My extrapolation of media_temporal_latent_multiplier is linear

[querbesd]

Hello pymc-marketing community 👋

I'm trying to understand the influence of the media_temporal_latent_multiplier when I do sample_posterior_predictive on X_test. I'm not very familiar with Gaussian Processes but I expected the extrapolation to be non linear. Am I correct?

Here is the sampled media_temporal_latent_multiplier:

Here is my code:

import xarray as xr

sampled_mtvp = mmm.sample_posterior_predictive(
    X_pred=X_test,
    include_last_observations=True,  # Focus only on future predictions
    original_scale=False,
    var_names=["media_temporal_latent_multiplier"],  # Focus on latent multipliers
    progressbar=True,
)

# Extract the media latent factors from the training data
media_latent_factor_train = mmm.fit_result["media_temporal_latent_multiplier"].quantile(
    [0.025, 0.50, 0.975], dim=["chain", "draw"]
)

# Extract the media latent factors from the sampled data
media_latent_factor_test = sampled_mtvp["media_temporal_latent_multiplier"].quantile(
    [0.025, 0.5, 0.975], dim="sample"
)

# Combine training and test dates
train_dates = mmm.fit_result.coords["date"]
test_dates = sampled_mtvp.coords["date"]
all_dates = np.concatenate([mmm.fit_result.coords["date"].values, sampled_mtvp.coords["date"].values])

# Combine training and test latent factors
media_latent_factor_combined = xr.concat(
    [media_latent_factor_train, media_latent_factor_test], dim="date"
)

# Plot the combined latent factors
fig, ax = plt.subplots(nrows=1, ncols=1, figsize=(15, 10))

# Median latent factor
sns.lineplot(
    x=all_dates,
    y=media_latent_factor_combined[1],  
    label="Predicted Median",
    color="blue",
    ax=ax
)

# Fill between the 2.5% and 97.5% quantiles
ax.fill_between(
    all_dates,
    media_latent_factor_combined[0], 
    media_latent_factor_combined[2], 
    alpha=0.3,
    color="blue"
)

separation_date = test_dates[0]["date"].values

# Add a vertical line to separate training and test data
ax.axvline(separation_date, color="red", linestyle="--", label="Train-Test Split")

# Set labels, title
ax.set_title("HSGP Media Latent Factor with Future Predictions")
ax.set_xlabel("Date")
ax.set_ylabel("Latent Factor")
ax.tick_params(axis="x", rotation=45)
ax.legend()

plt.show()

Thanks upfront 🙏

[querbesd]

I figured out the extrapolation was not linear but only apparent in my example. The apparent linear (or non-linear) behaviour of sampling on the test set was inherent to the parameters of the Gaussian process (especially ls_mu and L).

I also improved my code by computing media_latent_factor_test = mmm.posterior_predictive["media_temporal_latent_multiplier"].mean(axis=(0, 1)).values[:, None]

Thanks to this notebook 🙏