
Bimodality testing of the stochastic cusp model
Voříšek, Jan
Multimodal distributions are popular in many areas: biology (fish and shark population), engineering (material collapse under pressure, stability of ships), psychology (attitude transitions), physics (freezing of water) etc. There were a few attempts to utilize multimodal distributions in financial mathematics as well. Cobb et al. described a class of multimodal distributions belonging to the exponential family, which has unique maximum likelihood estimators and showed a connection to the stationary distribution of the stochastic cusp catastrophe model. Moreover was shown, how to identify bimodality for given parameters of the stochastic cusp model using the sign of Cardans discriminant. A statistical test for bimodality of the stochastic cusp model using maximum likelihood estimates is proposed in the paper as well as the necessary condition for bimodality which can be used for s simplified testing to reject bimodality. By proposed methods is tested the bimodality of exchange rate between USD and GBP in the periods within the years 1975  2014.


Approximate Transition Density Estimation of the Stochastic Cusp Model
Voříšek, Jan
Stochastic cusp model is defined by stochastic differential equation with cubic drift. Its stationary density allows for skewness, different tail shapes and bimodality. There are two stable equilibria in bimodality case and movement from one equilibrium to another is interpreted as a crash. Qualitative properties of the cusp model were employed to model crashes on financial markets, however, practical applications of the model employed the stationary distribution, which does not take into account the serial dependence between observations. Because closedform solution of the transition density is not known, one has to use approximate technique to estimate transition density. This paper extends approximate maximum likelihood method, which relies on the closedform expansion of the transition density, to incorporate timevarying parameters of the drift function to be driven by market fundamentals. A measure to predict endogenous crashes of the model is proposed using transition density estimates. Empirical example estimates Iceland Krona depreciation with respect to the British Pound in the year 2001 using differential of interbank interest rates as a market fundamental.


Stochastic Catastrophe Model Cusp
Voříšek, Jan ; Vošvrda, Miloslav (advisor)
Title: Stochastic Catastrophe Model Cusp Author: Jan Voříšek Department: Department of Probability and Mathematical Statistics Supervisor: Prof. Ing. Miloslav Vošvrda, CSc., Czech Academy of Sciences, Institute of Information Theory and Automation Abstract: The goal of this thesis is to analyze the stochastic cusp model. This task is divided into two main topics. The first of them concentrates on the stationary density of the cusp model and statistical testing of its bimodality, where power and size of the proposed tests are simulated and compared with the dip test of unimodality. The second main topic deals with the transition density of the stochastic cusp model. Comparison of approximate maximum likelihood approach with traditional finite difference and numerical simulations indicates its advantage in terms of speed of estimation. An approximate Fisher information matrix of general stochastic process is derived. An application of the cusp model to the exchange rate with timevarying parameters is estimated, the extension of the cusp model into stochastic bimodality model is proposed, and the measure of probability of intrinsic crash of the cusp model is suggested. Keywords: stochastic cusp model, bimodality testing, transition density ap proximation


Stochastic Catastrophe Model Cusp
Voříšek, Jan ; Vošvrda, Miloslav (advisor) ; Lukáš, Ladislav (referee) ; Lachout, Petr (referee)
Title: Stochastic Catastrophe Model Cusp Author: Jan Voříšek Department: Department of Probability and Mathematical Statistics Supervisor: Prof. Ing. Miloslav Vošvrda, CSc., Czech Academy of Sciences, Institute of Information Theory and Automation Abstract: The goal of this thesis is to analyze the stochastic cusp model. This task is divided into two main topics. The first of them concentrates on the stationary density of the cusp model and statistical testing of its bimodality, where power and size of the proposed tests are simulated and compared with the dip test of unimodality. The second main topic deals with the transition density of the stochastic cusp model. Comparison of approximate maximum likelihood approach with traditional finite difference and numerical simulations indicates its advantage in terms of speed of estimation. An approximate Fisher information matrix of general stochastic process is derived. An application of the cusp model to the exchange rate with timevarying parameters is estimated, the extension of the cusp model into stochastic bimodality model is proposed, and the measure of probability of intrinsic crash of the cusp model is suggested. Keywords: stochastic cusp model, bimodality testing, transition density ap proximation


Granger's causality in financial time series
Marčiny, Jakub ; Voříšek, Jan (advisor) ; Lachout, Petr (referee)
The bachelor thesis discusses causality in multiple time series. Granger causality, along with its more general counterparts instantaneous causality and multistep causality, are utilized to study the mutual influence of the individual components of a multiple time series. These concepts are investigated within the framework of vector autoregressive models VAR. After the introduction of basic definitions and facts, the construction of VAR model is described including methods for order selection and verification. Subsequently, causal relations within the model are examined. Finally, empirical analysis of real financial market data is performed using tests procedures programmed with computational software Mathematica.


Comparison of logistic regression and decision trees
Raadová, Zuzana ; Voříšek, Jan (advisor) ; Komárek, Arnošt (referee)
In this thesis we describe a classification of the binary data. For discussing this problem we use two wellknown methods  logistic regression and decision trees. These methods deal with the problem in different way, so our aim is to compare a successfulness of their predictions. At first a model of logistic regression is introduced and we show how to estimate its parameters using a method of maximum likelihood. Then we describe decision trees as one of the most popular classification tools. There are discussed older classic algorithms CART and C4.5 and also two new algorithms GUEST and CRUISE. The predictions of both of the methods are shown on a real data example.

 
 

Stochastická teorie katastrof
Vošvrda, Miloslav ; Voříšek, Jan
The so called Cusp deterministic catastrophe model extends the classical linear regression adding nonlinearity into a model. A property of a stochatic catastrophe model connected with stochastic differential equation could be described by density, which is known in closedform only in stationary case. The approximation of the transition density is done here by finite difference metod.


Selected Sampling Methods in SAS Software
Voříšek, Jan ; Vrabec, Michal (advisor) ; Berka, Petr (referee)
In the present work we study methodology of different kinds of sample surveys and their design in SAS software. Creating of SAS Enterprise Guide AddIn was the fundamental creative part of this work. This AddIn enables to compute important statistics of sample surveys, without need of being familiar with SAS code. AddIn was created in MSFT Visual Studio 2003 in C # language using a tamplate for AddIns provided by SAS. This work contains a general description of the creation of an AddIn as well as the description of the created AddIn for handling the sample surveys and its usage.
