Mixed Effects (aka Scaled Effects)

Introduction

The revenue we get from our investments are composed of multiple parts. The revenue we get is not purely due to the amount we have invested but rather on multiple external factors. We have a baseline, which is the revenue we get when we don't make any investments nor any external factor affecting our revenue. The investments we make create small impact on top of our baseline, but the rest that make up our revenue is what mixed effects factors in. We need to represent many factors that affect sales other than our marketing actions (these are usually multiplicative). We have a prior that these factors affect the ROI of marketing actions.

NOTE: Currently all mixed effects are multiplicative in nature!

Types of Effects

Few examples are as follows:

Global Effects

Effects that occur on a region or globally. Typically, these are effects that are not within our control.

Brand Effects

Effects that act on each brand. Typically, these are effects that we control.

This is only a few examples of how we can group together different mixed effects in our model.

From these we can see that there are 2 types of mixed effects

1. Positive Effect

Positive Mixed Effects are effects that are postively correlated with revenue. The model learns a positive relationship/weight for these kind of effects, which indirectly leads to greater impact as the effect signal increases.

2. Negative Effect

Negative Mixed Effects are effects that are negatively correlated with revenue. The model learns a negative relationship/weight for these kind of effects, which indirectly leads to smaller impact as the effect signal increases.

How Mixed Effect affect the Baseline?

Mixed Effects can be multiplicative or additive in nature. What this means is, the baseline scales up multiplicatively/additively which totals to our revenue and impacts made on our investments. If they are positive effects, they tend to contribute to a higher impact while negative effects scales the baseline to very small amount.

NOTE: Impacts are always positive! Model learns by how much do we scale the baseline up.

Math

We have the \(effectSignal\), which is the actual signal input. We learn the relationship of how the \(effectSignal\) correlates with revenue. This is our trainable variables \(W\).

\[ rawEffect = effectSignal * W \]

We apply a non-linear activation over this on each effect signal.

\[ {salesEffect}_{signal} = e^{rawEffect} \]

We then get the impact for each granularity and time by multiplying over the effects

\[ impact = \underset{e = numEffects}{\overset{e}{\prod }}{salesEffect}_{signal \ e} \]

This \(impact\) is now applied multiplicatively or additively on the baseline.