lw-mmm 0.1.6
Package for Media-Mix-Modelling
License: Apache 2.0

<img align="left" width="120" height="120" src="https://raw.githubusercontent.com/google/lightweight_mmm/main/images/lightweight_mmm_logo_colored_250.png" alt="lightweight_mmm_logo_colored_250"></img>

# Lightweight (Bayesian) Marketing Mix Modeling
### LMMM is a python library that helps organisations understand and optimise marketing spend across media channels.
##### This is not an official Google product.

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[Docs](https://lightweight-mmm.readthedocs.io/en/latest/) •
[Introduction](#introduction) •
[Theory](#theory) •
[Getting Started](#getting-started) •
[References](#references) •
[Community Spotlight](#community-spotlight)

## Introduction

[Marketing Mix Modeling (MMM)](https://en.wikipedia.org/wiki/Marketing_mix_modeling) is used by advertisers to measure advertising effectiveness and inform budget allocation decisions across media channels. Measurement based on aggregated data allows comparison across online and offline channels in addition to being unaffected by recent ecosystem changes (some related to privacy) which may affect attribution modelling. MMM allows you to:

- Estimate the optimal budget allocation across media channels.
- Understand how media channels perform with a change in spend.
- Investigate effects on your target KPI (such as sales) by media channel.

Taking a Bayesian approach to MMM allows an advertiser to integrate prior information into modelling, allowing you to:

- Utilise information from industry experience or previous media mix models using Bayesian priors.
- Report on both parameter and model uncertainty and propagate it to your budget optimisation.
- Construct hierarchical models, with generally tighter credible intervals, using breakout dimensions such as geography.

The LightweightMMM package (built using [Numpyro](https://github.com/pyro-ppl/numpyro) and [JAX](https://github.com/google/jax)) helps advertisers easily build Bayesian MMM models by providing the functionality to appropriately scale data, evaluate models, optimise budget allocations and plot common graphs used in the field.

## Theory

### Simplified Model Overview

An MMM quantifies the relationship between media channel activity  and sales, while controlling for other factors. A simplified model overview is shown below and the full model is set out in the [model documentation](https://lightweight-mmm.readthedocs.io/en/latest/models.html). An MMM is typically run using weekly level observations (e.g. the KPI could be sales per week), however, it can also be run at the daily level.

$$kpi = \alpha + trend + seasonality + media\ channels + other\ factors$$

Where *kpi* is typically the volume or value of sales per time period, $\alpha$ is the model intercept, $trend$ is a flexible non-linear function that captures trends in the data, $seasonality$ is a sinusoidal function with configurable parameters that flexibly captures seasonal trends, $media\ channels$ is a matrix of different media channel activity (typically impressions or costs per time period) which receives transformations depending on the model used (see Media Saturation and Lagging section) and $other\ factors$ is a matrix of other factors that could influence sales.

### Standard and Hierarchical models

The LightweightMMM can either be run using data aggregated at the national level (standard approach) or using data aggregated at a geo level (sub-national hierarchical approach).

1. **National level (standard approach).** This approach is appropriate if the data available is only aggregated at the national level (e.g. The KPI could be national sales per time period). This is the most common format used in MMMs.

2. **Geo level (sub-national hierarchical approach).** This approach is appropriate if the data can be aggregated at a sub-national level (e.g. the KPI could be sales per time period for each state within a country). This approach can yield more accurate results compared to the standard approach because it uses more data points to fit the model. We recommend using a sub-national level model for larger countries such as the US if possible.

### Media Saturation and Lagging

It is likely that the effect of a media channel on sales could have a lagged effect which tapers off slowly over time. Our powerful Bayesian MMM model architecture is designed to capture this effect and offers three different approaches. We recommend users compare all three approaches and use the approach that works the best. The approach that works the best will typically be the one which has the best out-of-sample fit (which is one of the generated outputs). The functional forms of these three approaches are briefly described below and are fully expressed in our [model documentation](https://lightweight-mmm.readthedocs.io/en/latest/models.html).

- [Adstock](https://en.wikipedia.org/wiki/Advertising_adstock): Applies an infinite lag that decreases its weight as time passes.
- [Hill-Adstock](https://en.wikipedia.org/wiki/Hill_equation_(biochemistry)): Applies a sigmoid like function for diminishing returns to the output of the adstock function.
- [Carryover](https://static.googleusercontent.com/media/research.google.com/en//pubs/archive/46001.pdf): Applies a [causal convolution](https://paperswithcode.com/method/causal-convolution) giving more weight to the near values than distant ones.

### Flow chart

<img src="https://raw.githubusercontent.com/google/lightweight_mmm/main/images/flowchart.png" alt="flow_chart"></img>


## Getting started

### Installation

The recommended way of installing lightweight_mmm is through PyPi:

```bash
pip install --upgrade pip
pip install lightweight_mmm
```

If you want to use the most recent and slightly less stable version you can install it from github:

`pip install --upgrade git+https://github.com/google/lightweight_mmm.git`

### Preparing the data
Here we use simulated data but it is assumed you have your data cleaned at this
point. The necessary data will be:

- Media data: Containing the metric per channel and time span (eg. impressions
  per time period). Media values must not contain negative values.
- Extra features: Any other features that one might want to add to the analysis.
  These features need to be known ahead of time for optimization or you would need
  another model to estimate them.
- Target: Target KPI for the model to predict. For example, revenue amount, number of app installs. This will also be the metric
  optimized during the optimization phase.
- Costs: The total cost per media unit per channel.

```python
# Let's assume we have the following datasets with the following shapes (we use
the `simulate_dummy_data` function in utils for this example):
media_data, extra_features, target, costs = utils.simulate_dummy_data(
    data_size=160,
    n_media_channels=3,
    n_extra_features=2,
    geos=5) # Or geos=1 for national model
```
Scaling is a bit of an art, Bayesian techniques work well if the input data is
small scale. We should not center variables at 0. Sales and media should have a
lower bound of 0.

1. `y` can be scaled as `y / jnp.mean(y)`.
2. `media` can be scaled as `X_m / jnp.mean(X_m, axis=0)`, which means the new column mean will be 1.

We provide a `CustomScaler` which can apply multiplications and division scaling
in case the wider used scalers don't fit your use case. Scale your data
accordingly before fitting the model.
Below is an example of usage of this `CustomScaler`:

```python
# Simple split of the data based on time.
split_point = data_size - data_size // 10
media_data_train = media_data[:split_point, :]
target_train = target[:split_point]
extra_features_train = extra_features[:split_point, :]
extra_features_test = extra_features[split_point:, :]

# Scale data
media_scaler = preprocessing.CustomScaler(divide_operation=jnp.mean)
extra_features_scaler = preprocessing.CustomScaler(divide_operation=jnp.mean)
target_scaler = preprocessing.CustomScaler(
    divide_operation=jnp.mean)
# scale cost up by N since fit() will divide it by number of time periods
cost_scaler = preprocessing.CustomScaler(divide_operation=jnp.mean)

media_data_train = media_scaler.fit_transform(media_data_train)
extra_features_train = extra_features_scaler.fit_transform(
    extra_features_train)
target_train = target_scaler.fit_transform(target_train)
costs = cost_scaler.fit_transform(unscaled_costs)
```

In case you have a variable that has a lot of 0s you can also scale by the mean
of non zero values. For instance you can use a lambda function to do this:
`lambda x: jnp.mean(x[x > 0])`. The same applies for cost scaling.

### Training the model
The model requires the media data, the extra features, the costs of each media
unit per channel and the target. You can also pass how many samples you would
like to use as well as the number of chains.

For running multiple chains in parallel the user would need to set
`numpyro.set_host_device_count` to either the number of chains or the number of
CPUs available.

See an example below:

```python
# Fit model.
mmm = lightweight_mmm.LightweightMMM()
mmm.fit(media=media_data,
        extra_features=extra_features,
        media_prior=costs,
        target=target,
        number_warmup=1000,
        number_samples=1000,
        number_chains=2)
```

If you want to change any prior in the model (besides the media prior which you
are already specifying always), you can do so with `custom_priors`:

```python
# See detailed explanation on custom priors in our documentation.
custom_priors = {"intercept": numpyro.distributions.Uniform(1, 5)}

# Fit model.
mmm = lightweight_mmm.LightweightMMM()
mmm.fit(media=media_data,
        extra_features=extra_features,
        media_prior=costs,
        target=target,
        number_warmup=1000,
        number_samples=1000,
        number_chains=2,
        custom_priors=custom_priors)
```

Please refer to our [documentation on custom_priors](https://lightweight-mmm.readthedocs.io/en/latest/custom_priors.html)
for more details.

You can switch between daily and weekly data by enabling
`weekday_seasonality=True` and `seasonality_frequency=365` or
`weekday_seasonality=False` and `seasonality_frequency=52` (default). In case
of daily data we have two types of seasonality: discrete weekday and smooth
annual.

### Model diagnostics
#### Convergence Check
Users can check convergence metrics of the parameters as follows:

```python
mmm.print_summary()
```

The rule of thumb is that `r_hat` values for all parameters are less than 1.1.

#### Fitting check
Users can check fitting between true KPI and predicted KPI by:

```python
plot.plot_model_fit(media_mix_model=mmm, target_scaler=target_scaler)
```

If `target_scaler` used for `preprocessing.CustomScaler()` is given, the target
would be unscaled. Bayesian R-squared and MAPE are shown in the chart.

#### Predictive check
Users can get the prediction for the test data by:

```python
prediction = mmm.predict(
    media=media_data_test,
    extra_features=extra_data_test,
    target_scaler=target_scaler
)
```

Returned prediction are distributions; if point estimates are desired, users
can calculate those based on the given distribution. For example, if `data_size`
of the test data is 20, `number_samples` is 1000 and `number_of_chains` is 2,
`mmm.predict` returns 2000 sets of predictions with 20 data points. Users can
compare the distributions with the true value of the test data and calculate
the metrics such as mean and median.

#### Parameter estimation check
Users can get detail of the parameter estimation by:

```python
mmm.print_summary()
```

The above returns the mean, standard deviation, median and the credible interval
for each parameter. The distribution charts are provided by:

```python
plot.plot_media_channel_posteriors(media_mix_model=mmm, channel_names=media_names)
```

`channel_names` specifies media names in each chart.

#### Media insights
Response curves are provided as follows:

```python
plot.plot_response_curves(media_mix_model=mmm, media_scaler=media_scaler, target_scaler=target_scaler)
```

If `media_scaler` and `target_scaler` used for `preprocessing.CustomScaler()` are given, both the media and target values would be unscaled.

To extract the media effectiveness and ROI estimation, users can do the following:

```python
media_effect_hat, roi_hat = mmm.get_posterior_metrics()
```

`media_effect_hat` is the media effectiveness estimation and `roi_hat` is the ROI estimation. Then users can visualize the distribution of the estimation as follows:

```python
plot.plot_bars_media_metrics(metric=media_effect_hat, channel_names=media_names)
```

```python
plot.plot_bars_media_metrics(metric=roi_hat, channel_names=media_names)
```

### Running the optimization

For optimization we will maximize the sales changing the media inputs such that
the summed cost of the media is constant. We can also allow reasonable bounds
on each media input (eg +- x%). We only optimise across channels and not over
time. For running the optimization one needs the following main parameters:

- `n_time_periods`: The number of time periods you want to simulate (eg. Optimize
  for the next 10 weeks if you trained a model on weekly data).
- The model that was trained.
- The `budget` you want to allocate for the next `n_time_periods`.
- The extra features used for training for the following `n_time_periods`.
- Price per media unit per channel.
- `media_gap` refers to the media data gap between the end of training data and
  the start of the out of sample media given. Eg. if 100 weeks of data were used
  for training and prediction starts 2 months after training data finished we
  need to provide the 8 weeks missing between the training data and the
  prediction data so data transformations (adstock, carryover, ...) can take
  place correctly.

See below and example of optimization:

```python
# Run media optimization.
budget = 40 # your budget here
prices = np.array([0.1, 0.11, 0.12])
extra_features_test = extra_features_scaler.transform(extra_features_test)
solution = optimize_media.find_optimal_budgets(
    n_time_periods=extra_features_test.shape[0],
    media_mix_model=mmm,
    budget=budget,
    extra_features=extra_features_test,
    prices=prices)
```

### Save and load the model
Users can save and load the model as follows:

```python
utils.save_model(mmm, file_path='file_path')
```

Users can specify `file_path` to save the model.
To load a saved MMM model:

```python
utils.load_model(file_path: 'file_path')
```

## Citing LightweightMMM

To cite this repository:

```
@software{lightweight_mmmgithub,
  author = {Pablo Duque and Dirk Nachbar and Yuka Abe and Christiane Ahlheim and Mike Anderson and Yan Sun and Omri Goldstein and Tim Eck},
  title = {LightweightMMM: Lightweight (Bayesian) Marketing Mix Modeling},
  url = {https://github.com/google/lightweight_mmm},
  version = {0.1.6},
  year = {2022},
}
```


## References

- [Jin, Y., Wang, Y., Sun, Y., Chan, D., & Koehler, J. (2017). Bayesian Methods for Media Mix Modeling with Carryover and Shape Effects. Google Inc.](https://research.google/pubs/pub46001/)

- [Chan, D., & Perry, M. (2017). Challenges and Opportunities in Media Mix Modeling.](https://research.google/pubs/pub45998/)

- [Sun, Y., Wang, Y., Jin, Y., Chan, D., & Koehler, J. (2017). Geo-level Bayesian Hierarchical Media Mix Modeling.](https://research.google/pubs/pub46000/)

## Community Spotlight

- [How To Create A Marketing Mix Model With LightweightMMM](https://forecastegy.com/posts/how-to-create-a-marketing-mix-model-with-lightweightmmm/) by Mario Filho.

- [How Google LightweightMMM Works](https://getrecast.com/google-lightweightmmm/) and [A walkthrough of Google’s LightweightMMM](https://www.youtube.com/watch?v=oRYN0V6sPlM) by Mike Taylor.