Fitting a line to data (linear regression)

The forecast says 20 degrees tomorrow. The shop has to decide tonight how many bikes to bring out of the store.

A year of days shows warm days are busier. A straight line through those days turns "busier" into a number you can plan with, and says how far to trust it.

A straight line is hires = intercept + slope * temperature. The slope says how many more hires each extra degree brings. The intercept is the line's guess at 0 degrees.

scikit-learn's LinearRegression finds both from the data. line.py fits hires against temperature alone, on all 365 days.

docker exec jupyter python3 steps/line.py

Every day sits a little above or below the line. That gap is the residual. Square each residual, so a gap below counts as much as one above. Then add them up.

Least squares is the line with the smallest total. Two short sums give it: the slope from how x and y move together, then the intercept from the means.

# Least squares by hand, for three days.
temps = [5, 10, 15]
hires = [200, 250, 330]
mean_x = sum(temps) / len(temps)
mean_y = sum(hires) / len(hires)
top = 0
bottom = 0
for n in range(len(temps)):
    top = top + (temps[n] - mean_x) * (hires[n] - mean_y)
    bottom = bottom + (temps[n] - mean_x) ** 2
slope = top / bottom
intercept = mean_y - slope * mean_x
print("slope:", slope, "intercept:", intercept)
# Change 330 to 300 and run again.
# Does the slope go up or down?

import pandas as pd

hires = pd.read_csv("hires.csv")
x = hires["temp_c"]
y = hires["hires"]
slope = ((x - x.mean()) * (y - y.mean())).sum() / ((x - x.mean()) ** 2).sum()
intercept = y.mean() - slope * x.mean()
# Compare with what line.py printed.
round(slope, 1), round(intercept, 1)

A line always fits the days it learnt from quite well. The real question is how it does on new days.

So split the data. Fit on about three quarters, the training set, and keep the rest back as a test set. Then measure R squared and the mean error on the test set alone.

docker exec jupyter python3 steps/split.py

Temperature alone explains about 60 per cent of the change from day to day. Rain matters too.

model.py fits one line with two inputs, temperature and rain. Each input gets its own slope. A negative slope means that input pulls the answer down.

docker exec jupyter python3 steps/model.py

In your notebook, add a third input to the line: a weekend column of 1 and 0, made with hires["weekday"].isin(["Sat", "Sun"]).astype(int).

Fit it on a training set and score it on a test set, as split.py does. Does the test R squared rise? Say in a sentence what the weekend slope means for the shop.