# Quickstart¶

In this simple example we will generate some simulated data, and fit them with 3ML.

Let’s start by generating our dataset:

```
[1]:
```

```
import warnings
warnings.simplefilter("ignore")
import numpy as np
np.seterr(all="ignore")
```

```
[1]:
```

```
{'divide': 'warn', 'over': 'warn', 'under': 'ignore', 'invalid': 'warn'}
```

```
[2]:
```

```
%%capture
from threeML import *
```

```
[3]:
```

```
from jupyterthemes import jtplot
%matplotlib inline
jtplot.style(context="talk", fscale=1, ticks=True, grid=False)
silence_warnings()
set_threeML_style()
```

```
[4]:
```

```
# Let's generate some data with y = Powerlaw(x)
gen_function = Powerlaw()
# Generate a dataset using the power law, and a
# constant 30% error
x = np.logspace(0, 2, 50)
xyl_generator = XYLike.from_function(
"sim_data", function=gen_function, x=x, yerr=0.3 * gen_function(x)
)
y = xyl_generator.y
y_err = xyl_generator.yerr
```

We can now fit it easily with 3ML:

```
[5]:
```

```
fit_function = Powerlaw()
xyl = XYLike("data", x, y, y_err)
parameters, like_values = xyl.fit(fit_function)
```

```
Best fit values:
```

result | unit | |
---|---|---|

parameter | ||

source.spectrum.main.Powerlaw.K | (9.6 -0.8 +0.9) x 10^-1 | 1 / (cm2 keV s) |

source.spectrum.main.Powerlaw.index | -2.008 +/- 0.033 |

```
Correlation matrix:
```

1.00 | -0.86 |

-0.86 | 1.00 |

```
Values of -log(likelihood) at the minimum:
```

-log(likelihood) | |
---|---|

data | 23.314999 |

total | 23.314999 |

```
Values of statistical measures:
```

statistical measures | |
---|---|

AIC | 50.885318 |

BIC | 54.454045 |

Plot data and model:

```
[6]:
```

```
fig = xyl.plot(x_scale="log", y_scale="log")
```

```
findfont: Font family ['sans-serif'] not found. Falling back to DejaVu Sans.
findfont: Generic family 'sans-serif' not found because none of the following families were found: Helvetica
findfont: Font family ['sans-serif'] not found. Falling back to DejaVu Sans.
findfont: Generic family 'sans-serif' not found because none of the following families were found: Helvetica
findfont: Font family ['sans-serif'] not found. Falling back to DejaVu Sans.
findfont: Generic family 'sans-serif' not found because none of the following families were found: Helvetica
```

Compute the goodness of fit using Monte Carlo simulations (NOTE: if you repeat this exercise from the beginning many time, you should find that the quantity “gof” is a random number distributed uniformly between 0 and 1. That is the expected result if the model is a good representation of the data)

```
[7]:
```

```
gof, all_results, all_like_values = xyl.goodness_of_fit()
print("The null-hypothesis probability from simulations is %.2f" % gof["data"])
```

```
The null-hypothesis probability from simulations is 0.55
```

The procedure outlined above works for any distribution for the data (Gaussian or Poisson). In this case we are using Gaussian data, thus the log(likelihood) is just half of a \(\chi^2\). We can then also use the \(\chi^2\) test, which gives a close result without performing simulations:

```
[8]:
```

```
import scipy.stats
# Compute the number of degrees of freedom
n_dof = len(xyl.x) - len(fit_function.free_parameters)
# Get the observed value for chi2
# (the factor of 2 comes from the fact that the Gaussian log-likelihood is half of a chi2)
obs_chi2 = 2 * like_values["-log(likelihood)"]["data"]
theoretical_gof = scipy.stats.chi2(n_dof).sf(obs_chi2)
print("The null-hypothesis probability from theory is %.2f" % theoretical_gof)
```

```
The null-hypothesis probability from theory is 0.53
```

There are however many settings where a theoretical answer, such as the one provided by the \(\chi^2\) test, does not exist. A simple example is a fit where data follow the Poisson statistic. In that case, the MC computation can provide the answer.