Multiple Features

Goal

By the end of this lesson, you can fit a regression model with multiple features, read the feature-matrix shape, and interpret a coefficient as one part of a fitted prediction rule rather than automatic causation.

One prediction can use several pieces of information

A house price may depend on floor area, number of rooms, age, location, and other information.

With one numeric feature, a linear model can look like:

prediction = weight × feature + intercept

With three features, the same idea becomes:

prediction = w1*x1 + w2*x2 + w3*x3 + intercept

Each feature value contributes through a learned coefficient, and the contributions are added together.

Read the matrix before reading the coefficients

Suppose X has shape (8, 3).

That means:

• 8 rows = 8 examples;
• 3 columns = 3 features per example.

A linear regression model fitted to those three columns learns three feature coefficients plus an intercept.

This gives you an immediate debugging check: if you expected three features but the model receives shape (8, 2), something is missing before you even inspect predictions.

Coefficients are conditional model contributions

Imagine this toy fitted rule:

prediction = 2*size - 1*age + 10

If size stays fixed and age increases by 1, the model output falls by 1 according to this rule.

That does not prove that making a real house one year older causes its price to fall by exactly one unit. A coefficient describes the fitted model while the other included features are held fixed.

Interpretation becomes especially tricky when features are correlated or measured in very different units.

A coefficient of 500 on a feature measured in kilometers is not automatically “more important” than a coefficient of 0.8 on a feature measured in meters. The units change the numerical size of the weight.

Trace one feature change in the Lab

The Lab fits a model using size_m2, rooms, and age_years. The new example starts as:

new_example = np.array([[90.0, 3.0, 10.0]])

1. Click Run and inspect X shape:, coefficients:, and prediction:.
2. In the printed coefficient dictionary, find the coefficient for size_m2 and note its sign.
3. In new_example, change only the first value from 90.0 to 100.0. Keep rooms = 3.0 and age_years = 10.0 fixed.
4. Before running, use the sign of the size_m2 coefficient to predict whether the output should move up or down.
5. Click Run and compare prediction: with the first run.
6. Restore new_example to [[90.0, 3.0, 10.0]].

Loading lab…

After this guided pass, choose either rooms or age_years, change only that one value, and predict the direction from its coefficient before running.

This controlled change lets you connect a coefficient to actual model behavior without pretending that the coefficient proves a real-world causal relationship.

More features are not automatically better

Adding a feature can help when it contains useful information that the existing features lack.

But extra columns can also:

• add noise;
• duplicate information already present;
• increase variance;
• make interpretation harder;
• introduce leakage if their timing is wrong.

The question is not “How many features can I add?” It is “Which information is valid, available, and useful for the prediction task?”

Quick Check

1. What does one column of `X` represent?

One example containing all of its featuresOne learned coefficient shared by every inputOne feature across all examples

2. Why must feature order stay consistent?

Libraries always recover the intended feature order from the values aloneFeature order matters during fitting but can change at prediction timeThe model associates each column position with a learned weight

3. Does a large coefficient prove causation?

Yes, if the coefficient stays similar across several foldsNo; it describes an association inside this fitted model, not a causeYes, if all features were standardized before fitting

0 of 3 questions answered.

Key Takeaways

• Multiple-feature models use a matrix of examples by features.
• Linear regression learns one coefficient per feature plus an intercept.
• Feature order and units matter when interpreting weights.
• Coefficients describe the fitted prediction rule, not automatic causation.
• Extra features should earn their place through valid, fair evidence.

Next Lesson

You have reached the Level 1 mini checkpoint. After it, the level switches from predicting numbers to predicting categories.

References

• scikit-learn, Linear Models.