The 4 data points are nearly perfectly collinear: feature 2 is approximately a linear function of feature 1 (X₂ ≈ 1.63·X₁ + 0.4). Geometrically, these points lie very close to a single straight line in 2D space.

PCA finds the directions of maximum variance, ordered from most to least. When data lies near a 1D line:

• PC1 aligns with that line and captures almost all variance (R² ≈ 1.0 for this dataset).
• PC2 is orthogonal to that line and captures only the tiny residual noise — essentially zero.

Projecting onto PC1 and reconstructing back to 2D therefore reproduces the original data with negligible error (MSE ≈ 0).

General principle: PCA succeeds at dimensionality reduction when features are highly correlated. If d features lie near a k-dimensional linear subspace (k < d), PCA's first k components capture the true structure and the remaining d−k components capture only noise. Here, 2D data lies near a 1D line, so k=1 suffices.

Code

from sklearn.decomposition import PCA
from sklearn.preprocessing import scale as sk_scale
from sklearn.metrics import mean_squared_error as mse_fn

fig, axes = plt.subplots(1, 2, figsize=(14, 6))

for ax, (dname, fname) in zip(axes,
        [('4a','dataset-task-4a.csv'),('4b','dataset-task-4b.csv')]):
    X_pca    = sk_scale(pd.read_csv(fname).values)   # centre + unit variance
    pca_full = PCA().fit(X_pca)
    evr      = pca_full.explained_variance_ratio_
    cumevr   = np.cumsum(evr)
    k_min    = int(np.argmax(cumevr >= 0.80)) + 1     # smallest k ≥ 80%

    # R² / MSE table
    for k_iter in range(1, len(evr)+1):
        pca_k = PCA(n_components=k_iter)
        sc    = pca_k.fit_transform(X_pca)
        r2    = float(np.sum(evr[:k_iter]))
        mse   = mse_fn(X_pca, sc @ pca_k.components_)
        print(f'k={k_iter}  R²={r2:.4f}  MSE={mse:.4f}')

    # Visualise with k_min components
    sc_vis = PCA(n_components=k_min).fit_transform(X_pca)
    if k_min == 1:
        ax.scatter(sc_vis[:,0], np.zeros(len(sc_vis)), alpha=0.7, s=40)
        ax.set_xlabel(f'PC 1 ({evr[0]*100:.1f}% variance)')
    else:
        ax.scatter(sc_vis[:,0], sc_vis[:,1], alpha=0.7, s=40)
        ax.set_xlabel(f'PC 1 ({evr[0]*100:.1f}% variance)')
        ax.set_ylabel(f'PC 2 ({evr[1]*100:.1f}% variance)')
    ax.set_title(f'Dataset {dname}: {k_min}D PCA ({cumevr[k_min-1]*100:.1f}% var.)')
plt.tight_layout();  plt.show()

Why we scale first

sk_scale centres each feature on 0 and scales to unit variance. Without this, PCA would be dominated by whichever feature has the largest numeric range — not necessarily the most informative one. Scaling ensures all features are considered equally by PCA.

Result

R² and MSE tables

Dataset 4a (3 features):

k	R² (cumulative)	MSE
1	0.8813	0.1187	← ≥80%
2	0.9753	0.0247
3	1.0000	0.0000	full

Dataset 4b (5 features):

k	R² (cumulative)	MSE
1	0.5163	0.4837
2	0.8210	0.1790	← ≥80%
3	0.9210	0.0790
4	0.9902	0.0098
5	1.0000	0.0000	full

Which dimension to select?

Dataset	Min k ≥ 80% var.	Recommended k for plotting	Reason
4a	k=1 (88.1%)	k=2 (97.5%)	2D scatter more informative than 1D; cost = +9.4% info at zero plot complexity
4b	k=2 (82.1%)	k=2 (82.1%)	Already ≥80% and directly plotable as 2D scatter

Dataset 4a: k=1 technically satisfies the 80% threshold (88.1%), but a single-axis plot is much less useful than a 2D scatter. Using k=2 captures 97.5% of variance while producing a standard 2D visualisation — a clear win for minimal extra cost.

Dataset 4b: k=2 exactly satisfies the threshold (82.1%) and directly produces a 2D scatter plot. Adding k=3 gives 92.1% but is no longer directly plotable without further reduction. Therefore k=2 is optimal for both datasets when the goal is to maximise explained variance while remaining plottable.