Re-reading the Metaphor
Consumers spend most of their time below the surface .
Each "surfacing" is a purchase. How often someone surfaces is their true purchase rate λ. CRM systems log the surfacing and never see the daily life below the water.
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14 consumers · λ ~ Gamma(k=1.5, θ=0.8) · frame-driven Poisson surfacing
River of Time
Greene 1982 → Schmittlein / Fader 1987-2005 → Netzer 2008 → today. Each generation widened the reality we can see.
λ distribution · GAMMA(1.5, 0.8)
Between-person heterogeneity
Stage 0 / 6
SHARP · PARALLEL-TRACK FOOTNOTE
When the river runs shallow and even.
In mass FMCG categories, every consumer's λ clusters in a low band. Surfacing rates run close together and differences in "the Between" (命) all but vanish. Resources should go to penetration and mental availability.This framework fits: luxury, premium, subscription, life-stage categories.Sharp fits: FMCG, mass everyday goods. Both tracks are right; it depends on the category.
How to read · River of Time
Every consumer is a lily pad drifting on the River of Time.λ . The faster it drifts, the more often it surfaces.between-person heterogeneity .→ to move in and watch each lily pad's path in its lane.
Stage 1 · Greene 1982
Everyone surfaces at their own rhythm.
14 lanes, one consumer each. The middle horizontal line is the surface : below it is daily life, above it the surfacings are recorded purchases.between-person heterogeneity ."the Between" (命) . It cannot see how one person changes over time.
EHRENBERG 1959 · GREENE 1982
Stage 2 · Schmittlein 1987 / Fader 2005
Some lily pads never surface again.
Pareto/NBD brings permanent churn into the framework; BG/NBD simplifies the estimation.
SCHMITTLEIN/MORRISON/COLOMBO 1987 · FADER/HARDIE/LEE 2005
Stage 3 · Netzer 2008
The same lily pad changes how often it surfaces.
HMM lets a consumer move between several latent states.Same lily pad, different rhythm .
NETZER/LATTIN/SRINIVASAN, MARKETING SCIENCE 2008
Stage 4 · Modern frontier
Even the color of a surfacing changes.
State is not only how fast or slow someone surfaces. It is multidimensional: frequency, category, basket size, and channel are all part of the state.different color each time. The whole purchase category has flipped. A one-dimensional HMM cannot see this; it takes a modern sequence model.
Sequence models / deep learning · 2018+
Stage 5 · Sharp footnote
When everyone surfaces at about the same rate.
Byron Sharp is not on the main line; he is on a different track .penetration and mental availability . Both tracks are right; it depends on the category.
SHARP, EHRENBERG-BASS INSTITUTE · 2010, 2018
Synthesis
The lineage evolves, and so does the metaphor.
Each generation redefined what "the lily pad underwater" is.Greene : surfacing rate = λ, constantFader : a lily pad can sink for goodNetzer : the surfacing rate can changeToday : the color of a surfacing can change