# Basic Fitting Tutorial¶

This tutorial shows how to fit basic point-source/point-lens model to the data. Similar material can be found in example 2.

## Importing data¶

First things first - we need to import some modules:

```
import os
import numpy as np
import matplotlib.pyplot as plt
import MulensModel as mm
```

Then we import the data (downloaded together with the code) to the MulensData class:

```
file_name = os.path.join(mm.DATA_PATH,
"photometry_files", "OB08092", "phot_ob08092_O4.dat")
my_data = mm.MulensData(file_name=file_name)
print("{:} file was imported".format(file_name))
```

## Plotting data¶

Next step would be plotting these data using matplotlib package:

```
plt.errorbar(my_data.time, my_data.mag, yerr=my_data.err_mag, fmt='.')
plt.gca().invert_yaxis() # We need this to invert magnitude axis.
plt.show()
```

From the plot we see that the peak was around JD’ of 5380.0, the peak magnitude was about 0.8 mag brighter than baseline, and the event lasted dozens of days. We can turn these pieces of information into a very rough estimates of the event parameters:

```
t_0 = 5380.0
u_0 = 0.5
t_E = 20.0 # This is in days.
```

We guessed `u_0 = 0.5`

based on the peak amplitude. The magnitude difference
of 0.8 mag corresponds to flux ratio of slightly above 2. The magnification
*A* and the impact parameter *u_0* are very approximately related via *A=1/u_0*
so *u_0 = 0.5* should be a good choice.

## Preparing for fitting¶

The rough estimates of the event parameters allow us to define a Model and combined it with data in an instance of the Event Class. This allows us to plot the model and the data:

```
pspl_model = mm.Model({'t_0': t_0, 'u_0': u_0, 't_E': t_E})
my_event = mm.Event(datasets=[my_data], model=pspl_model)
my_event.plot_data()
my_event.plot_model()
plt.show()
```

To associate a dataset with a model we provided `Event`

cunstructor a list of
datasets. In the present case this list contains only
a single dataset. If you have more datasets, then just include all of them
in the list, e.g.,
`mm.Event(datasets=[my_data, my_friends_data], model=pspl_model)`

.

The plot looks seems fine, i.e., the peak is more or less where it should be. Hence, we can use our rough estimates as a starting point for fitting procedure.

You may want to learn more on plotting in MulensModel from example 5.

To fit the model parameters we will need to calculate chi^2:

```
chi2_initial = my_event.get_chi2()
print(my_event.model.parameters)
print("give chi^2 of {:.2f}.".format(chi2_initial))
```

We have the ability to get the goodness of fit and it turn it into a function:

```
parameters_to_fit = ["t_0", "u_0", "t_E"]
initial_guess = [t_0, u_0, t_E]
def chi2_for_model(theta, event, parameters_to_fit):
"""
for given event set attributes from parameters_to_fit
(list of str) to values from the theta list
"""
for (key, parameter) in enumerate(parameters_to_fit):
setattr(event.model.parameters, parameter, theta[key])
return event.get_chi2()
```

The chi2_for_model() function as a first argument has a sequence of float-type values. The second argument is an instance of the Event class. The third argument is a list that specifies the attributes of Event.model that will be changed. Note that the order of theta values and parameters_to_fit are the same.

## Fitting model parameters¶

Ok, finally we can fit the parameters. Here we will use the SciPy minimize() function from SciPy subpackage optimize. We encourage you to try other fitting routines.

```
import scipy.optimize as op
result = op.minimize(chi2_for_model, x0=initial_guess,
args=(my_event, parameters_to_fit), method='Nelder-Mead')
```

Fitting is done, so we can inspect the results. The function minimize() gives different output depending on method parameter. We will use just a few:

```
print("Fitting was successful? {:}".format(result.success))
if not result.success:
print(result.message)
print("Function evaluations: {:}".format(result.nfev))
if isinstance(result.fun, np.ndarray):
if result.fun.ndim == 0:
result_fun = float(result.fun)
else:
result_fun = result.fun[0]
else:
result_fun = result.fun
print("The smallest function value: {:.3f}".format(result_fun))
print("for parameters: {:.5f} {:.4f} {:.3f}".format(*result.x.tolist()))
```

The best-fitting function parameters are stored in `result.x`

, which is
of *numpy.ndarray* type. To have a nice output, we converted them to a list.
The smallest function value is returned in `result.fun`

, which can be of
a *float* or a *numpy.ndarray* type.
Let’s plot two different models:

```
# Initial model:
pspl_model.parameters.t_0 = t_0
pspl_model.parameters.u_0 = u_0
pspl_model.parameters.t_E = t_E
my_event.plot_model(label='initial', c='red')
# Best fitting model:
pspl_model.parameters.t_0 = result.x[0]
pspl_model.parameters.u_0 = result.x[1]
pspl_model.parameters.t_E = result.x[2]
my_event.plot_model(label='fitted')
# Finally: data, legend, and show the plot:
my_event.plot_data()
plt.legend(loc='best')
plt.show()
```

If you zoom-in on the peak, you will easily see that the fitted model is much better.

Congratulations! You have fitted the model to the data.

## Exercise¶

Try using different optimization routine, starting point,
or apply constraints on the fit. If
the minimize() function
is now your favourite fitting routine, then still you can call it differently.
Try changing `method`

parameter to one of:
‘Powell’, ‘CG’, ‘BFGS’, ‘TNC’, ‘COBYLA’.