title: Applying Sklearn Gaussian Mixture algorithm to fit GM curves
body: <p>I have been looking at the Sklearn library and it seems to be very accurate in fitting wide components in <a href="http://www.astroml.org/book_figures/chapter4/fig_GMM_1D.html" rel="nofollow">gaussian mixtures distributions</a>:</p>
  
  <p><a href="https://i.stack.imgur.com/OHg6H.png" rel="nofollow"><img src="https://i.stack.imgur.com/OHg6H.png" alt="enter image description here"></a></p>
  
  <p>I would like to try this methodology for my astronomical data (modifying it a bit since the previous example is deprecated and it will not work in the current version)</p>
  
  <p>However, in my data I have a curve of data and not a distribution of points. Consequently, I generate a distribution from the numpy random.choice function to generate a distribution weighted by the shape of my curve. Afterwards I run sklearn fit:</p>
  
  <pre><code>import numpy            as np
  from sklearn.mixture    import GMM, GaussianMixture
  import matplotlib.pyplot as plt
  from scipy.stats        import norm
  
  #Raw data
  data = np.array([[6535.62597656, 7.24362260936e-17],
          [6536.45898438, 6.28683338273e-17],
          [6537.29248047, 5.84596729207e-17],
          [6538.12548828, 8.13193914837e-17],
          [6538.95849609, 6.70583742068e-17],
          [6539.79199219, 7.8511483881e-17],
          [6540.625, 9.22121293063e-17],
          [6541.45800781, 7.81353615478e-17],
          [6542.29150391, 8.58095991639e-17],
          [6543.12451172, 9.30569784967e-17],
          [6543.95800781, 9.92541957936e-17],
          [6544.79101562, 1.1682282379e-16],
          [6545.62402344, 1.21238102142e-16],
          [6546.45751953, 1.51062780724e-16],
          [6547.29052734, 1.92193416858e-16],
          [6548.12402344, 2.12669644265e-16],
          [6548.95703125, 1.89356624109e-16],
          [6549.79003906, 1.62571112976e-16],
          [6550.62353516, 1.73262984876e-16],
          [6551.45654297, 1.79300635724e-16],
          [6552.29003906, 1.93990357551e-16],
          [6553.12304688, 2.15530881856e-16],
          [6553.95605469, 2.13273711105e-16],
          [6554.78955078, 3.03175829363e-16],
          [6555.62255859, 3.17610250166e-16],
          [6556.45556641, 3.75917668914e-16],
          [6557.2890625, 4.64631505826e-16],
          [6558.12207031, 6.9828152092e-16],
          [6558.95556641, 1.19680535606e-15],
          [6559.78857422, 2.18677945421e-15],
          [6560.62158203, 4.07692754678e-15],
          [6561.45507812, 5.89089137849e-15],
          [6562.28808594, 7.48005986578e-15],
          [6563.12158203, 7.49293900174e-15],
          [6563.95458984, 4.59418727426e-15],
          [6564.78759766, 2.25848015792e-15],
          [6565.62109375, 1.04438093017e-15],
          [6566.45410156, 6.61019482779e-16],
          [6567.28759766, 4.45881319808e-16],
          [6568.12060547, 4.1486649376e-16],
          [6568.95361328, 3.69435405178e-16],
          [6569.78710938, 2.63747028003e-16],
          [6570.62011719, 2.58619514057e-16],
          [6571.453125, 2.28424298265e-16],
          [6572.28662109, 1.85772271843e-16],
          [6573.11962891, 1.90082094593e-16],
          [6573.953125, 1.80158097764e-16],
          [6574.78613281, 1.61992695352e-16],
          [6575.61914062, 1.44038495311e-16],
          [6576.45263672, 1.6536593789e-16],
          [6577.28564453, 1.48634721076e-16],
          [6578.11914062, 1.28145245545e-16],
          [6578.95214844, 1.30889102898e-16],
          [6579.78515625, 1.42521644591e-16],
          [6580.61865234, 1.6919170778e-16],
          [6581.45166016, 2.35394744146e-16],
          [6582.28515625, 2.75400454352e-16],
          [6583.11816406, 3.42150435774e-16],
          [6583.95117188, 3.06301301529e-16],
          [6584.78466797, 2.01059337187e-16],
          [6585.61767578, 1.36484708427e-16],
          [6586.45068359, 1.26422274651e-16],
          [6587.28417969, 9.79250952203e-17],
          [6588.1171875, 8.77299287344e-17],
          [6588.95068359, 6.6478752208e-17],
          [6589.78369141, 4.95864370066e-17]])
  
  
  #Get the data
  obs_wave, obs_flux = data[:,0], data[:,1]
  
  #Center the x data in zero and normalized the y data to the area of the curve
  n_wave = obs_wave - obs_wave[np.argmax(obs_flux)]
  n_flux = obs_flux / sum(obs_flux) 
  
  #Generate a distribution of points matcthing the curve
  line_distribution   = np.random.choice(a = n_wave, size = 100000, p = n_flux)
  number_points       = len(line_distribution)
  
  #Run the fit
  gmm = GaussianMixture(n_components = 4)
  gmm.fit(np.reshape(line_distribution, (number_points, 1)))
  gauss_mixt = np.array([p * norm.pdf(n_wave, mu, sd) for mu, sd, p in zip(gmm.means_.flatten(), np.sqrt(gmm.covariances_.flatten()), gmm.weights_)])
  gauss_mixt_t = np.sum(gauss_mixt, axis = 0)  
  
  #Plot the data
  fig, axis = plt.subplots(1, 1, figsize=(10, 12))
  axis.plot(n_wave, n_flux, label = 'Normalized observed flux')
  axis.plot(n_wave, gauss_mixt_t, label = '4 components fit')
  
  for i in range(len(gauss_mixt)):
      axis.plot(n_wave, gauss_mixt[i], label = 'Gaussian '+str(i))
  
  axis.set_xlabel('normalized wavelength')
  axis.set_ylabel('normalized flux')
  axis.set_title('Sklearn fit GM fit')
  
  axis.legend()
  plt.show()
  </code></pre>
  
  <p>Which gives me:</p>
  
  <p><a href="https://i.stack.imgur.com/Bty2l.png" rel="nofollow"><img src="https://i.stack.imgur.com/Bty2l.png" alt="enter image description here"></a></p>
  
  <p>And zooming</p>
  
  <p><a href="https://i.stack.imgur.com/KKglc.png" rel="nofollow"><img src="https://i.stack.imgur.com/KKglc.png" alt="enter image description here"></a></p>
  
  <p>If anyone has attempted to use this library towards this purpose my questions are two:</p>
  
  <p>1) Is there a class in sklearn to perform this fit without generating the data distribution as an intermediate step?</p>
  
  <p>2) How should I improve the fit? Is there a method to constrain the variables? For example set all the narrow components with the same standard deviation?</p>
  
  <p>Thanks for any advice</p>