# Gaussian Process Regression¶

Let’s set some setting for this Jupyter Notebook.

In [1]:

%matplotlib inline
from warnings import filterwarnings
filterwarnings("ignore")
import os
os.environ['THEANO_FLAGS'] = 'device=cpu'

import numpy as np
import pandas as pd
import pymc3 as pm
import seaborn as sns
import matplotlib.pyplot as plt
np.random.seed(12345)
rc = {'xtick.labelsize': 20, 'ytick.labelsize': 20, 'axes.labelsize': 20, 'font.size': 20,
'legend.fontsize': 12.0, 'axes.titlesize': 10, "figure.figsize": [12, 6]}
sns.set(rc = rc)
from IPython.core.interactiveshell import InteractiveShell
InteractiveShell.ast_node_interactivity = "all"


Now, let’s import the GaussianProcessRegression algorithm from the pymc-learn package.

In [2]:

import pmlearn
from pmlearn.gaussian_process import GaussianProcessRegressor
print('Running on pymc-learn v{}'.format(pmlearn.__version__))

Running on pymc-learn v0.0.1.rc0


## Step 1: Prepare the data¶

Generate synthetic data.

In [3]:

n = 150 # The number of data points
X = np.linspace(start = 0, stop = 10, num = n)[:, None] # The inputs to the GP, they must be arranged as a column vector

# Define the true covariance function and its parameters
length_scale_true = 1.0
signal_variance_true = 3.0
cov_func = signal_variance_true**2 * pm.gp.cov.ExpQuad(1, length_scale_true)

# A mean function that is zero everywhere
mean_func = pm.gp.mean.Zero()

# The latent function values are one sample from a multivariate normal
# Note that we have to call eval() because PyMC3 built on top of Theano
f_true = np.random.multivariate_normal(mean_func(X).eval(),
cov_func(X).eval() + 1e-8*np.eye(n), 1).flatten()

# The observed data is the latent function plus a small amount of Gaussian distributed noise
# The standard deviation of the noise is sigma
noise_variance_true = 2.0
y = f_true + noise_variance_true * np.random.randn(n)

## Plot the data and the unobserved latent function
fig = plt.figure()
ax = fig.gca()
ax.plot(X, f_true, "dodgerblue", lw=3, label="True f");
ax.plot(X, y, 'ok', ms=3, label="Data");
ax.set_xlabel("X"); ax.set_ylabel("y"); plt.legend();

In [4]:

from sklearn.model_selection import train_test_split
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.3)


## Step 2: Instantiate a model¶

In [5]:

model = GaussianProcessRegressor()

In [6]:

model?

[0;31mType:[0m        GaussianProcessRegressor
[0;31mString form:[0m GaussianProcessRegressor(kernel=None, prior_mean=None)
[0;31mFile:[0m        ~/pymc-learn/pmlearn/gaussian_process/gpr.py
[0;31mDocstring:[0m
Gaussian Process Regression built using PyMC3.

Fit a Gaussian process model and estimate model parameters using
MCMC algorithms or Variational Inference algorithms

Parameters
----------
prior_mean : mean object
The mean specifying the mean function of the GP. If None is passed,
the mean "pm.gp.mean.Zero()" is used as default.

Examples
--------
>>> from sklearn.datasets import make_friedman2
>>> from pmlearn.gaussian_process import GaussianProcessRegressor
>>> from pmlearn.gaussian_process.kernels import DotProduct, WhiteKernel
>>> X, y = make_friedman2(n_samples=500, noise=0, random_state=0)
>>> kernel = DotProduct() + WhiteKernel()
>>> gpr = GaussianProcessRegressor(kernel=kernel).fit(X, y)
>>> gpr.score(X, y) # doctest: +ELLIPSIS
0.3680...
>>> gpr.predict(X[:2,:], return_std=True) # doctest: +ELLIPSIS
(array([653.0..., 592.1...]), array([316.6..., 316.6...]))

Reference
----------
Rasmussen and Williams (2006). Gaussian Processes for Machine Learning.



## Step 3: Perform Inference¶

In [7]:

model.fit(X_train, y_train, inference_args={"n": 1000})

Average Loss = 416.16: 100%|██████████| 1000/1000 [00:02<00:00, 474.85it/s]
Finished [100%]: Average Loss = 415.55

Out[7]:

GaussianProcessRegressor(kernel=None, prior_mean=None)


## Step 4: Diagnose convergence¶

In [8]:

model.plot_elbo()

In [11]:

pm.traceplot(model.trace);

In [12]:

pm.traceplot(model.trace, lines = {"signal_variance": signal_variance_true,
"noise_variance": noise_variance_true,
"length_scale": length_scale_true},
varnames=["signal_variance", "noise_variance", "length_scale"]);

In [12]:

pm.forestplot(model.trace, varnames=["signal_variance", "noise_variance", "length_scale"]);


## Step 5: Critize the model¶

In [13]:

pm.summary(model.trace, varnames=["signal_variance", "length_scale", "noise_variance"])

Out[13]:

mean sd mc_error hpd_2.5 hpd_97.5
signal_variance__0 1.069652 1.472790 0.014072 0.016441 3.442904
length_scale__0_0 2.252174 2.231719 0.025261 0.119457 6.266875
noise_variance__0 3.066997 0.231325 0.002249 2.622387 3.516474
In [14]:

pm.plot_posterior(model.trace, varnames=["signal_variance", "noise_variance", "length_scale"],
figsize = [14, 8]);

In [15]:

# collect the results into a pandas dataframe to display
# "mp" stands for marginal posterior
pd.DataFrame({"Parameter": ["length_scale", "signal_variance", "noise_variance"],
"Predicted Mean Value": [float(model.trace["length_scale"].mean(axis=0)),
float(model.trace["signal_variance"].mean(axis=0)),
float(model.trace["noise_variance"].mean(axis=0))],
"True value": [length_scale_true, signal_variance_true, noise_variance_true]})

Out[15]:

Parameter Predicted Mean Value True value
0 length_scale 2.252174 1.0
1 signal_variance 1.069652 3.0
2 noise_variance 3.066997 2.0

## Step 6: Use the model for prediction¶

In [9]:

y_predict1 = model.predict(X_test)

100%|██████████| 2000/2000 [00:14<00:00, 135.47it/s]

In [10]:

y_predict1

Out[10]:

array([ 0.00166453,  0.07415753,  0.07185864,  0.01505948,  0.02280044,
-0.00041549, -0.02338406,  0.01753743,  0.02065263,  0.00825294,
0.02449021,  0.06761137,  0.04990807,  0.01614856, -0.03135927,
-0.00813461,  0.04545187, -0.03770688,  0.06116857,  0.06864128,
0.04164327, -0.01700696,  0.01389948, -0.02395358, -0.01853882,
-0.02147422,  0.05869176, -0.02825002,  0.01058576,  0.04180517,
0.01563565, -0.0086748 ,  0.01048786, -0.02464047,  0.0639958 ,
-0.02110329, -0.03658159,  0.0552832 , -0.00030839,  0.03097778,
0.00415975,  0.05252889,  0.00894602,  0.06400553, -0.05004306])

In [ ]:

model.score(X_test, y_test)

In [12]:

model.save('pickle_jar/gpr')


### Use already trained model for prediction¶

In [13]:

model_new = GaussianProcessRegressor()
model_new.score(X_test, y_test)

100%|██████████| 2000/2000 [00:14<00:00, 136.18it/s]

Out[13]:

-0.0049724872177634438


## Multiple Features¶

In [14]:

num_pred = 2
X = np.random.randn(1000, num_pred)
noise = 2 * np.random.randn(1000,)
Y = X.dot(np.array([4, 5])) + 3 + noise

In [15]:

y = np.squeeze(Y)

In [16]:

model_big = GaussianProcessRegressor()

In [17]:

model_big.fit(X, y, inference_args={"n" : 1000})

Average Loss = 6,077.1: 100%|██████████| 1000/1000 [02:17<00:00,  7.11it/s]
Finished [100%]: Average Loss = 6,056.9

Out[17]:

GaussianProcessRegressor(prior_mean=0.0)

In [18]:

pm.summary(model_big.trace, varnames=["signal_variance", "length_scale", "noise_variance"])

Out[18]:

mean sd mc_error hpd_2.5 hpd_97.5
signal_variance__0 5.420972 4.049228 0.041386 0.635510 12.967287
length_scale__0_0 2.460546 2.034025 0.021939 0.192283 6.279051
length_scale__0_1 2.437830 1.994458 0.018703 0.267447 6.202378
noise_variance__0 7.173519 4.732447 0.042548 0.936711 16.368718

## MCMC¶

In [ ]:

model2 = GaussianProcessRegressor()
model2.fit(X_train, y_train, inference_type='nuts')

In [ ]:

pm.traceplot(model2.trace, lines = {"signal_variance": signal_variance_true,
"noise_variance": noise_variance_true,
"length_scale": length_scale_true},
varnames=["signal_variance", "noise_variance", "length_scale"]);

In [ ]:

pm.gelman_rubin(model2.trace, varnames=["signal_variance", "noise_variance", "length_scale"])

In [ ]:

pm.energyplot(model2.trace);

In [ ]:

pm.forestplot(model2.trace, varnames=["signal_variance", "noise_variance", "length_scale"]);

In [ ]:

pm.summary(model2.trace, varnames=["signal_variance", "length_scale", "noise_variance"])

In [ ]:

# collect the results into a pandas dataframe to display
# "mp" stands for marginal posterior
pd.DataFrame({"Parameter": ["length_scale", "signal_variance", "noise_variance"],
"Predicted Mean Value": [float(model2.trace["length_scale"].mean(axis=0)),
float(model2.trace["signal_variance"].mean(axis=0)),
float(model2.trace["noise_variance"].mean(axis=0))],
"True value": [length_scale_true, signal_variance_true, noise_variance_true]})

In [ ]:

pm.plot_posterior(model2.trace, varnames=["signal_variance", "noise_variance", "length_scale"],
figsize = [14, 8]);

In [ ]:

y_predict2 = model2.predict(X_test)

In [ ]:

y_predict2

In [ ]:

model2.score(X_test, y_test)

In [ ]:

model2.save('pickle_jar/gpr2')
model2_new = GaussianProcessRegressor