Methodology

The model

In tsaopy we assume that the time series can be modelled as a function $x(t)$ that satisfies a differential equation of the form

\[\ddot{x} + \sum_n a_n \dot{x}|\dot{x}|^{n-1} + \sum_m b_m x^m + \sum_{ij} c_{ij} x^i\dot{x}^j = F(t, \mathfrak{f}_1, \dotsc, \mathfrak{f}_{N_f})\]

and we want to find the $a_n$, $b_n$, $c_{ij}$, $\mathfrak{f}_n$ values that best fit our time series data.

Computing trajectories

By proposing a set of parameter values and initial conditions we can compute a trajectory by numerically solving the differential equation.

• We use Runge-Kutta implemented in Fortran and wrapped to be called from Python with numpy.f2py.

The error function

Once we compute a trajectory with the proposed parameter values ($\theta$), we can compare the simulated data points($f_\theta$) with the real measurements($Y$) with an error or cost function. By default tsaopy uses the ‘negative logarithmic likelihood’ of the measurements given the proposed parameters

\[- \log{p(Y|\theta)} = \frac{1}{2}\sum_{i}\frac{(f_{\theta, i}-Y_i)^2}{\sigma_i^2}\]

minimizing this function is equivalent to performing a maximum likelihood estimation (MLE). The function can be minimized with any multivariate minimizer such as minimize from scipy.optimize.

Finding the posterior distribution with MCMC

After MLE, we use the MCMC implementation emcee to find the posterior distribution of the parameters. This allows us to

• find a value as well as error bars for each parameter
• analyze correlations between parameters