Gamma-gamma Model#

In this notebook we show how to fit a Gamma-Gamma model in PyMC-Marketing. We compare the results with the lifetimes package (no longer maintained and last meaningful update was July 2020). The model is presented in the paper: Fader, P. S., & Hardie, B. G. (2013). The Gamma-Gamma model of monetary value. February, 2, 1-9.

Prepare Notebook#

import arviz as az
import matplotlib.pyplot as plt
import pandas as pd
from lifetimes import GammaGammaFitter

from pymc_marketing import clv

# Plotting configuration
az.style.use("arviz-darkgrid")
plt.rcParams["figure.figsize"] = [10, 6]
plt.rcParams["figure.dpi"] = 100
plt.rcParams["figure.facecolor"] = "white"

%load_ext autoreload
%autoreload 2
%config InlineBackend.figure_format = "retina"

Load Data#

We start by loading the CDNOW dataset.

data_path = "https://raw.githubusercontent.com/pymc-labs/pymc-marketing/main/data/clv_quickstart.csv"

summary_with_money_value = pd.read_csv(data_path)
summary_with_money_value["customer_id"] = summary_with_money_value.index
summary_with_money_value.head()

	frequency	recency	T	monetary_value	customer_id
0	2	30.43	38.86	22.35	0
1	1	1.71	38.86	11.77	1
2	0	0.00	38.86	0.00	2
3	0	0.00	38.86	0.00	3
4	0	0.00	38.86	0.00	4

For the Gamma-Gamma model, we need to filter out customers who have made only one purchase.

returning_customers_summary = summary_with_money_value.query("frequency > 0")

returning_customers_summary.head()

	frequency	recency	T	monetary_value	customer_id
0	2	30.43	38.86	22.35	0
1	1	1.71	38.86	11.77	1
5	7	29.43	38.86	73.74	5
6	1	5.00	38.86	11.77	6
8	2	35.71	38.86	25.55	8

Model Specification#

Here we briefly describe the assumptions and the parametrization of the Gamma-Gamma model from the paper above.

The model of spend per transaction is based on the following three general assumptions:

The monetary value of a customer’s given transaction varies randomly around their average transaction value.
Average transaction values vary across customers but do not vary over time for any given individual.
The distribution of average transaction values across customers is independent of the transaction process.

For a customer with x transactions, let \(z_1, z_2, \ldots, z_x\) denote the value of each transaction. The customer’s observed average transaction value by

\[ \bar{z} = \frac{1}{x} \sum_{i=1}^{x} z_i \]

Now let’s describe the parametrization:

We assume that \(z_i \sim \text{Gamma}(p, ν)\), with \(E(Z_i| p, ν) = \xi = p/ν\).

– Given the convolution properties of the gamma, it follows that total spend across x transactions is distributed \(\text{Gamma}(px, ν)\).

– Given the scaling property of the gamma distribution, it follows that \(\bar{z} \sim \text{Gamma}(px, νx)\).
We assume \(ν \sim \text{Gamma}(q, \gamma)\).

We are interested in estimating the parameters \(p\), \(q\) and \(ν\).

Note

The Gamma-Gamma model assumes that there is no relationship between the monetary value and the purchase frequency. We can check this assumption by calculating the correlation between the average spend and the frequency of purchases.

returning_customers_summary[["monetary_value", "frequency"]].corr()

	monetary_value	frequency
monetary_value	1.000000	0.113884
frequency	0.113884	1.000000

The value of this correlation is close to \(0.11\), which in practice is considered low enough to proceed with the model.

Lifetimes Implementation#

First, we fit the model using the lifetimes package.

ggf = GammaGammaFitter()
ggf.fit(
    returning_customers_summary["frequency"],
    returning_customers_summary["monetary_value"],
)

<lifetimes.GammaGammaFitter: fitted with 946 subjects, p: 6.25, q: 3.74, v: 15.45>

ggf.summary

	coef	se(coef)	lower 95% bound	upper 95% bound
p	6.248802	1.189687	3.917016	8.580589
q	3.744588	0.290166	3.175864	4.313313
v	15.447748	4.159994	7.294160	23.601336

Once the model is fitted we can use the following method to compute the conditional expectation of the average profit per transaction for a group of one or more customers.

avg_profit = ggf.conditional_expected_average_profit(
    summary_with_money_value["frequency"], summary_with_money_value["monetary_value"]
)
avg_profit.head(10)

  24.658616
  18.911480
  35.171002
  35.171002
  35.171002
  71.462851
  18.911480
  35.171002
  27.282408
  35.171002
dtype: float64

avg_profit.mean()

35.25295817604993

PyMC Marketing Implementation#

We can use the pre-built PyMC Marketing implementation of the Gamma-Gamma model, which also provides nice plotting and prediction methods:

We can build the model so that we can see the model specification:

model = clv.GammaGammaModel(data=returning_customers_summary)
model.build_model()
model

Gamma-Gamma Model (Mean Transactions)
         p ~ HalfFlat()
         q ~ HalfFlat()
         v ~ HalfFlat()
likelihood ~ Potential(f(q, p, v))

Note

It is not necessary to build the model before fitting it. We can fit the model directly.

Using MAP#

To begin with, lets use a numerical optimizer (L-BFGS-B) from scipy.optimize to find the maximum a posteriori (MAP) estimate of the parameters.

idata_map = model.fit(fit_method="map").posterior.to_dataframe()

idata_map

		p	q	v
chain	draw
0	0	6.248787	3.744591	15.447813

These values are very close to the ones obtained by the lifetimes package.

MCMC#

We can also use MCMC to sample from the posterior distribution of the parameters. MCMC is a more robust method than MAP and provides uncertainty estimates for the parameters.

sampler_kwargs = {
    "draws": 2_000,
    "target_accept": 0.9,
    "chains": 4,
    "random_seed": 42,
}

idata_mcmc = model.fit(**sampler_kwargs)

Auto-assigning NUTS sampler...
Initializing NUTS using jitter+adapt_diag...
Multiprocess sampling (4 chains in 2 jobs)
NUTS: [p, q, v]

Sampling 4 chains for 1_000 tune and 2_000 draw iterations (4_000 + 8_000 draws total) took 449 seconds.

idata_mcmc

arviz.InferenceData

We can see some statistics of the posterior distribution of the parameters.

model.fit_summary()

	mean	sd	hdi_3%	hdi_97%	mcse_mean	mcse_sd	ess_bulk	ess_tail	r_hat
p	6.403	1.303	4.217	8.710	0.032	0.023	1686.0	1783.0	1.0
q	3.781	0.290	3.238	4.338	0.007	0.005	1753.0	2187.0	1.0
v	16.051	4.243	8.050	23.907	0.107	0.076	1563.0	1746.0	1.0

Let’s visualize the posterior distributions and the rank plot:

axes = az.plot_trace(
    data=model.idata,
    compact=True,
    kind="rank_bars",
    backend_kwargs={"figsize": (12, 9), "layout": "constrained"},
)
plt.gcf().suptitle("Gamma-Gamma Model Trace", fontsize=18, fontweight="bold");

../../_images/ab636a9863d07ec6eac028ef7619214f79ea1fb3363903d64fcaeab72f805308.png

We can compare the results with the ones obtained by the lifetimes package and the MAP estimation.

fig, axes = plt.subplots(
    nrows=3, ncols=1, figsize=(12, 10), sharex=False, sharey=False, layout="constrained"
)

for i, var_name in enumerate(["p", "q", "v"]):
    ax = axes[i]
    az.plot_posterior(
        idata_mcmc.posterior[var_name].values.flatten(),
        color="C0",
        point_estimate="mean",
        hdi_prob=0.95,
        ref_val=ggf.summary["coef"][var_name],
        ax=ax,
        label="MCMC",
    )
    ax.axvline(
        x=ggf.summary["lower 95% bound"][var_name],
        color="C1",
        linestyle="--",
        label="lifetimes 95% CI",
    )
    ax.axvline(
        x=ggf.summary["upper 95% bound"][var_name],
        color="C1",
        linestyle="--",
    )
    ax.axvline(x=idata_map[var_name].item(), color="C2", linestyle="-.", label="MAP")
    ax.legend(loc="upper right")

plt.gcf().suptitle("Gamma-Gamma Model Parameters", fontsize=18, fontweight="bold");

../../_images/db1c36f5c4ceec3ddfd59ee56c07de2cad9dce23bb9a76ff10dffbdc5bc357c1.png

We see that the lifetimes and MAP estimates are essentially the same. Both of them are close to the mean of the posterior distribution obtained by MCMC.

Expected Customer Spend#

Once we have the posterior distribution of the parameters, we can use the expected_average_profit method to compute the conditional expectation of the average profit per transaction for a group of one or more customers.

expected_spend = model.expected_customer_spend(data=summary_with_money_value)

Let’s see how it looks for a subset of customers.

az.summary(expected_spend.isel(customer_id=range(10)), kind="stats")

	mean	sd	hdi_3%	hdi_97%
x[0]	24.706	0.512	23.839	25.762
x[1]	18.994	1.311	16.641	21.547
x[2]	35.195	0.924	33.447	36.887
x[3]	35.195	0.924	33.447	36.887
x[4]	35.195	0.924	33.447	36.887
x[5]	71.387	0.599	70.165	72.409
x[6]	18.994	1.311	16.641	21.547
x[7]	35.195	0.924	33.447	36.887
x[8]	27.318	0.394	26.627	28.107
x[9]	35.195	0.924	33.447	36.887

ax, *_ = az.plot_forest(
    data=expected_spend.isel(customer_id=(range(10))), combined=True, figsize=(8, 7)
)
ax.set(xlabel="Expected Spend (10 Customers)", ylabel="Customer ID")
ax.set_title("Expected Spend", fontsize=18, fontweight="bold");

../../_images/61af525a1b22cb28eb96c5b2b2d9df17ce79f5a1c55b79a2a44c677f41412ab5.png

Finally, lets look at some statistics and the distribution for the whole dataset.

az.summary(expected_spend.mean("customer_id"), kind="stats")

	mean	sd	hdi_3%	hdi_97%
x	35.268	0.629	34.107	36.442

fig, ax = plt.subplots()
az.plot_dist(expected_spend.mean("customer_id"), label="Mean over Customer ID", ax=ax)
ax.axvline(x=expected_spend.mean(), color="black", ls="--", label="Overall Mean")
ax.legend(loc="upper right")
ax.set(xlabel="Expected Spend", ylabel="Density")
ax.set_title("Expected Spend", fontsize=18, fontweight="bold");

../../_images/b5a6e84270a0845c1558d4f6609653c9140b901681c95fadf97f1a897d25084d.png

%load_ext watermark
%watermark -n -u -v -iv -w -p pymc,pytensor

Last updated: Mon Jul 01 2024

Python implementation: CPython
Python version       : 3.10.13
IPython version      : 8.22.2

pymc    : 5.14.0
pytensor: 2.20.0

arviz         : 0.17.1
pymc_marketing: 0.6.0
matplotlib    : 3.8.3
pandas        : 2.2.1

Watermark: 2.4.3