Good day,
I have to start working on data fitting problem, but I do not know how to start.
I have found this equation can work as the curve around which data will fit.
$y_t = A + B(t_c - t)^\beta[1+C \cos(\omega \log (t_c -t) + \phi)] $
where
$\beta = \frac{\log \mu}{\log \lambda} $,
$\omega = \frac{2\pi}{\log \lambda} $
Now question is that from a given dataset say $x=[1..1200]$ and $y \text{ is fluctuating values from 200 to 800}$
how can we find $\mu$ and $\lambda$ ?
To give an idea when I plot the data I get
EDIT: from the paper I read:(financial crash)
$y_t > 0 \ is \text{ price index or log of price at time t }$ $A>0 \text{ is the value y_t would have if bubble were to last until critical time}\ t_c$ $B< 0 \text{increase in }y_t \text{utill before crash} $
$C \text{ is the magnitude of the fluctuations around exponential growth }$
$t_c \text{ is critical time}$ , $t<t_c \text{ is time into the bubble}$, $\phi \text{ is shift parameter}$ , $\beta = 0.33 +\pm 0.18 \text{ exponent of power law growth}$
$\omega =6.36 \pm 1.56 \text{ frequency of fluctuation during the buble}$
When I draw this function from the paper.. it looks like this.(not exact. But gives idea about the curve)
NOTE: original paper can be found here.. https://www.researchgate.net/publication/45899349_Fitting_the_Log_Periodic_Power_Law_to_financial_crashes_a_critical_analysis