<SYSTEM_TASK:>
Given the following text description, write Python code to implement the functionality described below step by step
<END_TASK>
<USER_TASK:>
Description:
 Linear Regression Algorithms using Apache SystemML
This notebook shows
Step2:  Import SystemML API
Step3:  Import numpy, sklearn, and define some helper functions
Step5:  Example 1
Step6:  Load diabetes dataset from scikit-learn
Step8:  Example 2
Step10:  Algorithm 2
Step12:  Algorithm 3
Step13:  Example 3
Step14:  Example 4
<ASSISTANT_TASK:>
Python Code:
!pip uninstall systemml --y
!pip install --user https://repository.apache.org/content/groups/snapshots/org/apache/systemml/systemml/1.0.0-SNAPSHOT/systemml-1.0.0-20171201.070207-23-python.tar.gz
!pip show systemml
Explanation: Linear Regression Algorithms using Apache SystemML
This notebook shows:
- Install SystemML Python package and jar file
  - pip
  - SystemML 'Hello World'
- Example 1: Matrix Multiplication
  - SystemML script to generate a random matrix, perform matrix multiplication, and compute the sum of the output
  - Examine execution plans, and increase data size to obverve changed execution plans
- Load diabetes dataset from scikit-learn
- Example 2: Implement three different algorithms to train linear regression model
  - Algorithm 1: Linear Regression - Direct Solve (no regularization)
  - Algorithm 2: Linear Regression - Batch Gradient Descent (no regularization)
  - Algorithm 3: Linear Regression - Conjugate Gradient (no regularization)
- Example 3: Invoke existing SystemML algorithm script LinearRegDS.dml using MLContext API
- Example 4: Invoke existing SystemML algorithm using scikit-learn/SparkML pipeline like API
- Uninstall/Clean up SystemML Python package and jar file
Install SystemML Python package and jar file
End of explanation
from systemml import MLContext, dml, dmlFromResource
ml = MLContext(sc)
print "Spark Version:", sc.version
print "SystemML Version:", ml.version()
print "SystemML Built-Time:", ml.buildTime()
ml.execute(dml(s = 'Hello World!').output("s")).get("s")
Explanation: Import SystemML API
End of explanation
import matplotlib.pyplot as plt
import numpy as np
from sklearn import datasets
plt.switch_backend('agg')
Explanation: Import numpy, sklearn, and define some helper functions
End of explanation
script = 
    X = rand(rows=$nr, cols=1000, sparsity=0.5)
    A = t(X) %*% X
    s = sum(A)
prog = dml(script).input('$nr', 1e5).output('s')
s = ml.execute(prog).get('s')
print (s)
Explanation: Example 1: Matrix Multiplication
SystemML script to generate a random matrix, perform matrix multiplication, and compute the sum of the output
End of explanation
%matplotlib inline
diabetes = datasets.load_diabetes()
diabetes_X = diabetes.data[:, np.newaxis, 2]
diabetes_X_train = diabetes_X[:-20]
diabetes_X_test = diabetes_X[-20:]
diabetes_y_train = diabetes.target[:-20].reshape(-1,1)
diabetes_y_test = diabetes.target[-20:].reshape(-1,1)
plt.scatter(diabetes_X_train, diabetes_y_train,  color='black')
plt.scatter(diabetes_X_test, diabetes_y_test,  color='red')
diabetes.data.shape
Explanation: Load diabetes dataset from scikit-learn
End of explanation
script = 
    # add constant feature to X to model intercept
    X = cbind(X, matrix(1, rows=nrow(X), cols=1))
    A = t(X) %*% X
    b = t(X) %*% y
    w = solve(A, b)
    bias = as.scalar(w[nrow(w),1])
    w = w[1:nrow(w)-1,]
prog = dml(script).input(X=diabetes_X_train, y=diabetes_y_train).output('w', 'bias')
w, bias = ml.execute(prog).get('w','bias')
w = w.toNumPy()
plt.scatter(diabetes_X_train, diabetes_y_train,  color='black')
plt.scatter(diabetes_X_test, diabetes_y_test,  color='red')
plt.plot(diabetes_X_test, (w*diabetes_X_test)+bias, color='blue', linestyle ='dotted')
Explanation: Example 2: Implement three different algorithms to train linear regression model
Algorithm 1: Linear Regression - Direct Solve (no regularization)
Least squares formulation
w* = argminw ||Xw-y||2 = argminw (y - Xw)'(y - Xw) = argminw (w'(X'X)w - w'(X'y))/2
Setting the gradient
dw = (X'X)w - (X'y) to 0, w = (X'X)-1(X' y) = solve(X'X, X'y)
End of explanation
script = 
    # add constant feature to X to model intercepts
    X = cbind(X, matrix(1, rows=nrow(X), cols=1))
    max_iter = 100
    w = matrix(0, rows=ncol(X), cols=1)
    for(i in 1:max_iter){
        XtX = t(X) %*% X
        dw = XtX %*%w - t(X) %*% y
        alpha = -(t(dw) %*% dw) / (t(dw) %*% XtX %*% dw)
        w = w + dw*alpha
    }
    bias = as.scalar(w[nrow(w),1])
    w = w[1:nrow(w)-1,]    
prog = dml(script).input(X=diabetes_X_train, y=diabetes_y_train).output('w', 'bias')
w, bias = ml.execute(prog).get('w', 'bias')
w = w.toNumPy()
plt.scatter(diabetes_X_train, diabetes_y_train,  color='black')
plt.scatter(diabetes_X_test, diabetes_y_test,  color='red')
plt.plot(diabetes_X_test, (w*diabetes_X_test)+bias, color='red', linestyle ='dashed')
Explanation: Algorithm 2: Linear Regression - Batch Gradient Descent (no regularization)
Algorithm
Step 1: Start with an initial point 
while(not converged) { 
  Step 2: Compute gradient dw. 
  Step 3: Compute stepsize alpha.     
  Step 4: Update: wnew = wold + alpha*dw 
}
Gradient formula
dw = r = (X'X)w - (X'y)
Step size formula
Find number alpha to minimize f(w + alpha*r) 
alpha = -(r'r)/(r'X'Xr)
End of explanation
script = 
    # add constant feature to X to model intercepts
    X = cbind(X, matrix(1, rows=nrow(X), cols=1))
    m = ncol(X); i = 1; 
    max_iter = 20;
    w = matrix (0, rows = m, cols = 1); # initialize weights to 0
    dw = - t(X) %*% y; p = - dw;        # dw = (X'X)w - (X'y)
    norm_r2 = sum (dw ^ 2); 
    for(i in 1:max_iter) {
        q = t(X) %*% (X %*% p)
        alpha = norm_r2 / sum (p * q);  # Minimizes f(w - alpha*r)
        w = w + alpha * p;              # update weights
        dw = dw + alpha * q;           
        old_norm_r2 = norm_r2; norm_r2 = sum (dw ^ 2);
        p = -dw + (norm_r2 / old_norm_r2) * p; # next direction - conjugacy to previous direction
        i = i + 1;
    }
    bias = as.scalar(w[nrow(w),1])
    w = w[1:nrow(w)-1,]    
prog = dml(script).input(X=diabetes_X_train, y=diabetes_y_train).output('w', 'bias')
w, bias = ml.execute(prog).get('w','bias')
w = w.toNumPy()
plt.scatter(diabetes_X_train, diabetes_y_train,  color='black')
plt.scatter(diabetes_X_test, diabetes_y_test,  color='red')
plt.plot(diabetes_X_test, (w*diabetes_X_test)+bias, color='red', linestyle ='dashed')
Explanation: Algorithm 3: Linear Regression - Conjugate Gradient (no regularization)
Problem with gradient descent: Takes very similar directions many times
Solution: Enforce conjugacy
Step 1: Start with an initial point 
while(not converged) {
   Step 2: Compute gradient dw.
   Step 3: Compute stepsize alpha.
   Step 4: Compute next direction p by enforcing conjugacy with previous direction.
   Step 4: Update: w_new = w_old + alpha*p
}
End of explanation
prog = dmlFromResource('scripts/algorithms/LinearRegDS.dml').input(X=diabetes_X_train, y=diabetes_y_train).input('$icpt',1.0).output('beta_out')
w = ml.execute(prog).get('beta_out')
w = w.toNumPy()
bias=w[1]
plt.scatter(diabetes_X_train, diabetes_y_train,  color='black')
plt.scatter(diabetes_X_test, diabetes_y_test,  color='red')
plt.plot(diabetes_X_test, (w[0]*diabetes_X_test)+bias, color='red', linestyle ='dashed')
Explanation: Example 3: Invoke existing SystemML algorithm script LinearRegDS.dml using MLContext API
End of explanation
from pyspark.sql import SQLContext
from systemml.mllearn import LinearRegression
sqlCtx = SQLContext(sc)
regr = LinearRegression(sqlCtx)
# Train the model using the training sets
regr.fit(diabetes_X_train, diabetes_y_train)
predictions = regr.predict(diabetes_X_test)
# Use the trained model to perform prediction
%matplotlib inline
plt.scatter(diabetes_X_train, diabetes_y_train,  color='black')
plt.scatter(diabetes_X_test, diabetes_y_test,  color='red')
plt.plot(diabetes_X_test, predictions, color='black')
Explanation: Example 4: Invoke existing SystemML algorithm using scikit-learn/SparkML pipeline like API
mllearn API allows a Python programmer to invoke SystemML's algorithms using scikit-learn like API as well as Spark's MLPipeline API.
End of explanation
<END_TASK>