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# Linear Regression
import numpy as np
import pandas as pd
from sklearn import datasets
from sklearn.linear_model import LinearRegression


    

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# load the diabetes datasets
# datasets this is a sklearn utility which loads a dataset for you
# existing datasets include load_boston (boston house prices)
# or load_iris (flower data set)
# Here, we load the diabetes dataset
# To access the feature vector, call datasets.data => returns X, a 442x10 array
# To acess the target , call datasets.target => returns Y, an array with 442 entries
# Basically, we have 442 samples, each sample being describes by 10 variables (= features)
dataset = datasets.load_diabetes()


    

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#Let us now build a pandas dataframe hosting the data at hand

# We first need the list of feature names for our columns
# BMI is the Body Mass Index
# ABP is the Average Blood Pressure
lfeat = ["Age", "Sex", "BMI", "ABP", "S1", "S2", "S3", "S4", "S5", "S6"]


    

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# We now build the Dataframe, with the data as argument
# and the list of column names as keyword argument
df_diabetes = pd.DataFrame(dataset.data, columns = lfeat)


    

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# Let's have a look at the first few entries
print "Printing data up to the 5th sample"
df_diabetes.iloc[:5,:] # Look at the first 5 samples for all features.


    

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# We also want to add the regression target
# Let's create a new column :
df_diabetes["Target"] = dataset.target # Must have the correct size of course


    

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#Let's review our complete dataframe:
print
print "Printing data up to the 5th sample"
print "Also print the target"
df_diabetes.iloc[:5,:] # Look at the first 5 samples for all features incuding target


    

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# We will fit a linear regression model to the data.
# This is the most basic supervised learning method
# Let us first introduce some notation
# We write x = (x1, x2, ... x10) a row vector of X
# We model the response y as a linear combination of the columns of x :
# y = b0 + b1x1 + ... + b10x10 (b0 is called the intercept)
# We now want to find the vector b = (b0, b1, .. b10) which gives the best representation of the data.
# Best is generally understood with respect to a loss criterion.
# The most popular is "called least squares".
# We try to minimize the squared error i.e. (y-b0 + b1x1 + ... + b10x10)**2
# With several samples (more than one row vector of X),
# We minimize the sum of the squared errors.
# In vector notation, this means minimizing ||(Y-Xb)||**2 (1)
# where we have added a column of ones to X to account for the intercept
# The value of b which minimises (1) can be found analytically or with gradient descent.


    

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# fit a linear regression model to the data
# This creates an instance of the Linear Regression class of model
model = LinearRegression()


    

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#the model.fit command will automatically find the coefficient b we talked about earlier
# we only have to specify X = dataset.data and Y = dataset.target
# with pandas we have to split the df in two :
# the feature part (X) and the target part (Y)
# This is done below :

data = df_diabetes[lfeat].values
target = df_diabetes["Target"].values

model.fit(data, target)
print(model)


    

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# Once the model has been fitted, it can be used to make new predictions
# (A new prediction is simply computed as y = b0 + b1x1 + ... + b10x10 
#  where b was found in the fitting step and x is any feature vector/matrix
# to this end, we simply call model.predict(x)
expected = target
predicted = model.predict(data)


    

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# summarize the fit of the model
# we can estimate the performance of the fit using the mse metric
# in this case, we simply compute the mean of the squared error on the sample
# the lower, the better
mse = np.mean((predicted-expected)**2)
print("Residual sum of squares: %.2f" % mse)
# Explained variance score: 1 is perfect prediction
print('Variance score: %.2f' % model.score(data, target))


    

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%matplotlib inline

# Code source: Jaques Grobler
# License: BSD 3 clause

import matplotlib.pyplot as plt
import numpy as np
from sklearn import datasets, linear_model

# Load the diabetes dataset
diabetes = datasets.load_diabetes()


# Use only one feature
diabetes_X = diabetes.data[:, np.newaxis, 2]

# Split the data into training/testing sets
diabetes_X_train = diabetes_X[:-20]
diabetes_X_test = diabetes_X[-20:]

# Split the targets into training/testing sets
diabetes_y_train = diabetes.target[:-20]
diabetes_y_test = diabetes.target[-20:]

# Create linear regression object
regr = linear_model.LinearRegression()

# Train the model using the training sets
regr.fit(diabetes_X_train, diabetes_y_train)

# The coefficients
print('Coefficients: \n', regr.coef_)
# The mean square error
print("Residual sum of squares: %.2f"
      % np.mean((regr.predict(diabetes_X_test) - diabetes_y_test) ** 2))
# Explained variance score: 1 is perfect prediction
print('Variance score: %.2f' % regr.score(diabetes_X_test, diabetes_y_test))

# Plot outputs
plt.scatter(diabetes_X_test, diabetes_y_test,  color='black')
plt.plot(diabetes_X_test, regr.predict(diabetes_X_test), color='blue',
         linewidth=3)

plt.xticks(())
plt.yticks(())

plt.show()


    

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