New tutorial draft called 'A kernel overview'

doc/tuto_kernel_overview.rst

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+
+****************************
+tutorial : A kernel overview
+****************************
+
+First we import the libraries we will need ::
+
+    import pylab as pb
+    import numpy as np
+    import GPy
+    pb.ion()
+
+For most kernels, the dimension is the only mandatory parameter to define a kernel object. However, it is also possible to specify the values of the parameters. For example, the three following commands are valid for defining a squared exponential kernel (ie rbf or Gaussian) ::
+
+    ker1 = GPy.kern.rbf(D=1)  # Equivalent to ker1 = GPy.kern.rbf(D=1, variance=1., lengthscale=1.)
+    ker2 = GPy.kern.rbf(D=1, variance = 1.5, lengthscale=2.)
+    ker3 = GPy.kern.rbf(1, .5, .5)
+
+A `plot` and a `print` functions are implemented to represent kernel objects ::
+    
+    print ker1
+
+    ker1.plot()
+    ker2.plot()
+    ker3.plot()
+
+.. figure::  Figures/tuto_kern_overview_basicdef.png
+    :align:   center
+    :height: 350px
+
+::
+
+    import pylab as pb
+    import numpy as np
+    import GPy
+    pb.ion()
+
+    ker1 = GPy.kern.rbf(D=1)  # Equivalent to ker1 = GPy.kern.rbf(D=1, variance=1., lengthscale=1.)
+    ker2 = GPy.kern.rbf(D=1, variance = .75, lengthscale=3.)
+    ker3 = GPy.kern.rbf(1, .5, .25)
+
+    ker1.plot()
+    ker2.plot()
+    ker3.plot()
+    #pb.savefig("Figures/tuto_kern_overview_basicdef.png")
+
+    kernels = [GPy.kern.rbf(1), GPy.kern.exponential(1), GPy.kern.Matern32(1), GPy.kern.Matern52(1),  GPy.kern.Brownian(1), GPy.kern.bias(1), GPy.kern.linear(1), GPy.kern.spline(1), GPy.kern.periodic_exponential(1), GPy.kern.periodic_Matern32(1), GPy.kern.periodic_Matern52(1), GPy.kern.white(1)]
+    kernel_names = ["GPy.kern.rbf", "GPy.kern.exponential", "GPy.kern.Matern32", "GPy.kern.Matern52", "GPy.kern.Brownian", "GPy.kern.bias", "GPy.kern.linear", "GPy.kern.spline", "GPy.kern.periodic_exponential", "GPy.kern.periodic_Matern32", "GPy.kern.periodic_Matern52", "GPy.kern.white"]
+    
+    pb.figure(figsize=(16,12))
+    pb.subplots_adjust(wspace=.5, hspace=.5)
+    for i, kern in enumerate(kernels):
+       pb.subplot(3,4,i+1)
+       kern.plot(x=7.5,plot_limits=[0.00001,15.])
+       pb.title(kernel_names[i]+ '\n')
+       #pb.axes([.1,.1,.8,.7])
+       #pb.figtext(.5,.9,'Foo Bar', fontsize=18, ha='center')
+       #pb.figtext(.5,.85,'Lorem ipsum dolor sit amet, consectetur adipiscing elit',fontsize=10,ha='center')
+
+    # actual plot for the noise
+    i = 11
+    X = np.linspace(0.,15.,201)
+    WN = 0*X
+    WN[100] = 1.
+    pb.subplot(3,4,i+1)
+    pb.plot(X,WN,'b')
+
+Implemented kernels
+===================
+
+Many kernels are already implemented in GPy. Here is a summary of most of them:
+
+.. figure::  Figures/tuto_kern_overview_allkern.png
+    :align:  center
+    :height: 800px
+
+On the other hand, it is possible to use the `sympy` package to build new kernels. This will be the subject of another tutorial.
+
+Operations to combine kernel
+============================
+
+In ``GPy``, kernel objects can be combined with the usual ``+`` and ``*`` operators. ::
+    
+    k1 = GPy.kern.rbf(1,variance=1., lengthscale=2)
+    k2 = GPy.kern.Matern32(1,variance=1., lengthscale=2)
+
+    ker_add = k1 + k2
+    print ker_add
+
+    ker_prod = k1 * k2
+    print ker_prod
+
+Note that by default, the operator ``+`` adds kernels defined on the same input space whereas ``*`` assumes that the kernels are defined on different input spaces. ::
+    
+    ker_add.D
+    ker_prod.D
+
+In order to add kernels defined on the different input spaces, the required command is::
+
+    ker_add_orth = k1.add_orthogonal(k2)
+
+The resulting kernel is
+    ker_add_orth.plot(plot_limits=[[-10,-10],[10,10]])
+
+.. figure::  Figures/tuto_kern_overview_add_orth.png
+    :align:  center
+    :height: 350px
+
+Example : Building an ANOVA kernel
+==================================
+
+In two dimensions ANOVA kernels have the following form: 
+
+.. math::
+
+    k_{ANOVA}(x,y) = \prod_{i=1}^2 (1 + k_i(x_i,y_i)) = 1 + k_1(x_1,y_1) + k_2(x_2,y_2) + k_1(x_1,y_1) \times k_2(x_2,y_2).
+
+Let us assume that we want to define an ANOVA kernel with a Matern 3/2 kernel for :math:`k_i`. As seen previously, we can define this kernel as follow::
+
+    k_cst = GPy.kern.bias(1,variance=1.)
+    k_mat = GPy.kern.Matern52(1,variance=1., lengthscale=3)
+    Kanova = (k_cst + k_mat) * (k_cst + k_mat)
+    print Kanova
+
+Note the ties between the lengthscales of ``Kanova`` to keep the number of lengthscales equal to 2. On the other hand, there are four variance terms in the new parameterization: one for each term of the right hand sign of the equation above. We can illustrate the use of this kernel on a toy example::
+
+    # sample inputs and outputs
+    X = np.random.uniform(-3.,3.,(40,2))
+    Y = 0.5*X[:,:1] + 0.5*X[:,1:] + 2*np.sin(X[:,:1]) * np.sin(X[:,1:])
+
+    # Create GP regression model
+    m = GPy.models.GP_regression(X,Y,Kanova)
+    m.plot()
+
+
+.. figure::  Figures/tuto_kern_overview_mANOVA.png
+    :align:  center
+    :height: 350px
+
+As :math:`k_{ANOVA}` corresponds to the sum of 4 kernels, the best predictor can be splited in a sum of 4 functions 
+
+.. math::
+
+    bp(x) & = k(x)^t K^{-1} Y \\
+          & = (1 + k_1(x_1) +  k_2(x_2) +  k_1(x_1)k_2(x_2))^t K^{-1} Y \\
+          & = 1^t K^{-1} Y + k_1(x_1)^t K^{-1} Y + k_2(x_2)^t K^{-1} Y + (k_1(x_1)k_2(x_2))^t K^{-1} Y
+
+The submodels can be represented with the option ``which_function`` of ``plot``: ::
+    
+    pb.figure(figsize=(20,5))
+    pb.subplots_adjust(wspace=0.5)
+    pb.subplot(1,5,1)
+    m.plot()
+    pb.subplot(1,5,2)
+    pb.ylabel("=   ",rotation='horizontal',fontsize='30')
+    pb.subplot(1,5,3)
+    m.plot(which_functions=[False,True,False,False])
+    pb.ylabel("cst          +",rotation='horizontal',fontsize='30')
+    pb.subplot(1,5,4)
+    m.plot(which_functions=[False,False,True,False])
+    pb.ylabel("+   ",rotation='horizontal',fontsize='30')
+    pb.subplot(1,5,5)
+    pb.ylabel("+   ",rotation='horizontal',fontsize='30')
+    m.plot(which_functions=[False,False,False,True])
+
+
+.. figure::  Figures/tuto_kern_overview_mANOVAdec.png
+    :align:  center
+    :height: 200px