I have watched a lot of tutorials and passed a bunch of courses about machine learning and AI before, but other than utilizing those methods on class assignments and a few of them for my master’s thesis, I didn’t have the chance to use machine learning and specifically deep learning in a long-term practical project. Fortunately, due to the need I have in my research, I have started to study machine learning methods from scratch and this time more in depth. According to a famous qoute by Albert Einstein

“You do not really understand something unless you can explain it to your grandmother.”

It is quite important to be able to transfer your knowledge to others using plain and easily understandable descriptions which also helps to solidify your comprehension of the topic. So, I have decided to start a series of blog posts to build common machine learning algorithms from scratch inorder to clarify these methods for myself and make sure I corectly understand the mechanics behind each of them.

Let me be clear from the very begining: It’s all about fitting a function!

Consider we have a dataset containing the number of trucks crossed border from Mexico to U.S through Ottay Messa port. Note that it’s just a subset of the entire inbound crossings dataset available on Kaggle. First of all, it’s always better to plot the data which may help us have some insight. To load the data from a .CSV file, we are going use Pandas which is a well-known data analysis/manipulation python library. We can then plot the data using Matplotlib (another python library for data visualization).


df = pd.read_csv('./regression/dataset.csv')
x_training_set = pd.to_datetime(df.Date, infer_datetime_format=True).to_numpy()
y_training_set = df.Value.to_numpy()
number_of_training_examples = len(x_training_set)

#plot the raw data
fig, ax = plt.subplots(figsize=(8, 6))
year_locator = mdates.YearLocator(2)
year_month_formatter = mdates.DateFormatter("%Y-%m")
ax.xaxis.set_major_locator(year_locator)
ax.xaxis.set_minor_locator(mdates.YearLocator())
ax.xaxis.set_major_formatter(year_month_formatter)
ax.plot(x_training_set,y_training_set, ".")
fig.autofmt_xdate()
plt.show()

Note that in the original data, each value is corresponding to a month, so I mapped the date intervals into an integer representation.

What we are observing here obviosuly is not an exact linear function, but for the sake of simplicity we can model broder corossings using a linear function! As we already know, the equation of a line is a below:

\[f(x) = mx + c\]

where m stands for the slope and c is the intercep on the y axis. But we can have infinite number of possible values for these parameters. Let’s look at some possible arbitrary lines with values [m=70,c=40000], [m=100,c=40000], [m=140,c=40000] represented with orange, green, and red colors respectively.

#plot arbitrary lines
def plot_line(ax, m,c,xMax):
    y0 = (1*m)+c
    ymax = (xMax*m)+c
    ax.plot([x_training_set[0],x_training_set[xMax-1]],[y0,ymax])

fig, ax = plt.subplots(figsize=(8, 6))
year_locator = mdates.YearLocator(2)
year_month_formatter = mdates.DateFormatter("%Y-%m")
ax.xaxis.set_major_locator(year_locator)
ax.xaxis.set_minor_locator(mdates.YearLocator())
ax.xaxis.set_major_formatter(year_month_formatter)
ax.plot(x_training_set,y_training_set, ".")
fig.autofmt_xdate()

plot_line(ax,70,40000,number_of_training_examples) 
plot_line(ax,100,40000,number_of_training_examples) 
plot_line(ax,140,40000,number_of_training_examples) 

plt.show()

But, What are the parameter values we are choosing for our linear equation to properly fit our data points?

Analytical Approach to find the regression parameters

To find the proper fit to our data, basically we have to minimize the average distance of the data points to our arbitrary line. In other words, we are finding the difference between the predicted value with the actual traing data.

there are two ways to calculate the error:

Mean Squared Error (MSE): which considers the squared difference of the values.

Mean Absolute Error (MAE): which considers the absolute difference of the values.

Note that we need to sum up these error values as positive numbers. Otherwise negative values will compensate for positive ones which will make our optimization problem impossible.

Our objective is to find our linear equation parameters m and c to minimize average error over all training set data (know as the cost function) defined below:

\[Cost Function(m,c) = {\frac{1}{n}\sum_{i=1}^{n} (f(x_i)-y_i)^2}\]

where n is the number of training examples. Now let’s explore the parameters space and plot the cost function to see how it looks like. for the MSE cost function we have the parameters space-cost plot below

# visualize cost function
def line_equation(m,c,x):
    return (m*x)+ c

def cost_function(m,c,training_examples_x,training_examples_y):
    sum_of_errors = 0
    item_index = 0
    for example in training_examples_x:
        predicted_y = line_equation(m,c,item_index)
        sum_of_errors += (predicted_y - training_examples_y[item_index])**2
        #sum_of_errors += abs(predicted_y - training_examples_y[item_index])
        item_index+=1
    mse = (sum_of_errors / len(training_examples_x))
    return mse

fig = plt.figure()
fig.set_size_inches(8, 6)
ax = fig.add_subplot(projection='3d')
cost_func_x_points = []
cost_func_y_points = []
cost_func_z_points = []
for m in np.arange(-200,500,10):
    for c in np.arange(-10000,60000,200):
         cost = cost_function(m,c,x_training_set,y_training_set)
         cost_func_x_points.append(m)
         cost_func_y_points.append(c)
         cost_func_z_points.append(cost)
ax.scatter(cost_func_x_points, cost_func_y_points,
cost_func_z_points,c=cost_func_z_points,marker='.')
ax.set_xlabel('M')
ax.set_ylabel('C')
ax.set_zlabel('Cost')
plt.show()

and for the MAE we have the plot below:

As you can see both these cost functions are convex and they just have one minimum point at the bottom of the slope(global minima). Then based on calculus, it means that if we could find the point that the derivative of this function is zero, we have found the optimal parameters for our model. We can simply use the equation below to find the parameters.

\[theta = (X^T.X)^-1.(X^T.Y)\]

where X is the training features sample vector. Y is the output vector. and the result(theta) are the parameters for our regression model.

This equation is called Normal Equation and you can find the math behind it here.

So let’s run it on our dataset and see how it works.

#linear equation using numpy
x_training_set_numbers = np.arange(0,len(x_training_set),1)
ones_vector = np.ones((len(x_training_set), 1))
x_training_set_numbers = np.reshape(x_training_set_numbers, (len(x_training_set_numbers), 1))
x_training_set_numbers = np.append(ones_vector, x_training_set_numbers, axis=1)
theta_list = np.linalg.inv(x_training_set_numbers.T.dot(x_training_set_numbers)) \
.dot(x_training_set_numbers.T).dot(y_training_set)
print(theta_list)

#visualize raw data with fitted line
fig, ax = plt.subplots(figsize=(8, 6))
year_locator = mdates.YearLocator(2)
year_month_formatter = mdates.DateFormatter("%Y-%m")
ax.xaxis.set_major_locator(year_locator)
ax.xaxis.set_minor_locator(mdates.YearLocator())
ax.xaxis.set_major_formatter(year_month_formatter)
ax.plot(x_training_set,y_training_set, ".")
fig.autofmt_xdate()
plot_line(ax,theta_list[1],theta_list[0],number_of_training_examples) 
plt.show()