"""
title: Influence plot
summary: Studentized residuals vs leverage, with bubbles sized by Cook's distance.
r: car::influencePlot()
tags: [influence, Cook's distance, leverage, outliers, studentized residuals]
description: |
  An observation is **influential** when it is both unusual in its predictors
  (high leverage, x-axis) and badly fitted (large studentized residual, y-axis).
  Bubble area grows with Cook's distance (∝ √D, so extremes stay readable),
  which combines the two. Dashed lines mark the usual rules of thumb — leverage of 2p/n and 3p/n, studentized
  residuals of ±2 — and observations with Cook's D above 4/n are named and coloured.

  The model explains the murder rate of the 50 US states + DC
  (`sm.datasets.statecrime`) with **statsmodels** OLS. Studentized (deleted)
  residuals, hat values and Cook's distances all come from
  `model.get_influence()`; the subtitle checks the largest residual with a
  Bonferroni outlier test, as `car::outlierTest()` does.
"""
import matplotlib.pyplot as plt
import numpy as np
import statsmodels.api as sm
import statsmodels.formula.api as smf
from scipy import stats

df = sm.datasets.statecrime.load_pandas().data
model = smf.ols("murder ~ poverty + single + urban + hs_grad + white", data=df).fit()
infl = model.get_influence()
rstud, lev, cook = infl.resid_studentized_external, infl.hat_matrix_diag, infl.cooks_distance[0]
n, p = int(model.nobs), int(model.df_model + 1)
flag = cook > 4 / n

fig, ax = plt.subplots(figsize=(8.5, 5.6))

# Rule-of-thumb thresholds
for h in (2 * p / n, 3 * p / n):
    ax.axvline(h, color="#AEAEB2", lw=1, ls="--", zorder=0)
    ax.text(h, 1.0, f" {round(h * n / p)}p/n", transform=ax.get_xaxis_transform(),
            fontsize=8.5, color="#8E8E93", va="top")
for r in (-2, 0, 2):
    ax.axhline(r, color="#AEAEB2" if r else "#C7C7CC", lw=1, ls="--" if r else "-", zorder=0)

# Bubbles: area proportional to √Cook's D (keeps extreme points readable)
area = lambda d: 20 + 700 * np.sqrt(d)
size = area(cook)
ax.scatter(lev[~flag], rstud[~flag], s=size[~flag], color="#007AFF", alpha=0.35,
           edgecolor="#007AFF", lw=0.8)
ax.scatter(lev[flag], rstud[flag], s=size[flag], color="#FF3B30", alpha=0.35,
           edgecolor="#FF3B30", lw=1.2)
for i in np.flatnonzero(flag):
    right = lev[i] > 0.5                          # far-right points: label on the left
    ax.annotate(f"{df.index[i]}\nD = {cook[i]:.2f}", (lev[i], rstud[i]),
                xytext=((-1 if right else 1) * (np.sqrt(size[i]) / 2 + 6), 0),
                textcoords="offset points", ha="right" if right else "left", va="center",
                fontsize=8.5, color="#3A3A3C", linespacing=1.1)

# Bubble-size key
for d, x in zip((0.1, 0.5, 1, 2), (0.455, 0.50, 0.55, 0.61)):
    ax.scatter(x, 0.86, s=area(d), transform=ax.transAxes, color="none",
               edgecolor="#8E8E93", lw=0.8)
    ax.text(x, 0.77, f"{d:g}", transform=ax.transAxes, ha="center", fontsize=8.5, color="#6E6E73")
ax.text(0.53, 0.97, "Cook's D", transform=ax.transAxes, ha="center", va="top",
        fontsize=8.5, color="#6E6E73")

ax.set_ylim(rstud.min() - 0.8, rstud.max() + 1.2)
ax.set_xlim(0, lev.max() * 1.15)
ax.set_xlabel("Leverage (hat value)")
ax.set_ylabel("Studentized residual")
ax.set_title("DC and Hawaii combine high leverage with large residuals", pad=26)

# Bonferroni outlier test on the largest |studentized residual|
k = np.argmax(np.abs(rstud))
p_bonf = min(1, n * 2 * stats.t.sf(abs(rstud[k]), model.df_resid - 1))
ax.text(0, 1.02, f"OLS murder rate ~ 5 predictors · n = {n} · {flag.sum()} obs. with Cook's D > 4/n · "
        f"largest |t*| = {abs(rstud[k]):.2f} ({df.index[k]}), Bonferroni p = {p_bonf:.3f}",
        transform=ax.transAxes, fontsize=9.5, color="#6E6E73")
plt.show()