Last time we have written on the power spectral density and we have “analyzed” deterministic periodic time series. This time we will consider spectral densities of some stochastic processes. Continue reading “Power spectral density (part 2)” »
Here, on the Physics of Risk, we frequently talk about two essential statistical features of the time series – probability and spectral densities. The probability density function should well known to our readers – it is related to the distribution of time series values. On the Physics of Risk we have also a Lithuanian-only article on this topic (see it here). So the time has come to discuss the power spectral density. Continue reading “Power spectral density (part 1)” »
In classical physics differential equations is the main tool to mathematically describe dynamical systems. Having obtained the mathematical descriptions of the system we should be able to predict the evolution of the system. It is noted that the evolution of the classical systems is pretty trivial – no matter what the initial condition is the system will “find” the stable state. Usually dissipative forces (such as frictions) are to be blamed for this. Though not all systems are so simple… Continue reading “Randomly generated strange attractors” »
Reviewers of one of our most recent papers have asked some very interesting questions. One of them was about the numerical methods we use to solve the stochastic differential equations. The question was to be expected as, while we provide the final difference equations, we do not discuss how they were obtained. Thus here we will briefly review the most basic principles of the numerical solution of the stochastic differential equations. Continue reading “Numerical methods for the stochastic differential equations” »
Modern science strongly relies on the computer modeling. Most of the models in the complexity science, the object of the Physics of Risk, requires computer modeling and usually may not be dealt with analytically. For a person familiar with the computer modeling it should be known that the variety computer algorithms is very large and that there also is a variety of ways to understand these algorithms. Thus each person might solve the same complex task slightly differently and thus produce slightly different results. This brings us to the point that in order to comprehend what has been done by a certain scientist one should not only study the equations and assumptions made by him, but one also needs to have access to the source code of the software used by that certain scientist.
Yet there is still a problem that only few scientists to make the source code of their software public available. This behavior is not very desired as in order to reproduce the same results other scientists must make a lot of efforts. Some times the attempts to reproduce published results fail. This problem may be solved by encouraging adoption of the open source ideas by the scientific community.
We, the contributors of Physics of Risk, have already faced the negative effects of the closed source culture, thus most of our models made available on Physic of Risk are published together with their source code. Though it is well hidden inside the applet’s JAR archive (open it with any modern archiver, inside you should find java file, which contains the source code).
Read more on open source software in science in Nature Editorial “If you want reproducible science, the software needs to be open source”.
Imagine that you have to measure the surface area of the lake by using only a cannon! Let us assume that the geometric shape of the lake is too complex to be dealt with using simple formulas and that you have almost infinite supply of cannon balls. In such case you just have to hope that you are perfectly random shooter! Why so?
If your shots cover the hitting area of the cannon uniformly then you can obtain the area of the lake by estimating the probability to hit it:
here is an estimate of the surface area of certain geometric shape (lake for example), is an estimate of the probability to hit the shape, while is the hitting area of the cannon. The obtained are of course will be only approximate, but one can arrive reasonably near the actual answer.
Next we illustrate this method by applying it towards three geometric shapes – square, circle and Euclidean egg. Why the Euclidean egg? Well, there are numerous reasons for it, one of them being Easter. Happy Easter! Continue reading “Obtaining surface area using Monte Carlo method” »
In the last year we have already written that work in the context of Physics of Risk provides varying insights into very different complex systems. The previous article 1 contained brief review of Physics of Risk platform and discussions on some of the models published using it. This article received great response and was even awarded the Best Paper Award by the publisher IARIA. Continue reading “IARIA publication reviewing our different research directions” »
Working on Physics of Risk is very interesting and useful experience. This experience provides valuable insights into the mechanics behind various complex systems, well modeled by macroscopic models. Using our experience we are able to obtain qualitative and quantitative agreements between varying models. In our newest publication 1 we have used one-step formalism 2 to obtain macroscopic treatments of Kirman model 3.
Newton-Raphson, sometimes just Newton or Newton-Fourier, method is an approximate method in mathematical analysis for finding local roots of very complex functions (such as polynomials with large powers). Recall that root of the function is defined as a solution of . The essence of this method is to linearize function at the guessing point. The point where linearized function passes the abscissa axis is assumed to be a more precise estimate of the actual root. Continue reading “Newton-Raphson method” »