pyLinearRegression

# Linear Regression Analysis

# Mike Kerry - Feb 2021 - acclivity2@gmail.com

# This was a fun project. We are given the prices of pizzas at 5 different sizes (6, 8, 10, 14 and 18 inches)
# The task is to compute a straight line graph through these points, finding the line that gives the minimum value
# of sum-of-squares differences from the actual prices.
# Having found this line, use it to predict the price of a 12" pizza

# I started by finding the slope of the existing graph between each adjacent pair of points.
# I found the minimum and maximum slope, and I assumed that my final prediction line would lie between these slopes.

# Using these min and max slopes, I made initial "inspired guesses" at the mininum and maximum price of a 6" pizza
# based on my least-sum-of-squares line.

# I then computed the sum-of-squares for 500 starting prices (between min and max 6" pizza) and for 500 slopes for
# each of these starting prices. I recorded the line that gave me the least sum-of-squares of errors in price

# Using my best-fit line, I computed the price of a 12" pizza. This came out as $13.68 (which was the right answer!)

# All this was done without using any Python Library.

# (Interestingly, this is a simplified version of the first computer program I ever wrote,
# which was a Multiple Regression Analysis written while I was at college in 1961)

x = [6, 8, 10, 14, 18]          # Sizes of pizzas in inches
y = [7, 9, 13, 17.5, 18]        # Price of pizza in $ at each of the given sizes

num = len(x)                    # Number of points in the given graph
slopes = []                     # a list of the slopes of the graph for all adjacent points

for i in range(1, num):         # look at each x interval
    xdiff = x[i] - x[i-1]
    ydiff = y[i] - y[i-1]
    slopes.append(ydiff/xdiff)      # compute the slope and append to list of slopes

minslope = min(slopes)
maxslope = max(slopes)
# we will integrate between these two slope values

mid = num // 2                      # Find a pizza size and price around the middle of the range
# Apply our min and max slopes to this mid-point pizza, and hence find a min and max price for a 6" (starting) pizza
midxdif = x[mid] - x[0]
maxystart = y[mid] - (midxdif * minslope)
minystart = y[mid] - (midxdif * maxslope)
# We will integrate between these starting prices

deltastart = (maxystart - minystart) / 500          # We will compute for 500 values of price (Y) at x[0]
deltaslope = (maxslope - minslope) / 500            # And compute for 500 values of slope per Y start

leastssq = 999999.0                                  # This will be the least sum-of-squares value of all 250,000 loops
bestystart = 0.0
bestslope = 0.0
ystart = minystart
while ystart < maxystart:                           # integrate over 500 start values
    aslope = minslope
    while aslope <= maxslope:                       # integrate over 500 values of slope of graph
        sumssq = 0.0
        for i in range(num):
            predicted_y = ystart + (x[i] - x[0]) * aslope
            sumssq += (predicted_y - y[i]) ** 2
        if sumssq < leastssq:
            leastssq = sumssq                        # record the least sum-of-squares so far
            bestystart = ystart                     # record the corresponding starting price
            bestslope = aslope                      # and the corresponding graph slope

        aslope += deltaslope                        # bump to next slope value

    ystart += deltastart                            # bump to next starting price

# ----------------------------------------------------------------------------------

print('Best ystart = $%.2f   Best slope = %.2f   Least SSQ = %.2f' % (bestystart, bestslope, leastssq))
print()

# compute best fit Y values for each value of X
# (Not actually required, unless we wanted to plot this line)
bestylist = []
for j in range(num):
    dxj = x[j] - x[0]
    yj = bestystart + dxj * bestslope
    bestylist.append(yj)

# Compute prediction for x = 12  (price for 12" pizza)
px = 12
dx = px - x[0]
predict_y = bestystart + (dx * bestslope)

print('Predicted price of 12" pizza: %.2f ' % (predict_y))