Hi,everyone: I’d like to talk with you of the LPPL model for financial crash prediction. Because i can not find a right place to throw this topic, I chose to discuss in the general trading section... My main focus of this thread is not concerning the theoritical foundation of this model, but the exact realization of the parameters estimation by curve fitting, which is not an easy thing. I will give my rough and probabily wrong understanding of the maths logic behind the curve fitting procedure, and put forward (my) problems. I welcome everyone get involve in this thread, and anyone experienced in LPPL model, please give your advise and comment. Thank you! Also please forgive my poor english and little knowledge of maths… 1.The main logic of the curve fitting of LPPL As we can see, in the model there are 7 parameters needed to be estimated. Usually we use the curve fitting method to get the parameter values. But, fitting the stock market with such a highly complex formula involves a number of considerations. The most obvious being to secure is that the best possible fit is obtained. Because fitting a function to some data is nothing but a minimisation algorithm of some cost-function of the data and the fitting-function, so the main task here is to use the right procedure to get the global minima of the cost-function. Since with noisy data and a fitting-function with a large number of degrees of freedom, many local minima of the cost-function exist where the minimisation algorithm can get trapped, so we can not do the fitting directly with 7 parameters. Thus the first step is try to reduce the number of free parameters of the fit.In order to do so, the 3 linear variables have been ”slaved”. This was done by requiring that the cosfunction has zero derivative with respect to A,B,C in a minimum. Then by the parameter slaving(requiring that the cosfunction has zero derivative with respect to A,B,C), we can get equation set which can be solved using the LU decomposition algorithm thus expressing A, B and C as functions of the remaining 4 non-linear ones. But I’m doubting that: is it mathematically proven that the global minimun we got when fitting 4 parameters with other 3 slaved parameters fixed is also the the global minimun we can get through a 7 parameters fitting? I guess it can be proved, otherwise the whole parameter slaving thing is wrong. But because of my poor maths education, I can not understand the logic behind that. And those papers didn’t even talk about it. 2. The fitting procedure (which I’m confused) To solve for each choice of values for the non-linear parameters, we use the LU decomposition algorithm. This means that we have to choice the certain values for the non-linear parameters(tc, beta, sigma and fi). But what’s the rules for this value choice? I’ve read some papers, and didn’t find the details…… Assuming that we made the value choice, thus we are facing a 4 parameters curve fitting. The mainstream method is to first do the fitting by a so-called Taboo-search where only beta and fi was fitted for fixed values of the other parameters. All such scans, which converged with 0 < beta < 1 was then re-fitted with all non-linear parameters free(usually by a LM algo). The above procedure is from the paper”Crashes as Critical Points” (section 4.2 Fitting Stock Market Indices)by Didier Sornette and his team. And I wondering what is this fitting procedure (taboo+LM) related to the “parameters slaving” above? I mean If we doesn’t perform the slaving, we can also do taboo with some parameters pre-choiced and then LM to get a fitting result, right? So why bother? I really hope someone experienced could help. thank you
http://www.google.com/url?sa=t&rct=...QtYSJB0Wx4MhHgsbgLxpcCg&bvm=bv.95039771,d.aWw Above is a link to the article ”Crashes as Critical Points” for those interested.
Everything You Always Wanted to Know about Log Periodic Power Laws for Bubble Modelling but Were Afraid to Ask http://www.mathworks.com/matlabcentral/answers/uploaded_files/12237/frolova-paper-15.pdf
It was an interesting read. But if you look at the charts pre-crash, almost each of them were in the 5th Elliott wave.....pretty uncanny. Also, the prices were way "up there" relative to recent history....thus giving credence to the mean-reversion theory too.
There are some interesting papers around fitting LPPL and I had some moderate success some years ago in this front, so brought out the code from retirement seeing this thread. Please note that I'm not even sure how well the parameters are being selected. Also based on the fits and the live forward walk on 5 min YM data - not too sure of the actual utility. Perhaps I should re-read the paper for interpretations. So an open question, where the bubble? Sample run - daily up to 3/14/2016 Sp500 and for the fun of it, live market data dow futures forward walk - animated gif.