RegressionEasy5 min10 pts

House Price Regression

Estimate sale price from size, bedrooms, and age. Lower is better: predicting the training mean scores an MSE of 2879.57.

Target

MSE ↓ 400.00

Baseline

2879.57

Time

5 min

Points

Lower is better. Scored on 15 held-out rows.

Dataset

x: feature 1 · y: feature 2

Each dot is one training row: feature 1 against feature 2, coloured by class.

Training split · 30 rows · 3 features. Scored by MSE on 15 held-out rows; target ≤ 400, baseline 2879.57.

What this lab teaches

Solve a four-unknown least-squares system by hand - intercept plus three features - and beat the training-mean predictor on MSE.

The split in numbers

There are 30 training rows and 15 hidden rows. The features are house size (roughly 830 to 3000), bedrooms (1 to 5), and age (0 to 37), and the targets are prices in the 140 to 360 range. Predicting the training mean costs MSE 2879.57 on the hidden rows, which is the recorded baseline, and the target is 400.

The fitted surface

Ordinary least squares gives price = 33.46 + 0.0969*size + 10.78*bedrooms - 0.385*age. On the hidden 15 rows that scores MSE about 283 with R-squared 0.90, comfortably inside the target. The training residual standard deviation is about 12.7, so part of the price is genuinely noisy; do not expect a perfect fit, just a far better one than the mean.

Assembling the system

Prepend a constant 1 column to the features, build the 4x4 normal equations (X-transpose X) theta = X-transpose y, and solve with Gaussian elimination. The matrix diagonal holds n, the sums of squares of each feature, and the off-diagonals hold pairwise feature sums - all computable in one pass over the 30 rows with plain Python loops.

Common pitfalls

Fitting on unscaled features and then evaluating with a different scaling. Age spans 37 while size spans about 2000, so a scale change that touches only one column silently shifts the intercept and slope.
Reading the first coefficient as a feature weight. The design matrix's first column is the constant 1, so theta[0] is the intercept; mixing up the column order of bedrooms and age flips their weights.
Dividing by a zero or near-zero pivot during elimination without a fallback, which happens whenever a candidate column has no variance in the rows you keep.

Rules of the run

Define predict(train_X, train_y, test_X) and return exactly one value per test row.
Pure Python or numpy both run; no external data or network access.
Return a numeric prediction for every test row.

Hint

Least squares has a closed form: solve the normal equations (X-transpose X) theta = X-transpose y.

Reference solution

Revealing doesn't affect your record — but try the hint first.

Timed run

5 min limit

10 points

not started · 05:00

The clock starts when you press start. Your predict(train_X, train_y, test_X) is then scored on 15 held-out rows within 5 minutes.

Starter codePython

def predict(train_X, train_y, test_X):
    # Baseline: predict the mean training price.
    # Replace with least squares, then return one value per test row.
    mean = sum(train_y) / len(train_y)
    return [mean for _ in test_X]

Tab inserts 4 spaces · ⌘+Enter runs your code

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