Tweedie regression on insurance claims¶

df = load_mtpl2(n_samples=60000)

# Note: filter out claims with zero amount, as the severity model
# requires strictly positive target values.
df.loc[(df["ClaimAmount"] == 0) & (df["ClaimNb"] >= 1), "ClaimNb"] = 0

# Correct for unreasonable observations (that might be data error)
# and a few exceptionally large claim amounts
df["ClaimNb"] = df["ClaimNb"].clip(upper=4)
df["Exposure"] = df["Exposure"].clip(upper=1)
df["ClaimAmount"] = df["ClaimAmount"].clip(upper=200000)

log_scale_transformer = make_pipeline(
    FunctionTransformer(func=np.log),
    StandardScaler()
)

column_trans = ColumnTransformer(
    [
        ("binned_numeric", KBinsDiscretizer(n_bins=10),
            ["VehAge", "DrivAge"]),
        ("onehot_categorical", OneHotEncoder(),
            ["VehBrand", "VehPower", "VehGas", "Region", "Area"]),
        ("passthrough_numeric", "passthrough",
            ["BonusMalus"]),
        ("log_scaled_numeric", log_scale_transformer,
            ["Density"]),
    ],
    remainder="drop",
)
X = column_trans.fit_transform(df)

# Insurances companies are interested in modeling the Pure Premium, that is
# the expected total claim amount per unit of exposure for each policyholder
# in their portfolio:
df["PurePremium"] = df["ClaimAmount"] / df["Exposure"]

# This can be indirectly approximated by a 2-step modeling: the product of the
# Frequency times the average claim amount per claim:
df["Frequency"] = df["ClaimNb"] / df["Exposure"]
df["AvgClaimAmount"] = df["ClaimAmount"] / np.fmax(df["ClaimNb"], 1)

with pd.option_context("display.max_columns", 15):
    print(df[df.ClaimAmount > 0].head())

df_train, df_test, X_train, X_test = train_test_split(df, X, random_state=0)

# The parameters of the model are estimated by minimizing the Poisson deviance
# on the training set via a quasi-Newton solver: l-BFGS. Some of the features
# are collinear, we use a weak penalization to avoid numerical issues.
glm_freq = PoissonRegressor(alpha=1e-3, max_iter=400)
glm_freq.fit(X_train, df_train["Frequency"],
             sample_weight=df_train["Exposure"])

scores = score_estimator(
    glm_freq,
    X_train,
    X_test,
    df_train,
    df_test,
    target="Frequency",
    weights="Exposure",
)
print("Evaluation of PoissonRegressor on target Frequency")
print(scores)

fig, ax = plt.subplots(ncols=2, nrows=2, figsize=(16, 8))
fig.subplots_adjust(hspace=0.3, wspace=0.2)

plot_obs_pred(
    df=df_train,
    feature="DrivAge",
    weight="Exposure",
    observed="Frequency",
    predicted=glm_freq.predict(X_train),
    y_label="Claim Frequency",
    title="train data",
    ax=ax[0, 0],
)

plot_obs_pred(
    df=df_test,
    feature="DrivAge",
    weight="Exposure",
    observed="Frequency",
    predicted=glm_freq.predict(X_test),
    y_label="Claim Frequency",
    title="test data",
    ax=ax[0, 1],
    fill_legend=True
)

plot_obs_pred(
    df=df_test,
    feature="VehAge",
    weight="Exposure",
    observed="Frequency",
    predicted=glm_freq.predict(X_test),
    y_label="Claim Frequency",
    title="test data",
    ax=ax[1, 0],
    fill_legend=True
)

plot_obs_pred(
    df=df_test,
    feature="BonusMalus",
    weight="Exposure",
    observed="Frequency",
    predicted=glm_freq.predict(X_test),
    y_label="Claim Frequency",
    title="test data",
    ax=ax[1, 1],
    fill_legend=True
)

mask_train = df_train["ClaimAmount"] > 0
mask_test = df_test["ClaimAmount"] > 0

glm_sev = GammaRegressor(alpha=10., max_iter=10000)

glm_sev.fit(
    X_train[mask_train.values],
    df_train.loc[mask_train, "AvgClaimAmount"],
    sample_weight=df_train.loc[mask_train, "ClaimNb"],
)

scores = score_estimator(
    glm_sev,
    X_train[mask_train.values],
    X_test[mask_test.values],
    df_train[mask_train],
    df_test[mask_test],
    target="AvgClaimAmount",
    weights="ClaimNb",
)
print("Evaluation of GammaRegressor on target AvgClaimAmount")
print(scores)

print("Mean AvgClaim Amount per policy:              %.2f "
      % df_train["AvgClaimAmount"].mean())
print("Mean AvgClaim Amount | NbClaim > 0:           %.2f"
      % df_train["AvgClaimAmount"][df_train["AvgClaimAmount"] > 0].mean())
print("Predicted Mean AvgClaim Amount | NbClaim > 0: %.2f"
      % glm_sev.predict(X_train).mean())

fig, ax = plt.subplots(ncols=1, nrows=2, figsize=(16, 6))

plot_obs_pred(
    df=df_train.loc[mask_train],
    feature="DrivAge",
    weight="Exposure",
    observed="AvgClaimAmount",
    predicted=glm_sev.predict(X_train[mask_train.values]),
    y_label="Average Claim Severity",
    title="train data",
    ax=ax[0],
)

plot_obs_pred(
    df=df_test.loc[mask_test],
    feature="DrivAge",
    weight="Exposure",
    observed="AvgClaimAmount",
    predicted=glm_sev.predict(X_test[mask_test.values]),
    y_label="Average Claim Severity",
    title="test data",
    ax=ax[1],
    fill_legend=True
)
plt.tight_layout()

glm_pure_premium = TweedieRegressor(power=1.9, alpha=.1, max_iter=10000)
glm_pure_premium.fit(X_train, df_train["PurePremium"],
                     sample_weight=df_train["Exposure"])

tweedie_powers = [1.5, 1.7, 1.8, 1.9, 1.99, 1.999, 1.9999]

scores_product_model = score_estimator(
    (glm_freq, glm_sev),
    X_train,
    X_test,
    df_train,
    df_test,
    target="PurePremium",
    weights="Exposure",
    tweedie_powers=tweedie_powers,
)

scores_glm_pure_premium = score_estimator(
    glm_pure_premium,
    X_train,
    X_test,
    df_train,
    df_test,
    target="PurePremium",
    weights="Exposure",
    tweedie_powers=tweedie_powers
)

scores = pd.concat([scores_product_model, scores_glm_pure_premium],
                   axis=1, sort=True,
                   keys=('Product Model', 'TweedieRegressor'))
print("Evaluation of the Product Model and the Tweedie Regressor "
      "on target PurePremium")
with pd.option_context('display.expand_frame_repr', False):
    print(scores)

res = []
for subset_label, X, df in [
    ("train", X_train, df_train),
    ("test", X_test, df_test),
]:
    exposure = df["Exposure"].values
    res.append(
        {
            "subset": subset_label,
            "observed": df["ClaimAmount"].values.sum(),
            "predicted, frequency*severity model": np.sum(
                exposure * glm_freq.predict(X) * glm_sev.predict(X)
            ),
            "predicted, tweedie, power=%.2f"
            % glm_pure_premium.power: np.sum(
                exposure * glm_pure_premium.predict(X)),
        }
    )

print(pd.DataFrame(res).set_index("subset").T)

def lorenz_curve(y_true, y_pred, exposure):
    y_true, y_pred = np.asarray(y_true), np.asarray(y_pred)
    exposure = np.asarray(exposure)

    # order samples by increasing predicted risk:
    ranking = np.argsort(y_pred)
    ranked_exposure = exposure[ranking]
    ranked_pure_premium = y_true[ranking]
    cumulated_claim_amount = np.cumsum(ranked_pure_premium * ranked_exposure)
    cumulated_claim_amount /= cumulated_claim_amount[-1]
    cumulated_samples = np.linspace(0, 1, len(cumulated_claim_amount))
    return cumulated_samples, cumulated_claim_amount


fig, ax = plt.subplots(figsize=(8, 8))

y_pred_product = glm_freq.predict(X_test) * glm_sev.predict(X_test)
y_pred_total = glm_pure_premium.predict(X_test)

for label, y_pred in [("Frequency * Severity model", y_pred_product),
                      ("Compound Poisson Gamma", y_pred_total)]:
    ordered_samples, cum_claims = lorenz_curve(
        df_test["PurePremium"], y_pred, df_test["Exposure"])
    gini = 1 - 2 * auc(ordered_samples, cum_claims)
    label += " (Gini index: {:.3f})".format(gini)
    ax.plot(ordered_samples, cum_claims, linestyle="-", label=label)

# Oracle model: y_pred == y_test
ordered_samples, cum_claims = lorenz_curve(
    df_test["PurePremium"], df_test["PurePremium"], df_test["Exposure"])
gini = 1 - 2 * auc(ordered_samples, cum_claims)
label = "Oracle (Gini index: {:.3f})".format(gini)
ax.plot(ordered_samples, cum_claims, linestyle="-.", color="gray",
        label=label)

# Random baseline
ax.plot([0, 1], [0, 1], linestyle="--", color="black",
        label="Random baseline")
ax.set(
    title="Lorenz Curves",
    xlabel=('Fraction of policyholders\n'
            '(ordered by model from safest to riskiest)'),
    ylabel='Fraction of total claim amount'
)
ax.legend(loc="upper left")
plt.plot()