Analyzing Monofractal Short and Very Short Time Series: A Comparison of Detrended Fluctuation Analysis and Convolutional Neural Networks as Classifiers

Abstract

Time series data are a crucial information source for various natural and societal processes. Short time series can exhibit long-range correlations that reveal significant features not easily discernible in longer ones. Such short time series find utility in AI applications for training models to recognize patterns, make predictions, and perform classification tasks. However, traditional methods like DFA fail as classifiers for monofractal short time series, especially when the series are very short. In this study, we evaluate the performance of the traditional DFA method against the CNN-SVM approach of neural networks as classifiers for different monofractal models. We examine their performance as a function of the decreasing length of synthetic samples. The results demonstrate that CNN-SVM achieves superior classification rates compared to DFA. The overall accuracy rate of CNN-SVM ranges between $64\%$ and $98\%$, whereas DFA’s accuracy rate ranges between $16\%$ and $64\%$.

1. Introduction

Nature and society encompass numerous processes that exhibit fractal or multifractal behavior [1,2,3]. These processes yield valuable information through time series obtained from measurements or observations. However, these time series may be affected by non-stationary uncertainties arising from experiments or observations. Separating these uncertainties from the intrinsic fluctuations and correlations of the studied system is a highly intricate task, and several methods have been proposed to achieve this objective [4]. One such method, detrended fluctuation analysis (DFA), has proven effective in detecting long-range correlations in data with trends. It was introduced initially by [5]. Subsequently, in [6] the authors generalized this method to analyze multifractal time series (MFDFA), which has further been extended to multidimensional series [7], and to investigate power-law correlations between simultaneously recorded time series [8,9,10]. MFDFA has been favorably compared to other methods [11,12] and has found applications in various fields. For example, it has been utilized in geophysics [13], physiology [14,15,16], financial markets [17,18], and notably, the study of currency exchange rates [19,20]. In these cases, due to the non-linear characteristics, different techniques have been used to improve the characterization and interpretation of fluctuation and correlation for the interconnection between different physiological systems, including wavelet analysis (WT) [21], fractal dimension analysis [22,23], spectral analysis [24,25,26], detrended fluctuation analysis (DFA) [5,27], and multifractal detrended fluctuation analysis (MFDFA) [6]. Related to spectral analysis, if the time series is stationary, we can apply standard spectral analysis techniques and calculate the power spectrum $C\left(f\right)$ of the time series as a function of the frequency f to determine its self-affine scaling behavior [28]. DFA and MFDFA perform well for time series with a length of at least $2^{16}$ elements, but it is crucial to assess the methods performance on shorter time series [29]. Two main reasons drive the need for this evaluation: first, many relevant records are short in length, and second, there are processes where long records exist that exhibit changing behavior over time, prompting the study of short segments to gain significant insights.

Short time series are important in forecasting, anomaly detection, and trend analysis [30,31,32]. In [32], the authors describe a study of historical rainfall data from the Sindh River Basin, India, which were analyzed for monthly, seasonal, and annual trends. Likewise, Ref. [30] presents a short-term series analysis for the forecast of electricity demand in Singapore, where a multiplicative decomposition model and a seasonal ARIMA model are proposed to accurately predict short-term demand.

Short time series can reveal patterns and trends that can inform decision-making, for example, to identify short-term trends in stock prices or other market data [33]. In healthcare, short time series can be used to monitor patient health metrics, which can help doctors identify and respond to potential health issues before they become serious [34,35].

In machine learning applications, short time series can be useful to train models to recognize patterns and make predictions [36,37], which can be especially important in applications where data are scarce or expensive to collect. In [37], the authors implemented and compared several forecasting techniques and showed how these methods improve the root mean square error score, while [36] focused on the application of machine learning techniques, particularly recurrent neural networks (RNN), to predict pore water pressure time series data.

In fractal analysis, short time series can reveal important features of the fractal structure that might not be apparent in longer time series or estimate a measure of the degree of self-similarity or complexity of the time series [33,38,39,40]. Related to the length of short time series, making the distinction between short-term and very short-term depends on the context and the specific characteristics of the data. For example, in time-series analysis, very short-term variations typically refer to changes that occur over relatively brief time intervals, such as seconds or hours (market volatility, seismic activity, or heart rate variability). Now, by understanding the monofractal nature of a very short time series, better forecasting models can be developed. These models can be used for very-short-term predictions where rapid and accurate forecasting is essential [41,42]. Despite these advantages, analyzing monofractal very short time series also comes with challenges. The very short length of the series may limit the robustness of the fractal analysis, and care must be taken to ensure that the identified patterns are statistically significant.

As stated before, short time series can also be useful in machine learning applications to train models to recognize patterns, make predictions and perform classification tasks [43], while ref. [44] presents an overview of state-of-the-art AI-based electrocardiogram signal processing methods. Focusing on classification tasks, these are essential for extracting meaningful insights, making informed decisions, and automating tasks. Currently, with scientific and technological advancements, various methods and tools have emerged aimed at performing classification tasks. Classification processes using AI offer several advantages over traditional techniques, including higher accuracy, adaptability and learning, scalability, handling complex and non-linear relationships, and efficient automation, to name a few [45,46,47,48]. The choice of approach depends on the specific problem, the available data, the interpretability requirements, and other factors.

There are several AI methods used for the classification of time series that have shown promising results in natural language processing tasks and that are also applicable to time series analysis. In particular, deep learning has been used extensively in time series analysis, particularly for forecasting and anomaly detection [49,50,51,52]. Some deep learning models used for time series analysis include recurrent neural networks (RNNs) [53], long short-term memory (LSTM) networks [54], and convolutional neural networks (CNNs) [55,56,57,58,59,60]. Ref. [56] provides a comprehensive overview of deep learning techniques used for time series classification tasks. In addition, ref. [60] introduces a sequence-to-sequence (seq2seq) model, which utilizes RNNs for tasks like machine translation, discusses the encoder-decoder architecture, and demonstrates the effectiveness of RNNs in capturing sequential dependencies.

RNNs and LSTMs are particularly useful for modeling time series data because they are designed to handle sequences of varying lengths, while CNNs have also been used for time series analysis, particularly to analyze the structure and patterns within the data, and identify features that are relevant for prediction and classification tasks [61,62,63]. In [61], the authors introduce the Long Short-Term Memory (LSTM) architecture, a type of RNN that overcomes the vanishing gradient problem. It discusses the memory cell and the forget gate mechanism, which enable LSTMs to remember and forget information over long sequences, while in [63], the application of hidden Markov models to improve classification performance capturing the underlying structure and dependencies in time series was discussed.

To summarize the above, one source of information for many processes of interest in nature and society is time series. Short time series can reveal important features that might not be apparent in longer time series. They can be useful in AI applications for training models in classification tasks, forecasting, and anomaly detection. Additionally, analyzing monofractal short time series can make it easier to identify patterns and structures, be more efficient and effective in modeling and understanding the dynamics of the system of interest, and assist in the early detection of abnormalities or diseases.

Based on the highlighted antecedents, the contributions of this work are the following:

•

The significance of classifying monofractal time series lies in the insights it provides for classification, forecasting, and anomaly detection. While most studies focus on analyzing long-term series [29], short- and very-short-term series receive less attention. However, this research aims to expand the investigation into the performance and application of DFA and neural networks as classification methods for synthetic monofractal short-term series.

•

We compare two distinct approaches for classifying monofractal short and very short time series of different lengths: DFA and CNN-SVM.

•

Our findings show that CNN-SVM achieves higher classification rates than DFA, and both methods performance declines as the length of short and very short time series decreases.

The aim of this research, as a first approximation, focuses on monofractal models and their comparison with the expected outcomes based on analytical predictions when DFA and convolutional neural networks are applied as classifiers in synthetic short time series. The understanding of monofractal time series is of great importance as it can significantly enhance our ability to create better models, make more accurate predictions, and facilitate improved decision-making in a wide range of domains. This can be critical for understanding and predicting system behavior, especially in short time frames. In some cases, monofractal models may not adequately describe the system and multifractal models could be necessary. With this study, we aim to open new opportunities for the use of neural networks to identify monofractality and its changes in short and very short time series, where DFA and MFDFA fail, which could be of great help for the development of fast and accurate forecasting models.

2. Detrended Fluctuation Analysis (DFA)

Detrended fluctuation analysis (DFA) has emerged as a crucial method for investigating scaling characteristics and identifying long-range correlations within non-stationary and noisy time series [5,64]. The DFA technique, detailed in [5], can be outlined in five steps as follows:

• Take a finite time series $w\left(i\right)$ of length M, with a minor portion of $w\left(i\right)$ elements being zero and compute a new time series $W\left(j\right)$, where $j=1,\dots ,M$:

  $$W\left(j\right)=\sum _{i=0}^j\left[w\left(i\right)-<w>\right].$$

  (1)
• Divide the new series $W\left(j\right)$ into $M_s$ segments of size s from the beginning and repeat the process starting from the end to obtain $2M_s$ segments.
• Calculate the local trend of order m, denoted as $P_{\nu }^m$, for all segments $\nu$ and all sizes s, and compute the variance as:

  $$F^2(\nu ,s)={\displaystyle \frac{1}{s}}\sum _{i=1}^s{\left\{W\left[(\nu -1)s+i\right]-P_{\nu }^m\left(i\right)\right\}}^2.$$

  (2)
• Compute the second-order fluctuations as an average over all segments of a given size s:

  $$F_2\left(s\right)={\left\{{\displaystyle \frac{1}{2M_s}}\sum _{\nu =1}^{2M_s}\left[F^2(\nu ,s)\right]\right\}}^{1/2},$$

  (3)
• For a range of sizes, $s_{\min }<s<s_{\max }$, observe the relationship:

  $$F_2\left(s\right)\sim s^H.$$

  (4)

  where H is the output of the DFA algorithm, known as the Hurst exponent.

3. Machine Learning Classification Approaches

3.1. Support Vector Machines

Support vector machines (SVM) are supervised learning algorithms originally designed for binary classification tasks [65]. However, thanks to their flexibility and capability, they have been successfully extended to multiclass schemes [66]. The most common approach applied to this type of problem is to split the task into multiple binary classifiers [67]. Then, each result is combined to obtain the final classification. Numerically, given N training samples $({\mathit{x}}_1,y_1),\dots ,({\mathit{x}}_N,y_N)$, defined for k classes, with ${\mathit{x}}_i\in {\mathbb{R}}^m$ and $y_i\in \{1,\dots ,k\}$, with $y_i$ being the class label of ${\mathit{x}}_{\mathrm{i}}$, the quadratic programming problem of the r-th SVM is as follows:

$$\begin{array}{cc}\hfill \mathrm{minimize}:& {\displaystyle J({\mathit{w}}^r,{\zeta }^{\mathit{r}})=\frac{1}{2}{\left({\mathit{w}}^r\right)}^T{\mathit{w}}^r+C\sum _{j=1}^N{\zeta }_j^r}\hfill \\ \hfill \mathrm{subject} \mathrm{to}:& y_i[{\left({\mathit{w}}^{\mathit{r}}\right)}^T\varphi \left({\mathit{x}}_{\mathit{i}}\right)+d^r]\ge 1-{\zeta }_i,r=1,\dots ,k\hfill \\ \hfill & {\zeta }_i\ge 0,i=1,\dots ,N\hfill \end{array}$$

(5)

where ${\mathit{w}}^r\in {\mathbb{R}}^N$ is the weight vector, C is the penalty parameter, ${\zeta }_j^r$ is an error term between the misclassified point and the separation hyperplane, proportional to the C value, $\varphi (·)$ is an operator that projects the input data to a higher dimensional space, and d is a scalar. The index r of the SVM schema indicates that it will be trained with input attributes corresponding to class r, assigning positive labels, while the remaining examples will receive negative labels.

This multiclass technique, known as one-versus-all, transforms the original problem into k binary classifiers, as detailed in Equation (5). After constructing the k dual classifiers, each associated with the decision function $f_r\left(x\right)={\left({\mathit{w}}^{\mathit{r}}\right)}^T\varphi \left(\mathit{x}\right)+d^r$, the maximal value of the k grouped classifiers predicts the class y of a sample $\mathit{x}$, that is,

$$y=\mathrm{a}\mathrm{r}\mathrm{g}\underset{r}{max}{f}_r\left(\mathit{x}\right).$$

(6)

In addition to this multiple classification strategy, there are other popular variants, such as the one-versus-one approach and hierarchical support vector machines [68]. The SVM classifier, originally designed to process linearly separable data, can be inefficient for non-linear tasks. To solve this problem, the literature introduced classifier systems based on the kernel function. In practice, the most common kernel types are the linear, Gaussian, polynomial, and radial basis functions [68].

3.2. Convolutional Neural Network

The convolutional neural network, a variant of deep learning, can extract relevant features of data without requiring human activity. Initially designed for image processing, it was subsequently extended to various disciplines due to its ability to capture high-level information. Although there are multiple variants, they all share three similar basic components, known as the convolutional, pooling, and fully connected layers. These elements, detailed in [69], can be summarized as follows:

•

Convolutional layer. A convolutional layer is the main building block of a CNN. It includes a set of filters or kernels, parameters that learn local informative features of the image, controlled by the padding and the stride. The padding controls the edges of the image, while the stride determines the number of pixels considered in the filter at each step during the convolution. The kernel operator, a matrix of weights, is applied to local regions of the input to generate output features, multiplying and summing values at each movement of the convolutional block (sliding window). Mathematically, the latent output representation of the s-th feature map of the current layer, denoted as ${\mathit{F}}^s$, is as follows:

$${\mathit{F}}^s=g\left(\sum _{l\in L}{\mathit{X}}^l\otimes {\mathit{W}}^s+{\mathit{b}}^s\right),$$

(7)

where ${\mathit{W}}^s$ and ${\mathit{b}}^s$ are the filters and the bias of the s-th feature map of the current layer, respectively, ${\mathit{X}}^l$ is the l-th feature map of the total feature maps $\left(L\right)$ of the previous layer, ⊗ denotes the convolution operation, and $g(·)$ is an activation function, normally non-linear.

The common activation functions include sigmoid, tanh, and ReLU. The first convolutional layer captures simple features, such as lines, while subsequent layers extract more complex features, such as shapes and specific objects [70,71].

•

Pooling layer. This layer reduces the resolution of the feature map in each channel through a pooling operator, maintaining the most relevant spatial features of the current convolutional layer. The common types of pooling layers include max pooling, average pooling, and global pooling. The mathematical expression corresponding to this layer is as follows:

$${\mathit{P}}^s=h\left({\mathit{F}}^s\right),$$

(8)

where ${\mathit{P}}^s$ is the output of the current pooling layer associated with the s-th feature map ( ${\mathit{F}}^s$) of the convolutional layer, and $h(·)$ is the spatial reduction function.

•

Fully connected layers. These consist of a block of fully connected layers, trained with previously extracted features. At this stage, an optimization problem, which integrates a cost function, is the main support for the optimal configuration of the model parameters. The cost function quantifies the error between the model prediction ${\widehat{y}}_i$ and the true label $y_i$. To illustrate, consider the cross-entropy cost function. The associated minimization problem is defined as follows:

$$\begin{array}{cc}\hfill minimize:& {\displaystyle \widehat{J}\left(\theta \right)=-\frac{1}{N}\sum _{k=1}^N[y_{k}log\left({\widehat{y}}_k\right)+(1-y_k)log(1-{\widehat{y}}_k)],}\hfill \end{array}$$

(9)

where $\widehat{J}(·)$ is the objective function, N is the number of samples, and $\theta$ is the set of weights and biases. Other cost functions incorporated into its architecture are the mean square error, hinge, and Huber. In addition, to obtain a more stable and robust performance, different regularization strategies are incorporated in Equation (9). These techniques include the $L_p$-norm with $p\ge 0$, Dropout, and DropConnect. Mainly, they help to mitigate the problem of overfitting and deal with outlier data. In this classification or prediction stage, the multilayer perceptron network is the classical algorithm used. However, other machine learning models can be incorporated. Among the most popular are logistic regression, linear discriminant analysis, decision tree classifiers, SVM, and non-iterative methods.

4. Materials and Methods

4.1. Monofractal Time Series

A monofractal time series exhibits self-similarity at all time scales, where the scaling properties of the time series are described by a single fractal dimension which captures how the patterns and structures within the time series change as you change the scale [4,29,72]. This means that when you zoom in or out on different parts of the time series, you see similar patterns and structures as you would observe in the entire series. Monofractal time series are commonly found in many natural systems. They are often used in various fields of research, including economics, finance, and physics, to model, analyze, and compare the fractal properties of more complex time series [73,74,75,76]. In addition, monofractal time series appear in various systems like weather patterns, river flows, and physiological processes within the human body [77,78,79,80]. However, not all time series are monofractal; some time series may exhibit multi-fractal behavior, which means that their scaling properties are described by multiple fractal dimensions [6,29].

Monofractal Synthetic Data

Our primary goal is to understand the minimum series length that each model can reliably analyze and determine the precision achievable for these lengths. We concentrate exclusively on monofractal models as an initial step and compare the findings with their corresponding theoretical forecasts. Monofractal models are particularly intriguing for evaluating the methods performance due to their straightforward functional form for the Hurst exponent, which remains constant. We studied distinct monofractal models in a $0.1<H<0.9$ range with step 0.1. The models studied correspond to white noise with a Hurst exponent (H) of $0.5$, signifying the absence of long-range correlations, and models with H values higher than $0.5$ and less than $0.5$, indicating the presence of long-range correlations and anti-correlations in the data, respectively.

Artificial signals were created to evaluate the efficacy of various “neural network techniques” in categorizing monofractal time series, considering the signal’s length. For each of the synthetic models under investigation, we use series of various lengths, namely, $2^k$, where k takes on the values of 10, 9, 8, and 7. To ensure a robust analysis, we produced multiple independent realizations for each case according to [81,82]. Specifically, we used 10 realizations for $k=20$ from which short time series were obtained.

It is vital to emphasize that these investigations yield estimations of the potential shortest lengths and analysis precision. Nevertheless, they do not furnish definitive forecasts because real-world time series data are significantly more intricate and complex compared to the artificially generated signals employed in our study.

4.2. Environment

All the experiments in this study were performed in a controlled environment, using a laptop with the following characteristics: Server Supermicro 540A-TR 4U, equipped with the Windows 11 Operating System, 1CPU Xeon 6338 N 2 P 32 C/64 T, 2.2 G 24 M 11.2 Gt, RAM 8 × 32 GB DDR4-2Rx8 ECC REG DIMM, 1 Micron 7450 PRO 960 GB NVMe PCIe 4.0 M.2 × HDD 3.5” 4 TB, SATA 6 Cg/s 7.2 KRPM, 256 MB, 1 × GPU NVIDIA PNYQuadro TRXA4000 16 GB GDDR6 PCIe 4.0, KIT RAIL. Both the model configuration and the results analysis were executed using the official Matlab documentation.

4.3. Performance Metrics

We applied four different evaluation metrics to discuss the performance of the proposed multiple classification system on monofractal synthetic data. These metrics include accuracy (Acc), sensitivity (Sen), specificity (Spe), and positive predictive value (PPV). The following formulas summarize these machine learning statistics, defined in terms of true positive (TP), false positive (FP), true negative (TN), and false negative rates (FN):

$$\begin{array}{cc}\hfill & \mathrm{Acc}={\displaystyle \frac{TP+TN}{TP+TN+FP+FN}}\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & \mathrm{Sen}={\displaystyle \frac{TP}{TP+FN}}\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill & \mathrm{Spe}={\displaystyle \frac{TN}{FP+TN}}\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill & \mathrm{PPV}={\displaystyle \frac{TP}{TP+FP}}\hfill \end{array}$$

(13)

The Acc index is the relation between correct and incorrect predictions. It is an easy measure to interpret in classification tasks. The PPV provides the ratio of correct predictions to the total number of predictions. The higher the PPV, the better the model distinguishes between positive and negative classes. The Sen highlights its ability to provide the radius of correct predictions with respect to the actual number of cases. The Spe indicates its ability to correctly identify negative instances.

The following pseudo-code summarizes the proposed monofractal synthetic signal classification approach (Algorithm 1).

Algorithm 1 Monofractal synthetic signal classification process

1: Datasets defined according to data length: 128, 256, 512 and 1024 2: Convolutional layers of interest: 3, 4, 5, 6 and 7 3: Classifiers used: MLP and SVM. 4: Evaluation metrics: Acc, Sen, Spe, and PPV 5: for  $L=\{128,256,512,1024\}$  do 6:     Generation of monofractal synthetic signals of L length 7:     Split the dataset into training and testing 8:     for  $CL=\{3,4,5,6,7\}$ do 9:         CNN model trained with $CL$ convolutional layers 10:         Obtain the feature map ${\mathit{X}}_c$ of the convolutional layer $CL$ 11:         for  $M=\{\mathrm{MLP},\mathrm{SVM}\}$ do 12:            Train M with the corresponding ${\mathit{X}}_c$ characteristics map. 13:            Predict test labels 14:            for  $EM=\{\mathrm{Acc},\mathrm{Sen},\mathrm{Spe},\mathrm{PPV}\}$ do 15:                Calculate the evaluation metric $EM$ 16:                Report the evaluation metric $EM$ 17:            end for 18:         end for 19:     end for 20: end for

5. Results and Discussion

5.1. Performance of DFA on Monofractal Synthetic Data

Within this section, the performance of DFA as a classification method is evaluated using monofractal synthetic signals across varying series lengths. The primary aim here is to gain insights into the reliability of DFA when classifying the shortest synthetic series from each monofractal model. Specifically, we seek to determine the precision level achievable for such short lengths.

It is essential to emphasize that these studies provide estimations regarding the shortest possible lengths and the precision of the analysis. However, it is important to note that real-time series are significantly more complex than their synthetic counterparts. In the context of this study, all synthetic cases underwent detrending using a second-order polynomial in the third step of the DFA method.

The main assumption of the DFA method is that $F_2\left(s\right)$ behaves like $s^H$ within a specific range of s, allowing us to extract H using a log-log scale line fit. This principle has been confirmed through extensive mono-fractal series studies, as indicated by [7,29]. Our primary objective is to evaluate DFA’s ability to classify monofractal models while considering the diminishing length of the short series.

To ensure statistical reliability, we computed the average $F_2\left(s\right)$, denoted as $〈F_2\left(s\right)〉$, across all independent realizations of a particular model. Figure 1 shows the behavior of $〈F_2\left(s\right)〉$ for three representative H values: $H=0.5$ for white noise, $H=0.7$ for long-range correlations, and $H=0.3$ for anti-correlations across the four lengths studied.

To simplify the analysis and reduce the reliance on H and s correlation, we focused on $s>20$. In this defined s range, each series realization displayed the power-law behavior outlined as $F_2\left(s\right)\sim s^H$. We determined the Hurst exponents through linear least squares fits for each realization, categorizing them into a “Predicted Class” based on the extracted Hurst exponent (H) and comparing them with the “True Class” of the simulated monofractal models. The outcomes for all monofractal scenarios are illustrated in Figure 2, encompassing both the shorter and longer series lengths examined in this study. This visual representation shows that the DFA model faces greater difficulties in distinguishing samples of length 128 compared with signals of size 1024. This weakness is noticeable since the error spread outside the main diagonal of the H-index is wider for shorter signals.

The performance assessment of DFA as a classifier is summarized in

Table 1. Overall accuracy of DFA for the classification of short-time synthetic signals.

H-Index	Signal Length: 128	Signal Length: 256	Signal Length: 512	Signal Length: 1024
	Acc (%)	Acc (%)	Acc (%)	Acc (%)
$H_{0.1}$	28.2	33.1	36.8	44.3
$H_{0.2}$	25.2	52.6	68.2	80.2
$H_{0.3}$	20.2	49.2	64.8	81.0
$H_{0.4}$	17.2	39.4	56.2	74.1
$H_{0.5}$	14.0	33.9	50.9	67.4
$H_{0.6}$	12.2	30.3	44.7	61.2
$H_{0.7}$	10.8	25.6	41.2	56.3
$H_{0.8}$	9.9	25.8	40.1	56.0
$H_{0.9}$	8.9	24.5	35.8	52.2

. Specifically, the table presents the proportion of samples correctly classified under each H-index. Moreover, for the overall accuracy, the best performance was observed for length $L=1024$, achieving corresponding accuracy percentages of 16.29%, 34.92%, 48.73%, and 63.64% for lengths 128, 256, 512, and 1024, respectively.

5.2. Performance of CNN-SVM on Monofractal Synthetic Data

5.2.1. Training of the Deep Learning Model

The synthetic data were organized according to the length of the monofractal synthetic signal (128, 256, 512, and 1024), each containing 53,820 samples. Subsequently, we evaluated the performance of the CNN-SVM classifier following the classical five-fold cross-validation scheme. This training and testing methodology avoids the use of fixed parts of the dataset, an important condition for disseminating unbiased and precise results. Each designed dataset was divided into five folds of uniform size. At each iteration, one fold was reserved as test data and trained with the other four parts. The overall results were obtained as the average value of the five iterations.

The proposed CNN model includes seven depth levels, focused on deep feature extraction. The output of the convolutional layers was normalized by applying the batch-normalized technique, followed by the ReLu function. These powerful tools significantly improve the performance of the network, making the training faster and more stable.

Table 2. Structure of the CNN architecture for monofractal feature extraction.

Depth Level	Layer Type	Filter Size	Stride	Padding	Activation
1	Convolutional	$7\times 7\times 32$	1	same	ReLU
	Max Pooling	$2\times 2$	2	—	—
2	Convolutional	$5\times 5\times 64$	1	same	ReLU
	Max Pooling	$2\times 2$	2	—	—
3	Convolutional	$5\times 5\times 64$	1	same	ReLU
	Max Pooling	$2\times 2$	2	—	—
4	Convolutional	$5\times 5\times 64$	1	same	ReLU
	Max Pooling	$2\times 2$	2	—	—
5	Convolutional	$3\times 3\times 32$	1	same	ReLU
	Max Pooling	$2\times 2$	2	—	—
6	Convolutional	$3\times 3\times 16$	1	same	ReLU
	Max Pooling	$2\times 2$	2	—	—
7	Convolutional	$3\times 3\times 32$	1	same	ReLU

summarizes the structure of the designed CNN, including the layer type, filter size, padding, stride, and activation function. For feature extraction, the crossentropyex cost function was incorporated into its architecture, adapted to the nine H-index categories. During the training phase, we used the stochastic gradient descent with momentum (SGDM) algorithm as the optimizer, with a learning rate of 0.01 and a momentum factor of 0.0009. We also employed an $L_2$-regularizer of 0.0001, a batch size of 64, and 4 training epochs. After training the CNN, the SVM replaces the fully connected layers to classify the monofractal synthetic data. In this process, the SVM was trained with CNN features and evaluated with independent data, organized according to the cross-validation scheme. Hinge was the cost function, and SGD was the training algorithm, with a learning rate of 0.01. The linear kernel and the constant $C=1$ were other settings.

To evaluate the potential of the deep features extracted by the CNN network, we integrated the coded features of the last five depth levels with the SVM classifier. Precisely, the SVM was trained using the output of convolutional layers 3, 4, 5, 6, and 7. These generated models were named CNN-SVM3, CNN-SVM4, CNN-SVM5, CNN-SVM6, and CNN-SVM7. The network topology, detailed in Table 2, was invariant for all the databases designed in this study. To speed up the training, we adopted an empirical or manual approach for hyperparameter tuning. In all the experiments designed, the models showed good results, assigning the same setting of values.

In our study, we addressed the problem of overfitting and underfitting using various machine learning strategies, such as regularization techniques ( $L_2$-norm and batch normalization), evaluation of the CNN model with different convolutional layers, optimal hyperparameter search (learning rate, batch size, among others), and incorporation of the five-fold cross-validation scheme. The $L_2$-norm regulated the complexity of the model and improved its generalization capability, while batch normalization stabilized the training and accelerated the convergence of the optimization approach. To avoid underfitting, we evaluated the classification system by adding convolutional layers progressively, thus capturing complex patterns of the monofractal signals across various feature maps. We also validated the experiments thoroughly using the five-fold cross-validation scheme, reporting unbiased evaluation metrics, as shown in Figure 3.

5.2.2. Analysis of Classification Metrics of Monofractal Synthetic Data

Short-duration monofractals quantify the degree and distribution of irregularities in a signal. With the synthetic data designed, we start by discussing the overall accuracy of the CNN-SVM model, as shown in

Table 3. Overall accuracy of CNN-SVM for the classification of short-time synthetic signals.

Dataset	Signal Length	Overall Accuracy (%)
		CNN-SVM3	CNN-SVM4	CNN-SVM5	CNN-SVM6	CNN-SVM7
Short-time synthetic signals	128	64.00	67.06	67.89	68.12	68.01
	256	73.64	78.67	81.54	81.88	81.67
	512	90.31	90.48	91.60	92.35	92.44
	1024	97.66	97.98	97.83	97.85	98.22

. The deep learning system evaluated its performance by focusing on the depth of the feature map, with signals of lengths 128, 256, 512, and 1024. For the short synthetic series of size 128, the models from CNN-SVM3 to CNN-SVM7 obtained an overall accuracy of 64%, 67.06%, 67.89%, 68.12%, and 68.01%, respectively. Meanwhile, their accuracy rate with the larger length series was 97.66%, 97.98%, 97.83%, 97.85%, and 98.22%. The other results correspond to lengths of 256 and 512. The summarized table shows that network depth was a key factor in the generalization capacity of the model. Likewise, as expected, the CNN-SVM demonstrated better accuracy rates as the length of the synthetic signal increased. In general, both CNN-SVM6 and CNN-SVM7 tended to provide better performance on the multiple classification task. Although the overall accuracy of CNN-SVM ranged between 64% and 68% on shorter signals, these results remain relevant compared with the DFA scheme.

The bar chart in Figure 3 shows the performance of the CNN-SVM model on different training and test sets derived from the cross-validation scheme. Each bar represents the overall accuracy of the CNN-SVM models on each of the five folds, using signals of sizes 128 and 1024. Through this visualization, we conclude that the deep learning approach is not sensitive to data partitioning in the classification system. The stability of the performance supports the objectivity of the overall accuracy in Table 3, ensuring robust and reliable results.

Although, as a general rule, accuracy is a very good indicator of performance, a more thorough classification analysis requires careful examination of other evaluation metrics, such as Acc, PPV, Sen, and Spe. Following these evaluation criteria, in

Table 4. Evaluation metrics of CNN-SVM6 for the classification of short-time synthetic signals.

H-Index	Signal Length: 128				Signal Length: 256				Signal Length: 512				Signal Length: 1024
	Acc	PPV	Sen	Spe	Acc	PPV	Sen	Spe	Acc	PPV	Sen	Spe	Acc	PPV	Sen	Spe
$H_{0.1}$	95.2	77.2	80.0	97.1	97.2	87.5	87.6	98.4	98.9	94.8	95.2	99.3	99.7	98.4	98.6	99.8
$H_{0.2}$	91.1	59.9	59.6	95.0	94.7	75.9	76.5	96.9	97.9	90.5	90.8	98.8	99.4	97.5	97.1	99.7
$H_{0.3}$	91.3	60.4	62.0	94.9	94.9	76.7	78.1	97.0	97.9	91.0	90.7	98.9	99.4	97.5	97.4	99.7
$H_{0.4}$	91.0	59.6	59.7	94.9	94.7	76.3	75.9	97.1	97.8	89.8	90.2	98.7	99.4	97.1	97.3	99.6
$H_{0.5}$	90.7	58.6	57.1	94.9	94.6	75.9	75.3	97.0	97.6	89.6	89.0	98.7	99.4	97.1	97.1	99.6
$H_{0.6}$	91.2	61.1	58.5	95.3	95.1	78.1	77.2	97.3	97.7	89.8	89.7	98.7	99.4	97.4	97.2	99.7
$H_{0.7}$	92.8	67.2	67.8	95.9	96.2	82.9	82.8	97.9	98.3	92.6	92.4	99.1	99.6	98.0	98.2	99.8
$H_{0.8}$	94.9	77.4	77.3	97.2	97.4	88.4	88.5	98.6	98.9	95.0	95.4	99.4	99.7	98.5	98.6	99.8
$H_{0.9}$	98.0	90.8	91.3	98.9	98.9	95.2	95.1	99.4	99.5	97.9	97.8	99.7	99.8	99.2	99.1	99.9

, we present the performance of CNN-SVM6, while in

Table 5. Evaluation metrics of CNN-SVM7 for the classification of short-time synthetic signals.

H-Index	Signal Length: 128				Signal Length: 256				Signal Length: 512				Signal Length: 1024
	Acc	PPV	Sen	Spe	Acc	PPV	Sen	Spe	Acc	PPV	Sen	Spe	Acc	PPV	Sen	Spe
$H_{0.1}$	95.1	77.1	79.9	97.0	97.3	87.6	88.0	98.4	98.9	94.9	95.2	99.4	99.7	98.6	99.1	99.8
$H_{0.2}$	91.1	59.9	58.9	95.1	94.8	76.4	76.3	97.1	97.9	90.9	90.9	98.9	99.5	98.1	97.7	99.8
$H_{0.3}$	91.3	60.2	62.8	94.8	94.9	76.4	78.5	96.9	98.1	91.2	91.4	98.9	99.6	98.1	98.1	99.8
$H_{0.4}$	91.1	59.9	58.6	95.1	94.7	76.1	75.8	97.0	97.8	90.1	90.2	98.8	99.5	97.7	97.9	99.7
$H_{0.5}$	90.7	58.4	58.1	94.8	94.5	75.6	74.9	96.9	97.6	89.2	89.2	98.7	99.5	97.8	97.6	99.7
$H_{0.6}$	91.3	61.4	58.5	95.4	94.9	77.6	76.9	97.2	97.8	90.0	89.9	98.8	99.5	97.7	97.9	99.7
$H_{0.7}$	92.8	67.4	67.9	95.9	96.1	82.7	81.9	97.9	98.4	93.3	92.4	99.2	99.6	98.4	98.3	99.8
$H_{0.8}$	94.8	76.9	76.6	97.1	97.3	88.1	87.6	98.5	98.9	94.9	94.9	99.4	99.7	98.6	98.3	99.8
$H_{0.9}$	97.9	90.0	90.8	98.8	98.9	94.7	95.1	99.3	99.5	97.4	97.7	99.7	99.8	98.9	99.1	99.9

, we show the prediction statistics of CNN-SVM7. These tables organize the model information according to the length of the monofractal series, indicating in each row the value of the metric associated with one of the nine categories. The values listed are the average of the results obtained in the five folds of the cross-validation scheme. Indeed, the classifiers obtained better results with data of size 1024. In Table 4, for the signals with length 128, the CNN-SVM6 model achieved an average Acc of 92.9% in the nine categories. The average Sen for all categories was 68.1%, indicating that the model correctly identified 68.1%% of the total true positive cases. Likewise, the average Spe reached 96%, suggesting that the model was able to correctly identify 96% of the total true negative cases. In addition, an average PPV of 68% was observed, indicating that 68% of all positive predictions made by the model were correct. Similar reasoning can be applied to the results obtained by the CNN-SVM7 network. This confirms that the proposed CNN-SVM model provides an optimal classification process for monofractal synthetic signals.

The confusion matrix is a fundamental tool to summarize the correct and incorrect predictions in classification tasks, introduced in our study to deepen our previous analysis. Figure 4 and Figure 5 overview the performance of the CNN-SVM6 and CNN-SVM7 models, respectively, for data with sizes 128 and 1024. Each matrix was selected from the cross-validation scheme, considering the fold with the highest accuracy rate. These representations facilitate a more precise distinction between the indices TP, FP, TN, and FN. For the CNN-SVM7 approach with data of length 1024 in Figure 5, it is observed that, for the $H_{0.1}$ category, 0.51% of the signals were misclassified as $H_{0.2}$. Regarding the category $H_{0.2}$, 0.85% of the signals were misclassified as $H_{0.1}$ and 0.34% as $H_{0.3}$. Similarly, 1.53% and 1.02% of the signals in the $H_{0.3}$ category were incorrectly labeled as $H_{0.2}$ and $H_{0.4}$, respectively. For the $H_{0.4}$ index, 0.66% of the signals were incorrectly classified as $H_{0.3}$, and 1.49% as $H_{0.5}$ series. A similar criterion can be followed for the analysis of the categories $H_{0.5}$, $H_{0.6}$, $H_{0.7}$, $H_{0.8}$, and $H_{0.9}$. The discussion of the other confusion matrices follows a similar evaluation scheme. In addition, when the signal size increases, the H-index classification with the CNN-SVM6 and CNN-SVM7 models tends to concentrate on the main diagonal. Comparing Figure 2 with Figure 4 and Figure 5, the performance of the proposed CNN-SVM is remarkable versus the traditional DFA scheme.

The results presented in this study show the successful performance of the CNN-SVM method over the traditional DFA approach in the classification of short-length synthetic data. However, there are some limitations that we should consider in the analysis of future results. Below, we summarize some disadvantages of this research and make recommendations for future work.

•

The limited amount of monofractal data may limit the training and generalization capabilities of the DFA and CNN-SVM models. Likewise, due to their short length, they are more likely to be susceptible to noise, representing an additional challenge in capturing complex temporal patterns. Another inherent disadvantage of these classifiers is the processing time, especially when the complexity of the time series increases.

•

Our research focuses on the classification of short-length synthetic data, which may restrict the model’s ability to generalize to prediction tasks that require longer series. In addition, monofractal analysis may not be suitable for numerous application problems that demand multifractal models. This implies that the data may not fully reflect the complexity of the task, sometimes skewing the model’s performance.

•

The study compares the DFA and CNN-SVM schemes; nevertheless, it would be desirable to extend the evaluation of the classification system to other advanced machine learning models currently available.

•

In future work, we intend to link this research with applications that may be of interest to humanity. For example, through synthetic monofractal signals, we can detect types of heart disease and categorize the degree of the disease using deep learning techniques or machine learning algorithms. This implementation will be the basis for the analysis of more complex models, which could improve the classification rate by carefully tuning certain parameters.

6. Conclusions

Monofractal synthetic data have variable statistical properties that model a variety of natural phenomena, from fluid turbulence to cardiovascular signal analysis. The short-term series excel in this field, in part due to the lack of appropriate processing methods. In this study, we analyzed the performance of two machine learning approaches in the classification of short-duration monofractal data. Specifically, the traditional DFA scheme and a CNN-SVM neural network were considered. Both models were trained and validated using four datasets, defined according to the signal size. Compared with the DFA algorithm, the proposed CNN-SVM excelled as the best model in the classification task. The results exhibited remarkable overall accuracy in the four designed scenarios, with values that ranged between 64% and 98%. In particular, the classification rate of 64% corresponded to signals of size 128, while for signals of length 1024, it reached an impressive rate of 98%. In contrast, the DFA showed limited performance, with a maximum accuracy of 64% on longer signals and only 16% on shorter signals. Even the CNN-SVM performance with three convolutional layers obtained better evaluation statistics. These results underline the superiority of the proposed CNN-SVM method in the classification of synthetic data of short lengths, offering promising perspectives for practical applications in fields such as medicine and engineering.