Is there a way to make the time-varying intercept always positive?

[sangrey]

Hello,
I'm working on a model with a time-varying intercept, and I was wondering if there was a way to make it always positive? The intercept here works like an organic share, but that should not be negative.

In a sense, I want to turn off the demeaning here:

pymc-marketing/pymc_marketing/mmm/tvp.py

Line 178 in 24496a0

centered_f = f - f.mean(axis=0) + 1

Here's the model I'm running:

   mmm = MMM(
    model_config=my_model_config,
    sampler_config=my_sampler_config,
    date_column="date_week",
    channel_columns=channels,
    control_columns=[] + holiday_names,
    adstock_max_lag=26,
    yearly_seasonality=12,
    adstock='weibull',
    saturation='michaelis_menten',
    time_varying_intercept=True,
)

Here's the config:

my_model_config = {
    "likelihood": {
        "dist": "StudentT",
        "kwargs": {
            "sigma": {"dist": "HalfNormal", "kwargs": {"sigma": 2}},
            "nu": {"dist": "Gamma", "kwargs": {"alpha": 10, "beta": 1}},
        },
    },
    "gamma_control": {"dist": "Gamma", "kwargs": {"mu": 1, "sigma": 4}},
    "intercept": {"dist": "HalfNormal", "kwargs": {"sigma": 0.1}},
    "adstock_lam": {
        "dist": "Gamma",
        "kwargs": {
            "alpha": np.array([1] * 6 + [4] * (len(channels) - 6)),
            "beta": 2 * np.ones(len(channels)),
        },
        "dims": "channel",
    },
    "saturation_lam": {"dist": "Gamma", "kwargs": {"alpha": 1, "beta": 3}},
    "intercept_tvp_config": {
        "m": 20,
        "L": None,
        "eta_lam": 1,
        "ls_mu": 10,
        "ls_sigma": 5,
        "cov_func": None,
    },
}

[williambdean]

I believe it will always be positive. There is a softplus applied to the GP.
Are you seeing negative samples in either prior or posterior?

@ulfaslak correct me if I am wrong

[sangrey]

I am seeing negative values in the posterior. Wouldn't the demeaning operation on line 178 remove the effect of the softplus?

[sangrey]

It appears that you can inherit from the MMM class and reimplement the build_model method. Something like the following should work.

class NoomMMM(MMM):

    def __init__(self, *args, **kwargs):

      super(NoomMMM, self).__init__(*args, **kwargs)
    
    def build_model(
        self,
        X: pd.DataFrame,
        y: pd.Series | np.ndarray,
        **kwargs,
    ) -> None:
       
        self._generate_and_preprocess_model_data(X, y)
        with pm.Model(
            coords=self.model_coords,
            coords_mutable=self.coords_mutable,
        ) as self.model:
            channel_data_ = pm.Data(
                name="channel_data",
                value=self.preprocessed_data["X"][self.channel_columns],
                dims=("date", "channel"),
                mutable=True,
            )

            target_ = pm.Data(
                name="target",
                value=self.preprocessed_data["y"],
                dims="date",
                mutable=True,
            )
            if self.time_varying_intercept | self.time_varying_media:
                time_index = pm.Data(
                    "time_index",
                    self._time_index,
                    dims="date",
                )

            if self.time_varying_intercept:
                intercept_distribution = get_distribution(
                    name=self.model_config["intercept"]["dist"]
                )
                baseline_intercept = intercept_distribution(
                    name="baseline_intercept",
                    **self.model_config["intercept"]["kwargs"],
                )


                intercept_latent_process = create_time_varying_gp_multiplier(
                    name="intercept",
                    dims="date",
                    time_index=time_index,
                    time_index_mid=self._time_index_mid,
                    time_resolution=self._time_resolution,
                    model_config=self.model_config,
                )

                intercept = pm.Deterministic(
                    name="intercept",
                    var=pt.exp(baseline_intercept * intercept_latent_process),
                    dims="date",
                )
            else:
                intercept = create_distribution_from_config(
                    name="intercept", config=self.model_config
                )
        
            mu_var = intercept +  channel_contributions.sum(axis=-1)

            mu = pm.Deterministic(name="mu", var=mu_var, dims="date")

            create_likelihood_distribution(
                name=self.output_var,
                param_config=self.model_config["likelihood"],
                mu=mu,
                observed=target_,
                dims="date",
            )