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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 Initialize
Define the Training Data Set
Define the training dataset for the independent and dependent variables
Step1:  Define the Test Set
Define the training dataset for the independent variables. In this case it is a "continuous" curve
Step2:  Train the Model
Instantiate the kernels, instantiate the GPR with the kernel, and train the model.
Step3:  Regression
Perform the regression based on the set of training data. The best estimate of the prediction is given by the mean of the distribution from which the posterior samples are drawn.
Predict (Initial Hyperparameters)
Perform regression using the initial user-specified hyperparameters.
Step4:  Optimize Hyperparameters
Optimize over the hyperparameters.
Step5:  array([ 1.47895967,  3.99711988,  0.16295754])
array([ 1.80397587,  4.86011667,  0.18058626])
Predict (Optimized Hyperparameters)
Perform the regression from the hyperparameters that optimize the log marginal likelihood. Note the improvement in the fit in comparison to the actual function (red dotted line).
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Python Code:
x = np.random.RandomState(0).uniform(-5, 5, 20)
#x = np.random.uniform(-5, 5, 20)
y = x*np.sin(x)
#y += np.random.normal(0,0.5,y.size)
y += np.random.RandomState(34).normal(0,0.5,y.size)
Explanation: Initialize
Define the Training Data Set
Define the training dataset for the independent and dependent variables
End of explanation
x_star = np.linspace(-5,5,500)
Explanation: Define the Test Set
Define the training dataset for the independent variables. In this case it is a "continuous" curve
End of explanation
#Define the basic kernels
k1 = SqExp(0.45,2)
k2 = RQ(0.5,2,3)
k3 = ExpSine(0.1,2,30)
k4 = WhiteNoise(0.01)
#Define the combined kernel
k1 = k1+k4
#Instantiate the GP predictor object with the desired kernel
gp = GPR(k1)
#Train the model
gp.train(x,y)
Explanation: Train the Model
Instantiate the kernels, instantiate the GPR with the kernel, and train the model.
End of explanation
#Predict a new set of test data given the independent variable observations
y_mean1,y_var1 = gp.predict(x_star,False)
#Convert the variance to the standard deviation
y_err1 = np.sqrt(y_var1)
plt.scatter(x,y,s=30)
plt.plot(x_star,x_star*np.sin(x_star),'r:')
plt.plot(x_star,y_mean1,'k-')
plt.fill_between(x_star,y_mean1+y_err1,y_mean1-y_err1,alpha=0.5)
Explanation: Regression
Perform the regression based on the set of training data. The best estimate of the prediction is given by the mean of the distribution from which the posterior samples are drawn.
Predict (Initial Hyperparameters)
Perform regression using the initial user-specified hyperparameters.
End of explanation
gp.optimize('SLSQP')
Explanation: Optimize Hyperparameters
Optimize over the hyperparameters.
End of explanation
#Predict a new set of test data given the independent variable observations
y_mean2,y_var2 = gp.predict(x_star,False)
#Convert the variance to the standard deviation
y_err2 = np.sqrt(y_var2)
plt.scatter(x,y,s=30)
plt.plot(x_star,x_star*np.sin(x_star),'r:')
plt.plot(x_star,y_mean2,'k-')
plt.fill_between(x_star,y_mean2+y_err2,y_mean2-y_err2,alpha=0.5)
Explanation: array([ 1.47895967,  3.99711988,  0.16295754])
array([ 1.80397587,  4.86011667,  0.18058626])
Predict (Optimized Hyperparameters)
Perform the regression from the hyperparameters that optimize the log marginal likelihood. Note the improvement in the fit in comparison to the actual function (red dotted line).
End of explanation
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