Fractal Analysis Of Time Series Research Articles

Fractal Analysis of Air Pollution Time Series in Urban Areas in Astana, Republic of Kazakhstan

Fractal Analysis of Air Pollution Time Series in Urban Areas in Astana, Republic of Kazakhstan

Diagnosing the stability of large-scale processes using fractal structure analysis of time series

Diagnosing the stability of large-scale processes using fractal structure analysis of time series

Характеристики транспортного затора на основе данных системы фото- и видеофиксации

The relevance of the present paper is justified by the need to search for a scientifically based criterion for the formation, development and elimination of traffic congestion using modern highly efficient mathematical methods of information processing. It seems necessary to study the results of using the rescaled range, statistical, fractal, harmonic, and wavelet analysis of time series formed by the time (duration) of the movement of cars in a stream along a road section in the absence and presence of traffic congestion. The object of study is a section of the road network equipped with a software and hardware complex that allows measuring the characteristics of any car in the flow of transport. The subject of the study is the patterns of changes in the duration of movement of individual cars in the absence and presence of a congestion situation. The purpose of this study is a quantitative indicator of the formation, evolution, and elimination of traffic congestion. The basis of the theoretical and methodological approach is the use of modern highly efficient mathematical methods for processing information about traffic flows received from stationary complexes of photo and video recording of traffic violations. As a result of the study of time series, decompositions of the analyzed time series by harmonics and wavelets are constructed, statistical and fractal characteristics of the time series under consideration are determined. The analysis of the obtained results is carried out, the features of the obtained dependencies are revealed. The theoretical and practical significance of the work lies in the scientific substantiation of the quantitative characteristics of traffic congestion, which is of practical interest from the point of view of predicting anomalies in the movement of vehicles on the road network, making decisions on managing the operation of traffic lights, etc. The direction of further research is an in-depth analysis of the identified indicator and determination on its basis of a criterion for determining the stages of formation, evolution, and elimination of traffic congestion at intersections of the street and road network of a large industrial city.

Analysis of methods to optimize control systems for power supply of marine vessels using fuzzy logic and fractal analysis

The paper considers the features of promising methods to optimize the control systems for power supply of marine vessels using fuzzy logic and fractal analysis. In order to design an effective control contour for the power supply system (PSS), it is proposed to use synergistic mechantronic systems based on intelligent technologies with fundamentally new properties that allow for a more effective solution of control problems using fractal analysis of time series to increase the adequacy of forecasting through in-depth analysis of the causes of emergency situations. The synergistic effect SE of the control within such systems is a set of effects obtained as a result of their combination and synchronization in time and space. Practical aspects of fractal analysis are considered on the example of a two-cycle engine with supercharging and air cooling. In the study of fractal processes, a method is proposed for identifying and eliminating the short-term dependence of the value of the time series of the process S(t), which is characteristic of autoregressive processes, using regression with respect to S(t-1) and conducting the R/S analysis of the remainder X(t). Short-term dependence is eliminated provided that long-term dependence is maintained. Autoregressive AR(1) differences are analyzed at a certain time interval for various engine operating modes. The results of the R/S analysis of the engine operation and the determination of the Hurst exponent are used to increase the efficiency of forecasting and control of the PSS in the period of the detected chaotic behavior of the time series.

Multifractal Dynamics in Executive Control When Adapting to Concurrent Motor Tasks.

There is some evidence that an improved understanding of executive control in the human movement system could be gained from explorations based on scale-free, fractal analysis of cyclic motor time series. Such analyses capture non-linear fractal dynamics in temporal fluctuations of motor instances that are believed to reflect how executive control enlist a coordination of multiple interactions across temporal scales between the brain, the body and the task environment, an essential architecture for adaptation. Here by recruiting elite rugby players with high motor skills and submitting them to the execution of rhythmic motor tasks involving legs and arms concurrently, the main attempt was to build on the multifractal formalism of movement control to show a marginal need of effective adaptation in concurrent tasks, and a preserved adaptability despite complexified motor execution. The present study applied a multifractal analytical approach to experimental time series and added surrogate data testing based on shuffled, ARFIMA, Davies&Harte and phase-randomized surrogates, for assessing scale-free behavior in repeated motor time series obtained while combining cycling with finger tapping and with circling. Single-tasking was analyzed comparatively. A focus-based multifractal-DFA approach provided Hurst exponents (H) of individual time series over a range of statistical moments H(q), q = [−15 15]. H(2) quantified monofractality and H(-15)-H(15) provided an index of multifractality. Despite concurrent tasking, participants showed great capacity to keep the target rhythm. Surrogate data testing showed reasonable reliability in using multifractal formalism to decipher movement control behavior. The global (i.e., monofractal) behavior in single-tasks did not change when adapting to dual-task. Multifractality dominated in cycling and did not change when cycling was challenged by upper limb movements. Likewise, tapping and circling behaviors were preserved despite concurrent cycling. It is concluded that the coordinated executive control when adapting to dual-motor tasking is not modified in people having developed great motor skills through physical training. Executive control likely emerged from multiplicative interactions across temporal scales which puts emphasis on multifractal approaches of the movement system to get critical cues on adaptation. Extending such analyses to less skilled people is appealing in the context of exploring healthy and diseased movement systems.

Two‐phase flow‐patterns identification in oil/gas pipelines based on fractal analysis

AbstractFlow‐pattern identification in two‐phase pipelines is essential for energy reduction as well as for the selection of suitable operating conditions for effective flow transport. The flow pattern in conventional oil/gas pipelines exhibit different flow patterns due to the underlying complex interactions of transport processes at different time scales. Fractal analysis of complex time series has received significant attention in the past few years due to its ability to extract hidden useful information from underlying phenomena in the time‐series complexity. In this work, the potential of fractal analysis for flow‐pattern identification in oil/gas pipelines is investigated using voltage measurements. Experimental data from a downward‐inclined oil/gas pipeline system at different superficial velocities of gas were analyzed. Our results indicate that fractal parameters can be useful for flow‐pattern identification and to gain insights into the complex phenomena of multiphase systems.

ЯКІСНИЙ ФРАКТАЛЬНИЙ АНАЛІЗ ДОВГОТРИВАЛИХ ЧАСОВИХ РЯДІВ ПАРАМЕТРІВ ЕЛЕКТРОПРОВІДНОСТІ ГРУНТІВ СІЛЬСЬКОГОСПОДАРСЬКОГО ПРИЗНАЧЕННЯ: МЕТОДИ НЕЛІНІЙНОЇ ДИНАМІКИ, ТЕОРІЇ ХАОСУ, ФАЗОВИХ ТРАЄКТОРІЙ

Запропоновано процедуру якісного фрактального аналізу довготривалих часових рядів параметрів електропровідності ґрунтів сільськогосподарського призначення, для яких не підтверджується гіпотеза про наявність тренда, із застосуванням методу нелінійної динаміки, теорії хаосу та фазових траєкторій. Розглянуті реальні часові ряди, що характеризують згадані параметри електропровідності українських ґрунтів (сільськогосподарського призначення). Обґрунтуванням для подібних досліджень є теорія Такенса. Хаотичність досліджуваної динаміки системи, що задана часовими реалізаціями, встановлена за допомогою показника Ляпунова. Оцінка стійкості стану оцінювалась фрактальною розмірністю Хаусдорффа й індексом фрактальності. Візуальна оцінка часового ряду проводилась за допомогою процедури відновлення фазових траєкторій. У результаті аналізу фазових точок фазового простору виявлені ознаки розщепленого аттрактору, що дає можливість говорити про його біфуркацію. Застосування методів теорії нелінійних динамічних систем до аналізу часових рядів базується на гіпотезі про можливість опису поведінки досліджуваних систем подібним чином, й до того ж це єдина доступна інформація про систему. Згідно з добре відомою теоремою Такенса [1], одного часового ряду для адекватного опису динамічної системи цілком достатньо. Аналіз часових рядів методами теорії нелінійних динамічних систем зараз набуває широкого застосування. Згідно з термінологією цієї теорії, процес, який описується часовими рядами, може утримувати у собі детермінований хаос, або, іншими словами, стає хаотичним. З точки зору методу лінійного аналізу, такі процеси є хаотичними. Нелінійний аналіз демонструє те, що такі процеси можна розглядати або як детерміновані, або як абсолютно стохастичні. Іншими словами, тільки короткотривале прогнозування поведінки системи є можливим з певною точністю.

Fractal analysis of tensile deformation curves of epoxy polymers based on modified epoxy binders

The paper proposes an approach to study the mechanism of deformation of epoxy polymers based on the methods of fractal calculations. The advantages of the method for determining the fractality index of the deformation curves of polymer samples using the minimum coverage method are shown. An algorithm for the quantitative determination of the location of the “critical” points of the deformation curves of polymer samples under tension is proposed. It is shown that the use of the developed methodology for the fractal analysis of time series on the basis of a set of points of deformation curves of epoxy polymer samples under tension, provides valuable information about the processes occurring in the structure of composite materials affected by mechanical loads and various aggressive factors.

ADAPTED BOX-COUNTING METHOD FOR ASSESSMENT OF THE FRACTAL DIMENSION OF FINANCIAL TIME SERIES

This paper continues research within the framework of the scientific direction in econophysics at the Department of Information Systems and Mathematical Methods in Economics of the faculty of Economics of PSU. Modeling and prediction of financial time series is quite a perspective area of research, because it allows participants of financial processes to reduce risks and make effective decisions. For example, we could research financial processes with the help of fractal analysis. In the article there is studied and worked out in detail one of the methods of fractal analysis of financial time series – the box-counting method for assessment of the fractal dimension. This method is often used in studies conducted by domestic authors, but the authors do not delve into the characteristics and problems of using the box-counting method for analysis of time series, that means that the answers to the interested questions have not yet been given. The main problem is that, as a rule, the analyzed object in the tasks of applying the box-counting method to time series is a computer image of the plot of series. In the article there is proposed the procedure of adaptation of the box-counting method for assessment of the fractal dimension of time series, the procedure does not require the formation of a computer image of the plot. In the article there is considered following difficulties developed from this adaptation: 1) high sensitivity of the resulting estimation of the dimension to the input parameters of the method (the ratio of the sides of the covered by cells plane with the plot; the used range of lengths of the side of the cell; the number of partitions of the plane into cells); 2) the non-obviousness of choosing the optimal values ​​of these parameters. In the article there are analyzed approaches to the selection of these parameters that were proposed by other authors, and there are determined the most suitable approaches for the adapted box-counting method. Also there are developed unique methods for determining the ratio of the sides of the plane with the plot. In the paper there is written the computer program that implements the developed method, and this program is tested on the generated data. The study obtained the following results. The fact of sensitivity of the adapted box-counting method to input parameters is confirmed, that indicates the high importance of the correct choice of these parameters. According to the study, there is found out inability of the proposed methods of automatic determination the ratio of the sides of the plane in relation to artificial time series. There are obtained the most precise (in a statistical sense) estimates of fractal dimension, those found by means of the adapted box-counting method, with the fixed ratio of the sides 1:1. According to comparing the adapted box-counting method and R/S analysis, there are obtained the most precise estimates by the second method (R/S analysis). Finally in the paper there are formulated the possible directions for further research: 1) comparison of the accuracy of various methods for assessment of the fractal dimension on series of different lengths; 2) comparison of the methods of fractal analysis and p-adic analysis for modeling and prediction of financial time series; 3) determination of the conditions of applicability of various methods; 4) approbation of the developed methods for determining of the ratio of the sides of the plane with the plot on real economic data.

Fractal analysis of muscle activity patterns during locomotion: pitfalls and how to avoid them.

Time-dependent physiological data sets are often difficult to interpret objectively. Biosignals such as electromyogram, electroencephalogram, or single-neuron recordings can be interpreted using various linear and nonlinear methods. Each analysis technique aims at the explanation of different data features that might be visible or not to the naked eye. Here, we used linear decomposition based on machine learning to extract motor primitives (the time-dependent coefficients of muscle synergies) from the hindlimb electromyographic activity of mice during normal and mechanically perturbed locomotion. We set out to investigate the effects of calculation parameters and data quality on two nonlinear metrics derived from fractal analysis: the Higuchi's fractal dimension (HFD) and the Hurst exponent (H). Both HFD and H proved to be exceptionally sensitive to changes in motor primitives induced by external perturbations to locomotion. We discuss the potential pitfalls that might arise from fractal analysis by using examples based on surrogate data. We conclude giving some simple, data-driven suggestions to reduce the chance of misinterpretations when metrics such as HFD and H are applied to any biological signal containing elements of periodicity.NEW & NOTEWORTHY Despite the lack of consensus on how to perform fractal analysis of physiological time series, many studies rely on this technique. Here, we shed light on the potential pitfalls of using the Higuchi's fractal dimension and the Hurst exponent. We expose and suggest how to solve the drawbacks of such methods when applied to data from normal and perturbed locomotion by combining in vivo recordings and computational approaches.

A GAME MODEL OF SUPPLY CHAIN MANAGEMENT BASED ON FRACTAL ANALYSIS OF TIME SERIES

Fractal analysis of time series is an important tool for describing complex systems and solving nonlinear problems based on time series datebases, of which the core is believed that the geometric bodies composed of various parts of the system have self-similarity and scale invariance. Supply chain management uses integrated thinking, from the perspective of system, to design, plan and control the logistics, capital flow, and information flow in the supply chain for minimizing internal consumption, improving competitiveness or welfare levels, and achieving win–win cooperation. Whether or not the supply chain enterprises cooperate is essentially a game between the enterprises; according to the game theory, the profit of an enterprise depends not only on its own behavior, but also on the behavior of another enterprise with which it deals. On basis of summarizing and analyzing of previous research works, this paper expounded the current research status and significance of the game model of supply chain management, elaborated the development background, current status, and future challenges of the fractal analysis of time series, introduced the methods and principles of fractal dimension calculation and replicator dynamic equation, established a game model of supply chain management based on the fractal analysis of time series including scale-free area determination and model optimization solution, and conducted the fractal analysis of the game model of supply chain management and discussed the equilibrium and statistical similarity of this proposed model. The final simulation experiment showed that the game model of supply chain management based on the fractal analysis of time series applied the phase space reconstruction principle to the data sample selection of fractal prediction algorithm in supply chain, which solved the problem of low similarity between data samples taken by fractal prediction objects with fuzzy self-similar characteristics and periodic ambiguity, and improved the prediction accuracy of the fractal prediction algorithm when predicting objects in the game model of supply chain management.

Oceanic thermal structure mediates dive sequences in a foraging seabird.

Changes in marine ecosystems are easier to detect in upper‐level predators, like seabirds, which integrate trophic interactions throughout the food web.Here, we examined whether diving parameters and complexity in the temporal organization of diving behavior of little penguins (Eudyptula minor) are influenced by sea surface temperature (SST), water stratification, and wind speed—three oceanographic features influencing prey abundance and distribution in the water column.Using fractal time series analysis, we found that foraging complexity, expressed as the degree of long‐range correlations or memory in the dive series, was associated with SST and water stratification throughout the breeding season, but not with wind speed. Little penguins foraging in warmer/more‐stratified waters exhibited greater determinism (memory) in foraging sequences, likely as a response to prey aggregations near the thermocline. They also showed higher foraging efficiency, performed more dives and dove to shallower depths than those foraging in colder/less‐stratified waters.Reductions in the long‐term memory of dive sequences, or in other words increases in behavioral stochasticity, may suggest different strategies concerning the exploration–exploitation trade‐off under contrasting environmental conditions.

Fractal Analysis of Human Gait Variability via Stride Interval Time Series.

Fractal analysis of stride interval time series is a useful tool in human gait research which could be used as a marker for gait adaptability, gait disorder, and fall risk among patients with movement disorders. This study is designed to systematically and comprehensively investigate two practical aspects of fractal analysis which significantly affect the outcome: the series length and the parameters used in the algorithm. The Hurst exponent, scaling exponent, and/or fractal dimension are computed from both simulated and experimental data using three fractal methods, namely detrended fluctuation analysis, box-counting dimension, and Higuchi's fractal dimension. The advantages and drawbacks of each method are discussed, in terms of biases and variability. The results demonstrate that a careful selection of fractal analysis methods and their parameters is required, which is dependent on the aim of study (either analyzing differences between experimental groups or estimating an accurate determination of fractal features). A set of guidelines for the selection of the fractal methods and the length of stride interval time series is provided, along with the optimal parameters for a robust implementation for each method.

AN IMPROVED PRE-FORECASTING ANALYSIS OF ELECTRICAL LOADS OF PUMPING STATION

Relevance of research. In order to reduce energy losses, an accurate and timely forecast of the amount of consumed electricity is necessary. Accurate forecasting of electrical loads of industrial enterprises and their divisions (productions, workshops, departments etc.) allows planning of normal operating conditions, concluding contracts for the electricity supply with the electricity supply company under more favorable conditions, and improving the electricity quality, which ultimately affects the final cost of the products produced by an enterprise. So far, more than 150 forecasting methods of electrical loads have been developed. Usually, the most convenient one is selected based on the forecaster experience by creating and analyzing several forecasting models in order to identify the best. Therefore, in order to simplify the forecasting procedure, it is necessary to develop the methodology for forecasting analysis. This methodology should enable canceling forecasting algorithms that will create lower quality forecasts. The main objective is to develop the methodology for making a forecasting analysis of power consumption on the example of a pumping station of an enterprise with a continuous cycle of work to increase the efficiency of energy consumption and implementation of energy-saving measures. Objects of research: the process of forecasting electrical loads of a pumping station of the enterprise with a continuous cycle of work. Methods of research: fundamental principles of the theory of electrical engineering, statistical methods for power consumption forecasting, the method for detecting the trend of radio signals, and fractal analysis of time series. Research results. The methodology for forecasting analysis of power consumption, which makes it possible to apply the most appropriate methods to forecast the operational power consumption, is developed. For the first time, the radio signal trend detection method is applied to identify the trend of electrical loads. The variation ranges of the fractal parameters of time series of electrical loads are established depending on the variation coefficient of the time series for different periods of time. The Brown method of exponential smoothing that is used to forecast the electrical loads, in the case of identifying the smoothing constant α is in the beyond set ( ), is further improved. The regularity of changes in the fractal parameters of time series of power consumption of a pumping station with an increase in the time series duration and their field of application are explained.

AN IMPROVED PRE-FORECASTING ANALYSIS OF ELECTRICAL LOADS OF PUMPING STATION

Relevance of research. In order to reduce energy losses, an accurate and timely forecast of the amount of consumed electricity is necessary. Accurate forecasting of electrical loads of industrial enterprises and their divisions (productions, workshops, departments etc.) allows planning of normal operating conditions, concluding contracts for the electricity supply with the electricity supply company under more favorable conditions, and improving the electricity quality, which ultimately affects the final cost of the products produced by an enterprise. So far, more than 150 forecasting methods of electrical loads have been developed. Usually, the most convenient one is selected based on the forecaster experience by creating and analyzing several forecasting models in order to identify the best. Therefore, in order to simplify the forecasting procedure, it is necessary to develop the methodology for forecasting analysis. This methodology should enable canceling forecasting algorithms that will create lower quality forecasts. The main objective is to develop the methodology for making a forecasting analysis of power consumption on the example of a pumping station of an enterprise with a continuous cycle of work to increase the efficiency of energy consumption and implementation of energy-saving measures. Objects of research: the process of forecasting electrical loads of a pumping station of the enterprise with a continuous cycle of work. Methods of research: fundamental principles of the theory of electrical engineering, statistical methods for power consumption forecasting, the method for detecting the trend of radio signals, and fractal analysis of time series. Research results. The methodology for forecasting analysis of power consumption, which makes it possible to apply the most appropriate methods to forecast the operational power consumption, is developed. For the first time, the radio signal trend detection method is applied to identify the trend of electrical loads. The variation ranges of the fractal parameters of time series of electrical loads are established depending on the variation coefficient of the time series for different periods of time. The Brown method of exponential smoothing that is used to forecast the electrical loads, in the case of identifying the smoothing constant α is in the beyond set ( ), is further improved. The regularity of changes in the fractal parameters of time series of power consumption of a pumping station with an increase in the time series duration and their field of application are explained.

ПРЕДПРОГНОЗНЫЙ АНАЛИЗ ЛОГИСТИЧЕСКИХ ВРЕМЕННЫХ РЯДОВ НА ОСНОВАНИИ ПОКАЗАТЕЛЯ ХЕРСТА

Цель: статья посвящена применению фрактального анализа как инструментария предпрогнозного исследования временного ряда. Выбранный метод позволит сделать вывод о типе поведения системы, определить постоянства трендов и продолжительность циклов, если такие имеются, выявить эффекты долговременной памяти и сделать выводы о влиянии фактора сезонности. Обсуждение: для определения динамики рынка расшифруй (HOD) – одной из главных задач его участников, авторами предлагается применение предпрогнозного анализа методами нелинейной динамики. Решить ее с помощью стандартных инструментов технического анализа зачастую очень сложно. Комплексным решением многих проблем в области оценки состояния рынка может стать фрактальный анализ. Этим инструментом часто незаслуженно пренебрегают трейдеры и инвесторы, но фрактальный анализ временных рядов помогает эффективно оценить наличие и устойчивость тренда на рынке. Результаты: в результате проведённого предварительного фрактального анализа можно сделать вывод, о том, что региональный рынок HOD обладает свойством фрактальности, а именно наличие сезонности в исследуемом процессе открывает дополнительные возможности наполнения трафика перевозок внепиковые периоды пригодными только для переработки материалами.

Identification of Extreme Temperature Fluctuation in Blast Furnace Based on Fractal Time Series Analysis

Identification of Extreme Temperature Fluctuation in Blast Furnace Based on Fractal Time Series Analysis

Applying the Fractal Analysis Methods for the Study of the Mechanisms of Deformation and Destruction of Polymeric Material Samples Affected by Tensile Stresses

The paper proposes an approach to study the mechanism of deformation of epoxy polymers based on the methods of fractal calculations. The advantages of the method for determining the fractality index of the deformation curves of polymer samples using the minimum coverage method are shown. An algorithm for the quantitative determination of the location of the "critical" points of the deformation curves of polymer samples under tension is proposed. It is shown that the use of the developed methodology for the fractal analysis of time series on the basis of a set of points of deformation curves of epoxy polymer samples under tension, provides valuable information about the processes occurring in the structure of composite materials affected by mechanical loads and various aggressive factors.

Analysis of Heat And Mass Transfer and Deformation of Colloid Capillary-Porous Bodies Processes by Fractal Analysis and Discrete Nonlinear Dynamics Methods

The fractal analysis of long-term series of parameters of colloidal capillary-porous bodies in conditions of heat and mass transfer to the environment and the resulting deformation process are presented. A fractal estimation of relevant statistical information on the moisture content, temperature and deformation of the above bodies is carried out. The algorithm for calculating the Hurst exponent is based on the R/S analysis. On the basis of the methodology for pre-predictive fractal analysis of time series (based on the sequential R/S analysis), the level of persistence is determined and the parameters (average values) of aperiodic cycles of time series are calculated. Based on smoothing of V-statistics using the ordinary moving averages and Kaufman’s adaptive moving average, the criterion for determining the average length of periodic and aperiodic cycles is proposed. The procedure of qualitative analysis of time series for which the hypothesis about the presence of a trend is not confirmed, using methods of nonlinear dynamics and chaos theory is also proposed. The real time series representative of the heat and mass transfer parameters (temperature, moisture content), stress and deformation in colloidal capillary-porous bodies (model of artistic paintings) involved in convective heat and mass transfer to their environment (premises where the museum exhibition is located) are examined; the latter also includes the artificial climate systems for museum premises and the museum visitors flow being present in this area at this time. Tuckens’s theorem is the support for such studies. The chaotic nature of the dynamical system under study, as prescribed by the time realizations, is determined with Liapunov exponent. The estimation of the persistence was evaluated using Hausdorff fractal dimension and fractal index. The visual estimation of time series was carried out using procedure for the reconstruction of phase trajectories. As a result of the phase area’s phase points analysis, a split attractor is discovered allowing to suppose its bifurcation.

FRACTAL ANALYSIS OF TEMPERATURE TIME SERIES FROM BATCH SUGARCANE CRYSTALLIZATION

Batch crystallization is an important process in many industries, for example, fine chemicals, foods, and pharmaceuticals. Monitoring of the main process variables is essential for process understanding, diagnosis, and for product quality control. It is known that temperature has a critical effect on crystallization. Temperature measurements from crystallization systems display fluctuations with apparently random and complex behavior. Fractal analysis of complex time series has received significant attention in the last few years due to its capability for extraction of hidden useful information of the underlying phenomena behind the time-series complexity. In this work, the potential of fractal analysis of time series for diagnosis of industrial crystallization processes is investigated using temperature measurements from a typical batch sugarcane crystallization system. The crystallizer was operated at different cooling profiles, finding that fractal index is directly related to crystal mean diameter dynamics. Thus, we establish that fractal analysis is a simple and robust alternative for the characterization of batch crystallization.