Growth Models in Time Series

Logistic vs. Gompertz growth

Both are S-shaped (sigmoid) growth curves, often used to model adoption, population growth, or diffusion of innovations. They differ mainly in where the inflection point (maximum growth rate) occurs.

Logistic Growth

• Symmetric S-shape.

• Growth accelerates until halfway to saturation, then slows down.

• Formula:

  $$f(t)=\frac{K}{1+\exp (-r(t-t_0))}$$
  • $K$ = carrying capacity (max customers)
  • $r$ = growth rate
  • $t_0$ = midpoint (inflection point, where growth is fastest)

Gompertz Growth

• Asymmetric S-shape.

• Inflection point occurs earlier than in the logistic curve.

• Growth slows sooner, which can better reflect customer adoption curves in real-world settings.

• Formula:

  $$f(t)=K\cdot \exp (-\exp (-r(t-t_0)))$$
  • $K$ = asymptotic maximum (saturation level)
  • $r$ = growth rate
  • $t_0$ = time at which growth rate is maximal

Key Differences

• Logistic: Symmetric, assumes adoption accelerates until 50% of maximum.
• Gompertz: Skewed, assumes fastest growth happens earlier (common in customer/product growth where early adopters drive rapid initial uptake).

Visual intuition

If you plotted them:

• Logistic = balanced “S” around the midpoint.
• Gompertz = “front-loaded” S, with rapid rise at the start and slower taper toward saturation.

Why consider Gompertz for your case?

• Logistic might overestimate remaining growth (too symmetric), while Gompertz would more realistically taper earlier.

Saturation Check

When you use logistic or Gompertz models, both assume that customer growth will eventually plateau at some maximum value $K$ (the carrying capacity).

A saturation check is simply a way of asking:

• How close are we to that plateau already?
• At what time will growth essentially stop (reach, say, 90–95% of $K$)?

Why it matters

• Business planning: If you’re already close to saturation, forecasts should expect only small incremental growth.
• Model choice: If $K$ is unrealistically high compared to your market size, the logistic/Gompertz fit may not be appropriate.
• Energy Demand Forecasting/Forecast horizon: If you hit 95% saturation in 18 months, forecasting 5 years ahead isn’t meaningful.

How to check saturation

1. Define a threshold (e.g., 95% of $K$)

For the fitted curve Logistic/Gompertz above. We solve for $t$ where $f(t)=0.95,K$.

2. Interpret the result

• If $t$ is in the past → you’re already basically saturated.
• If $t$ is within your 2-year forecast horizon → growth will flatten soon.
• If $t$ is well beyond forecast horizon → you’re still in growth mode.