Note

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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.

```
# Authors: Christian Lorentzen <lorentzen.ch@gmail.com>
# Roman Yurchak <rth.yurchak@gmail.com>
# Olivier Grisel <olivier.grisel@ensta.org>
# License: BSD 3 clause
```

```
from functools import partial
import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
from sklearn.datasets import fetch_openml
from sklearn.metrics import mean_tweedie_deviance
from sklearn.metrics import mean_absolute_error
from sklearn.metrics import mean_squared_error
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, parser="pandas").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, parser="pandas").data
# sum ClaimAmount over identical IDs
df_sev = df_sev.groupby("IDpol").sum()
df = df_freq.join(df_sev, how="left")
df["ClaimAmount"].fillna(0, inplace=True)
# unquote string fields
for column_name in df.columns[df.dtypes.values == object]:
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.pipeline import make_pipeline
from sklearn.preprocessing import FunctionTransformer, OneHotEncoder
from sklearn.preprocessing import StandardScaler, KBinsDiscretizer
from sklearn.compose import ColumnTransformer
df = load_mtpl2()
# 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, subsample=int(2e5), 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.model_selection import train_test_split
from sklearn.linear_model import PoissonRegressor
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 0.0201 0.0219
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 == 0`

as the Gamma distribution has support on \((0, \infty)\), not \([0, \infty)\).We use

`ClaimNb`

as`sample_weight`

to 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 2.400000e-03 2.700000e-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 slighly 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.