From 38375f4f19c692bd51fc4b300d81a754c00dbee7 Mon Sep 17 00:00:00 2001 From: Nicolas Date: Thu, 21 Feb 2013 11:13:51 +0000 Subject: [PATCH] Tutorial improved --- doc/Figures/tuto_kern_overview_basicdef.png | Bin 41365 -> 0 bytes ...f.png => tuto_kern_overview_basicplot.png} | Bin doc/Figures/tuto_kern_overview_multadd.png | Bin 0 -> 62861 bytes .../tuto_kern_overview_multperdecay.png | Bin 0 -> 37033 bytes doc/tuto_kernel_overview.rst | 68 +++++++++++++++--- 5 files changed, 58 insertions(+), 10 deletions(-) delete mode 100644 doc/Figures/tuto_kern_overview_basicdef.png rename doc/Figures/{kern-def.png => tuto_kern_overview_basicplot.png} (100%) create mode 100644 doc/Figures/tuto_kern_overview_multadd.png create mode 100644 doc/Figures/tuto_kern_overview_multperdecay.png diff --git a/doc/Figures/tuto_kern_overview_basicdef.png b/doc/Figures/tuto_kern_overview_basicdef.png deleted file mode 100644 index 7a96322840c72bdfafff901030e20b8f1572fdb1..0000000000000000000000000000000000000000 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should return:: rbf_variance | 0.7500 | | rbf_lengthscale | 2.0000 | | -.. figure:: Figures/tuto_kern_overview_basicdef.png +.. figure:: Figures/tuto_kern_overview_basicplot.png :align: center :height: 350px @@ -51,19 +51,67 @@ Operations to combine kernel In ``GPy``, two kernel objects can be added or multiplied. In both cases, two kinds of operations are possible since one can assume that the kernels to add/multiply are defined on the same space or on different subspaces. In other words, it is possible to use two kernels :math:`k_1,\ k_2` over :math:`\mathbb{R} \times \mathbb{R}` to create - * a kernel over :math:`\mathbb{R} \times \mathbb{R}`: :math:`k(x,y) = k_1(x,y) \times k_2(x,y)` - * a kernel over .. :math:`\mathbb{R}^2 \times \mathbb{R}^2`: :math:`k(x,y) = k_1(x_1,y_1) \times k_2(x_2,y_2)` + * a kernel over :math:`\mathbb{R} \times \mathbb{R}`: :math:`k(x,y) = k_1(x,y) \times k_2(x,y)` + * a kernel over :math:`\mathbb{R}^2 \times \mathbb{R}^2`: :math:`k(x,y) = k_1(x_1,y_1) \times k_2(x_2,y_2)` - multiplied combined with the usual ``+`` and ``*`` operators. :: +These two options are available in GPy under the name ``prod`` and ``prod_orthogonal`` (resp ``add`` and ``add_orthogonal`` for the addition). Here is a quick example :: + + k1 = GPy.kern.rbf(1,1.,2.) + k2 = GPy.kern.Matern32(1, 0.5, 0.2) + + # Product of kernels + k_prod = k1.prod(k2) + k_prodorth = k1.prod_orthogonal(k2) + + # Sum of kernels + k_add = k1.add(k2) + k_addorth = k1.add_orthogonal(k2) + + # plots + pb.figure(figsize=(8,8)) + pb.subplot(2,2,1) + k_prod.plot() + pb.title('prod') + pb.subplot(2,2,2) + k_prodorth.plot() + pb.title('prod_orthogonal') + pb.subplot(2,2,3) + k_add.plot() + pb.title('add') + pb.subplot(2,2,4) + k_addorth.plot() + pb.title('add_orthogonal') + pb.subplots_adjust(wspace=0.3, hspace=0.3) + +.. figure:: Figures/tuto_kern_overview_multadd.png + :align: center + :height: 1500px + +A shortcut for ``add`` and ``prod`` is provided by the usual ``+`` and ``*`` operators. Here is another example where we create a periodic kernel with some decay :: - k1 = GPy.kern.rbf(1,variance=1., lengthscale=2) - k2 = GPy.kern.Matern32(1,variance=1., lengthscale=2) + k1 = GPy.kern.rbf(1,1.,2) + k2 = GPy.kern.periodic_Matern52(1,variance=1e3, lengthscale=1, period = 1.5, lower=-5., upper = 5) - ker_add = k1 + k2 - print ker_add + k = k1 * k2 # equivalent to k = k1.prod(k2) + print k + + # Simulate sample paths + X = np.linspace(-5,5,501)[:,None] + Y = np.random.multivariate_normal(np.zeros(501),k.K(X),1) + +.. # plot + pb.figure(figsize=(8,4)) + pb.subplot(1,2,1) + k.plot() + pb.subplot(1,2,2) + pb.plot(X,Y.T) + pb.ylabel("Sample path") + pb.subplots_adjust(wspace=0.3) + +.. figure:: Figures/tuto_kern_overview_multperdecay.png + :align: center + :height: 800px - 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. Here for example ``ker_add.D`` will return ``1`` whereas ``ker_prod.D`` will return ``2``.