Argimiro Arratia

Forecasting with twitter data

The dramatic rise in the use of social network platforms such as Facebook or Twitter has resulted in the availability of vast and growing user-contributed repositories of data. Exploiting this data by extracting useful information from it has become a great challenge in data mining and knowledge discovery. A recently popular way of extracting useful information from social network platforms is to build indicators, often in the form of a time series, of general public mood by means of sentiment analysis. Such indicators have been shown to correlate with a diverse variety of phenomena. In this article we follow this line of work and set out to assess, in a rigorous manner, whether a public sentiment indicator extracted from daily Twitter messages can indeed improve the forecasting of social, economic, or commercial indicators. To this end we have collected and processed a large amount of Twitter posts from March 2011 to the present date for two very different domains: stock market and movie box office revenue. For each of these domains, we build and evaluate forecasting models for several target time series both using and ignoring the Twitter-related data. If Twitter does help, then this should be reflected in the fact that the predictions of models that use Twitter-related data are better than the models that do not use this data. By systematically varying the models that we use and their parameters, together with other tuning factors such as lag or the way in which we build our Twitter sentiment index, we obtain a large dataset that allows us to test our hypothesis under different experimental conditions. Using a novel decision-tree-based technique that we call summary tree we are able to mine this large dataset and obtain automatically those configurations that lead to an improvement in the prediction power of our forecasting models. As a general result, we have seen that nonlinear models do take advantage of Twitter data when forecasting trends in volatility indices, while linear ones fail systematically when forecasting any kind of financial time series. In the case of predicting box office revenue trend, it is support vector machines that make best use of Twitter data. In addition, we conduct statistical tests to determine the relation between our Twitter time series and the different target time series.

A construction of continuous-time ARMA models by iterations of Ornstein-Uhlenbeck processes

We present a construction of a family of continuous-time ARMA processes based on p iterations of the linear operator that maps a Levy process onto an Ornstein-Uhlenbeck process. The construction resembles the procedure to build an AR(p) from an AR(1). We show that this family is in fact a subfamily of the well-known CARMA(p,q) processes, with several interesting advantages, including a smaller number of parameters. The resulting processes are linear combinations of Ornstein-Uhlenbeck processes all driven by the same Levy process. This provides a straightforward computation of covariances, a state-space model representation and methods for estimating parameters. Furthermore, the discrete and equally spaced sampling of the process turns to be an ARMA(p, p−1) process. We propose methods for estimating the parameters of the iterated Ornstein-Uhlenbeck process when the noise is either driven by a Wiener or a more general Levy process, and show simulations and applications to real data.

Expressive Power and Complexity of a Logic with Quantifiers that Count Proportions of Sets

We present a second-order logic of proportional quantifiers, SOLP, which is essentially a first-order language extended with quantifiers that act upon second-order variables of a given arity r and count the fraction of elements in a subset of r-tuples of a model that satisfy a formula. Our logic is capable of expressing proportional versions of different problems of complexity up to NP-hard as, for example, the problem of deciding if at least a fraction 1/n of the set of vertices of a graph form a clique; and fragments within our logic capture complexity classes as NL and P, with auxiliary ordering relation. When restricted to monadic second-order variables, our logic of proportional quantifiers admits a semantic approximation based on almost linear orders, which is not as weak as other known logics with counting quantifiers (restricted to almost orders), for it does not have the bounded number of degrees property. Moreover, we show that, in this almost-ordered setting, different fragments of this logic vary in their expressive power, and show the existence of an infinite hierarchy inside our monadic language. We extend our inexpressibility result of almost-ordered structure to a fragment of SOLP, which in the presence of full order captures P. To obtain all our inexpressibility results, we developed combinatorial games appropriate for these logics, whose application could go beyond the almost-ordered models and hence are interesting by themselves.

Approximating the expressive power of logics in finite models

We present a probability logic (essentially a first order language extended with quantifiers that count the fraction of elements in a model that satisfy a first order formula) which, on the one hand, captures uniform circuit classes such as AC 0 and TC 0 over arithmetic models, namely, finite structures with linear order and arithmetic relations, and, on the other hand, their semantics, with respect to our arithmetic models, can be closely approximated by giving interpretations of their formulas on finite structures where all relations (including the order) are restricted to be modular (i.e. to act subject to an integer modulo). In order to give a precise measure of the proximity between satisfaction of a formula in an arithmetic model and satisfaction of the same formula in the approximate model, we define the approximate formulas and work on a notion of approximate truth. We also indicate how to enhance the expressive power of our probability logic in order to capture polynomial time decidable queries, There are various motivations for this work. As of today, there is not known logical description of any computational complexity class below NP which does not requires a built-in linear order. Also, it is widely recognized that many model theoretic techniques for showing definability in logics on finite structures become almost useless when order is present. Hence, if we want to obtain significant lower bound results in computational complexity via the logical description we ought to find ways of by-passing the ordering restriction. With this work we take steps towards understanding how well can we approximate, without a true order, the expressive power of logics that capture complexity classes on ordered structures.

On the Descriptive Complexity of a Simplified Game of Hex

The game Whex is here defined, which is similar to Generalized Hex but the players are restricted to colour vertices adjacent to the vertex last coloured by one of the players. It is shown that the problem of deciding existence of winning strategies for one of the players in this game is complete for PSPACE, via quantifier free projections, and that the extension of first order logic with the corresponding generalized quantifier captures PSPACE and verifies a normal form. This problem is used to show that the problem of finding a proof in a proof system, like propositional resolution, in which the user is allowed to introduce auxiliary statements in order to help the system reach the theorem that he had set it to prove, is also complete for PSPACE via quantifier free projections. Also, it is established the complexity of the game Whex when restricted to graphs of outdegree at most 3, and, as a generalized quantifier, its expressive capabilities in the absence of ordering relation.

Multivariate Dynamic Kernels for Financial Time Series Forecasting

We propose a forecasting procedure based on multivariate dynamic kernels, with the capability of integrating information measured at different frequencies and at irregular time intervals in financial markets. A data compression process redefines the original financial time series into temporal data blocks, analyzing the temporal information of multiple time intervals. The analysis is done through multivariate dynamic kernels within support vector regression. We also propose two kernels for financial time series that are computationally efficient without a sacrifice on accuracy. The efficacy of the methodology is demonstrated by empirical experiments on forecasting the challenging S&P500 market.

First Order Extensions of Residue Classes and Uniform Circuit Complexity

The first order logic

A Graphical Tool for Describing the Temporal Evolution of Clusters in Financial Stock Markets

\mathcal{R}ing0,+,*, for finite residue class rings with order is presented, and extensions of this logic with generalized quantifiers are given. It is shown that this logic and its extensions capture DLOGTIME-uniform circuit complexity classes ranging from AC 0 to TC 0. Separability results are obtained for the hierarchy of these logics when order is not present, and for

Counting proportions of sets: expressive power with almost order

\mathcal{R}ing0,+,*, from the unordered version. These separations are obtained using tools from class field theory, adapting notions as the spectra of polynomials over finite fields to sets of sentences in this logic of finite rings, and studying asymptotic measures of these sets such as their relative densities. This framework of finite rings with order provides new algebraic tools and a novel perspective for descriptive complexity.

The Greedy Algorithm and the Cohen-Macaulay Property of Rings, Graphs and Toric Projective Curves

We propose a methodology for clustering financial time series of stocks’ returns, and a graphical set-up to quantify and visualise the evolution of these clusters through time. The proposed graphical representation allows for the application of well known algorithms for solving classical combinatorial graph problems, which can be interpreted as problems relevant to portfolio design and investment strategies. We illustrate this graph representation of the evolution of clusters in time and its use on real data from the Madrid Stock Exchange market.