The article considers classification task of model fractal time series by the methods of machine learning. To classify the series, it is proposed to use the meta algorithms based on decision trees. To modeling the fractal time series, binomial stochastic cascade processes are used. Classification of time series by the ensembles of decision trees models is carried out. The analysis indicates that the best results are obtained by the methods of bagging and random forest which use regression trees.
Many complex systems have a fractal structure and their dynamics is represented by time series that have fractal (self-similar) properties. Fractal time series take place in technical, physical, biological and information systems. The results of time series fractal analysis are widely used in various fields, in particular, in recognition and classification.
The tasks of classification of fractal realizations arise when medical diagnosis
clarification by ECG and EEG [
Key to correctly solving the classification problem is the choice of the classification method. To answer the question which method is best for analyzing multifractal properties, we present the results of study in which the classification of model realizations possessing fractal properties was carried out. The aim of the work is a comparative analysis of the classification of multifractal stochastic time series performed by methods based on decision trees.
The self-similarity of random process is to preserve distribution law when changing
the time scale. A stochastic process X (t) is self-similar with a parameter H if the
process aH X at is described by the same finite-dimensional distribution laws as
X (t) . The parameter H , 0 H 1, called the Hurst exponent, represents the
measure of self-similarity and the long-term dependence. Multifractal stochastic processes
are inhomogeneous self-similar ones and have more flexible scaling relations. The
multifractal properties of process are defined the scaling exponent (q) . In the
general case (q) is a nonlinear function for which the value ( 1) / 2 coincides with
the value of Hurst exponent H . For monofractal process (q) is linear. [
One of the frequently used models of the multifractal process is the conservative stochastic binomial multiplicative cascade. To its construction, an iterative procedure based on two main parts is used. The first represents geometric detailing by iterative partitioning of intervals, and the second guarantees randomness of the weighting coefficients. For each iteration n , n 1 , we have a time series (cascade) with multifractal properties.
The fractal characteristics of stochastic multiplicative cascade obtained using beta
distribution random variable Beta( , ) are completely determined by the
parameters , 0 [
The decision tree method is effective method of classification. It is applicable to solving classification problems arising in various fields. It consists in the process of partition the original data set into groups, until homogeneous subsets are obtained. The set of rules that give such a partition allows then make a conclusion for new data. However the decision tree models are unstable: a slightest change in the training set can bring to the essential changes in the tree structure. In this case, it is expedient to use ensembles of elementary classifiers. The components of the ensemble can be the same type or different.
The bagging method [
Random forest is also a method of bagging, but it has several features [
To build decision tree models, Python with libraries that implement machine learning
methods was used [
In the work, to determine the time series belonging to one of the 11 classes, the methods of bagging and random forest were used. In this case the objects were the cascade time series, and the features were the values of this series. In each of the methods, the ensembles of decision trees, both classifications and regressions, were involved. In the case of regression decision trees, the result of the classification is the probability of matching of multifractal cascade to given class. The models for each class were trained on five hundred examples of time series and were tested on fifty test ones.
The probabilities are calculated by the formula: Pi 1 mi C , where mi is the regression result for the i-th example, С is theoretically known class number. If the condition Pi 0.5;1 is met, the classification is considered as correct. If Pi 0.5 and mi C then the cluster number is overvalued, otherwise it is understated. In this paper, the comparative analysis of the classification of model multifractal stochastic time series using meta-algorithms based on decision trees has been performed. Binomial multiplicative stochastic cascades were used as input time series.
Time series were divided into classes depending on their fractal properties. Random forest and bagging methods were used to classify the series. In each method, ensembles of decision trees, both of classification and regression, were involved.
From the research that has been carried it is possible to conclude that the classification of series by fractal properties using decision trees methods gives good results. The best results were obtained with the use of regression trees. In the classification of the series with a small length random forest method showed greater accuracy.
The obtained results can be used for practical applications related to the classification or clustering of real time series with fractal properties. In our future researches we intend to concentrate on the classification of real series using additional features such as fractal characteristics.