""" trainset.py -- numerical companion to the entry 'training set'. The entry's weather narrative, carried out: three days of synthetic weather recordings (morning minimum temperature as feature, maximum daytime temperature as label) form the training set, and two hypotheses are learned from it -- a straight line and a degree-two polynomial. Every claim of the entry is measured on the result. Self-contained (numpy/matplotlib only), deterministic. Blocks ------ [B-trainset] Three days of recordings form the training set: for each day the morning minimum temperature (feature) and the maximum daytime temperature (label) are known, so the loss of any candidate hypothesis can be evaluated on them. [B-erm] Fit a straight line by minimizing the average squared error over the training set. The value of that minimum is the training error, and any other line incurs a larger average. [B-poly] Fit a degree-two polynomial to the same training set: it passes through all three data points, so its training error is zero -- smaller than the training error of the line. [B-misleading] A small training error can be misleading: on twenty held-back validation days, the polynomial's average loss (its validation error) far exceeds the line's, reversing the training-set ranking (overfitting). [B-picture] The training set and both hypotheses evaluated on a grid, written to CSV for the entry's figure. Outputs ------- pythondemos/trainset.png : preview figure (checking only). pythondemos/trainset_train.csv : the three training days (x = morning minimum temperature, y = maximum daytime temperature) pythondemos/trainset_line.csv : the fitted line on a two-point grid pythondemos/trainset_poly.csv : the fitted polynomial on a dense grid """ import numpy as np import matplotlib matplotlib.use("Agg") import matplotlib.pyplot as plt from pathlib import Path OUT_DIR = Path(__file__).parent report = [] def check(name, ok): report.append((name, bool(ok))) print(f" [{'ok' if ok else 'FAIL'}] {name}") def days(n, gen, lo=-15.0, hi=5.0, w1=0.8, w0=4.0, noise=2.0): """n days: morning minimum x and maximum daytime temperature y.""" x = np.sort(gen.uniform(lo, hi, n)) y = w0 + w1 * x + gen.normal(0.0, noise, n) return x, y def avg_sqerr(x, y, coeffs): return float(np.mean((y - np.polyval(coeffs, x)) ** 2)) # -------------------------------------------------------------- [B-trainset] print("[B-trainset] three days of recordings form the training set") x, y = days(3, np.random.default_rng(100)) for xr, yr in zip(x, y): print(f" morning minimum {xr:6.1f} C, maximum daytime {yr:6.1f} C") check("both temperatures of every training day are known", np.all(np.isfinite(x)) and np.all(np.isfinite(y))) # ------------------------------------------------------------------- [B-erm] print("\n[B-erm] the line minimizes the average squared error over the " "training set") c_line = np.polyfit(x, y, 1) trainerr_line = avg_sqerr(x, y, c_line) print(f" fitted line: max temperature = {c_line[1]:.2f} + {c_line[0]:.2f}" f" * min temperature; training error {trainerr_line:.3f}") gen = np.random.default_rng(7) others = [avg_sqerr(x, y, c_line + d) for d in gen.normal(0.0, 0.3, (200, 2))] check("every perturbed line incurs a larger average on the training set", all(o >= trainerr_line for o in others)) # ------------------------------------------------------------------ [B-poly] print("\n[B-poly] a degree-two polynomial passes through all three days") c_poly = np.polyfit(x, y, 2) trainerr_poly = avg_sqerr(x, y, c_poly) print(f" polynomial training error {trainerr_poly:.2e} " f"(line: {trainerr_line:.3f})") check("the polynomial's training error is zero (up to round-off)", trainerr_poly < 1e-10) check("the polynomial's training error is smaller than the line's", trainerr_poly < trainerr_line) # ------------------------------------------------------------ [B-misleading] print("\n[B-misleading] the training-set ranking of the two hypotheses is " "misleading") xv, yv = days(20, np.random.default_rng(1100)) valerr_line = avg_sqerr(xv, yv, c_line) valerr_poly = avg_sqerr(xv, yv, c_poly) print(f" validation error on 20 held-back days: line {valerr_line:.3f}, " f"polynomial {valerr_poly:.3f}") check("the polynomial's validation error far exceeds the line's", valerr_poly > 3 * valerr_line) check("the validation days are disjoint from the training days", len(set(np.round(xv, 6)) & set(np.round(x, 6))) == 0) # --------------------------------------------------------------- [B-picture] print("\n[B-picture] training set and both hypotheses, written to CSV") xs_line = np.array([-14.0, 5.0]) xs_poly = np.linspace(-14.0, 5.0, 200) def write_csv(name, cols, header): np.savetxt(OUT_DIR / name, np.column_stack(cols), delimiter=",", header=header, comments="", fmt="%.4f") write_csv("trainset_train.csv", (x, y), "x,y") write_csv("trainset_line.csv", (xs_line, np.polyval(c_line, xs_line)), "x,yhat") write_csv("trainset_poly.csv", (xs_poly, np.polyval(c_poly, xs_poly)), "x,yhat") print(f" wrote {len(x)} training days, the line and the polynomial to " f"3 CSV files") check("the two curves agree on the training days only (max gap elsewhere " "is large)", np.max(np.abs(np.polyval(c_poly, xs_poly) - np.polyval(c_line, xs_poly))) > 3.0) fig, ax = plt.subplots(figsize=(5.6, 3.6)) ax.plot(xs_poly, np.polyval(c_poly, xs_poly), "--", color="black", lw=1.2, label="polynomial (training error zero)") ax.plot(xs_line, np.polyval(c_line, xs_line), "-", color="black", lw=1.2, label="straight line") ax.plot(x, y, "o", ms=6, color="black", label="training set") ax.set_xlabel("morning minimum temperature") ax.set_ylabel("maximum daytime temperature") ax.set_title("three training days, two hypotheses learned from them", fontsize=9) ax.legend(frameon=False, fontsize=8) fig.tight_layout() fig.savefig(OUT_DIR / "trainset.png", dpi=110) n_ok = sum(ok for _, ok in report) print(f"\n{n_ok}/{len(report)} checks pass") print("wrote trainset.png") if n_ok != len(report): raise SystemExit(1)