<SYSTEM_TASK:>
Given the following text description, write Python code to implement the functionality described below step by step
<END_TASK>
<USER_TASK:>
Description:
 GPFlow first approximation
Step1:  Simulating Data
Simulate random uniform 4-d vector. Give N of this.
Step2:  Calculate distance
X can be interpreted as covariate matrix in which the first two columns are the longitud and latitude.
GPFlow requires that all the covariates (including spatio-temporal coordinates) are in X.
Step3:  Simulate the Gaussian Process $S$
Remmember that for a stationary Gaussian Process, the value at Z is independent of the betas (Covariate weights).
Mean 0's $\Sigma$ Correlation matrix
$$S = MVN(0,\Sigma) + \epsilon$$
$ \epsilon \sim N(0,\sigma^{2}) $
S is a realization of a spatial process.
Step4:  Simulate the Response Variable $y$     
$$y_1(x_1,x_2) = S(x_1,x_2) $$
$$y_2(x_1,x_2) = \beta_0 + x_3\beta_1 + x_4\beta_2 + S(x_1,x_2)$$
Step5:  GP Model !
This model is without covariates
Step6:  Model for $y_1$
Step7:  Like in tensorflow, m is a graph and has at least three nodes
Step8:  compare with original parameters (made from the simulation)
Step9:  it was close enough
GAUSSIAN PROCESS WITH LINEAR TREND
Defining the model
Step10:  Original parameters
phi = 0.05  ---> lengthscale
sigma2 = 1.0 ---> variance transform
nugget = 0.03 ---> likelihood variance
beta_0 = 10.0 ---> mean_function b
beta_1 = 1.5 ---> mean_fucntionA [2]
beta_2 = -1.0 ---> mean_functionA [3]
mean_functionA[0] and mean_functionA[1] are the betas for for x and y (coordinates respectively)
Without spatial coordinates as covariates
Step13:  Custom made mean function (Erick Chacón )
Step14:  Now we can use the special mean function without the coordinates (covariates).
Step15:  Only 2 parameters now!
Step16:  Original parameters
phi = 0.05  ---> lengthscale
sigma2 = 1.0 ---> variance transform
nugget = 0.03 ---> likelihood variance
beta_0 = 10.0 ---> mean_function b
beta_1 = 1.5 ---> mean_fucntionA [2]
beta_2 = -1.0 ---> mean_functionA [3]
<ASSISTANT_TASK:>
Python Code:
## Import modules
import numpy as np
import scipy.spatial.distance as sp
from matplotlib import pyplot as plt
plt.style.use('ggplot')
Explanation: GPFlow first approximation
End of explanation
## Parameter definitions
N = 1000
phi = 0.05
sigma2 = 1.0
beta_0 = 10.0
beta_1 = 1.5
beta_2 = -1.0
# AL NAGAT
nugget = 0.03
X = np.random.rand(N,4)
Explanation: Simulating Data
Simulate random uniform 4-d vector. Give N of this.
End of explanation
points = X[:,0:2]
dist_points = sp.pdist(points)
## Reshape the vector to square matrix 
distance_matrix = sp.squareform(dist_points)
correlation_matrix  = np.exp(- distance_matrix / phi)
covariance_matrix = correlation_matrix * sigma2
plt.imshow(covariance_matrix)
Explanation: Calculate distance
X can be interpreted as covariate matrix in which the first two columns are the longitud and latitude.
GPFlow requires that all the covariates (including spatio-temporal coordinates) are in X.
End of explanation
S = np.random.multivariate_normal(np.zeros(N), correlation_matrix) +\
        np.random.normal(size = N) * nugget
S.shape
# We convert to Matrix [1 column]
S = S.reshape(N,1)
## Plot x, y using as color the Gaussian process
plt.scatter(X[:, 0], X[:, 1], c = S)
plt.colorbar()
Explanation: Simulate the Gaussian Process $S$
Remmember that for a stationary Gaussian Process, the value at Z is independent of the betas (Covariate weights).
Mean 0's $\Sigma$ Correlation matrix
$$S = MVN(0,\Sigma) + \epsilon$$
$ \epsilon \sim N(0,\sigma^{2}) $
S is a realization of a spatial process.
End of explanation
# remmember index 0 is 1
mu = beta_0 + beta_1 * X[:, 2] + beta_2 * X[:, 3]
mu = mu.reshape(N,1)
Y1 = S
Y2 = mu + S
plt.scatter(X[:, 0], X[:, 1], c = Y2)
plt.colorbar()
S.shape
Explanation: Simulate the Response Variable $y$     
$$y_1(x_1,x_2) = S(x_1,x_2) $$
$$y_2(x_1,x_2) = \beta_0 + x_3\beta_1 + x_4\beta_2 + S(x_1,x_2)$$
End of explanation
# Import GPFlow
import GPflow as gf
# Defining the model Matern function with \kappa = 0.5 
k = gf.kernels.Matern12(2, lengthscales=1, active_dims = [0,1] )
type(k)
Explanation: GP Model !
This model is without covariates
End of explanation
m = gf.gpr.GPR(points, Y1, k)
## First guess
init_nugget = 0.001
m.likelihood.variance = init_nugget
print(m)
Explanation: Model for $y_1$
End of explanation
# Estimation using symbolic gradient descent
m.optimize()
print(m)
Explanation: Like in tensorflow, m is a graph and has at least three nodes: lengthscale, kern variance and likelihood variance
End of explanation
print(phi,sigma2,nugget)
print points.shape
print Y1.shape
Explanation: compare with original parameters (made from the simulation)
End of explanation
k = gf.kernels.Matern12(2, lengthscales=1, active_dims = [0,1] )
gf.mean_functions.Linear()
meanf = gf.mean_functions.Linear(np.ones((4,1)), np.ones(1))
m = gf.gpr.GPR(X, Y2, k, meanf)
m.likelihood.variance = init_nugget
print(m)
# Estimation
m.optimize()
print(m)
Explanation: it was close enough
GAUSSIAN PROCESS WITH LINEAR TREND
Defining the model
End of explanation
# Defining the model
k = gf.kernels.Matern12(2, lengthscales=1, active_dims = [0,1])
Explanation: Original parameters
phi = 0.05  ---> lengthscale
sigma2 = 1.0 ---> variance transform
nugget = 0.03 ---> likelihood variance
beta_0 = 10.0 ---> mean_function b
beta_1 = 1.5 ---> mean_fucntionA [2]
beta_2 = -1.0 ---> mean_functionA [3]
mean_functionA[0] and mean_functionA[1] are the betas for for x and y (coordinates respectively)
Without spatial coordinates as covariates
End of explanation
from GPflow.mean_functions import MeanFunction, Param
import tensorflow as tf
class LinearG(MeanFunction):
    
    y_i = A x_i + b
    
    def __init__(self, A=None, b=None):
        
        A is a matrix which maps each element of X to Y, b is an additive
        constant.
        If X has N rows and D columns, and Y is intended to have Q columns,
        then A must be D x Q, b must be a vector of length Q.
        
        A = np.ones((1, 1)) if A is None else A
        b = np.zeros(1) if b is None else b
        MeanFunction.__init__(self)
        self.A = Param(np.atleast_2d(A))
        self.b = Param(b)
    def __call__(self, X):
        Anew = tf.concat([np.zeros((2,1)),self.A],0)
        return tf.matmul(X, Anew) + self.b
Explanation: Custom made mean function (Erick Chacón )
End of explanation
meanf = LinearG(np.ones((2,1)), np.ones(1))
X.shape
Y2.shape
m = gf.gpr.GPR(X, Y2, k, meanf)a
m.likelihood.variance = 0.1
print(m)
Explanation: Now we can use the special mean function without the coordinates (covariates).
End of explanation
# Estimation
m.optimize()
print(m)
Explanation: Only 2 parameters now!
End of explanation
predicted_x = np.linspace(0.0,1.0,100)
from external_plugins.spystats.models import makeDuples
predsX = makeDuples(predicted_x)
pX = np.array(predsX)
tt = np.ones((10000,2)) *0.5
## Concatenate with horizontal stack
SuperX = np.hstack((pX,tt))
SuperX.shape
mean, variance = m.predict_y(SuperX)
minmean = min(mean)
maxmean = max(mean)
#plt.figure(figsize=(12, 6))
plt.scatter(pX[:,0], pX[:,1])
Xx, Yy = np.meshgrid(predicted_x,predicted_x)
plt.pcolor(Xx,Yy,mean.reshape(100,100),cmap=plt.cm.Accent)
Nn = 300
predicted_x = np.linspace(0.0,1.0,Nn)
Xx, Yy = np.meshgrid(predicted_x,predicted_x)
## Predict
from external_plugins.spystats.models import makeDuples
predsX = makeDuples(predicted_x)
pX = np.array(predsX)
tt = np.ones((Nn**2,2)) *0.5
SuperX = np.hstack((pX,tt))
mean, variance = m.predict_y(SuperX)
minmean = min(mean)
maxmean = max(mean)
width = 12
height = 8
minz = minmean
maxz = maxmean
plt.figure(figsize=(width, height))
plt.subplot(1,2,1)
scat = plt.scatter(X[:, 0], X[:, 1], c = Y2)
#plt.axis('equal')
plt.xlim((0,1))
plt.ylim((0,1))
plt.clim(minz,maxz)
#plt.colorbar()
plt.subplot(1,2,2)
#field = plt.imshow(mean.reshape(100,100).transpose().transpose(),interpolation=None)
plt.pcolor(Xx,Yy,mean.reshape(Nn,Nn).transpose())
plt.colorbar()
plt.clim(minz,maxz)
fig, axes = plt.subplots(nrows=1, ncols=2)
scat = plt.scatter(X[:, 0], X[:, 1], c = Y2)
field = plt.imshow(mean.reshape(100,100),interpolation=None)
#fig.subplots_adjust(right=0.8)
#cbar_ax = fig.add_axes([0.85, 0.05])
fig.colorbar(field, ax=axes.ravel().tolist())
plt.imshow(mean.reshape(100,100),interpolation=None)
plt.colorbar()
Explanation: Original parameters
phi = 0.05  ---> lengthscale
sigma2 = 1.0 ---> variance transform
nugget = 0.03 ---> likelihood variance
beta_0 = 10.0 ---> mean_function b
beta_1 = 1.5 ---> mean_fucntionA [2]
beta_2 = -1.0 ---> mean_functionA [3]
End of explanation
<END_TASK>