Linear regression

In short: Linear regression predicts a number as a sum of the features with coefficients; MAE says how far off it is on average, and RMSE weighs big errors more, both in the same units.

Problem

“How much money do I expect on Friday?”

Accounting plans the money for the next weeks. It asks how much revenue to expect on Friday, 6 November, and on Sunday, 8 November. Here is the revenue by day of the year, without the cancelled orders, because they bring no money:

Pythondata: the shop tables

import numpy as np
import pandas as pd
from sklearn.linear_model import LinearRegression

orders = pd.read_csv('year_orders.csv')
items = pd.read_csv('year_items.csv')
orders = orders[orders['status'] != 'cancelled']
orders['ordered_at'] = pd.to_datetime(orders['ordered_at'])
items['revenue'] = items['quantity'] * items['unit_price']
lines = items.merge(orders, left_on='order_id', right_on='id')
daily = lines.set_index('ordered_at').resample('D')['revenue'].sum().to_frame()
daily['weekday'] = daily.index.dayofweek
daily['day'] = np.arange(len(daily))
daily['december'] = (daily.index.month == 12).astype(int)

print(len(daily), round(daily['revenue'].mean(), 2))
print(round(daily.loc[daily['december'] == 1, 'revenue'].mean(), 2))
daily.groupby('weekday')['revenue'].mean().round(2)

import numpy as np import pandas as pd from sklearn.linear_model import LinearRegression orders = pd.read_csv('year_orders.csv') items = pd.read_csv('year_items.csv') orders = orders[orders['status'] != 'cancelled'] orders['ordered_at'] = pd.to_datetime(orders['ordered_at']) items['revenue'] = items['quantity'] * items['unit_price'] lines = items.merge(orders, left_on='order_id', right_on='id') daily = lines.set_index('ordered_at').resample('D')['revenue'].sum().to_frame() daily['weekday'] = daily.index.dayofweek daily['day'] = np.arange(len(daily)) daily['december'] = (daily.index.month == 12).astype(int) print(len(daily), round(daily['revenue'].mean(), 2)) print(round(daily.loc[daily['december'] == 1, 'revenue'].mean(), 2)) daily.groupby('weekday')['revenue'].mean().round(2)

▶ RunRestore the code

A day brings €304.62 on average. Friday brings €350.79, and Sunday €250.85. December days bring €513.42 on average. The mean of every day of the week is a start, but December and the slow growth over the year are mixed into it. You need a model that puts them together.

The idea

A sum of features

Linear regression predicts as a sum: a starting number plus one number for each feature, multiplied by its value. These numbers are the coefficients. The features here are of three kinds:

• the day of the week, as columns of 0 and 1. With drop_first=True the column for Monday is dropped: Monday is the base, and every other day is “how much more than Monday”;
• day, the running number of the day: 0 on 1 November 2025. This is the trend;
• december: 1 for the days in December.

The model learns from November to June, 242 days. July–October, 123 days, are for the test. Why not shuffled is at the end of the lesson:

Pythondata: the shop tables

import numpy as np
import pandas as pd
from sklearn.linear_model import LinearRegression

orders = pd.read_csv('year_orders.csv')
items = pd.read_csv('year_items.csv')
orders = orders[orders['status'] != 'cancelled']
orders['ordered_at'] = pd.to_datetime(orders['ordered_at'])
items['revenue'] = items['quantity'] * items['unit_price']
lines = items.merge(orders, left_on='order_id', right_on='id')
daily = lines.set_index('ordered_at').resample('D')['revenue'].sum().to_frame()
daily['weekday'] = daily.index.dayofweek
daily['day'] = np.arange(len(daily))
daily['december'] = (daily.index.month == 12).astype(int)
X = pd.get_dummies(daily[['weekday', 'day', 'december']], columns=['weekday'], drop_first=True, dtype=int)
y = daily['revenue']
train = daily.index < '2026-07-01'
X_train, X_test = X[train], X[~train]
y_train, y_test = y[train], y[~train]
model = LinearRegression().fit(X_train, y_train)

print(len(X_train), len(X_test))
print(round(model.intercept_, 2), round(model.coef_[0] * 365, 2))
pd.Series(model.coef_, index=X.columns).round(2)

import numpy as np import pandas as pd from sklearn.linear_model import LinearRegression orders = pd.read_csv('year_orders.csv') items = pd.read_csv('year_items.csv') orders = orders[orders['status'] != 'cancelled'] orders['ordered_at'] = pd.to_datetime(orders['ordered_at']) items['revenue'] = items['quantity'] * items['unit_price'] lines = items.merge(orders, left_on='order_id', right_on='id') daily = lines.set_index('ordered_at').resample('D')['revenue'].sum().to_frame() daily['weekday'] = daily.index.dayofweek daily['day'] = np.arange(len(daily)) daily['december'] = (daily.index.month == 12).astype(int) X = pd.get_dummies(daily[['weekday', 'day', 'december']], columns=['weekday'], drop_first=True, dtype=int) y = daily['revenue'] train = daily.index < '2026-07-01' X_train, X_test = X[train], X[~train] y_train, y_test = y[train], y[~train] model = LinearRegression().fit(X_train, y_train) print(len(X_train), len(X_test)) print(round(model.intercept_, 2), round(model.coef_[0] * 365, 2)) pd.Series(model.coef_, index=X.columns).round(2)

▶ RunRestore the code

The starting number is €273.98: a Monday on day 0, outside December. Friday adds €63.58, and Sunday takes away €45.44. December adds €233.05 a day. Every day adds €0.06, that is about €21.33 a year. So the forecast for a Friday in December is 273.98 + 63.58 + 233.05 plus the trend.

More precisely, if you are curious

The coefficients are estimates and carry noise, like the mean in the lesson on samples. Trained on all 365 days, the model gives Friday +€44.70, not +€63.58. How regression finds the numbers (the smallest sum of squared errors) is in the lesson Linear regression of the AI course, without a library.

Try it

Which is the base

1. The coefficient of weekday_6 is −45.44. What does it mean?

On Sunday the model gives €45.44 less than on Monday for the same day of the yearOn Sunday the revenue is €45.44Every Sunday the revenue falls by €45.44 compared with the previous one

Check

The idea

How much it misses: MAE and RMSE

The error is measured on the test. There are two usual measures:

• MAE (mean absolute error): the mean error without the sign. In euros, and the manager understands it: “we miss by this much a day on average”.
• RMSE (root mean squared error): the errors are squared, the mean is taken and then the root. It is in euros too, but a big miss weighs more.

sklearn.metrics has both:

Pythondata: the shop tables

import numpy as np
import pandas as pd
from sklearn.linear_model import LinearRegression

orders = pd.read_csv('year_orders.csv')
items = pd.read_csv('year_items.csv')
orders = orders[orders['status'] != 'cancelled']
orders['ordered_at'] = pd.to_datetime(orders['ordered_at'])
items['revenue'] = items['quantity'] * items['unit_price']
lines = items.merge(orders, left_on='order_id', right_on='id')
daily = lines.set_index('ordered_at').resample('D')['revenue'].sum().to_frame()
daily['weekday'] = daily.index.dayofweek
daily['day'] = np.arange(len(daily))
daily['december'] = (daily.index.month == 12).astype(int)
X = pd.get_dummies(daily[['weekday', 'day', 'december']], columns=['weekday'], drop_first=True, dtype=int)
y = daily['revenue']
train = daily.index < '2026-07-01'
X_train, X_test = X[train], X[~train]
y_train, y_test = y[train], y[~train]
model = LinearRegression().fit(X_train, y_train)
from sklearn.metrics import mean_absolute_error, root_mean_squared_error

pred = model.predict(X_test)
print(round(mean_absolute_error(y_test, pred), 2), round(root_mean_squared_error(y_test, pred), 2))

for misses in ([10, 10], [0, 20]):
    e = np.array(misses)
    print(misses, np.abs(e).mean(), round(np.sqrt((e ** 2).mean()), 2))

import numpy as np import pandas as pd from sklearn.linear_model import LinearRegression orders = pd.read_csv('year_orders.csv') items = pd.read_csv('year_items.csv') orders = orders[orders['status'] != 'cancelled'] orders['ordered_at'] = pd.to_datetime(orders['ordered_at']) items['revenue'] = items['quantity'] * items['unit_price'] lines = items.merge(orders, left_on='order_id', right_on='id') daily = lines.set_index('ordered_at').resample('D')['revenue'].sum().to_frame() daily['weekday'] = daily.index.dayofweek daily['day'] = np.arange(len(daily)) daily['december'] = (daily.index.month == 12).astype(int) X = pd.get_dummies(daily[['weekday', 'day', 'december']], columns=['weekday'], drop_first=True, dtype=int) y = daily['revenue'] train = daily.index < '2026-07-01' X_train, X_test = X[train], X[~train] y_train, y_test = y[train], y[~train] model = LinearRegression().fit(X_train, y_train) from sklearn.metrics import mean_absolute_error, root_mean_squared_error pred = model.predict(X_test) print(round(mean_absolute_error(y_test, pred), 2), round(root_mean_squared_error(y_test, pred), 2)) for misses in ([10, 10], [0, 20]): e = np.array(misses) print(misses, np.abs(e).mean(), round(np.sqrt((e ** 2).mean()), 2))

▶ RunRestore the code

The model misses by €59.88 a day on average (MAE), and the RMSE is €75.91. The two small calculations show the difference. Misses of 10 and 10 give an MAE of 10 and an RMSE of 10. Misses of 0 and 20 give the same MAE of 10, but an RMSE of 14.14. If one big miss costs you more than a few small ones, for example an empty till on Friday, you look at the RMSE. Otherwise the MAE is clearer for the manager.

Try it

And the baseline?

Does the model beat “always the average day”? Put in guess the mean revenue of the training days, and in mae_base and rmse_base the errors of this forecast on the test days.

Pythondata: the shop tables

import numpy as np
import pandas as pd
from sklearn.linear_model import LinearRegression

orders = pd.read_csv('year_orders.csv')
items = pd.read_csv('year_items.csv')
orders = orders[orders['status'] != 'cancelled']
orders['ordered_at'] = pd.to_datetime(orders['ordered_at'])
items['revenue'] = items['quantity'] * items['unit_price']
lines = items.merge(orders, left_on='order_id', right_on='id')
daily = lines.set_index('ordered_at').resample('D')['revenue'].sum().to_frame()
daily['weekday'] = daily.index.dayofweek
daily['day'] = np.arange(len(daily))
daily['december'] = (daily.index.month == 12).astype(int)
X = pd.get_dummies(daily[['weekday', 'day', 'december']], columns=['weekday'], drop_first=True, dtype=int)
y = daily['revenue']
train = daily.index < '2026-07-01'
X_train, X_test = X[train], X[~train]
y_train, y_test = y[train], y[~train]
model = LinearRegression().fit(X_train, y_train)

guess = ...
mae_base = ...
rmse_base = ...
print(round(guess, 2), round(mae_base, 2), round(rmse_base, 2))

import numpy as np import pandas as pd from sklearn.linear_model import LinearRegression orders = pd.read_csv('year_orders.csv') items = pd.read_csv('year_items.csv') orders = orders[orders['status'] != 'cancelled'] orders['ordered_at'] = pd.to_datetime(orders['ordered_at']) items['revenue'] = items['quantity'] * items['unit_price'] lines = items.merge(orders, left_on='order_id', right_on='id') daily = lines.set_index('ordered_at').resample('D')['revenue'].sum().to_frame() daily['weekday'] = daily.index.dayofweek daily['day'] = np.arange(len(daily)) daily['december'] = (daily.index.month == 12).astype(int) X = pd.get_dummies(daily[['weekday', 'day', 'december']], columns=['weekday'], drop_first=True, dtype=int) y = daily['revenue'] train = daily.index < '2026-07-01' X_train, X_test = X[train], X[~train] y_train, y_test = y[train], y[~train] model = LinearRegression().fit(X_train, y_train) guess = ... mae_base = ... rmse_base = ... print(round(guess, 2), round(mae_base, 2), round(rmse_base, 2))

Check▶ Run

The idea

The residuals

A residual is the actual value minus the forecast. Plus means the shop sold more than the model said. With a good model the residuals are scattered around 0, with no pattern. A pattern is something the model does not know:

Pythondata: the shop tables

import numpy as np
import pandas as pd
from sklearn.linear_model import LinearRegression

orders = pd.read_csv('year_orders.csv')
items = pd.read_csv('year_items.csv')
orders = orders[orders['status'] != 'cancelled']
orders['ordered_at'] = pd.to_datetime(orders['ordered_at'])
items['revenue'] = items['quantity'] * items['unit_price']
lines = items.merge(orders, left_on='order_id', right_on='id')
daily = lines.set_index('ordered_at').resample('D')['revenue'].sum().to_frame()
daily['weekday'] = daily.index.dayofweek
daily['day'] = np.arange(len(daily))
daily['december'] = (daily.index.month == 12).astype(int)
X = pd.get_dummies(daily[['weekday', 'day', 'december']], columns=['weekday'], drop_first=True, dtype=int)
y = daily['revenue']
train = daily.index < '2026-07-01'
X_train, X_test = X[train], X[~train]
y_train, y_test = y[train], y[~train]
model = LinearRegression().fit(X_train, y_train)
import matplotlib.pyplot as plt

res = y_test - model.predict(X_test)
print(res.idxmax().date(), round(res.max(), 2), round(y_test[res.idxmax()], 2))
print(res.groupby(res.index.month).mean().round(2))

fig, ax = plt.subplots(figsize=(8, 3.5))
ax.scatter(res.index, res, s=10)
ax.axhline(0, color='black', linewidth=1)
fig.autofmt_xdate()
ax.set_title('Test residuals: actual minus forecast')
ax.set_ylabel('€ per day')

import numpy as np import pandas as pd from sklearn.linear_model import LinearRegression orders = pd.read_csv('year_orders.csv') items = pd.read_csv('year_items.csv') orders = orders[orders['status'] != 'cancelled'] orders['ordered_at'] = pd.to_datetime(orders['ordered_at']) items['revenue'] = items['quantity'] * items['unit_price'] lines = items.merge(orders, left_on='order_id', right_on='id') daily = lines.set_index('ordered_at').resample('D')['revenue'].sum().to_frame() daily['weekday'] = daily.index.dayofweek daily['day'] = np.arange(len(daily)) daily['december'] = (daily.index.month == 12).astype(int) X = pd.get_dummies(daily[['weekday', 'day', 'december']], columns=['weekday'], drop_first=True, dtype=int) y = daily['revenue'] train = daily.index < '2026-07-01' X_train, X_test = X[train], X[~train] y_train, y_test = y[train], y[~train] model = LinearRegression().fit(X_train, y_train) import matplotlib.pyplot as plt res = y_test - model.predict(X_test) print(res.idxmax().date(), round(res.max(), 2), round(y_test[res.idxmax()], 2)) print(res.groupby(res.index.month).mean().round(2)) fig, ax = plt.subplots(figsize=(8, 3.5)) ax.scatter(res.index, res, s=10) ax.axhline(0, color='black', linewidth=1) fig.autofmt_xdate() ax.set_title('Test residuals: actual minus forecast') ax.set_ylabel('€ per day')

▶ RunRestore the code

The biggest miss is on 30 July 2026: €239.28 above the forecast. It is an unusually strong Thursday with €535.40 of revenue. By month, July, August and October are close to 0 (€3.99, −€8.03 and €20.42). September is −€58.85 a day: for a whole month the shop sold less than the day and the trend explain. This is not noise, something happened. What, we will find out in the course project.

The idea

What the model cannot do

The model knows only what its features tell it. Without the december column:

Pythondata: the shop tables

import numpy as np
import pandas as pd
from sklearn.linear_model import LinearRegression

orders = pd.read_csv('year_orders.csv')
items = pd.read_csv('year_items.csv')
orders = orders[orders['status'] != 'cancelled']
orders['ordered_at'] = pd.to_datetime(orders['ordered_at'])
items['revenue'] = items['quantity'] * items['unit_price']
lines = items.merge(orders, left_on='order_id', right_on='id')
daily = lines.set_index('ordered_at').resample('D')['revenue'].sum().to_frame()
daily['weekday'] = daily.index.dayofweek
daily['day'] = np.arange(len(daily))
daily['december'] = (daily.index.month == 12).astype(int)
X = pd.get_dummies(daily[['weekday', 'day', 'december']], columns=['weekday'], drop_first=True, dtype=int)
y = daily['revenue']
train = daily.index < '2026-07-01'
X_train, X_test = X[train], X[~train]
y_train, y_test = y[train], y[~train]
from sklearn.metrics import mean_absolute_error

no_dec = X.drop(columns='december')
blind = LinearRegression().fit(no_dec[train], y_train)
print(round(blind.coef_[0], 2), round(blind.coef_[0] * 365, 2))
print(round(mean_absolute_error(y_test, blind.predict(no_dec[~train])), 2))

import numpy as np import pandas as pd from sklearn.linear_model import LinearRegression orders = pd.read_csv('year_orders.csv') items = pd.read_csv('year_items.csv') orders = orders[orders['status'] != 'cancelled'] orders['ordered_at'] = pd.to_datetime(orders['ordered_at']) items['revenue'] = items['quantity'] * items['unit_price'] lines = items.merge(orders, left_on='order_id', right_on='id') daily = lines.set_index('ordered_at').resample('D')['revenue'].sum().to_frame() daily['weekday'] = daily.index.dayofweek daily['day'] = np.arange(len(daily)) daily['december'] = (daily.index.month == 12).astype(int) X = pd.get_dummies(daily[['weekday', 'day', 'december']], columns=['weekday'], drop_first=True, dtype=int) y = daily['revenue'] train = daily.index < '2026-07-01' X_train, X_test = X[train], X[~train] y_train, y_test = y[train], y[~train] from sklearn.metrics import mean_absolute_error no_dec = X.drop(columns='december') blind = LinearRegression().fit(no_dec[train], y_train) print(round(blind.coef_[0], 2), round(blind.coef_[0] * 365, 2)) print(round(mean_absolute_error(y_test, blind.predict(no_dec[~train])), 2))

▶ RunRestore the code

The trend becomes −€0.40 a day, −€147.35 a year. December is at the start of the training data and “lifts” the start of the line, and the model has no other way to explain it. The test error becomes €69.43, worse than the baseline (€66.24). The jump from the gift cards in December shows only if a feature says so.

Two more limits:

• No extrapolation. The line goes on forever. From 8 months the growth is €21.33 a year, but whether it continues next year the data does not say. We have seen only one December.
• A coefficient is not a cause. “+€63.58 on Friday” does not mean that a promotion moved to Friday will bring €63.58. The coefficient describes the data. How much an action brings, an experiment tells you, as in the lesson Correlation and cause.

Try it

Where the model is blind

Fit the model without the december column on the training days. Compute its residuals on the same days and the mean residual by month in by_month (group by res.index.strftime('%Y-%m')). Draw by_month as bars, with a title and a label on the vertical axis.

Pythondata: the shop tables

import numpy as np
import pandas as pd
from sklearn.linear_model import LinearRegression

orders = pd.read_csv('year_orders.csv')
items = pd.read_csv('year_items.csv')
orders = orders[orders['status'] != 'cancelled']
orders['ordered_at'] = pd.to_datetime(orders['ordered_at'])
items['revenue'] = items['quantity'] * items['unit_price']
lines = items.merge(orders, left_on='order_id', right_on='id')
daily = lines.set_index('ordered_at').resample('D')['revenue'].sum().to_frame()
daily['weekday'] = daily.index.dayofweek
daily['day'] = np.arange(len(daily))
daily['december'] = (daily.index.month == 12).astype(int)
X = pd.get_dummies(daily[['weekday', 'day', 'december']], columns=['weekday'], drop_first=True, dtype=int)
y = daily['revenue']
train = daily.index < '2026-07-01'
X_train, X_test = X[train], X[~train]
y_train, y_test = y[train], y[~train]
import matplotlib.pyplot as plt

no_dec = ...
blind = ...
res = ...
by_month = ...

fig, ax = plt.subplots()
...

import numpy as np import pandas as pd from sklearn.linear_model import LinearRegression orders = pd.read_csv('year_orders.csv') items = pd.read_csv('year_items.csv') orders = orders[orders['status'] != 'cancelled'] orders['ordered_at'] = pd.to_datetime(orders['ordered_at']) items['revenue'] = items['quantity'] * items['unit_price'] lines = items.merge(orders, left_on='order_id', right_on='id') daily = lines.set_index('ordered_at').resample('D')['revenue'].sum().to_frame() daily['weekday'] = daily.index.dayofweek daily['day'] = np.arange(len(daily)) daily['december'] = (daily.index.month == 12).astype(int) X = pd.get_dummies(daily[['weekday', 'day', 'december']], columns=['weekday'], drop_first=True, dtype=int) y = daily['revenue'] train = daily.index < '2026-07-01' X_train, X_test = X[train], X[~train] y_train, y_test = y[train], y[~train] import matplotlib.pyplot as plt no_dec = ... blind = ... res = ... by_month = ... fig, ax = plt.subplots() ...

Check▶ Run

The idea

Split by time, not shuffled

The model in this lesson learns from November to June and is checked on July–October. Why not with train_test_split? Here is a model with the day and the month, checked both ways. The second number on each line is the baseline:

Pythondata: the shop tables

import numpy as np
import pandas as pd
from sklearn.linear_model import LinearRegression

orders = pd.read_csv('year_orders.csv')
items = pd.read_csv('year_items.csv')
orders = orders[orders['status'] != 'cancelled']
orders['ordered_at'] = pd.to_datetime(orders['ordered_at'])
items['revenue'] = items['quantity'] * items['unit_price']
lines = items.merge(orders, left_on='order_id', right_on='id')
daily = lines.set_index('ordered_at').resample('D')['revenue'].sum().to_frame()
daily['weekday'] = daily.index.dayofweek
daily['day'] = np.arange(len(daily))
daily['december'] = (daily.index.month == 12).astype(int)
from sklearn.model_selection import train_test_split
from sklearn.metrics import mean_absolute_error

daily['month'] = daily.index.month
Xm = pd.get_dummies(daily[['weekday', 'month']], columns=['weekday', 'month'], drop_first=True, dtype=int)
y = daily['revenue']

Xa, Xb, ya, yb = train_test_split(Xm, y, test_size=0.25, random_state=42)
mixed = LinearRegression().fit(Xa, ya)
print('shuffled:', round(mean_absolute_error(yb, mixed.predict(Xb)), 2), round(np.abs(yb - ya.mean()).mean(), 2))
near = Xb.index.isin(Xa.index + pd.Timedelta(days=1)) | Xb.index.isin(Xa.index - pd.Timedelta(days=1))
print(near.sum(), len(Xb))

train = daily.index < '2026-07-01'
by_time = LinearRegression().fit(Xm[train], y[train])
print('by time:', round(mean_absolute_error(y[~train], by_time.predict(Xm[~train])), 2), round(np.abs(y[~train] - y[train].mean()).mean(), 2))

import numpy as np import pandas as pd from sklearn.linear_model import LinearRegression orders = pd.read_csv('year_orders.csv') items = pd.read_csv('year_items.csv') orders = orders[orders['status'] != 'cancelled'] orders['ordered_at'] = pd.to_datetime(orders['ordered_at']) items['revenue'] = items['quantity'] * items['unit_price'] lines = items.merge(orders, left_on='order_id', right_on='id') daily = lines.set_index('ordered_at').resample('D')['revenue'].sum().to_frame() daily['weekday'] = daily.index.dayofweek daily['day'] = np.arange(len(daily)) daily['december'] = (daily.index.month == 12).astype(int) from sklearn.model_selection import train_test_split from sklearn.metrics import mean_absolute_error daily['month'] = daily.index.month Xm = pd.get_dummies(daily[['weekday', 'month']], columns=['weekday', 'month'], drop_first=True, dtype=int) y = daily['revenue'] Xa, Xb, ya, yb = train_test_split(Xm, y, test_size=0.25, random_state=42) mixed = LinearRegression().fit(Xa, ya) print('shuffled:', round(mean_absolute_error(yb, mixed.predict(Xb)), 2), round(np.abs(yb - ya.mean()).mean(), 2)) near = Xb.index.isin(Xa.index + pd.Timedelta(days=1)) | Xb.index.isin(Xa.index - pd.Timedelta(days=1)) print(near.sum(), len(Xb)) train = daily.index < '2026-07-01' by_time = LinearRegression().fit(Xm[train], y[train]) print('by time:', round(mean_absolute_error(y[~train], by_time.predict(Xm[~train])), 2), round(np.abs(y[~train] - y[train].mean()).mean(), 2))

▶ RunRestore the code

Shuffled, the model misses by €59.81, and the baseline by €81.48: 27% better. But 87 of the 92 test days have a neighbouring day, the one before or the one after, in the training data. The model has learned how the days around the test day went, the later ones too.

Checked by time, the same model misses by €71.98, more than the baseline (€66.24). The months July–October are not in the training data. Their columns are 0 in all the training rows, and the model gives them the level of January, the weakest month.

A real forecast is always for days after all the examples. So the test must come after the training too. Shuffling days lets the future into the training. This is information leakage, and a whole lesson further on is about it.

Exercise

Friday and Sunday

Now the answer for accounting. Fit the model again on all 365 days. Make X_new for Friday 6 November 2026 and Sunday 8 November 2026 (day continues the count from 1 November 2025, december is 0) and put in forecast the forecast for the two days (a Series with the dates as the index).

Pythondata: the shop tables

import numpy as np
import pandas as pd
from sklearn.linear_model import LinearRegression

orders = pd.read_csv('year_orders.csv')
items = pd.read_csv('year_items.csv')
orders = orders[orders['status'] != 'cancelled']
orders['ordered_at'] = pd.to_datetime(orders['ordered_at'])
items['revenue'] = items['quantity'] * items['unit_price']
lines = items.merge(orders, left_on='order_id', right_on='id')
daily = lines.set_index('ordered_at').resample('D')['revenue'].sum().to_frame()
daily['weekday'] = daily.index.dayofweek
daily['day'] = np.arange(len(daily))
daily['december'] = (daily.index.month == 12).astype(int)
X = pd.get_dummies(daily[['weekday', 'day', 'december']], columns=['weekday'], drop_first=True, dtype=int)
y = daily['revenue']
train = daily.index < '2026-07-01'
X_train, X_test = X[train], X[~train]
y_train, y_test = y[train], y[~train]

final = ...
dates = pd.to_datetime(['2026-11-06', '2026-11-08'])
new = pd.DataFrame({'weekday': dates.dayofweek, 'day': ..., 'december': 0}, index=dates)
X_new = ...
forecast = ...
print(forecast)

import numpy as np import pandas as pd from sklearn.linear_model import LinearRegression orders = pd.read_csv('year_orders.csv') items = pd.read_csv('year_items.csv') orders = orders[orders['status'] != 'cancelled'] orders['ordered_at'] = pd.to_datetime(orders['ordered_at']) items['revenue'] = items['quantity'] * items['unit_price'] lines = items.merge(orders, left_on='order_id', right_on='id') daily = lines.set_index('ordered_at').resample('D')['revenue'].sum().to_frame() daily['weekday'] = daily.index.dayofweek daily['day'] = np.arange(len(daily)) daily['december'] = (daily.index.month == 12).astype(int) X = pd.get_dummies(daily[['weekday', 'day', 'december']], columns=['weekday'], drop_first=True, dtype=int) y = daily['revenue'] train = daily.index < '2026-07-01' X_train, X_test = X[train], X[~train] y_train, y_test = y[train], y[~train] final = ... dates = pd.to_datetime(['2026-11-06', '2026-11-08']) new = pd.DataFrame({'weekday': dates.dayofweek, 'day': ..., 'december': 0}, index=dates) X_new = ... forecast = ... print(forecast)

Check▶ Run

Test

Check yourself

1. Two forecasts both have an MAE of €60. One misses by about €60 every day, and the other usually by a little, but sometimes by €300. Which has the bigger RMSE?

The first: it misses every dayThey are equal, since the MAE is equalThe second: RMSE squares the errors first, and the big ones weigh more

2. Why does shuffling days into training and test cheat?

The model learns from the neighbouring days, later ones too, and in a real forecast the future is not there yetBecause the test becomes too smallBecause train_test_split does not work with dates

3. The coefficient of december is +€233.05. What follows from it?

That a “December in July” campaign will bring €233 a dayThat December days are that much above the others, but not whyThat December causes a bigger revenue

4. The test residuals in September are −€58.85 a day on average. What does that mean?

September is always weakThe model got the formula wrongThe shop sold less than the day and the trend explain: there is something the model does not know

Check

Start the lesson over

Related lessons

• Builds on: The first chart · Distributions · Correlation and cause · Models: learning from examples
• Continues in: Classification · Data leakage · Cross-validation
• Tasks on this topic: Models with scikit-learn