From 98a281fa8cd66ef571cb44aa3a9fa8b4d069c8c4 Mon Sep 17 00:00:00 2001 From: Felipe Angelim Date: Mon, 16 Dec 2024 08:42:28 -0300 Subject: [PATCH 1/3] Lift test example --- docs/mmm/index.md | 72 ++ docs/mmm/lifttest.ipy | 262 ------ docs/mmm/lifttest/index.ipy | 292 +++---- docs/mmm/lifttest/index.md | 808 ++++++++++++++++++ docs/mmm/lifttest/index_files/output_14_1.png | Bin 0 -> 60313 bytes docs/mmm/lifttest/index_files/output_15_1.png | Bin 0 -> 71350 bytes docs/mmm/lifttest/index_files/output_1_4.png | Bin 0 -> 47704 bytes docs/mmm/lifttest/index_files/output_2_1.png | Bin 0 -> 157086 bytes docs/mmm/lifttest/index_files/output_7_1.png | Bin 0 -> 59340 bytes docs/mmm/lifttest/index_files/output_8_1.png | Bin 0 -> 47141 bytes mkdocs.yml | 5 +- src/prophetverse/datasets/_mmm/lifttest.py | 268 ++++++ 12 files changed, 1271 insertions(+), 436 deletions(-) create mode 100644 docs/mmm/index.md delete mode 100644 docs/mmm/lifttest.ipy create mode 100644 docs/mmm/lifttest/index.md create mode 100644 docs/mmm/lifttest/index_files/output_14_1.png create mode 100644 docs/mmm/lifttest/index_files/output_15_1.png create mode 100644 docs/mmm/lifttest/index_files/output_1_4.png create mode 100644 docs/mmm/lifttest/index_files/output_2_1.png create mode 100644 docs/mmm/lifttest/index_files/output_7_1.png create mode 100644 docs/mmm/lifttest/index_files/output_8_1.png create mode 100644 src/prophetverse/datasets/_mmm/lifttest.py diff --git a/docs/mmm/index.md b/docs/mmm/index.md new file mode 100644 index 0000000..3331c04 --- /dev/null +++ b/docs/mmm/index.md @@ -0,0 +1,72 @@ +# Marketing Mix Modeling (MMM) + +Marketing Mix Modeling (MMM) is a statistical analysis technique that helps obtaining +insights and planning marketing strategies. It is tighly related to Time Series Analysis - +we can think of MMM as a special case of Time Series forecasting, where the goal is to +understand the incrementality of different exogenous variables on the target variable. + +When Prophetverse was created, the objective was to provide a more up-to-date implementation +of Facebook's Prophet model, and add some features and customization options that were +not available in the original implementation. However, as the library evolved, it became +clear that it could be used for more than just forecasting, and that it could be a powerful +tool for MMM. + +Prophetverse has the following features that make it a great choice for MMM: + +* __Modularity__: Prophetverse allows one to create additive bayesian models in a modular +fashion. Users can easily use different effects to account different relationships +between the exogenous variables and the target variable, and even create their own effects. + +* __Versatility__: The effects API can be used for not only adding new components, +but also adding new likelihood terms, such as the ones used for lift test. + + +The following effects may be of interest if you are working on MMM: + +* [**GeometricAdstockEffect**](/reference/effects/#prophetverse.effects.GeometricAdstockEffect): The geometric adstock effect is a widely used technique in MMM to account +for the lagged effect of advertising on sales. It is based on the idea that the effect +of an ad on sales decays over time, and that the decay follows a geometric progression. + +* [**HillEffect**](/reference/effects/#prophetverse.effects.HillEffect): The hill curve accounts for diminishing returns in the effect of an +exogenous variable on the target variable. + +* [**ChainedEffects**](/reference/effects/#prophetverse.effects.ChainedEffects): The chained effect is a way to combine multiple effects into a single +one. For example, you can use adstock and hill together. + +* [**LiftExperimentLikelihood**](/reference/effects/#prophetverse.effects.LiftExperimentLikelihood): The lift experiment likelihood is a likelihood term that can be +used to account for the effect of a lift test on the target variable. It is useful if you +want to understand the incrementality of a variable, and you have already run a lift test +to understand how variations in the input affect the output. + +* [**ExactLikelihood**](/reference/effects/#prophetverse.effects.ExactLikelihood): + The exact likelihood is a likelihood term that can be used to +use a reference value as the incrementality of an exogenous variable. It is useful +if another team in your company has already calculated the incrementality of a +variable and wants to use it in your MMM model. + +## Related libraries + +I invite you to also check other libraries for MMM. Two of them are: + +* [PyMC-Marketing](https://www.pymc-marketing.io/en/stable/index.html): this is an +amazing work by PyMC's developers. It is a library that provides a set of tools for +building Bayesian models for marketing analytics. The documentation is very complete +and it's a good source of information. + +* [Lightweight-MMM](https://lightweight-mmm.readthedocs.io/en/latest/index.html): this +was a library, as far as I know, created by google developers, based on Numpro. Now, +they are developing a new one, called [Meridian](https://developers.google.com/meridian?hl=fr). + + +## Future work + +In future releases of Prophetverse, we aim at providing more tools for MMM. +Particularly, a modular interface for running budget optimization in MMM models. +If you are interested in other features, please let us know by opening an issue in the +[repository](https://github.com/felipeangelimvieira/prophetverse) + + + + + + diff --git a/docs/mmm/lifttest.ipy b/docs/mmm/lifttest.ipy deleted file mode 100644 index c11eb16..0000000 --- a/docs/mmm/lifttest.ipy +++ /dev/null @@ -1,262 +0,0 @@ -# %% [markdown] -# # Univariate timeseries and exogenous effects - -# %% -# Disable warnings -import warnings - -warnings.simplefilter(action="ignore") -import matplotlib.pyplot as plt -import numpy as np -import numpyro -import pandas as pd -from numpyro import distributions as dist -from sktime.forecasting.compose import ForecastingPipeline -from sktime.transformations.series.fourier import FourierFeatures - -numpyro.enable_x64() - -# %% [markdown] -# # Import dataset -# -# We import a dataset from Prophet's original repository. We then put it into sktime-friendly format, where the index is a `pd.PeriodIndex` and the colums are the time series. - -# %% -df = pd.read_csv( - "https://raw.githubusercontent.com/facebook/prophet/main/examples/example_wp_log_peyton_manning.csv" -) -df["ds"] = pd.to_datetime(df["ds"]).dt.to_period("D") -y = df.set_index("ds") -display(y) - -# %% -y2 = y.copy() -# Set numpy seed -np.random.seed(0) - -# Create random input -X = pd.DataFrame( - np.abs(np.random.rand(len(y2), 1)) ** 4, - index=y2.index, - columns=["exog"], -) -true_exog_effect = np.log(1.5 * X["exog"].values.reshape((-1, 1)) + 1) * 0.8 -y = y + true_exog_effect -ax = y.rename(columns={"y": "New series"}).plot.line() -(y - true_exog_effect).rename(columns={"y": "Original series"}).plot.line(ax=ax) - -# %% - -from prophetverse.effects import HillEffect -from prophetverse.effects.trend import PiecewiseLinearTrend -from prophetverse.engine import MAPInferenceEngine -from prophetverse.sktime import Prophetverse -from prophetverse.utils.regex import exact - -model = ( - Prophetverse() - >> PiecewiseLinearTrend( - changepoint_interval=100, - changepoint_prior_scale=0.1, - changepoint_range=-100, - ) - >> MAPInferenceEngine(num_steps=10_000) -) >> ( - "exog_effect", - HillEffect( - half_max_prior=dist.Delta(0.5), - slope_prior=dist.Delta(1), - max_effect_prior=dist.Delta(2), - ), - exact("exogenous"), -) -# %% -from sktime.forecasting.model_selection import temporal_train_test_split - -y_train, y_test, X_train, X_test = temporal_train_test_split(y, X, test_size=365 * 2) - -# %% [markdown] -# ## Custom effect -# -# In the last example, we used an effect that was imported from Prophetverse. You may, however, be interested in your own effects. Lets create a custom "Hill" effect, that is used in Marketing Mix Modelling applications. - -import jax.numpy as jnp - -# %% -from prophetverse.effects.base import BaseAdditiveOrMultiplicativeEffect - - -class HillEffect(BaseAdditiveOrMultiplicativeEffect): - """ - Represents a Hill effect in a time series model. - - Attributes: - half_max_prior: Prior distribution for the half-maximum parameter. - slope_prior: Prior distribution for the slope parameter. - max_effect_prior: Prior distribution for the maximum effect parameter. - effect_mode: Mode of the effect (either "additive" or "multiplicative"). - """ - - def __init__( - self, - half_max_prior=None, - slope_prior=None, - max_effect_prior=None, - min_slope=0, - effect_mode="multiplicative", - ): - - if half_max_prior is None: - half_max_prior = dist.Gamma(2, 1) - if slope_prior is None: - slope_prior = dist.HalfNormal(2) - if max_effect_prior is None: - max_effect_prior = dist.Gamma(2, 1) - - self.half_max_prior = half_max_prior - self.slope_prior = slope_prior - self.min_slope = min_slope - self.max_effect_prior = max_effect_prior - self.effect_mode = effect_mode - super().__init__() - - def _predict(self, data, previous_effects=None): - """ - Computes the effect using the log transformation. - - Args: - trend: The trend component. - data: The input data. - - Returns: - The computed effect. - """ - - half_max = numpyro.sample("half_max", self.half_max_prior) - slope = numpyro.sample("slope", self.slope_prior) + self.min_slope - max_effect = numpyro.sample("max_effect", self.max_effect_prior) - - effect = max_effect * (1 / (1 + (data / half_max) ** -slope)) - effect = jnp.clip(effect, 0, max_effect) - return effect - - -import numpyro - -from prophetverse.effects.lift_experiment import LiftExperimentLikelihood -from prophetverse.effects.linear import LinearEffect -# %% -from prophetverse.effects.log import LogEffect -from prophetverse.sktime import Prophetverse -from prophetverse.sktime.seasonality import seasonal_transformer -from prophetverse.utils.regex import starts_with - -exogenous_effects = [ - ( - "seasonality", - LinearEffect( - prior=dist.Normal(0, 0.1), - effect_mode="multiplicative", - ), - starts_with(["sin", "cos"]), - ), - ("exog", HillEffect(effect_mode="additive"), starts_with("exog")), -] - -model = Prophetverse( - trend="linear", - changepoint_interval=300, - changepoint_prior_scale=0.0001, - exogenous_effects=exogenous_effects, - noise_scale=0.05, - optimizer_steps=50000, - optimizer_name="Adam", - optimizer_kwargs={"step_size": 0.0001}, - inference_method="map", -) -model.fit(y=y_train, X=X_train) - -# %% -sites = model.predict_all_sites(fh=X.index, X=X) - -fig, ax = plt.subplots(figsize=(4, 4)) - -ax.scatter(sites["exog"], true_exog_effect, s=2) -# 45 degree line -ax.plot([0, 1], [0, 1], "k--") -ax.set_xlabel("Predicted effect") -ax.set_ylabel("True effect") -ax.set_title("Effect estimation") -fig.show() - -# %% [markdown] -# ## Leveraging A/B tests results to better detect the effect -# -# In many cases, such as Marketing Mix Modeling, we have access to A/B tests or other experiments that allow us to estimate the effect of a given intervention. We can use this information to tune the output of our variable's effect. In Prophetverse, this can be achieved with `prophetverse.effects.LiftExperimentLikelihood`, that adds a likelihood term using the expected effect for a given date. - -# %% -mocked_lift_test_experiment = pd.DataFrame( - data=np.random.normal(true_exog_effect, 0.1), - columns=["lift_results"], - index=y.index, -) -mocked_lift_test_experiment = mocked_lift_test_experiment.loc[y.index] -# Remove some samples, since we may only have A/B tests for a subset of the data - -mocked_lift_test_experiment = mocked_lift_test_experiment.loc[ - np.random.choice(mocked_lift_test_experiment.index, 100, replace=False) -].reindex(mocked_lift_test_experiment.index) - -display(mocked_lift_test_experiment.head(), mocked_lift_test_experiment.dropna().head()) - -# %% -from prophetverse.effects import LiftExperimentLikelihood - -model_with_lift = model.clone() - - -model_with_lift.set_params( - exogenous_effects=[ - ( - "seasonality", - LinearEffect( - prior=dist.Normal(0, 0.1), - effect_mode="multiplicative", - ), - starts_with(["sin", "cos"]), - ), - ( - "exog", - LiftExperimentLikelihood( - HillEffect(effect_mode="additive"), - lift_test_results=mocked_lift_test_experiment, - prior_scale=0.001, - ), - starts_with("exog"), - ), - ] -) - -model_with_lift.fit(y=y_train, X=X_train) - -# %% -sites_with_lift = model_with_lift.predict_all_sites(fh=X.index, X=X) -sites_with_lift.head() - -# %% -fig, ax = plt.subplots(figsize=(4, 4)) - -ax.scatter( - X["exog"], sites["exog"], s=2, label="Predicted effect without A/B test data" -) -ax.scatter(X["exog"], true_exog_effect, s=5, label="True effect") -ax.scatter( - X["exog"], - sites_with_lift["exog"], - s=2, - label="Predicted effect with A/B test data", -) -ax.set_xlabel("Input value") -ax.set_ylabel("Predicted effect") -fig.legend() -fig.show() diff --git a/docs/mmm/lifttest/index.ipy b/docs/mmm/lifttest/index.ipy index 56eff42..150b3db 100644 --- a/docs/mmm/lifttest/index.ipy +++ b/docs/mmm/lifttest/index.ipy @@ -1,31 +1,55 @@ # %% [markdown] # # Lift test - +# In this example, we will show how you can leverage lift tests executed in the past to improve the detection of the effect of an intervention. +# This example is inspired by the one in [PyMC-Marketing documentation](https://www.pymc-marketing.io/en/stable/notebooks/mmm/mmm_lift_test.html). +# +# ## Get data from simulated dataset +# In this exercise, we will load a synthetic dataset +# composed of a target timeseries (sales, for example) and +# two exogenous variables (investment1 and investment2). +# The two variables are highly correlated. +# +# From this dataset, we also obtain the results of a lift test. +# %% import matplotlib.pyplot as plt -import numpy as np + +from prophetverse.datasets._mmm.lifttest import get_dataset + +y, X, lift_test, true_effect, model = get_dataset() + + # %% -import pandas as pd -index = pd.period_range("2000-01-01", "2005-01-01", freq="D") -rng = np.random.default_rng(0) +fig, ax = plt.subplots(figsize=(10, 15), nrows=3, sharex=True) +y.plot.line(ax=ax[0]) +ax[0].set_title("Sales") +X.plot.line(alpha=0.9, ax=ax[1]) +ax[1].set_title("Exogenous variables") +true_effect.plot.line(ax=ax[2]) +ax[2].set_title("True effect of investment1 and investment2") +fig.show() + +# %% [markdown] +# The lift test dataframe looks like below, with the information +# of how a delta in the exogenous variable (investment1 and investment2) +# affects the target variable (sales). Note that the lift test +# is assigned to dates, since the effect of the intervention can +# vary with time. -X = pd.DataFrame( - { - "investment1": np.cumsum(rng.normal(0, 1, size=len(index))), - "investment2": np.cumsum(rng.normal(0, 1, size=len(index))), - }, - index=index, -) -X -= X.min() -X /= X.max() -X += 0.05 +# %% -X["investment2"] = X["investment1"] + 0.1 + rng.normal(0, 0.01, size=len(index)) -X.plot.line(alpha=0.9) +lift_test1, lift_test2 = lift_test +lift_test1 +# %% [markdown] +# ## Creating the model +# We create the model to estimate the effect of the exogenous variables. +# Since this is an exercise, we use the effects that are the ground truth. +# However, we will see that even with a correct specification, the correlated +# exogenous variables can lead to bad estimates. +# %% import numpyro.distributions as dist -# %% from prophetverse.effects import (HillEffect, LinearEffect, LinearFourierSeasonality) from prophetverse.effects.trend import PiecewiseLinearTrend @@ -56,7 +80,7 @@ model = Prophetverse( "investment1", HillEffect( half_max_prior=dist.HalfNormal(0.2), - slope_prior=dist.Gamma(2,1), + slope_prior=dist.Gamma(2, 1), max_effect_prior=dist.HalfNormal(1.5), effect_mode="additive", ), @@ -71,146 +95,73 @@ model = Prophetverse( exact("investment2"), ), ], - inference_engine=MAPInferenceEngine(num_steps=1), + inference_engine=MAPInferenceEngine( + num_steps=1000, optimizer=LBFGSSolver(memory_size=100, max_linesearch_steps=100) + ), ) - -from prophetverse.experimental.simulate import simulate - -samples = simulate( - model=model, - fh=X.index, - X=X, -) +model.fit(y=y, X=X) +components = model.predict_components(fh=X.index, X=X) # %% -y = pd.DataFrame(data={"sales" : samples["obs"][0].flatten()}, index=index) -true_effect = pd.DataFrame( - data={ - "investment1": samples["investment1"][0].flatten(), - "investment2": samples["investment2"][0].flatten(), - }, - index=index, - -) - -fig, ax = plt.subplots(figsize=(10, 5)) -y.plot.line(ax=ax) -fig.show() - fig, ax = plt.subplots(figsize=(10, 5)) -true_effect.plot.line(ax=ax) +y.plot.line(ax=ax, color="black", label="Sales") +components["obs"].to_frame("Forecast").plot.line(ax=ax) fig.show() - # %% -import logging +fig, axs = plt.subplots(figsize=(10, 10), nrows=2, sharex=True) -from prophetverse.engine.optimizer import LBFGSSolver -from prophetverse.logger import logger +ax = axs[0] -logging.basicConfig() -logging.root.setLevel(logging.DEBUG) - -model = model.set_params( - inference_engine=MAPInferenceEngine( - num_steps=1000, - optimizer=LBFGSSolver(memory_size=100, - max_linesearch_steps=100) -)) -model.fit(y=y, X=X) - -components = model.predict_components(fh=index, X=X) - - -# %% - -fig, ax = plt.subplots(figsize=(10, 5)) -components["obs"].plot.line(ax=ax) -y.plot.line(ax=ax, color="black") -# %% - -plt.figure() -plt.scatter(X["investment1"], components["investment1"]) -plt.scatter(X["investment1"], true_effect["investment1"], color="black") +ax.scatter(X["investment1"], + components["investment1"], + label="Inferred effect") +ax.scatter(X["investment1"], + true_effect["investment1"], + label="True effect", color="black") +ax.set_title("Investment1") +ax.legend() +ax = axs[1] +ax.scatter(X["investment2"], components["investment2"]) +ax.scatter(X["investment2"], true_effect["investment2"], color="black") +ax.set_title("Investment2") -plt.figure() -plt.scatter(X["investment2"], components["investment2"]) -plt.scatter(X["investment2"], true_effect["investment2"], color="black") +fig.show() +# %% [markdown] -# %% -def get_simulated_lift_test(X, n=10): - X_b = X.copy() - - for col in ["investment1", "investment2"]: - - - X_b[col] = X_b[col]*rng.uniform(0.1, 0.9, size=X.shape[0]) - - samples_b = simulate( - model=model.clone().set_params(inference_engine__num_steps=1), - fh=X.index, - X=X_b, - do={k: v[0] for k, v in samples.items()}, - ) - - true_effect_b = pd.DataFrame( - index=X_b.index, - data={ - "investment1": samples_b["investment1"][0].flatten(), - "investment2": samples_b["investment2"][0].flatten(), - }, - ) - - lift = np.abs(true_effect_b - true_effect) - - outs = [] - - for col in ["investment1", "investment2"]: - lift_test_dataframe = pd.DataFrame( - index=X.index, - data={ - "lift": lift[col], - "x_start": X.loc[:, col], - "x_end": X_b.loc[:, col], - "y_start": true_effect.loc[:, col], - "y_end": true_effect_b.loc[:, col], - }, - ) - outs.append(lift_test_dataframe.sample(n=n, replace=False)) - - return tuple(outs) - - -lift_test_dataframe1, lift_test_dataframe2 = get_simulated_lift_test(X, n=7) +# ## Using lift test to improve the estimation +# We will use the lift test to improve the estimation of the effect of the exogenous variables. +# We wrap the original effects of `investment1` and `investment2` in a `LiftExperimentLikelihood` effect. +# This effect will use the lift test data to add a new likelihood term to the model. +# +# ### Creating the effects # %% from prophetverse.effects.lift_likelihood import LiftExperimentLikelihood lift_experiment_effect1 = LiftExperimentLikelihood( effect=model.get_params()["investment1"], - lift_test_results=lift_test_dataframe1, + lift_test_results=lift_test1, prior_scale=1e-2, likelihood_scale=1, ) lift_experiment_effect2 = LiftExperimentLikelihood( effect=model.get_params()["investment2"], - lift_test_results=lift_test_dataframe2, + lift_test_results=lift_test2, prior_scale=1e-2, likelihood_scale=1, ) -# %% - -from numpyro.infer.initialization import init_to_feasible, init_to_sample - -from prophetverse.engine.optimizer import LBFGSSolver +# %% [markdown] +# ### Fitting the new model +# %% new_model = model.clone() new_model.set_params( investment1=lift_experiment_effect1, @@ -219,7 +170,7 @@ new_model.set_params( new_model.fit(y=y, X=X) # %% -new_components = new_model.predict_components(fh=index, X=X) +new_components = new_model.predict_components(fh=X.index, X=X) # %% @@ -229,50 +180,47 @@ y.plot.line(ax=ax, color="black") new_components["obs"].plot.line(ax=ax) # %% - -plt.figure() -plt.scatter(X["investment1"], components["investment1"]) -plt.scatter(X["investment1"], true_effect["investment1"], color="black") -plt.scatter(X["investment1"], new_components["investment1"], color="tab:orange") - -for date in lift_test_dataframe1.index: - baseline = true_effect.loc[date, "investment1"] - plt.plot( - [ - lift_test_dataframe1.loc[date, "x_start"], - lift_test_dataframe1.loc[date, "x_end"], - ], - [ - lift_test_dataframe1.loc[date, "y_start"], - lift_test_dataframe1.loc[date, "y_end"], - ], - color="red", - alpha=0.8, - ) -plt.show() - - -plt.figure() -plt.scatter(X["investment2"], components["investment2"]) -plt.scatter(X["investment2"], true_effect["investment2"], color="black") -plt.scatter(X["investment2"], new_components["investment2"], color="tab:orange") - - -for date in lift_test_dataframe2.index: - baseline = true_effect.loc[date, "investment1"] - plt.plot( - [ - lift_test_dataframe2.loc[date, "x_start"], - lift_test_dataframe2.loc[date, "x_end"], - ], - [ - lift_test_dataframe2.loc[date, "y_start"], - lift_test_dataframe2.loc[date, "y_end"], - ], - color="red", - alpha=0.9, - ) -plt.show() +fig, axs = plt.subplots(figsize=(10, 10), nrows=2, sharex=True) + +ax = axs[0] + +ax.scatter(X["investment1"], + components["investment1"], + label="Previous inferred effect", alpha=0.5) +ax.scatter(X["investment1"], + new_components["investment1"], + label="New inferred effect", alpha=0.5) +ax.scatter(X["investment1"], + true_effect["investment1"], + label="True effect", color="black") +ax.set_title("Investment1") +ax.legend() + +ax = axs[1] +ax.scatter(X["investment2"], + components["investment2"], + label="Previous inferred effect", alpha=0.5) +ax.scatter(X["investment2"], + new_components["investment2"], + label="New inferred effect", alpha=0.5) +ax.scatter(X["investment2"], + true_effect["investment2"], + color="black", label="True effect") +ax.set_title("Investment2") +ax.legend() +fig.show() -# %% +# %% [markdown] +# +# ## Conclusion +# +# In this example, we showed how you can use lift tests to improve the estimation of the effect of exogenous variables. +# The highlights are the following: +# +# - We used a synthetic dataset with two exogenous variables that are highly correlated. +# - We showed that even with a correct specification of the model, the correlated exogenous variables can lead to bad estimates. +# - We then used the lift test to improve the estimation of the effect of the exogenous variables. +# - We wrapped the original effects in a `LiftExperimentLikelihood` effect that uses the lift test data to add a new likelihood term to the model. + +# %% \ No newline at end of file diff --git a/docs/mmm/lifttest/index.md b/docs/mmm/lifttest/index.md new file mode 100644 index 0000000..e71cd10 --- /dev/null +++ b/docs/mmm/lifttest/index.md @@ -0,0 +1,808 @@ +# Lift test +In this example, we will show how you can leverage lift tests executed in the past to improve the detection of the effect of an intervention. +This example is inspired by the one in [PyMC-Marketing documentation](https://www.pymc-marketing.io/en/stable/notebooks/mmm/mmm_lift_test.html). + +## Get data from simulated dataset +In this exercise, we will load a synthetic dataset +composed of a target timeseries (sales, for example) and +two exogenous variables (investment1 and investment2). +The two variables are highly correlated. + +From this dataset, we also obtain the results of a lift test. + + + +```python +import matplotlib.pyplot as plt + +from prophetverse.datasets._mmm.lifttest import get_dataset + +y, X, lift_test, true_effect, model = get_dataset() + + + +``` +

Output: [1]

+ + + +![png](index_files/output_1_4.png) + + + + +```python + +fig, ax = plt.subplots(figsize=(10, 15), nrows=3, sharex=True) +y.plot.line(ax=ax[0]) +ax[0].set_title("Sales") +X.plot.line(alpha=0.9, ax=ax[1]) +ax[1].set_title("Exogenous variables") +true_effect.plot.line(ax=ax[2]) +ax[2].set_title("True effect of investment1 and investment2") +fig.show() + + +``` +

Output: [2]

+ + + +![png](index_files/output_2_1.png) + + + +The lift test dataframe looks like below, with the information +of how a delta in the exogenous variable (investment1 and investment2) +affects the target variable (sales). Note that the lift test +is assigned to dates, since the effect of the intervention can +vary with time. + + + + +```python + +lift_test1, lift_test2 = lift_test +lift_test1 + + +``` +

Output: [3]

+ + + + +
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
liftx_startx_endy_starty_end
2003-09-230.0238340.4383790.2964440.4760270.452193
2003-06-120.0205120.2710270.2152510.4453010.424789
2003-03-030.0266420.2924200.2147010.4511790.424538
2002-06-160.0098630.5751730.4542390.4876100.477747
2000-06-180.0024720.9640470.8585560.5017800.499307
2000-07-150.0339130.9381530.3729000.5012290.467317
2002-03-130.0271970.5948510.3398610.4888150.461617
2001-04-020.0045810.8299850.6978270.4985180.493937
2002-11-040.0456510.1755350.1232260.4030700.357418
2002-12-120.0762290.1044700.0652670.3330040.256775
+
+ + + +## Creating the model +We create the model to estimate the effect of the exogenous variables. +Since this is an exercise, we use the effects that are the ground truth. +However, we will see that even with a correct specification, the correlated +exogenous variables can lead to bad estimates. + + + +```python +import numpyro.distributions as dist + +from prophetverse.effects import (HillEffect, LinearEffect, + LinearFourierSeasonality) +from prophetverse.effects.trend import PiecewiseLinearTrend +from prophetverse.engine import MAPInferenceEngine +from prophetverse.engine.optimizer import LBFGSSolver +from prophetverse.sktime import Prophetverse +from prophetverse.utils.regex import exact, no_input_columns + +model = Prophetverse( + trend=PiecewiseLinearTrend( + changepoint_interval=100, + changepoint_prior_scale=0.001, + changepoint_range=-100, + ), + exogenous_effects=[ + ( + "seasonality", + LinearFourierSeasonality( + freq="D", + sp_list=[365.25], + fourier_terms_list=[3], + prior_scale=0.1, + effect_mode="multiplicative", + ), + no_input_columns, + ), + ( + "investment1", + HillEffect( + half_max_prior=dist.HalfNormal(0.2), + slope_prior=dist.Gamma(2, 1), + max_effect_prior=dist.HalfNormal(1.5), + effect_mode="additive", + ), + exact("investment1"), + ), + ( + "investment2", + LinearEffect( + prior=dist.HalfNormal(0.5), + effect_mode="additive", + ), + exact("investment2"), + ), + ], + inference_engine=MAPInferenceEngine( + num_steps=1000, optimizer=LBFGSSolver(memory_size=100, max_linesearch_steps=100) + ), +) + +model.fit(y=y, X=X) +components = model.predict_components(fh=X.index, X=X) + + +``` +

Output: [4]

+ + +```python + +fig, ax = plt.subplots(figsize=(10, 5)) +y.plot.line(ax=ax, color="black", label="Sales") +components["obs"].to_frame("Forecast").plot.line(ax=ax) +fig.show() + + +``` +

Output: [5]

+ + + +![png](index_files/output_7_1.png) + + + + +```python + +fig, axs = plt.subplots(figsize=(10, 10), nrows=2, sharex=True) + +ax = axs[0] + +ax.scatter(X["investment1"], + components["investment1"], + label="Inferred effect") +ax.scatter(X["investment1"], + true_effect["investment1"], + label="True effect", color="black") +ax.set_title("Investment1") +ax.legend() + +ax = axs[1] +ax.scatter(X["investment2"], components["investment2"]) +ax.scatter(X["investment2"], true_effect["investment2"], color="black") +ax.set_title("Investment2") + +fig.show() + + +``` +

Output: [6]

+ + + +![png](index_files/output_8_1.png) + + + + +## Using lift test to improve the estimation +We will use the lift test to improve the estimation of the effect of the exogenous variables. +We wrap the original effects of `investment1` and `investment2` in a `LiftExperimentLikelihood` effect. +This effect will use the lift test data to add a new likelihood term to the model. + +### Creating the effects + + + +```python + +from prophetverse.effects.lift_likelihood import LiftExperimentLikelihood + +lift_experiment_effect1 = LiftExperimentLikelihood( + effect=model.get_params()["investment1"], + lift_test_results=lift_test1, + prior_scale=1e-2, + likelihood_scale=1, +) + +lift_experiment_effect2 = LiftExperimentLikelihood( + effect=model.get_params()["investment2"], + lift_test_results=lift_test2, + prior_scale=1e-2, + likelihood_scale=1, +) + + +``` + +### Fitting the new model + + + + +```python +new_model = model.clone() +new_model.set_params( + investment1=lift_experiment_effect1, + investment2=lift_experiment_effect2, +) +new_model.fit(y=y, X=X) + + +``` +

Output: [8]

+ + + + +
Prophetverse(exogenous_effects=[('seasonality',
+                                 LinearFourierSeasonality(effect_mode='multiplicative',
+                                                          fourier_terms_list=[3],
+                                                          freq='D',
+                                                          prior_scale=0.1,
+                                                          sp_list=[365.25]),
+                                 '^$'),
+                                ('investment1',
+                                 LiftExperimentLikelihood(effect=HillEffect(effect_mode='additive',
+                                                                            half_max_prior=<numpyro.distributions.continuous.HalfNormal object at 0x31b44d650>,
+                                                                            max_effect_prior...
+2004-06-27  0.420033  0.793289  0.281071  0.650520  0.230487
+2002-01-17  0.171978  0.665964  0.456242  0.546110  0.374131,
+                                                          prior_scale=0.01),
+                                 '^investment2$')],
+             inference_engine=MAPInferenceEngine(num_steps=1000,
+                                                 optimizer=LBFGSSolver(max_linesearch_steps=100,
+                                                                       memory_size=100)),
+             trend=PiecewiseLinearTrend(changepoint_interval=100,
+                                        changepoint_prior_scale=0.001,
+                                        changepoint_range=-100))
Please rerun this cell to show the HTML repr or trust the notebook.
+ + + + +```python +new_components = new_model.predict_components(fh=X.index, X=X) + + +``` + + +```python + +fig, ax = plt.subplots(figsize=(10, 5)) +components["obs"].plot.line(ax=ax) +y.plot.line(ax=ax, color="black") +new_components["obs"].plot.line(ax=ax) + + +``` +

Output: [10]

+ + + + + + + + + + +![png](index_files/output_14_1.png) + + + + +```python +fig, axs = plt.subplots(figsize=(10, 10), nrows=2, sharex=True) + +ax = axs[0] + +ax.scatter(X["investment1"], + components["investment1"], + label="Previous inferred effect", alpha=0.5) +ax.scatter(X["investment1"], + new_components["investment1"], + label="New inferred effect", alpha=0.5) +ax.scatter(X["investment1"], + true_effect["investment1"], + label="True effect", color="black") +ax.set_title("Investment1") +ax.legend() + +ax = axs[1] +ax.scatter(X["investment2"], + components["investment2"], + label="Previous inferred effect", alpha=0.5) +ax.scatter(X["investment2"], + new_components["investment2"], + label="New inferred effect", alpha=0.5) +ax.scatter(X["investment2"], + true_effect["investment2"], + color="black", label="True effect") +ax.set_title("Investment2") +ax.legend() +fig.show() + + + +``` + + + + +![png](index_files/output_15_1.png) + + + + +## Conclusion + +In this example, we showed how you can use lift tests to improve the estimation of the effect of exogenous variables. +The highlights are the following: + +- We used a synthetic dataset with two exogenous variables that are highly correlated. +- We showed that even with a correct specification of the model, the correlated exogenous variables can lead to bad estimates. +- We then used the lift test to improve the estimation of the effect of the exogenous variables. +- We wrapped the original effects in a `LiftExperimentLikelihood` effect that uses the lift test data to add a new likelihood term to the model. + + diff --git a/docs/mmm/lifttest/index_files/output_14_1.png b/docs/mmm/lifttest/index_files/output_14_1.png new file mode 100644 index 0000000000000000000000000000000000000000..de2fdee2233b346b969bbdc77df45bab08df7541 GIT binary patch literal 60313 zcmagGbyQZ}7d86OlF~?nVt~>T(%k~$LpMlwgEUCDNOyO4mw*V;DIwh*((vs^eShQL zF}`u{AE3ZFhkf>5d+j;rnu{QLS#eA>5;O<|f+;B>q5y#)Kp_yg;s?m!J3sl?!Dq-D 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Marketing Mix Modeling 🚀: + - Overview: mmm/index.md + - Lift test calibration: mmm/lifttest/index.md - Mathematical formulation 📖: - Theory: the-theory.md diff --git a/src/prophetverse/datasets/_mmm/lifttest.py b/src/prophetverse/datasets/_mmm/lifttest.py new file mode 100644 index 0000000..4e80d3f --- /dev/null +++ b/src/prophetverse/datasets/_mmm/lifttest.py @@ -0,0 +1,268 @@ +"""Dataset for lifttest example.""" + +import numpy as np +import numpyro.distributions as dist +import pandas as pd + +from prophetverse.effects import HillEffect, LinearEffect, LinearFourierSeasonality +from prophetverse.effects.trend import PiecewiseLinearTrend +from prophetverse.engine import MAPInferenceEngine +from prophetverse.experimental.simulate import simulate +from prophetverse.sktime import Prophetverse +from prophetverse.utils.regex import exact, no_input_columns + + +def get_index(): + """ + Generate a time index ranging from 2000-01-01 to 2005-01-01 with daily frequency. + + Returns + ------- + pd.PeriodIndex + The generated time index. + """ + index = pd.period_range("2000-01-01", "2005-01-01", freq="D") + return index + + +def get_X(index): + """ + Create a DataFrame of two simulated investments with daily data. + + Parameters + ---------- + index : pd.PeriodIndex + The time index for the DataFrame. + + Returns + ------- + pd.DataFrame + DataFrame containing normalized simulated investments. + """ + rng = np.random.default_rng(0) + + X = pd.DataFrame( + { + "investment1": np.cumsum(rng.normal(0, 1, size=len(index))), + "investment2": np.cumsum(rng.normal(0, 1, size=len(index))), + }, + index=index, + ) + X -= X.min() + X /= X.max() + X += 0.05 + + X["investment2"] = X["investment1"] + 0.1 + rng.normal(0, 0.01, size=len(index)) + X.plot.line(alpha=0.9) + + return X + + +def get_groundtruth_model(): + """ + Define and configure a Prophetverse model with custom components. + + Returns + ------- + Prophetverse + Configured Prophetverse model. + """ + model = Prophetverse( + trend=PiecewiseLinearTrend( + changepoint_interval=100, + changepoint_prior_scale=0.001, + changepoint_range=-100, + ), + exogenous_effects=[ + ( + "seasonality", + LinearFourierSeasonality( + freq="D", + sp_list=[365.25], + fourier_terms_list=[3], + prior_scale=0.1, + effect_mode="multiplicative", + ), + no_input_columns, + ), + ( + "investment1", + HillEffect( + half_max_prior=dist.HalfNormal(0.2), + slope_prior=dist.Gamma(2, 1), + max_effect_prior=dist.HalfNormal(1.5), + effect_mode="additive", + ), + exact("investment1"), + ), + ( + "investment2", + LinearEffect( + prior=dist.HalfNormal(0.5), + effect_mode="additive", + ), + exact("investment2"), + ), + ], + inference_engine=MAPInferenceEngine(num_steps=1), + ) + + return model + + +def get_samples(model, X): + """ + Simulate samples from the provided model and input data. + + Parameters + ---------- + model : Prophetverse + The Prophetverse model to use for simulation. + X : pd.DataFrame + Exogenous data for the simulation. + + Returns + ------- + dict + Simulated samples from the model. + """ + samples = simulate( + model=model, + fh=X.index, + X=X, + ) + return samples + + +def get_y(samples, index): + """ + Extract observed sales data from the simulated samples. + + Parameters + ---------- + samples : dict + Simulated samples from the model. + index : pd.PeriodIndex + Time index for the observed data. + + Returns + ------- + pd.DataFrame + DataFrame containing observed sales data. + """ + return pd.DataFrame(data={"sales": samples["obs"][0].flatten()}, index=index) + + +def get_true_effect(samples, index): + """ + Extract the true effects for the exogenous variables from the simulated samples. + + Parameters + ---------- + samples : dict + Simulated samples from the model. + index : pd.PeriodIndex + Time index for the true effects. + + Returns + ------- + pd.DataFrame + DataFrame containing true effects for the exogenous variables. + """ + true_effect = pd.DataFrame( + data={ + "investment1": samples["investment1"][0].flatten(), + "investment2": samples["investment2"][0].flatten(), + }, + index=index, + ) + return true_effect + + +def get_simulated_lift_test(X, model, samples, true_effect, n=10): + """ + Perform a simulated lift test by perturbing exogenous variables. + + Parameters + ---------- + X : pd.DataFrame + Original exogenous data. + model : Prophetverse + The Prophetverse model to use for simulation. + samples : dict + Simulated samples from the model. + true_effect : pd.DataFrame + True effects of the exogenous variables. + n : int, optional + Number of samples to return for the lift test, by default 10. + + Returns + ------- + tuple of pd.DataFrame + Lift test results for each exogenous variable. + """ + rng = np.random.default_rng(1) + + X_b = X.copy() + + for col in ["investment1", "investment2"]: + + X_b[col] = X_b[col] * rng.uniform(0.1, 0.9, size=X.shape[0]) + + samples_b = simulate( + model=model.clone().set_params(inference_engine__num_steps=1), + fh=X.index, + X=X_b, + do={k: v[0] for k, v in samples.items()}, + ) + + true_effect_b = pd.DataFrame( + index=X_b.index, + data={ + "investment1": samples_b["investment1"][0].flatten(), + "investment2": samples_b["investment2"][0].flatten(), + }, + ) + + lift = np.abs(true_effect_b - true_effect) + + outs = [] + + for col in ["investment1", "investment2"]: + lift_test_dataframe = pd.DataFrame( + index=X.index, + data={ + "lift": lift[col], + "x_start": X.loc[:, col], + "x_end": X_b.loc[:, col], + "y_start": true_effect.loc[:, col], + "y_end": true_effect_b.loc[:, col], + }, + ) + outs.append(lift_test_dataframe.sample(n=n, replace=False)) + + return tuple(outs) + + +def get_dataset(): + """ + Generate a complete dataset. + + Includes time index, exogenous data, model, + simulated samples, observed sales, true effects, and lift test results. + + Returns + ------- + tuple + Contains observed sales, exogenous data, lift test results, true effects, + and the Prophetverse model. + """ + index = get_index() + X = get_X(index) + model = get_groundtruth_model() + samples = get_samples(model, X) + y = get_y(samples, index) + true_effect = get_true_effect(samples, index) + lift_test = get_simulated_lift_test(X, model, samples, true_effect, n=10) + + return y, X, lift_test, true_effect, model From 64c97233ce1c2ad568b86e13e92a65eb949a0aaa Mon Sep 17 00:00:00 2001 From: Felipe Angelim Date: Mon, 16 Dec 2024 08:44:38 -0300 Subject: [PATCH 2/3] Fix some typos --- docs/mmm/index.md | 67 +++++++++++------------------------------------ 1 file changed, 15 insertions(+), 52 deletions(-) diff --git a/docs/mmm/index.md b/docs/mmm/index.md index 3331c04..30278e5 100644 --- a/docs/mmm/index.md +++ b/docs/mmm/index.md @@ -1,72 +1,35 @@ # Marketing Mix Modeling (MMM) -Marketing Mix Modeling (MMM) is a statistical analysis technique that helps obtaining -insights and planning marketing strategies. It is tighly related to Time Series Analysis - -we can think of MMM as a special case of Time Series forecasting, where the goal is to -understand the incrementality of different exogenous variables on the target variable. +Marketing Mix Modeling (MMM) is a statistical analysis technique that helps in obtaining insights and planning marketing strategies. It is tightly related to Time Series Analysis — we can think of MMM as a special case of Time Series forecasting, where the goal is to understand the incrementality of different exogenous variables on the target variable. -When Prophetverse was created, the objective was to provide a more up-to-date implementation -of Facebook's Prophet model, and add some features and customization options that were -not available in the original implementation. However, as the library evolved, it became -clear that it could be used for more than just forecasting, and that it could be a powerful -tool for MMM. +When Prophetverse was created, the objective was to provide a more up-to-date implementation of Facebook's Prophet model and to add features and customization options that were not available in the original implementation. However, as the library evolved, it became clear that it could be used for more than just forecasting and that it could be a powerful tool for MMM. Prophetverse has the following features that make it a great choice for MMM: -* __Modularity__: Prophetverse allows one to create additive bayesian models in a modular -fashion. Users can easily use different effects to account different relationships -between the exogenous variables and the target variable, and even create their own effects. - -* __Versatility__: The effects API can be used for not only adding new components, -but also adding new likelihood terms, such as the ones used for lift test. +* __Modularity__: Prophetverse allows users to create additive Bayesian models in a modular fashion. Users can easily include different effects to account for various relationships between the exogenous variables and the target variable, and even create their own effects. +* __Versatility__: The effects API can be used not only for adding new components but also for adding new likelihood terms, such as those used for lift tests. The following effects may be of interest if you are working on MMM: -* [**GeometricAdstockEffect**](/reference/effects/#prophetverse.effects.GeometricAdstockEffect): The geometric adstock effect is a widely used technique in MMM to account -for the lagged effect of advertising on sales. It is based on the idea that the effect -of an ad on sales decays over time, and that the decay follows a geometric progression. - -* [**HillEffect**](/reference/effects/#prophetverse.effects.HillEffect): The hill curve accounts for diminishing returns in the effect of an -exogenous variable on the target variable. - -* [**ChainedEffects**](/reference/effects/#prophetverse.effects.ChainedEffects): The chained effect is a way to combine multiple effects into a single -one. For example, you can use adstock and hill together. - -* [**LiftExperimentLikelihood**](/reference/effects/#prophetverse.effects.LiftExperimentLikelihood): The lift experiment likelihood is a likelihood term that can be -used to account for the effect of a lift test on the target variable. It is useful if you -want to understand the incrementality of a variable, and you have already run a lift test -to understand how variations in the input affect the output. - -* [**ExactLikelihood**](/reference/effects/#prophetverse.effects.ExactLikelihood): - The exact likelihood is a likelihood term that can be used to -use a reference value as the incrementality of an exogenous variable. It is useful -if another team in your company has already calculated the incrementality of a -variable and wants to use it in your MMM model. - -## Related libraries - -I invite you to also check other libraries for MMM. Two of them are: - -* [PyMC-Marketing](https://www.pymc-marketing.io/en/stable/index.html): this is an -amazing work by PyMC's developers. It is a library that provides a set of tools for -building Bayesian models for marketing analytics. The documentation is very complete -and it's a good source of information. +* [**GeometricAdstockEffect**](/reference/effects/#prophetverse.effects.GeometricAdstockEffect): The geometric adstock effect is a widely used technique in MMM to account for the lagged effect of advertising on sales. It is based on the idea that the effect of an ad on sales decays over time and that the decay follows a geometric progression. -* [Lightweight-MMM](https://lightweight-mmm.readthedocs.io/en/latest/index.html): this -was a library, as far as I know, created by google developers, based on Numpro. Now, -they are developing a new one, called [Meridian](https://developers.google.com/meridian?hl=fr). +* [**HillEffect**](/reference/effects/#prophetverse.effects.HillEffect): The Hill curve accounts for diminishing returns in the effect of an exogenous variable on the target variable. +* [**ChainedEffects**](/reference/effects/#prophetverse.effects.ChainedEffects): The chained effect is a way to combine multiple effects into a single one. For example, you can use adstock and Hill together. -## Future work +* [**LiftExperimentLikelihood**](/reference/effects/#prophetverse.effects.LiftExperimentLikelihood): The lift experiment likelihood is a likelihood term that can be used to account for the effect of a lift test on the target variable. It is useful if you want to understand the incrementality of a variable and have already run a lift test to analyze how variations in the input affect the output. -In future releases of Prophetverse, we aim at providing more tools for MMM. -Particularly, a modular interface for running budget optimization in MMM models. -If you are interested in other features, please let us know by opening an issue in the -[repository](https://github.com/felipeangelimvieira/prophetverse) +* [**ExactLikelihood**](/reference/effects/#prophetverse.effects.ExactLikelihood): The exact likelihood is a likelihood term that can be used to incorporate a reference value as the incrementality of an exogenous variable. It is useful if another team in your company has already calculated the incrementality of a variable and you want to use it in your MMM model. +## Related Libraries +I invite you to check out other libraries for MMM. Two of them are: +* [PyMC-Marketing](https://www.pymc-marketing.io/en/stable/index.html): This is an amazing project by PyMC's developers. It is a library that provides a set of tools for building Bayesian models for marketing analytics. The documentation is very comprehensive and a great source of information. +* [Lightweight-MMM](https://lightweight-mmm.readthedocs.io/en/latest/index.html): This library, as far as I know, was created by Google developers based on NumPyro. Now, they are developing a new one called [Meridian](https://developers.google.com/meridian?hl=fr). +## Future Work +In future releases of Prophetverse, we aim to provide more tools for MMM, particularly a modular interface for running budget optimization in MMM models. If you are interested in other features, please let us know by opening an issue in the [repository](https://github.com/felipeangelimvieira/prophetverse). From f9bf9afc0238739e34836bf4d2d4928aad4dff48 Mon Sep 17 00:00:00 2001 From: Felipe Angelim Date: Mon, 16 Dec 2024 08:45:51 -0300 Subject: [PATCH 3/3] Bump to version 0.5.1 --- pyproject.toml | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/pyproject.toml b/pyproject.toml index 3b78fc3..7d70162 100644 --- a/pyproject.toml +++ b/pyproject.toml @@ -1,6 +1,6 @@ [tool.poetry] name = "prophetverse" -version = "0.5.0" +version = "0.5.1" description = "" authors = ["Felipe Angelim "] readme = "README.md"