Rocio Kiman, Lia Corrales, and Zé Vinícius.

- Know basic models in Astropy Modeling
- Learn common functions to fit
- Be able to make a quick fit of your data
- Visualize the fit

modeling, model fitting, astroquery, matplotlib, astrostatistics

In this tutorial, we will become familiar with the models available in
`astropy.modeling`

and learn how to make a quick fit to our data.

Check http://docs.astropy.org/en/stable/modeling/ for more information

In[1]:

```
import numpy as np
import matplotlib.pyplot as plt
from astropy.modeling import models, fitting
from astroquery.vizier import Vizier
import scipy.optimize
# Make plots display in notebooks
%matplotlib inline
```

We are going to start with a **linear fit to real data**. The data comes
from the paper Bhardwaj et al.
2017.
This is a catalog of **Type II Cepheids**, which is a type of **variable
stars** that pulsate with a period between 1 and 50 days. In this part
of the tutorial, we are going to measure the **Cepheids
Period-Luminosity** relation using `astropy.modeling`

. This relation
states that if a star has a longer period, the luminosity we measure is
higher.

To get it, we are going to import it from Vizier using astroquery.

In[2]:

```
catalog = Vizier.get_catalogs('J/A+A/605/A100')
```

This catalog has a lot of information, but for this tutorial we are
going to work only with periods and magnitudes. Let’s grab them using
the keywords `'Period'`

and `__Ksmag__`

. Note that `'e__Ksmag_'`

refers to the error bars in the magnitude measurements.

In[3]:

```
period = np.array(catalog[0]['Period'])
log_period = np.log10(period)
k_mag = np.array(catalog[0]['__Ksmag_'])
k_mag_err = np.array(catalog[0]['e__Ksmag_'])
```

Let’s take a look at the magnitude measurements as a function of period:

In[4]:

```
plt.errorbar(log_period, k_mag, k_mag_err, fmt='k.')
plt.xlabel(r'$\log_{10}$(Period [days])')
plt.ylabel('Ks')
```

Out[4]:

```
<matplotlib.text.Text at 0x7f96d926aac8>
```

One could say that there is a linear relationship between log period and
magnitudes. To probe it, we want to make a fit to the data. This is
where `astropy.modeling`

is useful. We are going to understand how in
three simple lines we can make any fit we want. We are going to start
with the linear fit, but first, let’s understand what a model and a
fitter are.

Models in Astropy are known parametrized functions. With this format they are easy to define and to use, given that we do not need to write the function expression every time we want to use a model, just the name. They can be linear or non-linear in the variables. Some examples of models are:

- Gaussian1D
- Trapezoid1D
- Polynomial1D
- Sine1D
- Linear1D
- The list continues.

Fitters in Astropy are the classes resposable for making the fit. They can be linear or non-linear in the parameters (no the variable, like models). Some examples are:

- LevMarLSQFitter() Levenberg-Marquardt algorithm and least squares statistic.
- LinearLSQFitter() A class performing a linear least square fitting.
- SLSQPLSQFitter() SLSQP optimization algorithm and least squares statistic.
- SimplexLSQFitter() Simplex algorithm and least squares statistic.
- More detailles here

Now we continue with our fitting.

First we need to choose which model we are going to use to fit to our data. As we said before, our data looks like a linear relation, so we are going to use a linear model.

In[5]:

```
model = models.Linear1D()
```

Second we are going to choose the fitter we want to use. This choice is basically which method we want to use to fit the model to the data. In this case we are going to use the Linear Least Square Fitting. In the next exercise we are going to analyze how to choose the fitter.

In[6]:

```
fitter = fitting.LinearLSQFitter()
```

Finally, we give to our **fitter** (method to fit the data) the
**model** and the **data** to perform the fit. Note that we are
including weights: This means that values with higher error will have
smaller weight (less importance) in the fit, and the contrary for data
with smaller errors. This way of fitting is called *Weighted Linear
Least Squares* and you can find more information about it
here
or
here.

In[7]:

```
best_fit = fitter(model, log_period, k_mag, weights=1.0/k_mag_err**2)
print(best_fit)
```

Out[7]:

```
Model: Linear1D
Inputs: ('x',)
Outputs: ('y',)
Model set size: 1
Parameters:
slope intercept
------------------- ------------------
-2.0981402468153076 13.418358848855155
```

And that’s it!

We can evaluate the fit at our particular x axis by doing
`best_fit(x)`

.

In[8]:

```
plt.errorbar(log_period,k_mag,k_mag_err,fmt='k.')
plt.plot(log_period, best_fit(log_period), color='g', linewidth=3)
plt.xlabel(r'$\log_{10}$(Period [days])')
plt.ylabel('Ks')
```

Out[8]:

```
<matplotlib.text.Text at 0x7f96d8953be0>
```

**Conclusion:** Remember, you can fit data with three lines of code:

Use the model `Polynomial1D(degree=1)`

to fit the same data and
compare the results.

In[None]:

For our second example, let’s fit a polynomial of degree more than 1. In
this case, we are going to create fake data to make the fit. Note that
we’re adding gaussian noise to the data with the function
`np.random.normal(0,2)`

which gives a random number from a gaussian
distribution with mean 0 and standard deviation 2.

In[9]:

```
N = 100
x1 = np.linspace(0, 4, N) # Makes an array from 0 to 4 of N elements
y1 = x1**3 - 6*x1**2 + 12*x1 - 9
# Now we add some noise to the data
y1 += np.random.normal(0, 2, size=len(y1)) #One way to add random gaussian noise
sigma = 1.5
y1_err = np.ones(N)*sigma
```

Let’s plot it to see how it looks:

In[10]:

```
plt.errorbar(x1, y1, yerr=y1_err,fmt='k.')
plt.xlabel('$x_1$')
plt.ylabel('$y_1$')
```

Out[10]:

```
<matplotlib.text.Text at 0x7f96d88a66a0>
```

To fit this data let’s remember the three steps: model, fitter and perform fit.

In[11]:

```
model_poly = models.Polynomial1D(degree=3)
fitter_poly = fitting.LinearLSQFitter()
best_fit_poly = fitter_poly(model_poly, x1, y1, weights = 1.0/y1_err**2)
```

In[12]:

```
print(best_fit_poly)
```

Out[12]:

```
Model: Polynomial1D
Inputs: ('x',)
Outputs: ('y',)
Model set size: 1
Degree: 3
Parameters:
c0 c1 c2 c3
------------------ ----------------- ------------------ ------------------
-8.014437168535283 8.781440335690885 -4.285200880424714 0.7526943607330109
```

What would happend if we use a different fitter (method)? Let’s use the
same model but with `SimplexLSQFitter`

as fitter.

In[13]:

```
fitter_poly_2 = fitting.SimplexLSQFitter()
best_fit_poly_2 = fitter_poly_2(model_poly, x1, y1, weights = 1.0/y1_err**2)
```

Out[13]:

```
WARNING: Model is linear in parameters; consider using linear fitting methods. [astropy.modeling.fitting]
WARNING: The fit may be unsuccessful; Maximum number of iterations reached. [astropy.modeling.optimizers]
```

In[14]:

```
print(best_fit_poly_2)
```

Out[14]:

```
Model: Polynomial1D
Inputs: ('x',)
Outputs: ('y',)
Model set size: 1
Degree: 3
Parameters:
c0 c1 c2 c3
------------------- ------------------- ------------------- -------------------
0.05175165167650991 -0.8953161279614488 -0.9172403816612804 0.40090404966332005
```

Note that we got a warning after using `SimplexLSQFitter`

to fit the
data. The first line says:

`WARNING: Model is linear in parameters; consider using linear fitting methods. [astropy.modeling.fitting]`

If we look at the model we chose:
\(y = c_0 + c_1\times x + c_2\times x^2 + c_3\times x^3\), it is
linear in the parameters \(c_i\). The warning means that
`SimplexLSQFitter`

works better with models that are not linear in the
parameters, and that we should use a linear fitter like
`LinearLSQFitter`

. The second line says:

`WARNING: The fit may be unsuccessful; Maximum number of iterations reached. [astropy.modeling.optimizers]`

So it’s not surprising that the results are different, because this
means that the fitter is not working properly. Let’s discuss a method of
choosing between fits and remember to **pay attention** when you choose
the **fitter**.

One way to check which model parameters are a better fit is calculating the Reduced Chi Square Value. Let’s define a function to do that because we’re going to use it several times.

In[15]:

```
def calc_reduced_chi_square(fit, x, y, yerr, N, n_free):
'''
fit (array) values for the fit
x,y,yerr (arrays) data
N total number of points
n_free number of parameters we are fitting
'''
return 1.0/(N-n_free)*sum(((fit - y)/yerr)**2)
```

In[16]:

```
reduced_chi_squared = calc_reduced_chi_square(best_fit_poly(x1), x1, y1, y1_err, N, 4)
print('Reduced Chi Squared with LinearLSQFitter: {}'.format(reduced_chi_squared))
```

Out[16]:

```
Reduced Chi Squared with LinearLSQFitter: 1.6261547637196885
```

In[17]:

```
reduced_chi_squared = calc_reduced_chi_square(best_fit_poly_2(x1), x1, y1, y1_err, N, 4)
print('Reduced Chi Squared with SimplexLSQFitter: {}'.format(reduced_chi_squared))
```

Out[17]:

```
Reduced Chi Squared with SimplexLSQFitter: 4.382415444504842
```

As we can see, the *Reduced Chi Square* for the first fit is closer to
one, which means this fit is better. Note that this is what we expected
after the discussion of the warnings.

We can also compare the two fits visually:

In[18]:

```
plt.errorbar(x1, y1, yerr=y1_err,fmt='k.')
plt.plot(x1, best_fit_poly(x1), color='r', linewidth=3, label='LinearLSQFitter()')
plt.plot(x1, best_fit_poly_2(x1), color='g', linewidth=3, label='SimplexLSQFitter()')
plt.xlabel(r'$\log_{10}$(Period [days])')
plt.ylabel('Ks')
plt.legend()
```

Out[18]:

```
<matplotlib.legend.Legend at 0x7f96d875ddd8>
```

Results are as espected, the fit performed with the linear fitter is better than the second, non linear one.

**Conclusion:** Pay attention when you choose the fitter.

Scipy has the function scipy.optimize.curve_fit to fit in a similar way that we are doing. Let’s compare the two methods with fake data in the shape of a Gaussian.

In[19]:

```
mu, sigma, amplitude = 0.0, 10.0, 10.0
N2 = 100
x2 = np.linspace(-30, 30, N)
y2 = amplitude * np.exp(-(x2-mu)**2 / (2*sigma**2))
y2 = np.array([y_point + np.random.normal(0, 1) for y_point in y2]) #Another way to add random gaussian noise
sigma = 1
y2_err = np.ones(N)*sigma
```

In[20]:

```
plt.errorbar(x2, y2, yerr=y2_err, fmt='k.')
plt.xlabel('$x_2$')
plt.ylabel('$y_2$')
```

Out[20]:

```
<matplotlib.text.Text at 0x7f96d87756d8>
```

Let’s do our three steps to make the fit we want. For this fit we’re
going to use a non-linear fitter, `LevMarLSQFitter`

, because the model
we need (`Gaussian1D`

) is non-linear in the parameters.

In[21]:

```
model_gauss = models.Gaussian1D()
fitter_gauss = fitting.LevMarLSQFitter()
best_fit_gauss = fitter_gauss(model_gauss, x2, y2, weights=1/y2_err**2)
```

In[22]:

```
print(best_fit_gauss)
```

Out[22]:

```
Model: Gaussian1D
Inputs: ('x',)
Outputs: ('y',)
Model set size: 1
Parameters:
amplitude mean stddev
----------------- -------------------- -----------------
9.792572202996084 -0.36976461574406116 9.901721618244407
```

We can get the covariance
matrix from
`LevMarLSQFitter`

, which provides an error for our fit parameters by
doing `fitter.fit_info['param_cov']`

. The elements in the diagonal of
this matrix are the square of the errors. We can check the order of the
parameters using:

In[23]:

```
model_gauss.param_names
```

Out[23]:

```
('amplitude', 'mean', 'stddev')
```

In[24]:

```
cov_diag = np.diag(fitter_gauss.fit_info['param_cov'])
print(cov_diag)
```

Out[24]:

```
[0.04822573 0.06570863 0.06588353]
```

Then:

In[25]:

```
print('Amplitude: {} +\- {}'.format(best_fit_gauss.amplitude.value, np.sqrt(cov_diag[0])))
print('Mean: {} +\- {}'.format(best_fit_gauss.mean.value, np.sqrt(cov_diag[1])))
print('Standard Deviation: {} +\- {}'.format(best_fit_gauss.stddev.value, np.sqrt(cov_diag[2])))
```

Out[25]:

Amplitude: 9.792572202996084 +- 0.219603571049431 Mean: -0.36976461574406116 +- 0.25633694663326045 Standard Deviation: 9.901721618244407 +- 0.2566778629058843

We can apply the same method with `scipy.optimize.curve_fit`

, and
compare the results using again the *Reduced Chi Square Value*.

In[26]:

```
def f(x,a,b,c):
return a * np.exp(-(x-b)**2/(2.0*c**2))
```

In[27]:

```
p_opt, p_cov = scipy.optimize.curve_fit(f,x2, y2, sigma=y1_err)
a,b,c = p_opt
best_fit_gauss_2 = f(x2,a,b,c)
```

In[28]:

```
print(p_opt)
```

Out[28]:

```
[ 9.79257198 -0.36976456 9.90172207]
```

In[29]:

```
print('Amplitude: {} +\- {}'.format(p_opt[0], np.sqrt(p_cov[0,0])))
print('Mean: {} +\- {}'.format(p_opt[1], np.sqrt(p_cov[1,1])))
print('Standard Deviation: {} +\- {}'.format(p_opt[2], np.sqrt(p_cov[2,2])))
```

Out[29]:

Amplitude: 9.79257197735967 +- 0.21960357812910236 Mean: -0.36976455810330805 +- 0.25633691780111767 Standard Deviation: 9.9017220749291 +- 0.2566778434710096

In[30]:

```
reduced_chi_squared = calc_reduced_chi_square(best_fit_gauss(x2), x2, y2, y2_err, N2, 3)
print('Reduced Chi Squared using astropy.modeling: {}'.format(reduced_chi_squared))
```

Out[30]:

```
Reduced Chi Squared using astropy.modeling: 0.9302611815542323
```

In[31]:

```
reduced_chi_squared = calc_reduced_chi_square(best_fit_gauss_2, x2, y2, y2_err, N2, 3)
print('Reduced Chi Squared using scipy: {}'.format(reduced_chi_squared))
```

Out[31]:

```
Reduced Chi Squared using scipy: 0.9302611815554019
```

As we can see there is a very small difference in the *Reduced Chi
Squared*. This actually needed to happen, because the fitter in
`astropy.modeling`

uses scipy to fit. The advantage of using
`astropy.modeling`

is you only need to change the name of the fitter
and the model to perform a completely different fit, while scipy require
us to remember the expression of the function we wanted to use.

In[32]:

```
plt.errorbar(x2, y2, yerr=y2_err, fmt='k.')
plt.plot(x2, best_fit_gauss(x2), 'g-', linewidth=6, label='astropy.modeling')
plt.plot(x2, best_fit_gauss_2, 'r-', linewidth=2, label='scipy')
plt.xlabel('$x_2$')
plt.ylabel('$y_2$')
plt.legend()
```

Out[32]:

```
<matplotlib.legend.Legend at 0x7f96d85d5fd0>
```

**Conclusion:** Choose the method most convenient for every case you
need to fit. We recomend `astropy.modeling`

because is easier to write
the name of the function you want to fit than to remember the expression
every time we want to use it. Also, `astropy.modeling`

becomes useful
with more complicated models like two
gaussians
plus a black
body,
but that is another tutorial.

Let’s review the conclusion we got in this tutorial:

- You can fit data with
**three lines of code**:- model
- fitter
- perform fit to data

**Pay attention**when you choose the**fitter**.- Choose the method most convenient for every case you need to fit. We
recomend
`astropy.modeling`

to make**quick fits of known functions**.

For the next data: * Choose model and fitter to fit this data * Compare different options

In[33]:

```
N3 = 100
x3 = np.linspace(0, 3, N3)
y3 = 5.0 * np.sin(2 * np.pi * x3)
y3 = np.array([y_point + np.random.normal(0, 1) for y_point in y3])
sigma = 1.5
y3_err = np.ones(N)*sigma
```

In[34]:

```
plt.errorbar(x3, y3, yerr=y3_err, fmt='k.')
plt.xlabel('$x_3$')
plt.ylabel('$y_3$')
```

Out[34]:

```
<matplotlib.text.Text at 0x7f96d854a438>
```

In[None]: