Curve Formulas
Curve Formula (ATAN)
ATAN Arctangent Saturation Transformation
What is the ATAN Formula?
The ATAN (Arctangent) curve uses the inverse tangent function to create smooth saturation transformations. Unlike the 3-parameter CDR formula, ATAN achieves saturation effects with just 2 parameters, making it faster to compute and easier to interpret.
Why Use ATAN?
Simplicity: Only 2 parameters (Alpha, Power) vs 3 parameters (Alpha, Beta, Gamma)
Speed: Faster computation, especially for large datasets and Bayesian models
Interpretability: Fewer parameters mean clearer business insights
Sufficient: For most marketing channels, 2 parameters capture saturation adequately
The Mathematical Formula
The ATAN transformation in MixModeler applies:
Transformed Value = (2/π) × arctan(alpha × x^power)
Where:
x = original variable value (e.g., media spend)
alpha (α) = scaling parameter controlling the inflection point
power = shape parameter controlling curve form
arctan = inverse tangent function (arctangent)
π = mathematical constant pi (≈3.14159)
(2/π) = normalization factor to scale output between 0 and 1
Key Properties:
Output range: 0 to 1 (normalized saturation)
Always monotonically increasing
Smooth, continuous curve
Asymptotically approaches 1 as spend increases
Formula Behavior by Spend Level
Low Spend Region
Transformation is approximately linear
Effect proportional to spend
No saturation yet
Medium Spend Region
Curve begins bending
Rate of return starts decreasing
Saturation effects emerge
High Spend Region
Curve approaches horizontal asymptote at y=1
Severe diminishing returns
Near-complete saturation
Parameter Roles
Alpha (α) - Inflection Point Parameter
What it controls: Where the curve transitions from linear to saturated
Range: Typically 0.1 to 1.0
Low alpha (0.1-0.3): Early saturation, curve bends quickly
Medium alpha (0.4-0.6): Moderate saturation point
High alpha (0.7-1.0): Late saturation, stays linear longer
Business meaning: The spend level at which diminishing returns become significant
Power - Shape Parameter
What it controls: Whether curve is concave or S-shaped
Key values:
Power = 1.0: Pure concave curve (ADBUG behavior - immediate diminishing returns)
Power = 1.5-2.0: S-shaped curve (ICP behavior - threshold effect)
Power > 2.0: Very steep S-shape with strong threshold
Business meaning: Whether the channel shows immediate diminishing returns or requires threshold spend
ATAN vs CDR Comparison
Parameters
2 (Alpha, Power)
3 (Alpha, Beta, Gamma)
Complexity
Simple
Complex
Speed
Fast
Slower
Flexibility
Moderate
High
Best For
Most channels
Complex patterns
Practical Example
TV advertising for brand awareness:
Original variable: TV_Spend = [0, 10000, 20000, 30000, 40000, 50000]
ATAN parameters: Alpha = 0.0001, Power = 1.8 (S-shaped)
Transformed values (conceptual):
$0 → 0.00 (no spend, no effect)
$10,000 → 0.15 (below threshold)
$20,000 → 0.45 (crossing threshold, rapid growth)
$30,000 → 0.70 (strong effect, starting to saturate)
$40,000 → 0.85 (diminishing returns evident)
$50,000 → 0.92 (near saturation)
Business Insight: TV needs ~$20K weekly to cross effectiveness threshold, optimal range $20K-$35K before severe saturation.
Variable Naming Convention
When you create ATAN curve variables, they follow this pattern:
Format: OriginalVariable|CurveType_ATAN aX.X_powerY.Y
Examples:
TV_Spend|ICP_ATAN a0.5_power1.8Search_Spend_adstock_70|ADBUG_ATAN a0.3_power1.0Display_Impressions|ICP_ATAN a0.7_power2.0
When to Use ATAN
✅ Use ATAN When:
You want faster model fitting (especially Bayesian)
Dataset is smaller
Simpler interpretation is valuable
Computational resources are limited
⚠️ Use CDR When:
You need maximum flexibility for complex patterns
Dataset is large enough to fit 3 parameters
Initial ATAN tests show poor fit
Key Takeaways
ATAN uses 2 parameters for efficient saturation modeling
Alpha controls WHERE saturation begins
Power controls SHAPE: 1.0 for concave, >1.5 for S-shape
Faster computation than CDR
Sufficient for most marketing channels
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