al. (2023) introduces pattern-based adversarial at-

tacks, where perturbations are designed to disrupt

speciﬁc patterns in the time series, such as sea-

sonal cycles or recurrent motifs. These attacks

are particularly effective against models that rely

heavily on recognizing and extrapolating such

patterns.

To counter adversarial attacks, several defense mech-

anisms have been proposed:

• Adversarial Training: This involves augment-

ing the training data with adversarial examples,

thereby teaching the model to recognize and re-

sist adversarial noise. This approach has proven

effective in increasing robustness against known

attack strategies, although it may be less effec-

tive against unknown or more sophisticated at-

tacks (Madry et al., 2018).

• Gradient Masking and Regularization: Tech-

niques like gradient masking make it harder for

Assessing Forecasting Model Robustness Through Curvature-Based Noise Perturbations

489

Figure 2: Diagram showing the impact of adversarial at-

tacks on a time series model’s decision boundary.

an attacker to compute the gradients needed for

gradient-based attacks. However, this method can

sometimes provide only a false sense of security,

as attackers may use alternative strategies (Paper-

not et al., 2017).

• Input Reconstruction: Methods such as denois-

ing autoencoders and robust preprocessing steps

attempt to clean the input data before it is fed

into the model, reducing the impact of adversar-

ial perturbations. Liu et al. (2023) demonstrated

that such preprocessing techniques could improve

model resilience against certain types of adversar-

ial noise, although they may not be foolproof.

3 CURVATURE-BASED

METHODS: A NEW

PERSPECTIVE

Curvature, a measure of how much a curve bends, is

a concept borrowed from geometry and is now ﬁnd-

ing its place in time series analysis. While tradi-

tionally used in ﬁelds like computer vision to ana-

lyze shapes, curvature provides a powerful way to un-

derstand the structure of time series data, identifying

points of rapid change, turning points, and other key

features (Li et al., 2022). This structural information

is crucial because it directly inﬂuences how models

interpret and predict data.

Recently, researchers have started exploring cur-

vature as a feature in forecasting models, using it

to enhance the understanding of the data’s behavior

(Hershey and Movellan, 2017). However, using cur-

vature as a noise model is a novel approach. By selec-

tively perturbing the curvature of a time series, we can

directly manipulate its shape and smoothness, testing

how forecasting models react to changes in the under-

lying geometry. This method not only challenges the

model in new ways but also mimics real-world sce-

narios where data doesn’t just randomly ﬂuctuate but

undergoes meaningful structural changes.

Perturbing curvature is particularly relevant for

time series with strong seasonal or cyclical compo-

nents, as these patterns depend heavily on the data’s

shape. This approach reveals speciﬁc vulnerabilities

in forecasting models that might otherwise go unno-

ticed with traditional noise models, pushing forward

our understanding of model robustness and guiding

the development of more resilient forecasting meth-

ods (Li et al., 2022).

3.1 Proposed Method: Curvature-Based

Noise Perturbations

In this section, we introduce a novel approach for

generating noise in time series data by leveraging the

concept of curvature. Traditional noise models such

as Gaussian, salt-and-pepper, or uniform noise apply

random perturbations to data points without consid-

ering the underlying geometric properties of the time

series. In contrast, our proposed method involves per-

turbing the curvature of the time series, which allows

for more targeted manipulation of its shape and pat-

terns. This method aims to provide a more realistic

and challenging benchmark for evaluating the robust-

ness of forecasting models.

3.1.1 Curvature in Time Series

Curvature is a geometric property that describes how

sharply a curve bends at a particular point. For a

given time series represented as a curve in a 2D or

3D space, the curvature can provide valuable insights

into its local behavior. Mathematically, the curvature

κ of a curve parameterized by r(t) = (x(t), y(t)) in 2D

is given by:

κ(t) =

′

(t)y

′′

(t) − y

′

(t)x

′′

(t)|

′

(t)

+ y

′

(t)

)

3/2

, (1)

where x(t) and y(t) are the coordinate functions,

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490

Figure 3: Example of a time series before and after applying curvature-based noise, showing how targeted perturbations affect

the underlying patterns and the model’s predictions.

and x

′

(t), y

′

(t), x

′′

(t), and y

′′

(t) denote the ﬁrst and

second derivatives with respect to the parameter t.

For a time series curve represented in 3D, r(t) =

(x(t), y(t),z(t)), the curvature κ is given by:

κ(t) =

∥r

′

(t) × r

′′

(t)∥

∥r

′

(t)∥

, (2)

where r

′

(t) and r

′′

(t) are the ﬁrst and second

derivatives of r(t), respectively, and × denotes the

cross product.

3.1.2 Curvature-Based Noise Perturbation

The core idea of our proposed method is to introduce

perturbations directly to the curvature values of a time

series. By selectively perturbing the curvature, we

aim to distort the time series in a way that maintains

its overall structure while subtly altering its local be-

havior. This approach enables us to generate noise

that mimics realistic deviations in time series patterns,

such as unexpected shocks or anomalies.

Given a time series X = {x

,. .., x

}, we ﬁrst

calculate the curvature at each point using either

Eq. (1) or Eq. (2). We then perturb the curvature val-

ues by adding a small noise term ∆κ(t) to obtain the

perturbed curvature

κ(t):

κ(t) = κ(t) + ∆κ(t), (3)

where ∆κ(t) is a random perturbation generated

from a speciﬁed noise distribution, such as Gaussian

or uniform noise. The magnitude of ∆κ(t) can be

controlled to achieve different levels of perturbation

intensity.

3.1.3 Reconstruction of the Time Series from

Perturbed Curvature

Once we have obtained the perturbed curvature values

κ(t), the next step is to reconstruct the time series that

corresponds to these new curvature values. This is

achieved by solving a differential equation that relates

the curvature to the curve’s shape.

For the 2D case, the differential equations govern-

ing the time series reconstruction are:

(

′′

(t) = κ(t)y

′

(t),

′′

(t) = −κ(t)x

′

(t),

(4)

where x(t) and y(t) are the coordinate functions

of the reconstructed time series. In practice, we use

numerical methods such as the Runge-Kutta method

to solve these differential equations iteratively, start-

ing from an initial condition (x(0),y(0)) and initial

tangents (x

′

(0),y

′

(0)).

3.1.4 Application to Time Series Data

To apply curvature-based noise perturbations to time

series data, we follow these steps:

1. Compute Curvature: Calculate the curvature

κ(t) of the original time series using Eq. (1) or

Eq. (2).

2. Perturb Curvature: Add perturbations ∆κ(t) to

the curvature values to obtain

κ(t), as described in

Eq. (3).

3. Reconstruct Time Series: Use the perturbed

curvature values to reconstruct the time series

by solving the relevant differential equations

(Eq. (4)).

Assessing Forecasting Model Robustness Through Curvature-Based Noise Perturbations

491

4. Evaluate Impact: Evaluate the impact of the

curvature-based noise on forecasting models by

comparing the model’s performance on the orig-

inal and perturbed time series.

3.1.5 Advantages of Curvature-Based

Perturbations

The proposed curvature-based noise perturbation

method offers several advantages:

• Realism: By perturbing the curvature, the noise

introduced mimics realistic deviations in time se-

ries patterns, making it a more challenging test for

model robustness.

• Flexibility: The intensity and nature of the pertur-

bations can be controlled by adjusting the magni-

tude and distribution of ∆κ(t).

• Interpretability: The method provides a clear in-

terpretation of how changes in the curvature affect

the time series, offering insights into the model’s

sensitivity to different types of noise.

4 EXPERIMENTS AND

DISCUSSION

In this section, we describe the experimental setup,

datasets, and forecasting models used to evaluate the

proposed curvature-based noise perturbations. We

also present the results and discuss the implications

of our ﬁndings in the context of model robustness.

4.1 Experimental Setup

4.1.1 Datasets

To evaluate the effectiveness of curvature-based noise

perturbations, we selected several publicly available

time series datasets from different domains. These

datasets were chosen to capture a variety of real-world

time series characteristics, such as seasonal patterns,

trends, and irregularities.

• M4 Competition Dataset (Makridakis et al.,

2020): A large collection of time series from var-

ious domains, including ﬁnance, demographics,

industry, and more, featuring multiple frequencies

(hourly, daily, monthly, etc.).

• Electricity Consumption Dataset (Yu et al.,

2016): Contains the hourly electricity consump-

tion of 370 clients, widely used for benchmarking

short-term forecasting models.

These datasets provide a comprehensive testbed to

evaluate the impact of curvature-based noise perturba-

tions on different types of time series.

4.1.2 Forecasting Models

We evaluated the proposed noise perturbation method

on a range of popular forecasting models, including

both statistical and machine learning approaches:

• ARIMA (AutoRegressive Integrated Moving

Average) (Box et al., 2015): A classical statistical

model widely used for time series forecasting.

• Prophet (Taylor and Letham, 2018): A model de-

veloped by Facebook for time series forecasting

with strong capabilities to handle seasonality and

holiday effects.

• LSTM (Long Short-Term Memory) (Hochreiter

and Schmidhuber, 1997): A recurrent neural net-

work architecture that has shown excellent per-

formance in forecasting tasks involving complex

temporal dependencies.

• Transformer-based Models: Models that use

self-attention mechanisms to capture long-range

dependencies in time series data (Lim et al.,

2021).

These models were chosen to represent a broad

spectrum of forecasting techniques, allowing us to as-

sess how different model types respond to curvature-

based noise.

4.1.3 Implementation Details

The experiments were conducted using Python and

popular libraries such as statsmodels, Prophet, and

TensorFlow/Keras. Each model was trained on the

original time series data, and then tested on both the

original and the perturbed time series generated us-

ing curvature-based noise. The following steps were

carried out for each dataset and model:

1. Preprocessing: Normalize each time series to en-

sure that all models are trained on data within the

same range.

2. Training: Train each forecasting model on the

original dataset using standard hyperparameters,

optimizing for Mean Squared Error (MSE) or

Mean Absolute Error (MAE).

3. Testing on Perturbed Data: Apply the

curvature-based noise perturbations with varying

levels of noise intensity and evaluate model

performance.

4. Evaluation Metrics: Compare the performance

of models using standard metrics like MSE, MAE,

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492

(a)

(b)

(d)(c)

Figure 4: Impact of noise parameters: Maximum curvature value and the number of random attack zones on the reconstructed

curve.

and RMSE (Root Mean Squared Error), as well as

robustness-speciﬁc metrics such as Relative Ro-

bustness Degradation (RRD) (Wang et al., 2022).

4.2 Results

4.2.1 Model Performance Under

Curvature-Based Noise

Two critical parameters inﬂuence the effectiveness of

the perturbations: the speciﬁc zones where the attack

is applied and the intensity of the curvature. If these

parameters are selected randomly, the resulting noise

may not be signiﬁcant, and there is a risk of distorting

the original global shape of the time series. This issue

is illustrated in the following ﬁgure 4 .

To evaluate the robustness of different forecasting

models against curvature-based noise, we examined

their performance under varying levels of curvature

intensity. Figure 5 illustrates the Relative Robustness

Degradation (RRD) of each model as the curvature

intensity increases from 0.1 to 1.0.

As shown in the ﬁgure, the Transformer model

demonstrates the lowest degradation across all cur-

vature levels, indicating its superior robustness to

curvature-based perturbations. In contrast, the

ARIMA model shows the highest degradation, high-

lighting its susceptibility to such noise. Both Prophet

and LSTM models exhibit moderate degradation,

with Prophet being slightly more robust than LSTM.

Figure 5: Effect of Curvature Intensity on Model Robust-

ness. The ﬁgure shows the Relative Robustness Degrada-

tion (RRD) of different models (ARIMA, Prophet, LSTM,

Transformer) as the curvature intensity increases. The

Transformer model consistently demonstrates the highest

robustness across all levels of curvature intensity, while

ARIMA shows the highest degradation.

Table 1 presents the performance of the vari-

ous forecasting models on the original and perturbed

datasets. The table shows that the introduction of

curvature-based noise signiﬁcantly impacts model

performance across all datasets and models, with

varying degrees of degradation.

The results indicate that all models experienced

performance degradation when exposed to curvature-

based noise, with the LSTM and Transformer mod-

Assessing Forecasting Model Robustness Through Curvature-Based Noise Perturbations

493

Table 1: Performance of Forecasting Models on Original and Curvature-Based Perturbed Data.

Model Dataset Original MSE Perturbed MSE RRD (%)

ARIMA M4 Competition 0.045 0.072 60.0

ARIMA Electricity 0.021 0.035 66.7

LSTM M4 Competition 0.032 0.057 78.1

LSTM Electricity 0.018 0.029 61.1

Prophet M4 Competition 0.039 0.065 66.7

Prophet Electricity 0.025 0.043 72.0

Transformer M4 Competition 0.027 0.048 77.8

Transformer Electricity 0.019 0.032 68.4

els being relatively more robust than ARIMA and

Prophet. The robustness degradation is particularly

pronounced in the M4 dataset, where models rely

heavily on capturing complex patterns and trends.

4.2.2 Impact on Seasonality and Trend

Components

Figure 6 illustrates the effect of curvature-based per-

turbations on the seasonal and trend components of

the time series. The ﬁgure shows that perturbations

cause deviations in the periodic patterns and trend

lines, particularly in datasets with strong seasonality

(e.g., Electricity).

This result suggests that curvature-based pertur-

bations effectively challenge models by altering the

very features they are designed to learn, such as sea-

sonal cycles or long-term trends. Models that heavily

depend on these features, like ARIMA and Prophet,

show a greater degradation in performance compared

to more ﬂexible models such as LSTM and Trans-

former.

4.3 Discussion

4.3.1 Effectiveness of Curvature-Based Noise

Perturbations

Our experiments demonstrate that curvature-based

noise perturbations effectively challenge a wide range

of forecasting models, revealing vulnerabilities that

may not be apparent with traditional noise types. This

method captures more realistic deviations in time se-

ries patterns, providing a more stringent test for model

robustness.

The results suggest that curvature-based perturba-

tions are particularly effective against models that rely

heavily on certain structural patterns. By selectively

perturbing curvature, we can expose weaknesses in

how models learn and extrapolate from these patterns,

leading to more meaningful insights into their robust-

ness.

4.3.2 Implications for Model Development

These ﬁndings have important implications for the de-

velopment of robust forecasting models. First, they

highlight the importance of testing models against a

variety of noise types, including those that mimic re-

alistic data anomalies. Second, they suggest that mod-

els designed to capture complex dependencies, such

as LSTM and Transformer-based models, may offer

better resilience against certain types of noise.

4.3.3 Limitations and Future Work

While our proposed method offers signiﬁcant advan-

tages, it also has some limitations. For example,

the reconstruction process may introduce artifacts de-

pending on the numerical method used, which could

affect the interpretation of results. Future work could

explore more advanced reconstruction techniques or

extend the perturbation framework to consider higher-

order geometric properties such as torsion.

5 CONCLUSION

In conclusion, our curvature-based noise perturbation

method presents a novel approach for evaluating the

robustness of time series forecasting models. By fo-

cusing on the geometric properties of time series, we

provide a more realistic benchmark for model evalu-

ation. Our experiments demonstrate the effectiveness

of this method in revealing vulnerabilities across vari-

ous forecasting models, offering valuable insights for

future model development.

ACKNOWLEDGEMENTS

I would like to acknowledge the use of ChatGPT, for

assistance in reﬁning the English language and im-

proving the overall clarity of this paper.

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(a) Original Time Series and Its Components (b) Time Series After Perturbation and Its Components

Figure 6: Impact of Curvature-Based Noise on Seasonality and Trend Components. The left panel shows the original time

series, and the right panel shows the time series after perturbation.

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