Processus de régression quantile

import matplotlib.pyplot as plt
import numpy as np
import statsmodels.api as sm
import statsmodels.formula.api as smf

# Data: Engel food expenditure vs household income (Belgian francs)
df = sm.datasets.engel.load_pandas().data
quantiles = np.arange(0.05, 0.96, 0.05)

# Fit one quantile regression per τ and keep slope + 95% CI
rows = []
for q in quantiles:
    fit = smf.quantreg("foodexp ~ income", df).fit(q=q)
    lo, hi = fit.conf_int().loc["income"]
    rows.append((q, fit.params["Intercept"], fit.params["income"], lo, hi))
qr = np.array(rows)
ols = smf.ols("foodexp ~ income", df).fit()
ols_lo, ols_hi = ols.conf_int().loc["income"]

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(11, 5), gridspec_kw=dict(width_ratios=[1, 1.25]))

# Left: data with selected quantile lines and the OLS line
ax1.scatter(df.income, df.foodexp, s=14, color="#8E8E93", alpha=0.5, lw=0)
x = np.linspace(df.income.min(), df.income.max(), 50)
for q, a, s_, *_ in qr[[0, 4, 9, 14, 18]]:
    ax1.plot(x, a + s_ * x, color="#007AFF", lw=1.2, alpha=0.4 + 0.6 * (q == 0.5))
    ax1.text(x[-1] * 1.01, a + s_ * x[-1], f"τ = {q:.2f}", fontsize=8.5, color="#007AFF", va="center")
ax1.plot(x, ols.params["Intercept"] + ols.params["income"] * x, color="#FF9500", lw=2, ls="--")
ax1.set_xlim(right=x[-1] * 1.17)
ax1.set_xlabel("Household income")
ax1.set_ylabel("Food expenditure")
ax1.set_title("Quantile lines fan out", fontsize=12)

# Right: slope estimates across τ vs the OLS slope
ax2.fill_between(qr[:, 0], qr[:, 3], qr[:, 4], color="#007AFF", alpha=0.15, lw=0)
ax2.plot(qr[:, 0], qr[:, 2], "o-", color="#007AFF", ms=5, mec="white", mew=1, label="Quantile regression (95% CI)")
ax2.axhline(ols.params["income"], color="#FF9500", lw=2, label="OLS (95% CI)")
for v in (ols_lo, ols_hi):
    ax2.axhline(v, color="#FF9500", lw=1, ls="--")
ax2.set_xticks(np.arange(0.1, 1.0, 0.1))
ax2.set_xlabel("Quantile τ of food expenditure")
ax2.set_ylabel("Slope: extra food spending per franc of income")
ax2.legend(loc="upper left")
ax2.set_title("Income slope by quantile", fontsize=12)

slope = dict(zip(np.round(qr[:, 0], 2), qr[:, 2]))
fig.text(0.01, 1.07, "Income raises food spending most for the biggest spenders",
         fontsize=15, fontweight="semibold", va="bottom")
fig.text(0.01, 1.02, f"Engel data · n = {len(df)} · income slope at τ = .10: {slope[0.1]:.2f}, "
         f"τ = .50: {slope[0.5]:.2f}, τ = .90: {slope[0.9]:.2f} · OLS: {ols.params['income']:.2f} "
         f"[{ols_lo:.2f}, {ols_hi:.2f}]", fontsize=9.5, color="#6E6E73", va="bottom")
plt.show()