ILO-GMM: examples and cases of study

ILOGMM.ipynb

ILOGMM.py

# coding: utf-8

# **The following serves as a simple introduction to ILO-GMM. It also illustrate, using carefully selected examples, problems we might encounter when learning forward and, especially, inverse models.**
# 
# # Overview
# 
# We compare different inference method for ILO-GMM:
# - Weighted mean of the Gaussian center by their relative importance
# - Max of the probability distribution
# - Mean of the most important (bigger weight) Gaussian
# - Sampling the probability distribution
# 
# The data are sampled from a 1 period sinus with a small Gaussian noise.
# 
# We investigate the forward and inverse model. We show that the inverse problem, which illustrates the redundancy/multimodal/multimodes problem, poses problem to some infering methods.
# 
# We conclude that, for most practical robotic problems involving inverse models, the order of inference methods to use is as follow:
# - Max of the probability distribution
# - Mean of the most important (bigger weight) Gaussian
# - Sampling the probability distribution
# - Weighted mean of the Gaussian center by their relative importance
# 
# 
# We then consider the problem of redundancy using a 3 period sinus. This problem helps visualizing a limitation of ILO-GMM that is the fixed number of Gaussian. We explore briefly two options:
# - DPGMM: http://scikit-learn.org/stable/modules/generated/sklearn.mixture.DPGMM.html#sklearn.mixture.DPGMM
# - use of a clustering algortihm to identify the number of modes. DPGMM, wile showing promises in theroy, is not convincing experimentally on our simple example. The clustering shows more practical promises.
# 
# 
# # Before starting
# 
# The code below requires the following python modules:
# - http://www.numpy.org/
# - http://scikit-learn.org/stable/index.html
# - https://github.com/AlexanderFabisch/gmr
# - http://matplotlib.org/
# 
# 
# # Simple 1 period Sinus
# 
# ## Generate data

# In[1]:

import numpy as np
rng = np.random.RandomState(10)
X = np.linspace(0, 2 * np.pi, 500)[:, np.newaxis]
Y = np.sin(X).ravel() + rng.normal(0, 0.1, X.shape[0])
Y = Y[:, np.newaxis]


# ## Plot data

# In[2]:

import matplotlib.pyplot as plt
get_ipython().magic(u'matplotlib inline')


# In[3]:

fig = plt.figure(0)
ax = fig.add_subplot(111, aspect='equal')
ax.scatter(X, Y)
plt.xlim([-0.1,2 * np.pi + 0.1])
plt.ylim([-1.5,1.5])


# ## Build Tree to find NN

# In[4]:

from sklearn.neighbors import KDTree
X_tree = KDTree(X, leaf_size=2)
Y_tree = KDTree(Y, leaf_size=2)


# # Forward Model: example of unimodal/grouped data
# 
# ## Find local data for x=1

# In[5]:

_, ind = X_tree.query(1, k=50)     
local_X = X[ind]
local_Y = Y[ind]


# ## Plot local data at x=1

# In[6]:

fig = plt.figure(0)
ax = fig.add_subplot(111, aspect='equal')
ax.scatter(local_X, local_Y)
plt.xlim([-0.1,2 * np.pi + 0.1])
plt.ylim([-1.5,1.5])


# ## Fit local GMM with 3 components

# In[7]:

from sklearn.mixture import GMM


# In[8]:

local_gmm = GMM(n_components=3, covariance_type='full')
local_data = np.squeeze(np.concatenate((local_X, local_Y), axis=2))
local_gmm.fit(local_data)


# ## Plot local GMM
# 
# Here I switch to https://github.com/AlexanderFabisch/gmr because there is a simple interface for conditional distribution and ellipse drawing
# 
# However I fit the GMM using sklearn because I will later use the DPGMM from sklearn, so it is all consistent and it help see where to find the code to reimplement all this in a uniform python module.

# In[9]:

import gmr


# In[10]:

local_gmminf = gmr.gmm.GMM(3, 
                           local_gmm.weights_, 
                           local_gmm.means_, 
                           local_gmm.covars_)


# In[11]:

ellipses = local_gmminf.to_ellipses(2)


# In[12]:

from matplotlib.patches import Ellipse

fig = plt.figure()
ax = fig.add_subplot(111, aspect='equal')

for ellipse in ellipses:
    e = Ellipse(xy=ellipse[0], angle=ellipse[1][0]*180/np.pi, width=ellipse[1][1], height=ellipse[1][2])
    ax.add_artist(e)
    e.set_clip_box(ax.bbox)
    e.set_alpha(0.5)
    e.set_facecolor(rng.rand(3))

plt.xlim([-0.1,2 * np.pi + 0.1])
plt.ylim([-1.5,1.5])


# ## We compute and plot the conditional for x=1

# In[13]:

conditional_gmm = local_gmminf.condition([0], [1])


# In[14]:

YY = np.linspace(-1.5, 1.5, 500)[:, np.newaxis]
proba = conditional_gmm.to_probability_density(YY)
plt.plot(proba, YY)


# ### Method 1: Find the Max

# In[15]:

max_proba = np.max(proba)
max_id = np.where(proba == max_proba)[0]
max_value = YY[max_id][0,0]


# In[16]:

plt.plot(proba, YY)
plt.plot([0, max_proba], [max_value, max_value], 'r')


# In[17]:

fig = plt.figure(0)
ax = fig.add_subplot(111, aspect='equal')
ax.scatter(X, Y)
ax.scatter(1, max_value, 50, 'r')
plt.xlim([-0.1,2 * np.pi + 0.1])
plt.ylim([-1.5,1.5])


# ### Method 2: Sample from distribution

# In[18]:

sampled_values = conditional_gmm.sample(10)


# In[19]:

fig = plt.figure(0)
ax = fig.add_subplot(111, aspect='equal')
ax.scatter(X, Y)
ax.scatter(np.ones(sampled_values.shape), sampled_values, 50, 'r')
plt.xlim([-0.1,2 * np.pi + 0.1])
plt.ylim([-1.5,1.5])


# ### Method 3: Weighted mean

# In[20]:

weighted_mean = conditional_gmm.priors.dot(conditional_gmm.means)


# In[21]:

plt.plot(proba, YY)
plt.plot([0, max_proba], [weighted_mean, weighted_mean], 'r')


# **Note that it is not the max.**

# In[22]:

fig = plt.figure(0)
ax = fig.add_subplot(111, aspect='equal')
ax.scatter(X, Y)
ax.scatter(1, weighted_mean, 50, 'r')
plt.xlim([-0.1,2 * np.pi + 0.1])
plt.ylim([-1.5,1.5])


# ### Method 4: Mean of most important Gaussian

# In[23]:

max_weight = np.max(conditional_gmm.priors)
max_ind = np.where(conditional_gmm.priors == max_weight)[0][0]

fig = plt.figure(0)
ax = fig.add_subplot(111, aspect='equal')
ax.scatter(X, Y)
ax.scatter(1, conditional_gmm.means[max_ind], 50, 'r')
plt.xlim([-0.1,2 * np.pi + 0.1])
plt.ylim([-1.5,1.5])


# ### Ranking
# 
# - Max
# - Mean of most weighted Gaussian
# - Weighted Mean
# - Sample

# # Inverse Model: example of redundancy/multimodality/multimodes
# 
# ## Find local data for y=0.5

# In[24]:

_, ind = Y_tree.query(0.5, k=50)     
local_X = X[ind]
local_Y = Y[ind]


# ## Plot local data at y=0.5

# In[25]:

fig = plt.figure(0)
ax = fig.add_subplot(111, aspect='equal')
ax.scatter(local_X, local_Y)
plt.xlim([-0.1,2 * np.pi + 0.1])
plt.ylim([-1.5,1.5])


# **This is the problem of redundancy**

# ## Fit local GMM with 3 components

# In[26]:

local_gmm = GMM(n_components=3, covariance_type='full')
local_data = np.squeeze(np.concatenate((local_X, local_Y), axis=2))
local_gmm.fit(local_data)


# ## Plot local GMM

# In[27]:

local_gmminf = gmr.gmm.GMM(3, 
                           local_gmm.weights_, 
                           local_gmm.means_, 
                           local_gmm.covars_)


# In[28]:

ellipses = local_gmminf.to_ellipses(2)


# In[29]:

from matplotlib.patches import Ellipse

fig = plt.figure()
ax = fig.add_subplot(111, aspect='equal')

for ellipse in ellipses:
    e = Ellipse(xy=ellipse[0], angle=ellipse[1][0]*180/np.pi, width=ellipse[1][1], height=ellipse[1][2])
    ax.add_artist(e)
    e.set_clip_box(ax.bbox)
    e.set_alpha(0.5)
    e.set_facecolor(rng.rand(3))

plt.xlim([-0.1,2 * np.pi + 0.1])
plt.ylim([-1.5,1.5])


# **Note that the GMM model well the bi-modality of the data**

# ## We compute and plot the conditional for y=0.5

# In[30]:

conditional_gmm = local_gmminf.condition([1], [0.5])


# In[31]:

proba = conditional_gmm.to_probability_density(X)
plt.plot(X, proba)


# **Note that the GMM model well the bi-modality of the data**

# ### Method 1: Find the Max

# In[32]:

max_proba = np.max(proba)
max_id = np.where(proba == max_proba)[0]
max_value = X[max_id][0,0]


# In[33]:

plt.plot(X, proba)
plt.plot([max_value, max_value], [0, max_proba] , 'r')


# In[34]:

fig = plt.figure(0)
ax = fig.add_subplot(111, aspect='equal')
ax.scatter(X, Y)
ax.scatter(max_value, 0.5, 50, 'r')
plt.xlim([-0.1,2 * np.pi + 0.1])
plt.ylim([-1.5,1.5])


# ### Method 2: Sample from distribution

# In[35]:

sampled_values = conditional_gmm.sample(10)


# In[36]:

fig = plt.figure(0)
ax = fig.add_subplot(111, aspect='equal')
ax.scatter(X, Y)
ax.scatter(sampled_values, 0.5 * np.ones(sampled_values.shape), 50, 'r')
plt.xlim([-0.1,2 * np.pi + 0.1])
plt.ylim([-1.5,1.5])


# ### Method 3: Weighted mean

# In[37]:

weighted_mean = conditional_gmm.priors.dot(conditional_gmm.means)


# In[38]:

plt.plot(X, proba)
plt.plot([weighted_mean, weighted_mean],[0, max_proba], 'r')


# ** This is not desired **

# In[39]:

fig = plt.figure(0)
ax = fig.add_subplot(111, aspect='equal')
ax.scatter(X, Y)
ax.scatter(weighted_mean, 0.5, 50, 'r')
plt.xlim([-0.1,2 * np.pi + 0.1])
plt.ylim([-1.5,1.5])


# ** This is not desired **

# ### Method 4: Mean of most important Gaussian

# In[40]:

max_weight = np.max(conditional_gmm.priors)
max_ind = np.where(conditional_gmm.priors == max_weight)[0][0]

fig = plt.figure(0)
ax = fig.add_subplot(111, aspect='equal')
ax.scatter(X, Y)
ax.scatter(conditional_gmm.means[max_ind], 0.5, 50, 'r')
plt.xlim([-0.1,2 * np.pi + 0.1])
plt.ylim([-1.5,1.5])


# ### Ranking
# 
# - Max
# - Mean of most weighted Gaussian
# - Sample
# - Weighted Mean

# # Bigger redundancy: the need to adapt the number of gaussian

# ## Generate more redundant data

# In[41]:

rng = np.random.RandomState(10)
X = np.linspace(0, 6 * np.pi, 1000)[:, np.newaxis]
Y = 6*np.sin(X).ravel() + rng.normal(0, 0.5, X.shape[0])
Y = Y[:, np.newaxis]


# ## Plot data

# In[42]:

fig = plt.figure(0)
ax = fig.add_subplot(111, aspect='equal')
ax.scatter(X, Y)
plt.xlim([-0.1, 6 * np.pi + 0.1])
plt.ylim([-6.5,6.5])


# ## Build Tree to find NN

# In[43]:

from sklearn.neighbors import KDTree
X_tree = KDTree(X, leaf_size=2)
Y_tree = KDTree(Y, leaf_size=2)


# # Inverse Model
# 
# ## Find local data for y=3

# In[44]:

_, ind = Y_tree.query(3, k=100)     
local_X = X[ind]
local_Y = Y[ind]


# ## Plot local data at y=3

# In[45]:

fig = plt.figure(0)
ax = fig.add_subplot(111, aspect='equal')
ax.scatter(local_X, local_Y)
plt.xlim([-0.1, 6 * np.pi + 0.1])
plt.ylim([-6.5,6.5])


# ## Fit local GMM with 3 components

# In[46]:

local_gmm = GMM(n_components=3, covariance_type='full')
local_data = np.squeeze(np.concatenate((local_X, local_Y), axis=2))
local_gmm.fit(local_data)


# ## Plot local GMM

# In[47]:

local_gmminf = gmr.gmm.GMM(3, 
                           local_gmm.weights_, 
                           local_gmm.means_, 
                           local_gmm.covars_)


# In[48]:

ellipses = local_gmminf.to_ellipses(2)


# In[49]:

from matplotlib.patches import Ellipse

fig = plt.figure()
ax = fig.add_subplot(111, aspect='equal')
ax.scatter(local_X, local_Y)


for ellipse in ellipses:
    e = Ellipse(xy=ellipse[0], angle=ellipse[1][0]*180/np.pi, width=ellipse[1][1], height=ellipse[1][2])
    ax.add_artist(e)
    e.set_clip_box(ax.bbox)
    e.set_alpha(0.5)
    e.set_facecolor(rng.rand(3))

plt.xlim([-0.1, 6 * np.pi + 0.1])
plt.ylim([-6.5,6.5])


# **There is not enough gaussian to model this properly**
# 
# Such problem are arise quite often in our simple example, e.g. multidimensional arm
# 
# **Note that the same problem might arise with a simple NN model if we take the weighted mean of more than 1 neigborh**

# ## We compute and plot the conditional for y=3

# In[50]:

conditional_gmm = local_gmminf.condition([1], [3])


# In[51]:

proba = conditional_gmm.to_probability_density(X)
plt.plot(X, proba)


# ## Find the Max

# In[52]:

max_proba = np.max(proba)
max_id = np.where(proba == max_proba)[0]
max_value = X[max_id][0,0]


# In[53]:

plt.plot(X, proba)
plt.plot([max_value, max_value], [0, max_proba] , 'r')


# In[54]:

fig = plt.figure(0)
ax = fig.add_subplot(111, aspect='equal')
ax.scatter(X, Y)
ax.scatter(max_value, 3, 50, 'r')
plt.xlim([-0.1, 6 * np.pi + 0.1])
plt.ylim([-6.5,6.5])


# ** This is not desired **

# # Would DPGMM solve the problem?
# 
# Short answer: I did'nt succeed, on paper it is what we look for but in pratice it did not work on this simple example.
# 
# I tried as above, simply replace GMM by DPGMM. It looked to me some prior was pulling Gaussians towards [0,0] but I am not sure at all..
# 
# So I tried scaling the data, but this didn't help.
# 
# ## Fit local DPGMM

# In[55]:

from sklearn.mixture import DPGMM
local_gmm = DPGMM(n_components=10, 
                  covariance_type='full', 
                  alpha=1,
                  tol=1e-3, 
                  params='wmc',
                  init_params='wmc')
local_data = np.squeeze(np.concatenate((local_X, local_Y), axis=2))
local_gmm.fit(local_data)


# **Play a bit with the parameter to get a sense of what is a happening: http://scikit-learn.org/stable/modules/generated/sklearn.mixture.DPGMM.html#sklearn.mixture.DPGMM**
# 
# ## Plot local GMM

# In[56]:

local_gmminf = gmr.gmm.GMM(local_gmm.weights_.shape[0], 
                           local_gmm.weights_, 
                           local_gmm.means_, 
                           local_gmm._get_covars())


# In[57]:

ellipses = local_gmminf.to_ellipses(2)


# In[58]:

from matplotlib.patches import Ellipse

fig = plt.figure()
ax = fig.add_subplot(111) ##, aspect='equal')
ax.scatter(local_X, local_Y)


for ellipse in ellipses:
    e = Ellipse(xy=ellipse[0], angle=ellipse[1][0]*180/np.pi, width=ellipse[1][1], height=ellipse[1][2])
    ax.add_artist(e)
    e.set_clip_box(ax.bbox)
    e.set_alpha(0.5)
    e.set_facecolor(rng.rand(3))

plt.xlim([-0.1, 6 * np.pi + 0.1])
plt.ylim([-6.5,6.5])


# **Note that the Gaussian you see at [0,0] are the ones that are not in use, there is a way to know this. See here: http://scikit-learn.org/stable/auto_examples/mixture/plot_gmm_sin.html#example-mixture-plot-gmm-sin-py**

# ## Let's try to estimate the number of clusters with a clustering algorithm
# 
# See: http://scikit-learn.org/stable/modules/clustering.html
# 
# I picked: http://scikit-learn.org/stable/modules/generated/sklearn.cluster.AffinityPropagation.html#sklearn.cluster.AffinityPropagation
# 

# In[59]:

from sklearn.cluster import AffinityPropagation

af = AffinityPropagation(damping=0.6)
af.fit(local_data)
n_clusters = len(af.cluster_centers_)
print("We found {} clusters".format(n_clusters))


# ## Plot the clusters

# In[60]:

fig = plt.figure()
ax = fig.add_subplot(111) ##, aspect='equal')
ax.scatter(local_X, local_Y, 5)

for cs in af.cluster_centers_:
    ax.scatter(cs[0], cs[1], 100, rng.rand(3))

plt.xlim([-0.1, 6 * np.pi + 0.1])
plt.ylim([-6.5,6.5])


# ## Let's use this number ot fit the GMM

# In[61]:

local_gmm = GMM(n_components=n_clusters, covariance_type='full')
local_gmm.fit(local_data)


# In[62]:

local_gmminf = gmr.gmm.GMM(n_clusters, 
                           local_gmm.weights_, 
                           local_gmm.means_, 
                           local_gmm.covars_)


# In[63]:

ellipses = local_gmminf.to_ellipses(2)


# In[64]:


from matplotlib.patches import Ellipse

fig = plt.figure()
ax = fig.add_subplot(111, aspect='equal')
ax.scatter(local_X, local_Y)


for ellipse in ellipses:
    e = Ellipse(xy=ellipse[0], angle=ellipse[1][0]*180/np.pi, width=ellipse[1][1], height=ellipse[1][2])
    ax.add_artist(e)
    e.set_clip_box(ax.bbox)
    e.set_alpha(0.5)
    e.set_facecolor(rng.rand(3))

plt.xlim([-0.1, 6 * np.pi + 0.1])
plt.ylim([-6.5,6.5])


# ## Compute the conditional for y=3

# In[65]:

conditional_gmm = local_gmminf.condition([1], [3])


# In[66]:

proba = conditional_gmm.to_probability_density(X)
plt.plot(X, proba)


# ## Find the max

# In[67]:

max_proba = np.max(proba)
max_id = np.where(proba == max_proba)[0]
max_value = X[max_id][0,0]


# In[68]:

plt.plot(X, proba)
plt.plot([max_value, max_value], [0, max_proba] , 'r')


# In[69]:

fig = plt.figure(0)
ax = fig.add_subplot(111, aspect='equal')
ax.scatter(X, Y)
ax.scatter(max_value, 3, 50, 'r')
plt.xlim([-0.1, 6 * np.pi + 0.1])
plt.ylim([-6.5,6.5])


# **Look good!**
# 
# # Discussion
# 
# Finding the max of the conditional gmm is to be privileged. The weighted mean inevitably will give wrong answers. 
# 
# Finding the maximum is impossible analytically. Think of those cases when the maximum might not be unique (e.g. two Gaussians well separated of equal weights), it makes it really hard to find an analytical solution. I am sure talented mathematicians worked hard on this, if nothing is out there yet there must be a reason.
# 
# Some have tried with gradient descent sort of things:
# 
# -  Pulkkinen, Seppo, Marko Mikael Mäkelä, and Napsu Karmitsa. "A continuation approach to mode-finding of multivariate Gaussian mixtures and kernel density estimates." Journal of Global Optimization 56.2 (2013): 459-487. https://www.researchgate.net/profile/Seppo_Pulkkinen/publication/255996172_A_continuation_approach_to_mode-finding_of_multivariate_Gaussian_mixtures_and_kernel_density_estimates/links/00b495215e455d90ed000000.pdf
# 
# - Carreira-Perpiñán, Miguel Á. "Mode-finding for mixtures of Gaussian distributions." IEEE Transactions on Pattern Analysis & Machine Intelligence11 (2000): 1318-1323. http://faculty.ucmerced.edu/mcarreira-perpinan/papers/cs-99-03.pdf
# De le même personne: http://www.cs.toronto.edu/~miguel/research/GMmodes.html
# Code matlab: http://www.cs.toronto.edu/~miguel/research/software/gm_modes.tar.gz
# 
# - des slides: https://www.utu.fi/en/units/sci/units/math/Research/optimization/Documents/sovmat13012012.pdf
# 
# 
# But, as the function is generally very smooth, it should be easy to find the maximum using an optimization algorithm. CMA-ES should perfectly do the job, a quick study could be done on which initialization parameter to use.
# 
# I propose the following for initializing CMA-ES: "mean = centroid (pondéré) des Gaussiennes de la GMM à évaluer, sigma = covariance (pondéré) des centres par rapports au mean(centroid) + régularisation sur la diagonale"
# 
# Finally, in most scenario, it is way better to get stuck in a local maximum than sampling randomly or using the weighted mean. The next option is thus to take the mean of the Gaussian that has the largest weight.
# 
# 
# ## Calinon's work  and ILO-GMM paper
# 
# For Calinon's work and ILO-GMM paper, I believe the weighted mean works because of the properties of the problem, i.e. trajectories. You are never at two positions at the same time. Or, if time is not in the input space, the feature are made such that the model is unimodal (e.g. adding current speed (dx, dy)).
# 
# It would be worth checking Calinon litterature to see if they tried the following XP: A user move an obect from position A to B in straight line. We place an obstacle on the way. The user now decide randomly to go one side or the other of the obstacle. 
# 
# This creates a bimodal dataset, very similar to what we encounter in inverse model for robotics. Taking the weighted mean would make you run into the obstacle.
# 
# How would Calinon's work handle this? I guess they run very early into such problems and might have found a solution. It is worth investigating.
# 
# 
# # Conclusion
# 
# It is important to consider the problem of redundancy/multilodalities/multimodes when designing our algorithms. Such problem is omnipresent in our robotic experiments when considering inverse models. 
# 
# NN (with k > 1) is not solving this problem. ILO-GMM, which models the multimodality, is promising but could be improved by adapting the number of Gaussians. For this we proposed to use a clustering algorithm.
# 
# The clustering algorithm might fail when the data are not well separated but still non-linear. In such case, the clustering algorithm might tell us there is only one cluster, but modeling with one Gaussian won't be enough. To limit that problem we could:
# - start by enforcing a minimum in the number of Gaussian to use, e.g. 3.
# - use 3x the number of clusters as the number of components of the local GMM
# 
# It would be nice to make a proper benchmark with a similar problem in a more concrete scenario, e.g. multi-segment arm.
# 
# A future consideration should be the definition of locality, we consider each dimension as having the same local importance, that might not be the case in many practical problem. Can the robot/agent finds out by itself the weight to assign to each dimension for defining locality?

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<p><strong>The following serves as a simple introduction to ILO-GMM. It also illustrate, using carefully selected examples, problems we might encounter when learning forward and, especially, inverse models.</strong></p>
<h1 id="Overview">Overview<a class="anchor-link" href="#Overview">&#182;</a></h1><p>We compare different inference method for ILO-GMM:</p>
<ul>
<li>Weighted mean of the Gaussian center by their relative importance</li>
<li>Max of the probability distribution</li>
<li>Mean of the most important (bigger weight) Gaussian</li>
<li>Sampling the probability distribution</li>
</ul>
<p>The data are sampled from a 1 period sinus with a small Gaussian noise.</p>
<p>We investigate the forward and inverse model. We show that the inverse problem, which illustrates the redundancy/multimodal/multimodes problem, poses problem to some infering methods.</p>
<p>We conclude that, for most practical robotic problems involving inverse models, the order of inference methods to use is as follow:</p>
<ul>
<li>Max of the probability distribution</li>
<li>Mean of the most important (bigger weight) Gaussian</li>
<li>Sampling the probability distribution</li>
<li>Weighted mean of the Gaussian center by their relative importance</li>
</ul>
<p>We then consider the problem of redundancy using a 3 period sinus. This problem helps visualizing a limitation of ILO-GMM that is the fixed number of Gaussian. We explore briefly two options:</p>
<ul>
<li>DPGMM: <a href="http://scikit-learn.org/stable/modules/generated/sklearn.mixture.DPGMM.html#sklearn.mixture.DPGMM">http://scikit-learn.org/stable/modules/generated/sklearn.mixture.DPGMM.html#sklearn.mixture.DPGMM</a></li>
<li>use of a clustering algortihm to identify the number of modes. DPGMM, wile showing promises in theroy, is not convincing experimentally on our simple example. The clustering shows more practical promises.</li>
</ul>
<h1 id="Before-starting">Before starting<a class="anchor-link" href="#Before-starting">&#182;</a></h1><p>The code below requires the following python modules:</p>
<ul>
<li><a href="http://www.numpy.org/">http://www.numpy.org/</a></li>
<li><a href="http://scikit-learn.org/stable/index.html">http://scikit-learn.org/stable/index.html</a></li>
<li><a href="https://github.com/AlexanderFabisch/gmr">https://github.com/AlexanderFabisch/gmr</a></li>
<li><a href="http://matplotlib.org/">http://matplotlib.org/</a></li>
</ul>
<h1 id="Simple-1-period-Sinus">Simple 1 period Sinus<a class="anchor-link" href="#Simple-1-period-Sinus">&#182;</a></h1><h2 id="Generate-data">Generate data<a class="anchor-link" href="#Generate-data">&#182;</a></h2>
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<div class=" highlight hl-ipython2"><pre><span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
<span class="n">rng</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">RandomState</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span> <span class="mi">500</span><span class="p">)[:,</span> <span class="n">np</span><span class="o">.</span><span class="n">newaxis</span><span class="p">]</span>
<span class="n">Y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">X</span><span class="p">)</span><span class="o">.</span><span class="n">ravel</span><span class="p">()</span> <span class="o">+</span> <span class="n">rng</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="n">X</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="n">Y</span> <span class="o">=</span> <span class="n">Y</span><span class="p">[:,</span> <span class="n">np</span><span class="o">.</span><span class="n">newaxis</span><span class="p">]</span>
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<h2 id="Plot-data">Plot data<a class="anchor-link" href="#Plot-data">&#182;</a></h2>
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<div class=" highlight hl-ipython2"><pre><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="kn">as</span> <span class="nn">plt</span>
<span class="o">%</span><span class="k">matplotlib</span> inline
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<div class=" highlight hl-ipython2"><pre><span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">111</span><span class="p">,</span> <span class="n">aspect</span><span class="o">=</span><span class="s">&#39;equal&#39;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">([</span><span class="o">-</span><span class="mf">0.1</span><span class="p">,</span><span class="mi">2</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span> <span class="o">+</span> <span class="mf">0.1</span><span class="p">])</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">([</span><span class="o">-</span><span class="mf">1.5</span><span class="p">,</span><span class="mf">1.5</span><span class="p">])</span>
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<pre>(-1.5, 1.5)</pre>
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