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Multivariate Gradient Descent in Python
def multivariate_gradient_descent(training_examples, alpha=0.01):
"""
Apply gradient descent on the training examples to learn a line that fits through the examples
:param examples: set of all examples in (x,y) format
:param alpha = learning rate
:return:
"""
# initialize the weight and x_vectors
W = [0 for index in range(0, len(training_examples[0][0]))]
# W_0 is a constant
W_0 = 0
# repeat until "convergence", meaning that w0 and w1 aren't changing very much
# --> need to define what 'not very much' means, and that may depend on problem domain
convergence = False
while not convergence:
# initialize temporary variables, and set them to 0
deltaW_0 = 0
deltaW_n = [0 for x in range(0,len(training_examples[0][0]))]
for pair in training_examples:
# grab our data points from the example
x_i = pair[0]
y_i = pair[1]
# calculate a prediction, and find the error
# needs to be an element-wise plus
deltaW_0 += multivariate_prediction_error(W_0, y_i, W, x_i)
deltaW_n = numpy.multiply(numpy.add(deltaW_n, multivariate_prediction_error(W_0, y_i, W, x_i)), x_i)
#print "DELTA_WN = " + str(deltaW_n)
# store previous weighting values
prev_w0 = W_0
prev_Wn = W
# get new weighting values
W_0 = W_0 + alpha*deltaW_0
W = numpy.add(W,numpy.multiply(alpha,deltaW_n))
alpha -= 0.001
# every few iterations print out current model
# 1. --> (w0 + w1x1 + w2x2 + ... + wnxn)
variables = [( str(W[i]) + "*x" + str(i+1) + " + ") for i in range(0,len(W))]
var_string = ''.join(variables)
var_string = var_string[:-3]
print "Current model is: " + str(W_0)+" + "+var_string
# 2. --> averaged squared error over training set, using the current line
summed_error = sum_of_squared_error_over_entire_dataset(W_0, W, training_examples)
avg_error = summed_error/len(training_examples)
print "Average Squared Error="+str(sum(avg_error))
print ""
# check if we have converged
if abs(prev_w0 - W_0) < 0.00001 and abs(numpy.subtract(prev_Wn, W)).all() < 0.00001:
convergence = True
# after convergence, print out the parameters of the trained model (w0, ... wn)
variables = [( "w"+str(i+1)+"="+str(W[i])+", ") for i in range(0,len(W))]
var_string = ''.join(variables)
var_string = var_string[:-2]
print "RESULTS: "
print "\tParameters of trained model are: w0="+str(W_0)+", "+var_string
return W_0, W
################################
##### MULTIVARIATE HELPERS #####
################################
# generalize these to just take a w0, a vector of weights, and a vector x-values
def multivariate_model_prediction(w0, weights, xs):
return w0 + numpy.dot(weights, xs)
# again, this needs to take just a w0, vector of weights, and a vector of x-values
def multivariate_prediction_error(w0, y_i, weights, xs):
# basically, we just take the true value (y_i)
# and we subtract the predicted value from it
# this gives us an error, or J(w0,w1) value
return y_i - multivariate_model_prediction(w0, weights, xs)
# should be the same, but use the generalize functions above, and update the weights inside the vector titself
# also need to have a vector fo delta_Wn values to simplify
def multivariate_sum_of_squared_error_over_entire_dataset(w0, weights, training_examples):
# find the squared error over the whole training set
sum = 0
for pair in training_examples:
x_i = pair[0]
y_i = pair[1]
# cast back to values in range [1 --> 20]
prediction = multivariate_model_prediction(w0,weights,x_i) / (1/20.0)
actual = y_i / (1/20.0)
error = abs(actual - prediction)
error_sq = error ** 2
sum += error_sq
return sum
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