This dashboard is a follow up to some work I did a few years back looking at return on investment for investments. The core of the project came about because we didn’t know what the returns would look like to model them. So, I recreated it here as an advertising campaign. So the core context is the initial spend, expected ROI on the spend, and functional form of how that ROI arrives. It will also show the total return mapped by the function, and the amount of return that occurs within half given time. Example, if you have a 12 month timeline, how much ROI will occur in 6 months.

One of the core reasons I found value in this, was that decay functions can be very complex. The harmonic decay includes special constants and functions, the power decay used Euler's solution for the Basel Problem, and the exponential decay required working back from the integral solution since the process solves for remaining amount at a specific time, and not total decay over time. Hopefully this can help as a reference for solving decays for ROI over time.

Models Explored

For all formulas:
Expected Return: ER = Advertising Spend * (Advertising ROI + 1)
t: any given month within the function
T: total number of months
(μ,σ): Mean an standard deviation for normal distribution
(k,λ): scale and shape for Weibull distribution
Euler-Mascheroni: 0.5772156649

1. Linear Decay: A linear model, such that each month is the same step size change.
   Decay: $$ d(t) = ER * (T*(T+1)/2)/((T+1)-t) $$
2. Harmonic Decay: A model using 1/t, such that each month has a relatively smaller decline.
   Harmonic functions rely on the Hurwitz zeta function, and in this application uses a 3rd order Taylor Series.
   Hurwitz zeta function: $$log(T) + Euler-Mascheroni + 1/2n - 1/12n^2 + 1/120n^4 - e$$ Decay: $$ d(t) = (ER / Hurwitz-Zeta) / t $$
3. Power Decay: The strongest initial decay, but smaller steps after t = 1 and t = 2.
   Decay: $$ d(t) = (ER / (π^2 / 6 - 1/(1 + T))/t $$
4. Exponential Decay: Strongest decay over 50%, but more consistent step.
   Decay: $$d(t) = ER*e^{- (t * rate)}$$
5. Normal Distribution Decay: Standard normal distribution, assumes peak returns is at halfway.
   Decay: $$ d(t) = (k/λ(t/λ)^{k-1}e^{(t/λ)^k}) * ER $$
6. Weibull Distribution Decay: The Weibull distribution has a dynamic shape, using scale and shape variables. This is geared towards earlier adoption and than the Normal decay.
   Decay: $$ d(t) = (1/(σ(2π)^{1/2}) e^{-1/2 ((t-μ)/σ)^2}) * ER $$
7. Uniform: A simple straight line amount.
   Decay: $$ d(t) = ER / T $$