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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]))]

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def 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 w0 and w1 to some small value, here just using 0 for simplicity
    w0 = 0
    w1 = 0

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# first value in state tuple
height_category = 0
dist_to_pipe_bottom = pipe_bottom - bird.y
if dist_to_pipe_bottom < 8: # very close
   height_category = 0
elif dist_to_pipe_bottom < 20: # close
   height_category = 1
elif dist_to_pipe_bottom < 125: #mid
   height_category = 2
elif dist_to_pipe_bottom < 250: # far

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function [ energy_matrix ] = energy_image( image_matrix_input )
%ENERGY_IMAGE Computes the energy at each pixel in a matrix nxmx3 matrix
%   Outputs a 2D-matrix containing energy equation outputs, of datatype DBL

% convert image to grayscale first
G = rgb2gray(image_matrix_input);

% convert to double
G2 = im2double(G);

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-- Do Tree Traversals and Built a Visitation List for each

preorder :: BinaryTree a -> [a]
preorder Leaf = []
preorder (Node left root right) = root : preorder left ++ preorder right
-- NOTE: Need to use the ++ so each list gets built separately and then concatenated
-- after it hits bottom

inorder :: BinaryTree a -> [a]
inorder Leaf = []

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data BinaryTree a =
    Leaf
  | Node (BinaryTree a) a (BinaryTree a)
  deriving (Eq, Ord, Show)

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quicksort :: (Ord a) => [a] -> [a]  
quicksort [] = []  
quicksort (x:xs) =   
    let smallerSorted = quicksort [a | a <- xs, a <= x]  
        biggerSorted = quicksort [a | a <- xs, a > x]  
    in  smallerSorted ++ [x] ++ biggerSorted  

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# PROCEDURE, From ROSS: Simulation (5th Edition) Page 78
# Step 1: Generate Y1, an exponential random variable with rate 1
Y1 = gen_exponential_distro_rand_variable()
# Step 2: Generate Y2, an exponential random variable with rate 2
Y2 = gen_exponential_distro_rand_variable()
# Step 3: If Y2 - (Y1 - 1)^2/2 > 0, set Y = Y2 - (Y1 - 1)^2/2, and go to Step 4 (accept)
#         Otherwise, go to Step 1 (reject)
subtraction_value = ( math.pow( ( Y1 - 1 ), 2 ) ) / 2
critical_value = Y2 - subtraction_value
if critical_value > 0: