Search of Weights in the Problem of Weighted Finite-Rank Time-Series Approximation

Today, huge amounts of time series data are sensed continuously by AIoT devices, transmitted to edge nodes, and to data centers. It costs a lot of energy to transmit these data, store them, and process them. Data compression technologies are commonly used to reduce the data size and thus save energy. When a certain level of data accuracy is sacrificed, lossy compression technologies can achieve better compression ratios. However, different applications may have different requirements for data accuracy. Instead of keeping multiple compressed versions of a time series w.r.t. different error bounds, HIRE hierarchically maintains a tree, where the root records a constant function to approximate the whole time series, and each other node records a constant function to approximate a part of the residual function of its parent for a particular time period. To retrieve data w.r.t. a specific error bound, it traverses the tree from the root down to certain levels according to the requested error bound and aggregates the constant functions on the visited nodes to generate a new bounded error compressed version dynamically. However, the number of nodes to be visited is unknown before the tree traversal completes, and thus the data size of the new version. In this paper, a time series is progressively decomposed into multiple piecewise linear functions. The first function is an approximation of the original time series w.r.t. the largest error bound. The second function is an approximation of the residual function between the original time series and the first function w.r.t. the second largest error bound, and so forth. The sum of the first, second, …, and m-th functions is an approximation of the original time series w.r.t. the m-th error bound. For each iteration, Swing-RR is used to generate a Bounded Error Piecewise Linear Approximation (BEPLA). Resolution Reduction (RR) plays an important role. Eight real-world datasets are used to evaluate the proposed method. For each dataset, approximations w.r.t. three typical error bounds, 5%, 1%, and 0.5%, are requested. Three BEPLAs are generated accordingly, which can be summed up to form three approximations w.r.t. the three error bounds. For all datasets, the total data size of the three BEPLAs is almost the same, with the size used to store just one version w.r.t. the smallest error bound and significantly smaller than the size used to keep three independent versions. The experiment result shows that the proposed method, referred to as PBEPLA-RR, can achieve very good compression ratios and provide multiple approximations w.r.t. different error bounds.

This paper presents a wavelet neural-network for learning and approximation of chaotic time series. Wavelet networks are a class of neural network that take advantage of good localization and approximation properties of multiresolution analysis. These networks use wavelets as activation functions in the hidden layer and a hierarchical method is used for learning. Comparisons are made between a wavelet network, tested with two different wavelets, and the typical feedforward network trained with the back-propagation algorithm. The results reported in this paper show that wavelet networks have better approximation properties than back-propagation networks.

A method is proposed to approximate the main features or patterns including interventions that may occur in a time series. Collision data from the Ontario Ministry of Transportation illustrate the approach using monthly collision counts from police reports over a 10-year period from 1990 to 1999. The domain of the time series is partitioned into nonoverlapping subdomains. The major condition on the approximation requires that the series and the approximation have the same average value over each subdomain. To obtain a smooth approximation, based on the second difference of the series, a few iterations are necessary since an iteration over one subdomain is affected by the previous iteration over the adjacent subdomains.

There are several applications of time series forecasting for which accurate knowledge of it is not required. In those cases it is enough to deal with the approximation of time series by intervals i.e. interval-valued time series (ITS). In this paper we propose a new method for the forecasting of univariate ITS. A part of the theoretical contribution of the paper is the development of the forecasting model which is based on fuzzy cognitive maps (FCMs). Instead of fuzzy sets used in standard FCMs we apply interval-valued intuitionistic fuzzy sets as their concepts. In this way we get interval-valued intuitionistic fuzzy cognitive maps (IVI-FCMs) which we use for the forecasting of ITS. To validate IVI-FCMs we apply them for the forecasting of the ITS made up by the daily minima and maxima of Nasdaq-100 stock index. Experimental evaluation proved high efficiency of the proposed approach.

For increments of time series from increments of the fractal Brownian motion (fBm) we proposed the method of approximation by power function. For approximating fBm we performed estimation of parameters by means of the algorithm, proposed by the author.

For approximation of some well-known time series of Paramecia caudatun population dynamics (G. F. Gause, The Struggle for Existence, 1934) Verhulst and Gompertz models were used. The parameters were estimated for each of the models in two different ways: with the least squares method (global fitting) and non-traditional approach (a method of extreme points). The results obtained were compared and also with those represented by G. F. Gause. Deviations of theoretical (model) trajectories from experimental time series were tested using various non-parametric statistical tests. It was shown that the least square method-estimations lead to the results which not always meet the requirements imposed for a "fine" model. But in some cases a small modification of the least square method-estimations is possible allowing for satisfactory representations of experimental data set for approximation.

Over recent years the popularity of time series has soared. Given the widespread use of modern information technology, a large number of time series may be collected during business, medical or biological operations, for example. As a consequence there has been a dramatic increase in the amount of interest in querying and mining such data, which in turn has resulted in a large number of works introducing new methodologies for indexing, classification, clustering and approximation of time series. In particular, many new distance measures between time series have been introduced. In this paper, we propose a new distance function based on a derivative. In contrast to well-known measures from the literature, our approach considers the general shape of a time series rather than point-to-point function comparison. The new distance is used in classification with the nearest neighbor rule. In order to provide a comprehensive comparison, we conducted a set of experiments, testing effectiveness on 20 time series datasets from a wide variety of application domains. Our experiments show that our method provides a higher quality of classification on most of the examined datasets.

fABBA: A Python library for the fast symbolic approximation of time series

The paper considers peculiarities of forecasting in the field of railroad facilities based on approximation of time series and neural networks. The objects of traction power supply of the Transbaikal Railway are considered as the object of the study. Forecasting on the basis of approximation for consumers with maximum power consumption increases the accuracy of forecasting the electric energy consumption by 4…7%. The application of neural networks in forecasting of power consumption allows reducing the value of error to 2%, thus leading to significant reduction of costs for fuel and energy balance of a structural division or an enterprise as a whole.

Utilization of multiple trajectories of a dynamical system model provides us with several benefits in approximation of time series. For short term predictions a high accuracy can be achieved via switches to new trajectory at any time. Different long term trends (tendency to different stationary points) of the phase portrait characterize various scenarios of the process realization influenced by externalities. The dynamical system's phase portrait analysis helps to see if the equations properly describe the reality. We also extend the dynamical systems approach (discussed in \cite{R5}) to the dynamical systems with external control. We illustrate these ideas with the help of new examples of the rental properties HOMES.mil platform data. We also compare the qualitative properties of HOMES.mil and Wikipedia.org platforms' phase portraits and the corresponding differences of the two platforms' users. In our last example with COVID-19 data we discuss the high accuracy of the short term prediction of confirmed infection cases, recovery cases and death cases in various countries.

In this paper we propose two problems which related to fractional Brownian motion. First problem- simultaneous estimation of two parameters-Hurst exponent and the volatility, that describes this random process. Numerical tests for the simulated fBm provided an efficient method. Second problem- approximation of the increments of observed time series with power function by increments from the fractional Brownian motion. Approximation and estimation have shown on the example of real data- daily deposit interest rates.

The huge amount of data chronologically collected in short periods of time by different devices and technologies is an important challenge in the analysis of times series. This problem has produced the development of new automatic techniques to reduce the number of points in the resulting time series, in order to facilitate their processing and analysis. In this paper, we propose a new modification of a coral reefs optimization algorithm (CRO) to tackle the problem of reducing the size of the time series minimizing the approximation error. The modification includes a memetization procedure (hybridization with a local search procedure) of the standard algorithm to improve its quality when finding a promising solution in a given searching area. The memetization process is applied to the worse individuals of the algorithm at the beginning, and only to the best ones at the end of the algorithm’s convergence, resulting in a dynamical search approach called dynamic memetic CRO (DMCRO). The proposed DMCRO performance is compared in this paper against other state-of-the-art CRO algorithms, such as the standard one, its statistically driven version (SCRO) and two different hybrid versions (HCRO and HSCRO, respectively), and the standard memetic version (MCRO). All the algorithms compared have been tested in 15 time series approximation, collected from different sources, including financial problems, oceanography data, and cardiology signals, among others, showing that the best results are obtained by DMCRO.

This paper presents a new method in order to predict the monthly electricity peak load of a country based on the prediction of Discrete Fourier Transform (DFT) of monthly peak electricity demand variation using the ARIMA methodology. For validation, the result of this method was used to predict monthly peak load variation of the recent two years in Iranian national grid. The primary goal of this article is to show the application and implementation of Discrete Fourier Transform to predict monthly variation of electricity peak load in national electric power systems. Furthermore, it is elaborated to demonstrate the benefits and shortcomings of DFT approach comparing to the commonly used methodologies known by time series approximation. Comparing the predicted and real value of monthly peak load in the recent years indicates a good and reliable prediction by the new applied methodology.

At first, we discuss the basic structure of the fuzzy system as a simple yet powerful fuzzy modeling technique. Neural networks and fuzzy logic models are based on very similar underlying mathematics. The similarity between RBF networks and fuzzy models is noted in detail. Then, we propose the extension of RBF neural networks by the cloud model. Time series approximation and prediction by applying RBF neural networks or fuzzy models and comparisons between the various types of RBF networks and statistical models are discussed at length.

Analgorithm has beon develowd to find exwnential time series approximations to unsteq aerodynamic data at discrete frequencies using a least squares fit. The method differs from previous methods in that the pole locations of the exponential series approximation we explicitly included in the fit search. and that the fit simultmously minimizes the error in both the real and imaginary parts of the approximation A NewtonRaphsonsearch algorithm is used to find the minim of the weighted square error in the parameter space of the approximation while constraining the poles to be in the left half plane. The results reported here demonstrate the accuracy of the fit achieved by including the poles as free parameters. However, the minima of the cost f~tion found we neither unique nor necessarily global, anddepend on the number of poles in the approximatioo, the initial trial minimum, and the details of the cost minimization algorithm. While eachof the converged minima represents a goodapproximation to the aerodyramic data. the poles found in the search are not necessarily the poles of the true aerodyMmic transfer function Example approximations of the Theodorsenfunction are presented to demonstrate this behavior.