""" datapoint.py — numerical companion to the glossary entry 'data point'. One block per paragraph of the entry (marked [P...]): each block verifies numerically what the corresponding statement asserts. Self-contained (numpy/matplotlib only), fixed seed. Blocks ------ [P-featlab] A data point carries two categories of properties: features (measurable/computable) and labels (higher-level facts). A synthetic image data point yields pixel-intensity features by computation, while its label (number of bright objects) is fixed by construction — known to the "expert" that generated the scene, not read off a sensor. [P-image] The image example: color intensities of all pixels serve as features x_1..x_d, and can be augmented with capture metadata (timestamp, location) as further features. [P-choice] Feature vs label is a design choice: the same attribute (body weight) acts as a feature when predicting disease and as the label when predicted from other attributes — both predictions run on the same patient table. [P-labelnoise] Labels are error-prone proxies: one-sided label noise (a fraction of positive training labels recorded as negative, as when a diagnosis is missed) monotonically depresses the fraction of correctly predicted positive test data points for the learned classifier on clean test data. [P-featnoise] Features are error-prone too: adding measurement noise to the features at prediction time increases the error of a fixed learned hypothesis monotonically in the sensor noise. Outputs ------- datapoint.png : preview figure (checking only). Data generated by pythondemos/datapoint.py. """ import numpy as np import matplotlib matplotlib.use("Agg") import matplotlib.pyplot as plt rng = np.random.default_rng(42) report = [] def check(name, ok): report.append((name, bool(ok))) print(f" [{'ok' if ok else 'FAIL'}] {name}") # -------------------------------------------------------- [P-featlab] print("[P-featlab] features are computed; labels are known facts") def make_scene(n_objects): img = 0.05 * rng.random((16, 16)) for _ in range(n_objects): i, j = rng.integers(2, 14, size=2) img[i - 1:i + 2, j - 1:j + 2] = 1.0 return img label_true = 3 # higher-level fact img = make_scene(label_true) features = np.array([img.mean(), img.std(), img.max()]) check("features are computed from the data point itself", np.isclose(features[0], img.mean())) check("the label is not a pixel statistic (needs scene knowledge)", label_true not in np.round(features).astype(int)[:1]) # ---------------------------------------------------------- [P-image] print("[P-image] pixel intensities + metadata as features") x_pixels = img.flatten() x_meta = np.array([1717.0, 47.5]) # timestamp, latitude x = np.concatenate([x_pixels, x_meta]) check("pixel features have length d = 256", x_pixels.size == 16 * 16) check("metadata extends the features to d + 2", x.size == 258) # --------------------------------------------------------- [P-choice] print("[P-choice] feature vs label is a design choice") m = 300 weight = 60 + 20 * rng.random(m) age = 20 + 50 * rng.random(m) disease = (0.03 * weight + 0.05 * age + rng.normal(0, 0.4, m) > 4.5) # design A: weight is a FEATURE for predicting the disease label XA = np.stack([weight, age], axis=1) wA = np.linalg.lstsq(np.c_[XA, np.ones(m)], disease.astype(float), rcond=None)[0] accA = np.mean((np.c_[XA, np.ones(m)] @ wA > 0.5) == disease) # design B: weight is the LABEL predicted from age and disease status XB = np.c_[age, disease.astype(float), np.ones(m)] wB = np.linalg.lstsq(XB, weight, rcond=None)[0] check("design A: weight used as a feature (prediction beats chance)", accA > 0.6) check("design B: weight used as the label (predicted better than by its average)", np.var(weight - XB @ wB) < np.var(weight)) # ----------------------------------------------------- [P-labelnoise] print("[P-labelnoise] label noise degrades the learned hypothesis") def train_test_acc(flip, reps=25): accs_r = [] for _ in range(reps): Xtr, Xte = XA[:200], XA[200:] ytr, yte = disease[:200].copy(), disease[200:] pos = np.nonzero(ytr)[0] # missed diagnoses: idx = rng.choice(pos, int(flip * pos.size), replace=False) ytr[idx] = False # positives recorded negative wn = np.linalg.lstsq(np.c_[Xtr, np.ones(200)], ytr.astype(float), rcond=None)[0] pred = np.c_[Xte, np.ones(100)] @ wn > 0.5 accs_r.append(np.mean(pred[yte])) # correct predictions on positive data points return float(np.mean(accs_r)) accs = [train_test_acc(f) for f in (0.0, 0.2, 0.4)] print(f" correctly predicted positives at flip = 0, 0.2, 0.4: " f"{accs[0]:.2f}, {accs[1]:.2f}, {accs[2]:.2f}") check("correct positive predictions decrease with the label-noise level", accs[0] > accs[1] > accs[2]) # ------------------------------------------------------ [P-featnoise] print("[P-featnoise] feature measurement noise degrades predictions") w_clean = np.linalg.lstsq(np.c_[XA, np.ones(m)], disease.astype(float), rcond=None)[0] errs = [] for s in (0.0, 5.0, 15.0): Xn = XA + rng.normal(0, s, XA.shape) # sensor uncertainty errs.append(np.mean((np.c_[Xn, np.ones(m)] @ w_clean > 0.5) != disease)) print(f" error at sensor noise 0, 5, 15: " f"{errs[0]:.2f}, {errs[1]:.2f}, {errs[2]:.2f}") check("prediction error grows with feature noise", errs[0] < errs[1] < errs[2]) # ------------------------------------------------------------ preview fig, ax = plt.subplots(1, 2, figsize=(8.2, 3.0)) ax[0].imshow(img, cmap="gray") ax[0].set_title(f"[P-featlab] data point (label = {label_true})") ax[1].plot([0, 0.2, 0.4], accs, "o-") ax[1].set_xlabel("label-noise level"); ax[1].set_ylabel("correct positive predictions") ax[1].set_title("[P-labelnoise]") fig.tight_layout() fig.savefig("datapoint.png", dpi=110) print(f"\n{sum(ok for _, ok in report)}/{len(report)} checks passed") assert all(ok for _, ok in report)