Dinamička modalna dekompozicija u analizi financijskih transakcija i algoritamskom trgovanju

master's thesis

• Author: Ivan Kulušić
• Mentor: Zlatko Drmač (mentor)
• Committee member: Zlatko Drmač (predsjednik povjerenstva)
• Committee member: Miljenko Marušić (član povjerenstva)
• Committee member: Mladen Jurak (član povjerenstva)
• Committee member: Slaven Kožić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2019-03-07, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu opisujemo DMD metodu koju koristimo za aproksimaciju rješenja nelinearnog dinamičkog sustava. Analiziramo teorijsku pozadinu, intuitivnu interpretaciju i algoritam za računanje metode. Također opisujemo POD metodu koja se nalazi u pozadini DMD algoritma. U kratkim crtama ističemo važnost SVD dekompozicije matrica koja nam daje POD modove i generalizirani inverz matrice. POD metoda nalazi, u smislu najmanjih kvadrata, najbolji potprostor za dani skup vektora. Takav potprostor je pogodan za aproksimaciju sustava većeg ranga sustavom nižeg ranga. DMD metoda pronalazi koherentne prostorno-vremenske modove koji tvore gibanje. Oni su dani svojstvenim vektorima matrice \(A\) koja opisuje lokalno lineariziranu dinamiku sustava. Operator preslikavanja \(A\) općenito je nepoznat, no poznata su izmjerena stanja sustava koja su nam dovoljna za rekonstrukciju dinamike. Iz izmjerenih stanja možemo odrediti operator \(\tilde{A}\) koji je aproksimacija nižeg reda operatora \(A\). Svojstvene vrijednosti operatora \(\tilde{A}\) čine podskup svojstvenih vrijednosti operatora \(A\), a iz svojstvenih vektora operatora \(\tilde{A}\) lako rekonstruiramo pripadne svojstvene vektore operatora \(A\) koje nazivamo DMD modovima. Aproksimaciju stanja sustava dobivamo iz DMD modova, pripadnih svojstvenih vrijednosti i početnog stanja. DMD metodu primjenjujemo na cijene dionica. Hvatamo dinamiku kretanja cijena i predviđamo kuda će cijene ići slijedećih nekoliko trenutaka. Koristeći ta predviđanja razvijamo strategiju ulaganja i u konačnici dobivamo uspješnu metodu algoritamskog trgovanja. U posljednjem poglavlju DMD metodu stavljamo u širi teoretski kontekst. Uvodimo pojam Koopmanovog operatora i opisujemo EDMD metodu pomoću koje računamo svojstvene vrijednosti, svojstvene funkcije i modove Koopmanovog operatora. Pokazujemo vezu između DMD algoritma i EDMD algoritma te zaključujemo da je DMD metoda samo specijalan slučaj EDMD metode.

Abstract (english)

In this paper we have studied DMD method used to approximate nonlinear dynamics system solution. We analyze the theoretical background, intuitive interpretation and a numerical algorithm. We also describe the POD method which appears in the background of the DMD algorithm. We emphasize the importance of the SVD matrix decomposition which gives us POD modes and the generalized inverse matrix. POD method finds, in terms of least squares, the best subspace for a given set of vectors. Such a subspace is suitable for the approximation of a higher ranking system with a lower ranking system. The DMD method finds coherent spatio-temporal modes that form the motion. They are given by the eigenvectors of the matrix \(A\) that describes the locally linearized dynamics of the system. The mapping operator \(A\) is generally unknown, but we know the measured system states that are sufficient for the reconstruction of the dynamics. We can determine the operator \(\tilde{A}\) (lower rank approximation of \(A\)) using only measured states. The eigenvalues of the operator \(\tilde{A}\) are a subset of the eigenvalues of the operator \(A\), and from the eignevectors of operator \(\tilde{A}\) we easily reconstruct the corresponding eigenvectors of operator \(A\). That eignevectors are called DMD modes. The proximal state of the system is obtained from DMD modes, associated eigenvalues and initial states. We apply the DMD method to stock prices. We capture the dynamics of price movements and predict where the price will go in the next few moments. Using that forecasting, we develop an investment strategy and ultimately get a successful method of algorithmic trading. In the last chapter we put the DMD method into a broader theoretical context. We introduce the Koopman operator and describe the EDMD method by which we calculate eigenvalues, eigenfunctions and modes of Koopman operator. We show the link between the DMD algorithm and the EDMD algorithm and we conclude that DMD method is only a special case of EDMD method.

• Keywords: DMD metoda; nelinearni dinamički sustav; POD metoda; SVD dekompozicija matrica; Koopmanov operator; EDMD metoda
• Keywords (english): DMD method; nonlinear dynamics system; POD method; SVD matrix decomposition; Koopman operator; EDMD method
• Language: croatian
• URN:NBN: urn:nbn:hr:217:755841
• Study programme: Title: Finance and Business Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-07-19 12:30:29