BACRY Emmanuel

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We consider the problem of unveiling the implicit network structure of node interactions (such as user interactions in a social network), based only on high-frequency timestamps. Our inference is based on the minimization of the least-squares loss associated with a multivariate Hawkes model, penalized by L1 and trace norm of the interaction tensor. We provide a first theoretical analysis for this problem, that includes sparsity and low-rank inducing penalizations. This result involves a new data-driven concentration inequality for matrix martingales in continuous time with observable variance, which is a result of independent interest and a broad range of possible applications since it extends to matrix martingales former results restricted to the scalar case. A consequence of our analysis is the construction of sharply tuned L1 and trace-norm penalizations, that leads to a data-driven scaling of the variability of information available for each users. Numerical experiments illustrate the significant improvements achieved by the use of such data-driven penalizations.

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The common thread of this thesis is the study of Hawkes processes. These point processes decipher the inter-causality that can occur between several series of events. Concretely, they determine the influence that the events of one series have on the future events of all other series. For example, in the context of social networks, they describe how likely a user's action, such as a Tweet, will be to trigger reactions from others.The first chapter is a brief introduction to point processes followed by a deeper look at Hawkes processes and in particular the properties of the most commonly used exponential kernel parameterization. In the next chapter, we introduce an adaptive penalty to model, with Hawkes processes, the propagation of information in social networks. This penalty is able to take into account a priori knowledge of the characteristics of these networks, such as sparse interactions between users or community structure, and reflect them on the estimated model. Our technique uses weighted penalties whose weights are determined by a fine-grained analysis of the generalization error.Next, we discuss convex optimization and the progress made with first order stochastic methods with variance reduction. The fourth chapter is dedicated to the adaptation of these techniques to optimize the data attachment term most commonly used with Hawkes processes. Indeed, this function does not verify the gradient-Lipschitz hypothesis usually used. Thus, we work with another regularity assumption, and obtain a linear convergence rate for a lagged version of Stochastic Dual Coordinate Ascent that improves the state of the art. Moreover, such functions have many linear constraints that are frequently violated by classical first-order algorithms, but in their dual version these constraints are much easier to satisfy. Thus, the robustness of our algorithm is more comparable to that of second-order methods which are prohibitively expensive in high dimensions.Finally, the last chapter presents a new statistical learning library for Python 3 with a particular focus on temporal models. Called tick, this library relies on a C++ implementation and state-of-the-art optimization algorithms to perform very fast estimates in a multi-core environment. Published on Github, this library has been used throughout this thesis to perform experiments.

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With the increased availability of large electronic health records databases comes the chance of enhancing health risks screening. Most post-marketing detection of adverse drug reaction (ADR) relies on physicians' spontaneous reports, leading to under-reporting. To take up this challenge, we develop a scalable model to estimate the effect of multiple longitudinal features (drug exposures) on a rare longitudinal outcome. Our procedure is based on a conditional Poisson regression model also known as self-controlled case series (SCCS). To overcome the need of precise risk periods specification, we model the intensity of outcomes using a convolution between exposures and step functions, which are penalized using a combination of group-Lasso and total-variation. Up to our knowledge, this is the first SCCS model with flexible intensity able to handle multiple longitudinal features in a single model. We show that this approach improves the state-of-the-art in terms of mean absolute error and computation time for the estimation of relative risks on simulated data. We apply this method on an ADR detection problem, using a cohort of diabetic patients extracted from the large French national health insurance database (SNIIRAM), a claims database containing medical reimbursements of more than 53 million people. This work has been done in the context of a research partnership between Ecole Polytechnique and CNAMTS (in charge of SNIIRAM).

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No comparative data is available to report on the effect of online self-exclusion. The aim of this study was to assess the effect of self-exclusion in online poker gambling as compared to matched controls, after the end of the self-exclusion period. Methods: We included all gamblers who were first-time self-excluders over a 7-year period (n = 4887) on a poker website, and gamblers matched for gender, age and account duration (n = 4451). We report the effects over time of self-exclusion after it ended, on money (net losses) and time spent (session duration) using an analysis of variance procedure between mixed models with and without the interaction of time and self-exclusion. Analyzes were performed on the whole sample, on the sub-groups that were the most heavily involved in terms of time or money (higher quartiles) and among short-duration self-excluders (<3 months). Results: Significant effects of self-exclusion and short-duration self-exclusion were found for money and time spent over 12 months. Among the gamblers that were the most heavily involved financially, no significant effect on the amount spent was found. Among the gamblers who were the most heavily involved in terms of time, a significant effect was found on time spent. Short-duration self-exclusions showed no significant effect on the most heavily involved gamblers. Conclusions: Self-exclusion seems efficient in the long term. However, the effect on money spent of self-exclusions and of short-duration self-exclusions should be further explored among the most heavily involved gamblers.

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We design a new nonparametric method that allows one to estimate the matrix of integrated kernels of a multivariate Hawkes process. This matrix not only encodes the mutual influences of each node of the process, but also disentangles the causality relationships between them. Our approach is the first that leads to an estimation of this matrix without any parametric modeling and estimation of the kernels themselves. As a consequence, it can give an estimation of causality relationships between nodes (or users), based on their activity timestamps (on a social network for instance), without knowing or estimating the shape of the activities lifetime. For that purpose, we introduce a moment matching method that fits the second-order and the third-order integrated cumulants of the process. A theoretical analysis allows us to prove that this new estimation technique is consistent. Moreover, we show, on numerical experiments, that our approach is indeed very robust with respect to the shape of the kernels and gives appealing results on the MemeTracker database and on financial order book data.

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The goal of this thesis is to show that the arsenal of new optimization methods allows us to solve difficult estimation problems based on event models.These dated events are ordered chronologically and therefore cannot be considered as independent.This simple fact justifies the use of a particular mathematical tool called point process to learn a certain structure from these events. The first is the point process behind the Cox proportional hazards model: its conditional strength allows to define the hazard ratio, a fundamental quantity in the survival analysis literature.The Cox regression model relates the time to the occurrence of an event, called a failure, to the covariates of an individual.This model can be reformulated using the point process framework. The second is the Hawkes process which models the impact of past events on the probability of future events.The multivariate case allows to encode a notion of causality between the different dimensions considered.This theme is divided into three parts.The first part is concerned with a new optimization algorithm that we have developed.It allows to estimate the parameter vector of the Cox regression when the number of observations is very large.Our algorithm is based on the SVRG (Stochastic Variance Reduced Gradient) algorithm and uses an MCMC (Monte Carlo Marker Model) method.We have proved convergence speeds for our algorithm and have shown its numerical performance on simulated and real-world data sets.The second part shows that causality in the Hawkes sense can be reduced to a minimum. The second part shows that the causality in the Hawkes sense can be estimated in a non-parametric way thanks to the integrated cumulants of the multivariate point process.We have developed two methods for estimating the integrals of the kernels of the Hawkes process, without making any assumption on the shape of these kernels. Our methods are faster and more robust, with respect to the shape of the kernels, compared to the state of the art. We have demonstrated the statistical consistency of the first method, and have shown that the second one can be applied to a convex optimization problem.The last part highlights the order book dynamics using the first non-parametric estimation method introduced in the previous part.We have used data from the EUREX futures market, defined new order book models (e.g., the order book of the same day), and developed a new method for the estimation of the order book.We have used data from the EUREX futures market, developed new order book models (based on the previous work of Bacry et al.) and applied the estimation method on these point processes.The results obtained are very satisfactory and consistent with an economic analysis.Such a work proves that the method we have developed allows to extract a structure from data as complex as those from high-frequency finance.

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We show that multivariate Hawkes processes coupled with the nonparametric estimation procedure first proposed in Bacry and Muzy (2015) can be successfully used to study complex interactions between the time of arrival of orders and their size, observed in a limit order book market. We apply this methodology to high-frequency order book data of futures traded at EUREX. Specifically, we demonstrate how this approach is amenable not only to analyze interplay between different order types (market orders, limit orders, cancellations) but also to include other relevant quantities, such as the order size, into the analysis, showing also that simple models assuming the independence between volume and time are not suitable to describe the data.

We propose a fast and efficient estimation method that is able to accurately recover the parameters of a d-dimensional Hawkes point-process from a set of observations. We exploit a mean-field approximation that is valid when the fluctuations of the stochastic intensity are small. We show that this is notably the case in situations when interactions are sufficiently weak, when the dimension of the system is high or when the fluctuations are self-averaging due to the large number of past events they involve. In such a regime the estimation of a Hawkes process can be mapped on a least-squares problem for which we provide an analytic solution. Though this estimator is biased, we show that its precision can be comparable to the one of the Maximum Likelihood Estimator while its computation speed is shown to be improved considerably. We give a theoretical control on the accuracy of our new approach and illustrate its efficiency using synthetic datasets, in order to assess the statistical estimation error of the parameters.

We present a modified version of the non parametric Hawkes kernel estimation procedure studied in Bacry and Muzy [arXiv:1401.0903, 2014] that is adapted to slowly decreasing kernels. We show on numerical simulations involving a reasonable number of events that this method allows us to estimate faithfully a power-law decreasing kernel over at least six decades. We then propose a eight-dimensional Hawkes model for all events associated with the first level of some asset order book. Applying our estimation procedure to this model, allows us to uncover the main properties of the coupled dynamics of trade, limit and cancel orders in relationship with the mid-price variations.

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In this thesis, we study two applications of stochastic processes to Internet marketing. The first chapter focuses on the scoring of Internet users for real-time auctions. This problem consists in finding the probability that a given Internet user performs an action of interest, called conversion, within a few days after the display of an advertising banner. We show that Hawkes processes are a natural model of this phenomenon but that state-of-the-art algorithms are not applicable to the size of data typically used in industrial applications. We therefore develop two new non-parametric inference algorithms that are several orders of magnitude faster than previous methods. We show empirically that the first one performs better than the state-of-the-art competitors, and that the second one can be applied to even larger datasets without paying too high a price in terms of predictive power. The resulting algorithms have been implemented with very good performances for several years at 1000 mercy, the leading marketing agency being the industrial partner of this CIFRE thesis, where they have become an important production asset. The second chapter focuses on diffusive processes on graphs which are an important tool to model the propagation of a viral marketing operation on social networks. We establish the first theoretical bounds on the total number of nodes reached by a contagion under any graph and diffusion dynamics, and show the existence of two distinct regimes: the sub-critical regime where at most $O(sqrt{n})$ nodes will be infected, where $n$ is the size of the network, and the over-critical regime where $O(n)$ nodes can be infected. We also study the behavior with respect to the observation time $T$ and highlight the existence of critical times below which a diffusion, even an over-critical one in the long run, behaves in a sub-critical way. Finally, we extend our work to percolation and epidemiology, where we improve existing results.

We propose a fast and efficient estimation method that is able to accurately recover the parameters of a d -dimensional Hawkes point-process from a set of observations. We exploit a mean-field approximation that is valid when the fluctuations of the stochastic intensity are small. We show that this is notably the case in situations when interactions are sufficiently weak, when the dimension of the system is high or when the fluctuations are self-averaging due to the large number of past events they involve. In such a regime the estimation of a Hawkes process can be mapped on a least-squares problem for which we provide an analytic solution. Though this estimator is biased, we show that its precision can be comparable to the one of the maximum likelihood estimator while its computation speed is shown to be improved considerably. We give a theoretical control on the accuracy of our new approach and illustrate its efficiency using synthetic datasets, in order to assess the statistical estimation error of the parameters.

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Scattering moments provide nonparametric models of random processes with stationary increments. They are expected values of random variables computed with a nonexpansive operator, obtained by iteratively applying wavelet transforms and modulus nonlinearities, which preserves the variance. First- and second-order scattering moments are shown to characterize intermittency and self-similarity properties of multiscale processes. Scattering moments of Poisson processes, fractional Brownian motions, Lévy processes and multifractal random walks are shown to have characteristic decay. The Generalized Method of Simulated Moments is applied to scattering moments to estimate data generating models. Numerical applications are shown on financial time-series and on energy dissipation of turbulent flows.

This thesis studies some aspects of stochastic modeling of order books. In the first part, we analyze a model in which the order arrival times are Poissonian independent. We show that the order book is stable (in the sense of Markov chains) and that it converges to its stationary distribution exponentially fast. We deduce that the price generated in this framework converges to a Brownian motion at large time scales. We illustrate the results numerically and compare them to market data, highlighting the successes of the model and its limitations. In a second part, we generalize the results to a framework where arrival times are governed by self- and mutually-existing processes, under assumptions about the memory of these processes. The last part is more applied and deals with the identification of a realistic multivariate model from the order flows. We detail two approaches: the first one by likelihood maximization and the second one from the covariance density, and succeed in having a remarkable agreement with the data. We apply the estimated model to two concrete algorithmic trading problems, namely the measurement of the execution probability and the cost of a limit order.

We introduce a multivariate Hawkes process that accounts for the dynamics of market prices through the impact of market order arrivals at microstructural level. Our model is a point process mainly characterized by four kernels associated with, respectively, the trade arrival self-excitation, the price changes mean reversion, the impact of trade arrivals on price variations and the feedback of price changes on trading activity. It allows one to account for both stylized facts of market price microstructure (including random time arrival of price moves, discrete price grid, high-frequency mean reversion, correlation functions behaviour at various time scales) and the stylized facts of market impact (mainly the concave-square-root-like/relaxation characteristic shape of the market impact of a meta-order). Moreover, it allows one to estimate the entire market impact profile from anonymous market data. We show that these kernels can be empirically estimated from the empirical conditional mean intensities. We provide numerical examples, application to real data and comparisons to former approaches.

In this paper, we use a database of around 400,000 metaorders issued by investors and electronically traded on European markets in 2010 in order to study market impact at different scales. At the intraday scale we confirm a square root temporary impact in the daily participation, and we shed light on a duration factor in $1/T^{\gamma}$ with $\gamma \simeq 0.25$. Including this factor in the fits reinforces the square root shape of impact. We observe a power-law for the transient impact with an exponent between $0.5$ (for long metaorders) and $0.8$ (for shorter ones). Moreover we show that the market does not anticipate the size of the meta-orders. The intraday decay seems to exhibit two regimes (though hard to identify precisely): a "slow" regime right after the execution of the meta-order followed by a faster one. At the daily time scale, we show price moves after a metaorder can be split between realizations of expected returns that have triggered the investing decision and an idiosynchratic impact that slowly decays to zero. Moreover we propose a class of toy models based on Hawkes processes (the Hawkes Impact Models, HIM) to illustrate our reasoning. We show how the Impulsive-HIM model, despite its simplicity, embeds appealing features like transience and decay of impact. The latter is parametrized by a parameter $C$ having a macroscopic interpretation: the ratio of contrarian reaction (i.e. impact decay) and of the "herding" reaction (i.e. impact amplification).

The development of organized electronic markets puts constant pressure on academic research in finance. The price impact of a stock market transaction involving a large quantity of shares over a short period of time is a central topic. Controlling and monitoring the price impact is of great interest to practitioners, and its modeling has thus become a central focus of quantitative finance research. Historically, stochastic calculus has gradually been imposed in finance, under the implicit assumption that asset prices satisfy diffusive dynamics. But these hypotheses do not hold at the level of "price formation", i.e. when one considers the fine scales of market participants. New mathematical techniques derived from the statistics of point processes are therefore progressively imposed. The observables (processed price, middle price) appear as events taking place on a discrete network, the order book, and this at very short time scales (a few tens of milliseconds). The approach of prices seen as Brownian diffusions satisfying equilibrium conditions becomes rather a macroscopic description of complex phenomena arising from price formation. In the first chapter, we review the properties of electronic markets. We recall the limitations of diffusive models and introduce Hawkes processes. In particular, we review the research on the maket impact and present the progress of this thesis. In a second part, we introduce a new continuous time and discrete space impact model using Hawkes processes. We show that this model takes into account the microstructure of markets and is able to reproduce recent empirical results such as the concavity of the temporary impact. In the third chapter, we study the impact of a large volume of action on the price formation process at the daily scale and at a larger scale (several days after the execution). Furthermore, we use our model to highlight new stylized facts discovered in our database. In a fourth part, we focus on a non-parametric estimation method for a one-dimensional Hawkes process. This method relies on the link between the self-covariance function and the kernel of the Hawkes process. In particular, we study the performance of this estimator in the direction of the squared error on Sobolev spaces and on a certain class containing "very" smooth functions.

This thesis explores theoretically and empirically some aspects of the formation and evolution of financial asset prices observed in high frequency. We begin by studying the joint dynamics of the option and its underlying. Since high-frequency data make the realized volatility process of the underlying observable, we investigate whether this information is used to price options. We find that the market does not exploit it. Stochastic volatility models are therefore to be considered as reduced-form models. Nevertheless, this study allows us to test the relevance of an empirical hedging measure that we call effective delta. It is the slope of the regression of the option price returns on those of the underlying. It provides a fairly satisfactory indicator of hedging that is independent of any modeling. For price dynamics, we turn in the following chapters to more explicit models of the market microstructure. One of the characteristics of market activity is its clustering. Hawkes processes, which are point processes with this characteristic, therefore provide an adequate mathematical framework for the study of this activity. The Markovian representation of these processes, as well as their affine character when the kernel is exponential, allow us to use the powerful analytical tools of the infinitesimal generator and Dynkin's formula to compute various quantities related to them, such as the moments or autocovariances of the number of events on a given interval. We start with a one-dimensional framework, simple enough to illuminate the approach, but rich enough to allow applications such as grouping order arrival times, predicting future market activity knowing past activity, or characterizing unusual, but nevertheless observed, forms of signature plot where the measured volatility decreases as the sampling frequency increases. Our calculations also allow us to make the calibration of Hawkes processes instantaneous by using the method of moments. The generalization to the multidimensional case then allows us to capture, with clustering, the mean reversion phenomenon that also characterizes the market activity observed at high frequency. General formulas for the signature plot are then obtained and allow us to link its shape to the relative importance of clustering or mean reversion. Our calculations also allow us to obtain the explicit form of the volatility associated with the diffusive limit, connecting the microscopic level dynamics to the volatility observed macroscopically, for example on a daily scale. Moreover, the modeling of buying and selling activities by Hawkes processes allows to compute the impact of a meta order on the asset price. We then find and explain the concave shape of this impact as well as its temporal relaxation. The analytical results obtained in the multidimensional case then provide the appropriate framework for the study of correlation. We then present general results on the Epps effect, as well as on the formation of the correlation and the lead lag.

Abstract In the context of statistics for random processes, we prove a law of large numbers and a functional central limit theorem for multivariate Hawkes processes observed over a time interval [ 0 , T ] when T ? ? . We further exhibit the asymptotic behaviour of the covariation of the increments of the components of a multivariate Hawkes process, when the observations are imposed by a discrete scheme with mesh ? over [ 0 , T ] up to some further time shift ? . The behaviour of this functional depends on the relative size of ? and ? with respect to T and enables to give a full account of the second-order structure. As an application, we develop our results in the context of financial statistics. We introduced in Bacry et al. (2013) [7] a microscopic stochastic model for the variations of a multivariate financial asset, based on Hawkes processes and that is confined to live on a tick grid. We derive and characterise the exact macroscopic diffusion limit of this model and show in particular its ability to reproduce the important empirical stylised fact such as the Epps effect and the lead?lag effect. Moreover, our approach enables to track these effects across scales in rigorous mathematical terms.

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Multifractal models and random cascades have been successfully used to model asset returns. In particular, the log-normal continuous cascade is a parsimonious model that has proven to reproduce most observed stylized facts. In this paper, several statistical issues related to this model are studied. We first present a quick, but extensive, review of its main properties and show that most of these properties can be studied analytically. We then develop an approximation theory in the limit of small intermittency λ-super-2 ≪ 1, i.e. when the degree of multifractality is small. This allows us to prove that the probability distributions associated with these processes possess some very simple aggregation properties across time scales. Such a control of the process properties at different time scales allows us to address the problem of parameter estimation. We show that one has to distinguish two different asymptotic regimes: the first, referred to as the ‘low-frequency asymptotics’, corresponds to taking a sample whose overall size increases, whereas the second, referred to as the ‘high-frequency asymptotics’, corresponds to sampling the process at an increasing sampling rate. The first case leads to convergent estimators, whereas in the high-frequency asymptotics, the situation is much more intricate: only the intermittency coefficient λ-super-2 can be estimated using a consistent estimator. However, we show that, in practical situations, one can detect the nature of the asymptotic regime (low frequency versus high frequency) and consequently decide whether the estimations of the other parameters are reliable or not. We apply our results to equity market (individual stocks and indices) daily return series and illustrate a possible application to the prediction of volatility and conditional value at risk.

We introduce a new stochastic model for the variations of asset prices at the tick-by-tick level in dimension 1 (for a single asset) and 2 (for a pair of assets). The construction is based on marked point pro- cesses and relies on linear self and mutually exciting stochastic inten- sities as introduced by Hawkes. We associate a counting process with the positive and negative jumps of an asset price. By coupling suitably the stochastic intensities of upward and downward changes of prices for several assets simultaneously, we can reproduce microstructure noise (i.e. strong microscopic mean reversion at the level of seconds to a few minutes) and the Epps effect (i.e. the decorrelation of the increments in microscopic scales) while preserving a standard Brownian diffusion behaviour on large scales. More effectively, we obtain analytical closed-form formulae for the mean signature plot and the correlation of two price increments that enable to track across scales the effect of the mean-reversion up to the diffusive limit of the model. We show that the theoretical results are consistent with empirical fits on futures Euro-Bund and Euro-Bobl in several situations.

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The aim of this thesis is to deepen the academic knowledge on the joint variations of high-frequency financial assets by analyzing them from a novel perspective. We take advantage of a tick-by-tick price database to highlight new stylistic facts about high-frequency correlation, and also to test the empirical validity of multivariate models. In Chapter 1, we discuss why high-frequency correlation is of paramount importance to trading. Furthermore, we review the empirical and theoretical literature on correlation at small time scales. Then we describe the main characteristics of the dataset we use. Finally, we state the results obtained in this thesis. In chapter 2, we propose an extension of the subordination model to the multivariate case. It is based on the definition of a global event time that aggregates the financial activity of all the assets considered. We test the ability of our model to capture notable properties of the empirical multivariate distribution of returns and observe convincing similarities. In Chapter 3, we study high-frequency lead/lag relationships using a correlation function estimator fit to tick-by-tick data. We illustrate its superiority over the standard correlation estimator in detecting the lead/lag phenomenon. We draw a parallel between lead/lag and classical liquidity measures and reveal an arbitrage to determine the optimal pairs for lead/lag trading. Finally, we evaluate the performance of a lead/lag based indicator to forecast short-term price movements. In Chapter 4, we focus on the seasonal profile of intraday correlation. We estimate this profile over four stock universes and observe striking similarities. We attempt to incorporate this stylized fact into a tick-by-tick price model based on Hawkes processes. The model thus constructed captures the empirical correlation profile quite well, despite its difficulty to reach the absolute correlation level.

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This paper is composed of two independent parts. In the first part, we are interested in the study of fractal objects using the continuous wavelet transform. In particular, we apply the multifractal formalism of singular measurements to signals. The new formalism obtained allows us to study the relative importance of the different types of singularities involved in a singular signal. Numerical applications on computer-generated signals as well as on experimental signals from fully developed turbulence experiments illustrate our remarks. In the second part of this paper, we present a space and time adaptive numerical scheme for the solution of partial differential equations. This scheme is based on the use of orthogonal wavelet bases. The multiresolution structure generated by such bases allows, in a natural way, to adapt the fineness of the spatial grid of a numerical scheme to the local regularity of the solution and thus to obtain a space adaptive scheme. The aim is to adapt not only the fineness of the spatial grid but also that of the temporal grid in order to concentrate the numerical effort in the regions of space where strong singularities appear. Numerical tests concerning the stability, the complexity and the accuracy are performed on the burgers equation.