Hill Function Visualizer

By Rohit Ravi

Explore how saturation parameters affect response curves in marketing mix models. Adjust the sliders to see how K, S, and max transform your media spend data.

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f(Spend) = 100 × Spend1.50 / (Spend1.50 + 501.50)
50
Spend at 50% response
1.50
Controls curve steepness
100
Maximum response ceiling

Response Curve

The hill function transforms any spend input into a saturation-adjusted response.

Transformed Time Series

See how the hill function transforms a sample media spend series with always-on activation and two peaks.

Saturation Profiles

Different channels exhibit distinct saturation behaviors. Click an example to apply those parameters.

TV Advertising
High Half-Saturation, High Shape
K: 70 S: 2.0 Max: 100
TV builds awareness slowly, requiring significant spend to reach effectiveness. The high K and S values create a pronounced S-curve, reflecting delayed brand-building effects and long tail of influence.
Digital Video
Moderate Saturation
K: 40 S: 1.2 Max: 100
Digital video captures intent-driven attention with faster response than traditional media. Lower K allows earlier effectiveness, while moderate S provides a balanced response curve reflecting engagement patterns.
Paid Search
Low Half-Saturation, Low Shape
K: 20 S: 0.8 Max: 100
Search captures immediate purchase intent, delivering rapid returns on spend. Very low K means effectiveness kicks in quickly, while S below 1 creates a diminishing returns curve that flattens early.
OOH / Billboard
High K, Low S
K: 80 S: 0.8 Max: 100
Out-of-home advertising reaches a broad audience but requires substantial spend for impact. The high K delays effectiveness, while low S produces a gradual concave curve—moderate spend yields proportional reach, but benefits diminish slowly as exposure saturates.
Social Retargeting
Low K, High S
K: 25 S: 2.0 Max: 100
Retargeting reaches users already familiar with your brand, so effectiveness kicks in at lower spend. The high S creates an S-curve—minimal spend yields little, then rapid acceleration, then plateau as the audience exhausts. Good for conversion-focused campaigns.
Email Marketing
Medium K, Medium S, 60% Max
K: 50 S: 1.0 Max: 60
Email marketing provides reliable but bounded returns—once your list is engaged, additional frequency has diminishing impact. Medium K and S near 1 create a standard diminishing returns curve, while the lower max reflects email's limited incremental contribution to overall conversions.