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Tweedie regression on insurance claims#
This example illustrates the use of Poisson, Gamma and Tweedie regression on the French Motor Third-Party Liability Claims dataset, and is inspired by an R tutorial [1].
In this dataset, each sample corresponds to an insurance policy, i.e. a contract within an insurance company and an individual (policyholder). Available features include driver age, vehicle age, vehicle power, etc.
A few definitions: a claim is the request made by a policyholder to the insurer to compensate for a loss covered by the insurance. The claim amount is the amount of money that the insurer must pay. The exposure is the duration of the insurance coverage of a given policy, in years.
Here our goal is to predict the expected value, i.e. the mean, of the total claim amount per exposure unit also referred to as the pure premium.
There are several possibilities to do that, two of which are:
Model the number of claims with a Poisson distribution, and the average claim amount per claim, also known as severity, as a Gamma distribution and multiply the predictions of both in order to get the total claim amount.
Model the total claim amount per exposure directly, typically with a Tweedie distribution of Tweedie power \(p \in (1, 2)\).
In this example we will illustrate both approaches. We start by defining a few helper functions for loading the data and visualizing results.
from functools import partial
import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
from sklearn.datasets import fetch_openml
from sklearn.metrics import (
mean_absolute_error,
mean_squared_error,
mean_tweedie_deviance,
)
def load_mtpl2(n_samples=None):
"""Fetch the French Motor Third-Party Liability Claims dataset.
Parameters
----------
n_samples: int, default=None
number of samples to select (for faster run time). Full dataset has
678013 samples.
"""
# freMTPL2freq dataset from https://www.openml.org/d/41214
df_freq = fetch_openml(data_id=41214, as_frame=True).data
df_freq["IDpol"] = df_freq["IDpol"].astype(int)
df_freq.set_index("IDpol", inplace=True)
# freMTPL2sev dataset from https://www.openml.org/d/41215
df_sev = fetch_openml(data_id=41215, as_frame=True).data
# sum ClaimAmount over identical IDs
df_sev = df_sev.groupby("IDpol").sum()
df = df_freq.join(df_sev, how="left")
df["ClaimAmount"] = df["ClaimAmount"].fillna(0)
# unquote string fields
for column_name in df.columns[[t is object for t in df.dtypes.values]]:
df[column_name] = df[column_name].str.strip("'")
return df.iloc[:n_samples]
def plot_obs_pred(
df,
feature,
weight,
observed,
predicted,
y_label=None,
title=None,
ax=None,
fill_legend=False,
):
"""Plot observed and predicted - aggregated per feature level.
Parameters
----------
df : DataFrame
input data
feature: str
a column name of df for the feature to be plotted
weight : str
column name of df with the values of weights or exposure
observed : str
a column name of df with the observed target
predicted : DataFrame
a dataframe, with the same index as df, with the predicted target
fill_legend : bool, default=False
whether to show fill_between legend
"""
# aggregate observed and predicted variables by feature level
df_ = df.loc[:, [feature, weight]].copy()
df_["observed"] = df[observed] * df[weight]
df_["predicted"] = predicted * df[weight]
df_ = (
df_.groupby([feature])[[weight, "observed", "predicted"]]
.sum()
.assign(observed=lambda x: x["observed"] / x[weight])
.assign(predicted=lambda x: x["predicted"] / x[weight])
)
ax = df_.loc[:, ["observed", "predicted"]].plot(style=".", ax=ax)
y_max = df_.loc[:, ["observed", "predicted"]].values.max() * 0.8
p2 = ax.fill_between(
df_.index,
0,
y_max * df_[weight] / df_[weight].values.max(),
color="g",
alpha=0.1,
)
if fill_legend:
ax.legend([p2], ["{} distribution".format(feature)])
ax.set(
ylabel=y_label if y_label is not None else None,
title=title if title is not None else "Train: Observed vs Predicted",
)
def score_estimator(
estimator,
X_train,
X_test,
df_train,
df_test,
target,
weights,
tweedie_powers=None,
):
"""Evaluate an estimator on train and test sets with different metrics"""
metrics = [
("D² explained", None), # Use default scorer if it exists
("mean abs. error", mean_absolute_error),
("mean squared error", mean_squared_error),
]
if tweedie_powers:
metrics += [
(
"mean Tweedie dev p={:.4f}".format(power),
partial(mean_tweedie_deviance, power=power),
)
for power in tweedie_powers
]
res = []
for subset_label, X, df in [
("train", X_train, df_train),
("test", X_test, df_test),
]:
y, _weights = df[target], df[weights]
for score_label, metric in metrics:
if isinstance(estimator, tuple) and len(estimator) == 2:
# Score the model consisting of the product of frequency and
# severity models.
est_freq, est_sev = estimator
y_pred = est_freq.predict(X) * est_sev.predict(X)
else:
y_pred = estimator.predict(X)
if metric is None:
if not hasattr(estimator, "score"):
continue
score = estimator.score(X, y, sample_weight=_weights)
else:
score = metric(y, y_pred, sample_weight=_weights)
res.append({"subset": subset_label, "metric": score_label, "score": score})
res = (
pd.DataFrame(res)
.set_index(["metric", "subset"])
.score.unstack(-1)
.round(4)
.loc[:, ["train", "test"]]
)
return res
Loading datasets, basic feature extraction and target definitions#
We construct the freMTPL2 dataset by joining the freMTPL2freq table,
containing the number of claims (ClaimNb), with the freMTPL2sev table,
containing the claim amount (ClaimAmount) for the same policy ids
(IDpol).
from sklearn.compose import ColumnTransformer
from sklearn.pipeline import make_pipeline
from sklearn.preprocessing import (
FunctionTransformer,
KBinsDiscretizer,
OneHotEncoder,
StandardScaler,
)
df = load_mtpl2()
# 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)
# If the claim amount is 0, then we do not count it as a claim. The loss function
# used by the severity model needs strictly positive claim amounts. This way
# frequency and severity are more consistent with each other.
df.loc[(df["ClaimAmount"] == 0) & (df["ClaimNb"] >= 1), "ClaimNb"] = 0
log_scale_transformer = make_pipeline(
FunctionTransformer(func=np.log), StandardScaler()
)
column_trans = ColumnTransformer(
[
(
"binned_numeric",
KBinsDiscretizer(n_bins=10, random_state=0),
["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())
ClaimNb Exposure Area VehPower VehAge DrivAge BonusMalus VehBrand \
IDpol
139 1 0.75 F 7 1 61 50 B12
190 1 0.14 B 12 5 50 60 B12
414 1 0.14 E 4 0 36 85 B12
424 2 0.62 F 10 0 51 100 B12
463 1 0.31 A 5 0 45 50 B12
VehGas Density Region ClaimAmount PurePremium Frequency \
IDpol
139 'Regular' 27000 R11 303.00 404.000000 1.333333
190 'Diesel' 56 R25 1981.84 14156.000000 7.142857
414 'Regular' 4792 R11 1456.55 10403.928571 7.142857
424 'Regular' 27000 R11 10834.00 17474.193548 3.225806
463 'Regular' 12 R73 3986.67 12860.225806 3.225806
AvgClaimAmount
IDpol
139 303.00
190 1981.84
414 1456.55
424 5417.00
463 3986.67
Frequency model – Poisson distribution#
The number of claims (ClaimNb) is a positive integer (0 included).
Thus, this target can be modelled by a Poisson distribution.
It is then assumed to be the number of discrete events occurring with a
constant rate in a given time interval (Exposure, in units of years).
Here we model the frequency y = ClaimNb / Exposure, which is still a
(scaled) Poisson distribution, and use Exposure as sample_weight.
from sklearn.linear_model import PoissonRegressor
from sklearn.model_selection import train_test_split
df_train, df_test, X_train, X_test = train_test_split(df, X, random_state=0)
Let us keep in mind that despite the seemingly large number of data points in this dataset, the number of evaluation points where the claim amount is non-zero is quite small:
len(df_test)
169504
len(df_test[df_test["ClaimAmount"] > 0])
6237
As a consequence, we expect a significant variability in our evaluation upon random resampling of the train test split.
The parameters of the model are estimated by minimizing the Poisson deviance
on the training set via a Newton solver. Some of the features are collinear
(e.g. because we did not drop any categorical level in the OneHotEncoder),
we use a weak L2 penalization to avoid numerical issues.
glm_freq = PoissonRegressor(alpha=1e-4, solver="newton-cholesky")
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)
Evaluation of PoissonRegressor on target Frequency
subset train test
metric
D² explained 0.0448 0.0427
mean abs. error 0.1379 0.1378
mean squared error 0.2441 0.2246
Note that the score measured on the test set is surprisingly better than on the training set. This might be specific to this random train-test split. Proper cross-validation could help us to assess the sampling variability of these results.
We can visually compare observed and predicted values, aggregated by the
drivers age (DrivAge), vehicle age (VehAge) and the insurance
bonus/malus (BonusMalus).
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,
)
According to the observed data, the frequency of accidents is higher for
drivers younger than 30 years old, and is positively correlated with the
BonusMalus variable. Our model is able to mostly correctly model this
behaviour.
Severity Model - Gamma distribution#
The mean claim amount or severity (AvgClaimAmount) can be empirically
shown to follow approximately a Gamma distribution. We fit a GLM model for
the severity with the same features as the frequency model.
Note:
We filter out
ClaimAmount == 0as the Gamma distribution has support on \((0, \infty)\), not \([0, \infty)\).We use
ClaimNbassample_weightto account for policies that contain more than one claim.
from sklearn.linear_model import GammaRegressor
mask_train = df_train["ClaimAmount"] > 0
mask_test = df_test["ClaimAmount"] > 0
glm_sev = GammaRegressor(alpha=10.0, solver="newton-cholesky")
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)
Evaluation of GammaRegressor on target AvgClaimAmount
subset train test
metric
D² explained 3.900000e-03 4.400000e-03
mean abs. error 1.756746e+03 1.744042e+03
mean squared error 5.801770e+07 5.030677e+07
Those values of the metrics are not necessarily easy to interpret. It can be insightful to compare them with a model that does not use any input features and always predicts a constant value, i.e. the average claim amount, in the same setting:
from sklearn.dummy import DummyRegressor
dummy_sev = DummyRegressor(strategy="mean")
dummy_sev.fit(
X_train[mask_train.values],
df_train.loc[mask_train, "AvgClaimAmount"],
sample_weight=df_train.loc[mask_train, "ClaimNb"],
)
scores = score_estimator(
dummy_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 a mean predictor on target AvgClaimAmount")
print(scores)
Evaluation of a mean predictor on target AvgClaimAmount
subset train test
metric
D² explained 0.000000e+00 -0.000000e+00
mean abs. error 1.756687e+03 1.744497e+03
mean squared error 5.803882e+07 5.033764e+07
We conclude that the claim amount is very challenging to predict. Still, the
GammaRegressor is able to leverage some
information from the input features to slightly improve upon the mean
baseline in terms of D².
Note that the resulting model is the average claim amount per claim. As such, it is conditional on having at least one claim, and cannot be used to predict the average claim amount per policy. For this, it needs to be combined with a claims frequency model.
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()
)
print(
"Predicted Mean AvgClaim Amount (dummy) | NbClaim > 0: %.2f"
% dummy_sev.predict(X_train).mean()
)
Mean AvgClaim Amount per policy: 71.78
Mean AvgClaim Amount | NbClaim > 0: 1951.21
Predicted Mean AvgClaim Amount | NbClaim > 0: 1940.95
Predicted Mean AvgClaim Amount (dummy) | NbClaim > 0: 1978.59
We can visually compare observed and predicted values, aggregated for
the drivers age (DrivAge).
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()
Overall, the drivers age (DrivAge) has a weak impact on the claim
severity, both in observed and predicted data.
Pure Premium Modeling via a Product Model vs single TweedieRegressor#
As mentioned in the introduction, the total claim amount per unit of exposure can be modeled as the product of the prediction of the frequency model by the prediction of the severity model.
Alternatively, one can directly model the total loss with a unique
Compound Poisson Gamma generalized linear model (with a log link function).
This model is a special case of the Tweedie GLM with a “power” parameter
\(p \in (1, 2)\). Here, we fix apriori the power parameter of the
Tweedie model to some arbitrary value (1.9) in the valid range. Ideally one
would select this value via grid-search by minimizing the negative
log-likelihood of the Tweedie model, but unfortunately the current
implementation does not allow for this (yet).
We will compare the performance of both approaches.
To quantify the performance of both models, one can compute
the mean deviance of the train and test data assuming a Compound
Poisson-Gamma distribution of the total claim amount. This is equivalent to
a Tweedie distribution with a power parameter between 1 and 2.
The sklearn.metrics.mean_tweedie_deviance depends on a power
parameter. As we do not know the true value of the power parameter, we here
compute the mean deviances for a grid of possible values, and compare the
models side by side, i.e. we compare them at identical values of power.
Ideally, we hope that one model will be consistently better than the other,
regardless of power.
from sklearn.linear_model import TweedieRegressor
glm_pure_premium = TweedieRegressor(power=1.9, alpha=0.1, solver="newton-cholesky")
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)
Evaluation of the Product Model and the Tweedie Regressor on target PurePremium
Product Model TweedieRegressor
subset train test train test
metric
D² explained NaN NaN 1.640000e-02 1.370000e-02
mean Tweedie dev p=1.5000 7.669930e+01 7.617050e+01 7.640770e+01 7.640880e+01
mean Tweedie dev p=1.7000 3.695740e+01 3.683980e+01 3.682880e+01 3.692270e+01
mean Tweedie dev p=1.8000 3.046010e+01 3.040530e+01 3.037600e+01 3.045390e+01
mean Tweedie dev p=1.9000 3.387580e+01 3.385000e+01 3.382120e+01 3.387830e+01
mean Tweedie dev p=1.9900 2.015716e+02 2.015414e+02 2.015347e+02 2.015587e+02
mean Tweedie dev p=1.9990 1.914573e+03 1.914370e+03 1.914538e+03 1.914387e+03
mean Tweedie dev p=1.9999 1.904751e+04 1.904556e+04 1.904747e+04 1.904558e+04
mean abs. error 2.730119e+02 2.722128e+02 2.739865e+02 2.731249e+02
mean squared error 3.295040e+07 3.212197e+07 3.295505e+07 3.213056e+07
In this example, both modeling approaches yield comparable performance metrics. For implementation reasons, the percentage of explained variance \(D^2\) is not available for the product model.
We can additionally validate these models by comparing observed and predicted total claim amount over the test and train subsets. We see that, on average, both model tend to underestimate the total claim (but this behavior depends on the amount of regularization).
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)
subset train test
observed 3.917618e+07 1.299546e+07
predicted, frequency*severity model 3.916555e+07 1.313276e+07
predicted, tweedie, power=1.90 3.951751e+07 1.325198e+07
Finally, we can compare the two models using a plot of cumulative claims: for each model, the policyholders are ranked from safest to riskiest based on the model predictions and the cumulative proportion of claim amounts is plotted against the cumulative proportion of exposure. This plot is often called the ordered Lorenz curve of the model.
The Gini coefficient (based on the area between the curve and the diagonal) can be used as a model selection metric to quantify the ability of the model to rank policyholders. Note that this metric does not reflect the ability of the models to make accurate predictions in terms of absolute value of total claim amounts but only in terms of relative amounts as a ranking metric. The Gini coefficient is upper bounded by 1.0 but even an oracle model that ranks the policyholders by the observed claim amounts cannot reach a score of 1.0.
We observe that both models are able to rank policyholders by riskiness significantly better than chance although they are also both far from the oracle model due to the natural difficulty of the prediction problem from a few features: most accidents are not predictable and can be caused by environmental circumstances that are not described at all by the input features of the models.
Note that the Gini index only characterizes the ranking performance of the model but not its calibration: any monotonic transformation of the predictions leaves the Gini index of the model unchanged.
Finally one should highlight that the Compound Poisson Gamma model that is directly fit on the pure premium is operationally simpler to develop and maintain as it consists of a single scikit-learn estimator instead of a pair of models, each with its own set of hyperparameters.
from sklearn.metrics import auc
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]
cumulative_claim_amount = np.cumsum(ranked_pure_premium * ranked_exposure)
cumulative_claim_amount /= cumulative_claim_amount[-1]
cumulative_exposure = np.cumsum(ranked_exposure)
cumulative_exposure /= cumulative_exposure[-1]
return cumulative_exposure, cumulative_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),
]:
cum_exposure, cum_claims = lorenz_curve(
df_test["PurePremium"], y_pred, df_test["Exposure"]
)
gini = 1 - 2 * auc(cum_exposure, cum_claims)
label += " (Gini index: {:.3f})".format(gini)
ax.plot(cum_exposure, cum_claims, linestyle="-", label=label)
# Oracle model: y_pred == y_test
cum_exposure, cum_claims = lorenz_curve(
df_test["PurePremium"], df_test["PurePremium"], df_test["Exposure"]
)
gini = 1 - 2 * auc(cum_exposure, cum_claims)
label = "Oracle (Gini index: {:.3f})".format(gini)
ax.plot(cum_exposure, 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=(
"Cumulative proportion of exposure\n"
"(ordered by model from safest to riskiest)"
),
ylabel="Cumulative proportion of claim amounts",
)
ax.legend(loc="upper left")
plt.plot()
[]
Total running time of the script: (0 minutes 8.568 seconds)
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