make hill pass through the origin (#920)

Author: juanitorduz

pymc_marketing/mmm/components/saturation.py

@@ -367,9 +367,9 @@ class HillSaturation(SaturationTransformation):
     function = hill_saturation
 
     default_priors = {
-        "sigma": Prior("HalfNormal", sigma=2),
-        "beta": Prior("HalfNormal", sigma=2),
-        "lam": Prior("HalfNormal", sigma=2),
+        "sigma": Prior("HalfNormal", sigma=1.5),
+        "beta": Prior("HalfNormal", sigma=1.5),
+        "lam": Prior("HalfNormal", sigma=1.5),
     }
 
 

pymc_marketing/mmm/transformers.py

@@ -908,10 +908,10 @@ def hill_saturation(
     r"""Hill Saturation Function
 
     .. math::
-        f(x) = \frac{\sigma}{1 + e^{-\beta(x - \lambda)}}
+        f(x) = \frac{\sigma}{1 + e^{-\beta(x - \lambda)}} - \frac{\sigma}{1 + e^{\beta\lambda}}
 
     where:
-     - :math:`\sigma` is the maximum value (upper asymptote)
+     - :math:`\sigma` is the upper asymptote
      - :math:`\beta` is the slope parameter
      - :math:`\lambda` is the transition point on the X-axis
      - :math:`x` is the independent variable
@@ -920,7 +920,9 @@ def hill_saturation(
     used to describe the saturation effect in biological systems. The curve is
     characterized by its sigmoidal shape, representing a gradual transition from
     a low, nearly zero level to a high plateau, the maximum value the function
-    will approach as the independent variable grows large.
+    will approach as the independent variable grows large. In this implementation,
+    we add an offset to the sigmoid function to ensure that the function always passes
+    through the origin as we expect zero spend to result in zero contribution.
 
     .. plot::
         :context: close-figs
@@ -968,6 +970,7 @@ def hill_saturation(
         plt.ylabel('Hill Saturation')
         plt.tight_layout()
         plt.show()
+
     Parameters
     ----------
     x : float or array-like
@@ -987,7 +990,7 @@ def hill_saturation(
     float or array-like
         The value of the Hill function for each input value of x.
     """
-    return sigma / (1 + pt.exp(-beta * (x - lam)))
+    return sigma / (1 + pt.exp(-beta * (x - lam))) - sigma / (1 + pt.exp(beta * lam))
 
 
 def root_saturation(

tests/mmm/test_transformers.py

@@ -466,11 +466,23 @@ def test_michaelis_menten(self, x, alpha, lam, expected):
             (3, 2, -1),
         ],
     )
-    def test_monotonicity(self, sigma, beta, lam):
+    def test_hill_monotonicity(self, sigma, beta, lam):
         x = np.linspace(-10, 10, 100)
         y = hill_saturation(x, sigma, beta, lam).eval()
         assert np.all(np.diff(y) >= 0), "The function is not monotonic."
 
+    @pytest.mark.parametrize(
+        "sigma, beta, lam",
+        [
+            (1, 1, 0),
+            (2, 0.5, 1),
+            (3, 2, -1),
+        ],
+    )
+    def test_hill_zero(self, sigma, beta, lam):
+        y = hill_saturation(0, sigma, beta, lam).eval()
+        assert y == pytest.approx(0.0)
+
     @pytest.mark.parametrize(
         "x, sigma, beta, lam",
         [
@@ -479,7 +491,7 @@ def test_monotonicity(self, sigma, beta, lam):
             (-3, 3, 2, -1),
         ],
     )
-    def test_sigma_upper_bound(self, x, sigma, beta, lam):
+    def test_hill_sigma_upper_bound(self, x, sigma, beta, lam):
         y = hill_saturation(x, sigma, beta, lam).eval()
         assert y <= sigma, f"The output {y} exceeds the upper bound sigma {sigma}."
 
@@ -491,11 +503,13 @@ def test_sigma_upper_bound(self, x, sigma, beta, lam):
             (-1, 3, 2, -1, 1.5),
         ],
     )
-    def test_behavior_at_lambda(self, x, sigma, beta, lam, expected):
+    def test_hill_behavior_at_lambda(self, x, sigma, beta, lam, expected):
         y = hill_saturation(x, sigma, beta, lam).eval()
+        offset = sigma / (1 + np.exp(beta * lam))
+        expected_with_offset = expected - offset
         np.testing.assert_almost_equal(
             y,
-            expected,
+            expected_with_offset,
             decimal=5,
             err_msg="The function does not behave as expected at lambda.",
         )
@@ -508,7 +522,7 @@ def test_behavior_at_lambda(self, x, sigma, beta, lam, expected):
             (np.array([1, 2, 3]), 3, 2, 2),
         ],
     )
-    def test_vectorized_input(self, x, sigma, beta, lam):
+    def test_hill_vectorized_input(self, x, sigma, beta, lam):
         y = hill_saturation(x, sigma, beta, lam).eval()
         assert (
             y.shape == x.shape
@@ -522,12 +536,14 @@ def test_vectorized_input(self, x, sigma, beta, lam):
             (3, 2, -1),
         ],
     )
-    def test_asymptotic_behavior(self, sigma, beta, lam):
+    def test_hill_asymptotic_behavior(self, sigma, beta, lam):
         x = 1e6  # A very large value to approximate infinity
         y = hill_saturation(x, sigma, beta, lam).eval()
+        offset = sigma / (1 + np.exp(beta * lam))
+        expected = sigma - offset
         np.testing.assert_almost_equal(
             y,
-            sigma,
+            expected,
             decimal=5,
             err_msg="The function does not approach sigma as x approaches infinity.",
         )