Smoothing

269. Smoothing

Operators that compute smoothed (averaged) versions of input signals. Implemented as exponential delays but used to model perception rather than physical transit.

269.1. First-order smoothing (exponential smoothing)

𝑑 \frac{\hat{𝑋}}{𝑑} 𝑡 = \frac{𝑋 - \hat{𝑋}}{𝜏}

$\hat{𝑋}$ tracks $𝑋$ with smoothing time $𝜏$ — equivalent to single-exponential smoothing.

In discrete time: ${\hat{𝑋}}_{𝑡} = 𝛼 𝑋_{𝑡} + (1 - 𝛼) {\hat{𝑋}}_{𝑡 - 1}$ with $𝛼 = Δ \frac{𝑡}{𝜏}$ .

In Vensim: SMOOTH(input, tau).

269.2. Higher-order smoothing

$𝑛$ first-order smooths in series — each one smoother. SMOOTH3 is third-order (very common default in SD models). Removes more noise but adds more phase lag.

269.3. TREND function

Computes a smoothed fractional rate of change:

TREND (𝑋, 𝜏_{𝑝}, 𝜏_{𝑡}) = \frac{𝑋 - {\hat{𝑋}}_{𝑝}}{{\hat{𝑋}}_{𝑝} \cdot 𝜏_{𝑡}}

where ${\hat{𝑋}}_{𝑝}$ is $𝑋$ smoothed with time $𝜏_{𝑝}$ . Gives “perceived growth rate” — used to model planners’ expectations.

In Vensim: TREND(input, perception_time, trend_time).

269.4. FORECAST function

Extrapolates the input forward using TREND:

FORECAST (𝑋, ℎ) = 𝑋 \cdot (1 + ℎ \cdot TREND)

Forecast $ℎ$ time units ahead based on current trend.

Used to model planning under expectations — what someone thinks future demand will be.

269.5. Why smoothing models decision-making

People (and organizations) don’t react to raw data; they react to smoothed versions:

Expected demand: smoothed over weeks / months
Perceived inflation: long-term average, not daily news
Expected supply lead time: averaged over past orders

Smoothing operators in SD capture this adaptive expectation behavior. Combined with delays and stocks, they generate the “policy resistance” and stability problems that define Sterman’s beer-game style dynamics.

269.6. Phase lag

Smoothing introduces a delay in perception:

First-order smooth: lag is approximately $𝜏$
Third-order smooth: lag approximately $3 \frac{𝜏}{3} = 𝜏$ but with much sharper response
Higher orders: tighter pulse, slower mean response

Phase lag = part of why supply-chain bullwhip happens: each echelon’s reaction is delayed relative to actual demand.

269.7. See also

Delays — same mechanism, different framing
Stock Management — uses smoothing for expected demand
Exponential Smoothing (ETS) — discrete-time analog