# Gaussian Processes

(c) 2016 by Chris Fonnesbeck

Example of simple GP fit, adapted from Stan's example-models repository.

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In [1]:

%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
sns.set()

import pymc3 as pm
from pymc3 import Model, MvNormal, HalfCauchy, sample, traceplot, summary, find_MAP, NUTS, Deterministic
import theano.tensor as T
from theano import shared
from theano.tensor.nlinalg import matrix_inverse

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In [2]:

x = np.array([-5, -4.9, -4.8, -4.7, -4.6, -4.5, -4.4, -4.3, -4.2, -4.1, -4,
-3.9, -3.8, -3.7, -3.6, -3.5, -3.4, -3.3, -3.2, -3.1, -3, -2.9,
-2.8, -2.7, -2.6, -2.5, -2.4, -2.3, -2.2, -2.1, -2, -1.9, -1.8,
-1.7, -1.6, -1.5, -1.4, -1.3, -1.2, -1.1, -1, -0.9, -0.8, -0.7,
-0.6, -0.5, -0.4, -0.3, -0.2, -0.1, 0, 0.1, 0.2, 0.3, 0.4, 0.5,
0.6, 0.7, 0.8, 0.9, 1, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8,
1.9, 2, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 3, 3.1,
3.2, 3.3, 3.4, 3.5, 3.6, 3.7, 3.8, 3.9, 4, 4.1, 4.2, 4.3, 4.4,
4.5, 4.6, 4.7, 4.8, 4.9, 5])

y = np.array([1.04442478194401, 0.948306088493654, 0.357037759697332, 0.492336514646604,
0.520651364364746, 0.112629866592809, 0.470995468454158, -0.168442254267804,
0.0720344402575861, -0.188108980535916, -0.0160163306512027,
-0.0388792158617705, -0.0600673630622568, 0.113568725264636,
0.447160403837629, 0.664421188556779, -0.139510743820276, 0.458823971660986,
0.141214654640904, -0.286957663528091, -0.466537724021695, -0.308185884317105,
-1.57664872694079, -1.44463024170082, -1.51206214603847, -1.49393593601901,
-2.02292464164487, -1.57047488853653, -1.22973445533419, -1.51502367058357,
-1.41493587255224, -1.10140254663611, -0.591866485375275, -1.08781838696462,
-0.800375653733931, -1.00764767602679, -0.0471028950122742, -0.536820626879737,
-0.151688056391446, -0.176771681318393, -0.240094952335518, -1.16827876746502,
-0.493597351974992, -0.831683011472805, -0.152347043914137, 0.0190364158178343,
-1.09355955218051, -0.328157917911376, -0.585575679802941, -0.472837120425201,
-0.503633622750049, -0.0124446353828312, -0.465529814250314,
-0.101621725887347, -0.26988462590405, 0.398726664193302, 0.113805181040188,
0.331353802465398, 0.383592361618461, 0.431647298655434, 0.580036473774238,
0.830404669466897, 1.17919105883462, 0.871037583886711, 1.12290553424174,
0.752564860804382, 0.76897960270623, 1.14738839410786, 0.773151715269892,
0.700611498974798, 0.0412951045437818, 0.303526087747629, -0.139399513324585,
-0.862987735433697, -1.23399179134008, -1.58924289116396, -1.35105117911049,
-0.990144529089174, -1.91175364127672, -1.31836236129543, -1.65955735224704,
-1.83516148300526, -2.03817062501248, -1.66764011409214, -0.552154350554687,
-0.547807883952654, -0.905389222477036, -0.737156477425302, -0.40211249920415,
0.129669958952991, 0.271142753510592, 0.176311762529962, 0.283580281859344,
0.635808289696458, 1.69976647982837, 1.10748978734239, 0.365412229181044,
0.788821368082444, 0.879731888124867, 1.02180766619069, 0.551526067300283])

N = len(y)

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In [3]:

squared_distance = lambda x, y: np.array([[(x[i] - y[j])**2 for i in range(len(x))] for j in range(len(y))])

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In [4]:

with Model() as gp_fit:

μ = np.zeros(N)

η_sq = HalfCauchy('η_sq', 5)
ρ_sq = HalfCauchy('ρ_sq', 5)
σ_sq = HalfCauchy('σ_sq', 5)

D = squared_distance(x, x)

# Squared exponential
Σ = T.fill_diagonal(η_sq * T.exp(-ρ_sq * D), η_sq + σ_sq)

obs = MvNormal('obs', μ, Σ, observed=y)

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This is what our initial covariance matrix looks like. Intuitively, every data point's Y-value correlates with points according to their squared distances.

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In [5]:

sns.heatmap(Σ.tag.test_value, xticklabels=False, yticklabels=False)

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Out[5]:

<matplotlib.axes._subplots.AxesSubplot at 0x7fa0aca4cdd8>

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The following generates predictions from the GP model in a grid of values:

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In [6]:

with gp_fit:

# Prediction over grid
xgrid = np.linspace(-6, 6)
D_pred = squared_distance(xgrid, xgrid)
D_off_diag = squared_distance(x, xgrid)

# Covariance matrices for prediction
Σ_pred = η_sq * T.exp(-ρ_sq * D_pred)
Σ_off_diag = η_sq * T.exp(-ρ_sq * D_off_diag)

# Posterior mean
μ_post = Deterministic('μ_post', T.dot(T.dot(Σ_off_diag, matrix_inverse(Σ)), y))
# Posterior covariance
Σ_post = Deterministic('Σ_post', Σ_pred - T.dot(T.dot(Σ_off_diag, matrix_inverse(Σ)), Σ_off_diag.T))

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In [7]:

with gp_fit:
gp_trace = pm.variational.svgd(n=300, n_particles=50)

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100%|██████████| 300/300 [00:22<00:00, 13.43it/s]

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In [8]:

traceplot(gp_trace, varnames=['η_sq', 'ρ_sq', 'σ_sq']);

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Sample from the posterior GP

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In [9]:

y_pred = [np.random.multivariate_normal(m, S) for m, S in zip(gp_trace['μ_post'], gp_trace['Σ_post'])]

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/home/wiecki/miniconda3/lib/python3.5/site-packages/ipykernel/__main__.py:1: RuntimeWarning: covariance is not positive-semidefinite.
if __name__ == '__main__':

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In [10]:

for yp in y_pred:
plt.plot(np.linspace(-6, 6), yp, 'c-', alpha=0.1);
plt.plot(x, y, 'r.')

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Out[10]:

[<matplotlib.lines.Line2D at 0x7fa08b5d38d0>]

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