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zV{(fJ;J--9As3!}f61^z@+wR5Bt`r6mM<(ImuZi1!YX(uS0IrJj(rSGvtae^pV z1Zp)0mGSqH^vtL08uc%KmdSX9QLnw@SBF?gg$O@D`EbooHEC9^$cFcq(HU46B13p#{jVMMEx$O( zKLs`121u(maxJrn>%`)D%e55fjeMKk!^v+s&VTjNN4?9>tW^51-ujj|%{{w%KRv4d n>EAjd6m0*ePvS$T!iNnq?;klAl5Ou!$6&8A{XTQ0!%zPLw+Wd* diff --git a/pymc_marketing/mmm/__init__.py b/pymc_marketing/mmm/__init__.py index 1c452e70..c636272a 100644 --- a/pymc_marketing/mmm/__init__.py +++ b/pymc_marketing/mmm/__init__.py @@ -25,6 +25,7 @@ ) from pymc_marketing.mmm.components.saturation import ( HillSaturation, + HillSaturationSigmoid, InverseScaledLogisticSaturation, LogisticSaturation, MichaelisMentenSaturation, @@ -50,6 +51,7 @@ "DelayedSaturatedMMM", "GeometricAdstock", "HillSaturation", + "HillSaturationSigmoid", "LogisticSaturation", "InverseScaledLogisticSaturation", "MMM", diff --git a/pymc_marketing/mmm/components/saturation.py b/pymc_marketing/mmm/components/saturation.py index b510dddb..7cba45aa 100644 --- a/pymc_marketing/mmm/components/saturation.py +++ b/pymc_marketing/mmm/components/saturation.py @@ -77,7 +77,8 @@ def function(self, x, b): from pymc_marketing.mmm.components.base import Transformation from pymc_marketing.mmm.transformers import ( - hill_saturation, + hill_function, + hill_saturation_sigmoid, inverse_scaled_logistic_saturation, logistic_saturation, michaelis_menten, @@ -343,7 +344,7 @@ class MichaelisMentenSaturation(SaturationTransformation): class HillSaturation(SaturationTransformation): """Wrapper around Hill saturation function. - For more information, see :func:`pymc_marketing.mmm.transformers.hill_saturation`. + For more information, see :func:`pymc_marketing.mmm.transformers.hill_function`. .. plot:: :context: close-figs @@ -364,7 +365,41 @@ class HillSaturation(SaturationTransformation): lookup_name = "hill" - function = hill_saturation + def function(self, x, slope, kappa, beta): + return beta * hill_function(x, slope, kappa) + + default_priors = { + "slope": Prior("HalfNormal", sigma=1.5), + "kappa": Prior("HalfNormal", sigma=1.5), + "beta": Prior("HalfNormal", sigma=1.5), + } + + +class HillSaturationSigmoid(SaturationTransformation): + """Wrapper around Hill saturation sigmoid function. + + For more information, see :func:`pymc_marketing.mmm.transformers.hill_saturation_sigmoid`. + + .. plot:: + :context: close-figs + + import matplotlib.pyplot as plt + import numpy as np + from pymc_marketing.mmm import HillSaturationSigmoid + + rng = np.random.default_rng(0) + + adstock = HillSaturationSigmoid() + prior = adstock.sample_prior(random_seed=rng) + curve = adstock.sample_curve(prior) + adstock.plot_curve(curve, sample_kwargs={"rng": rng}) + plt.show() + + """ + + lookup_name = "hill_sigmoid" + + function = hill_saturation_sigmoid default_priors = { "sigma": Prior("HalfNormal", sigma=1.5), @@ -415,6 +450,7 @@ def function(self, x, alpha, beta): TanhSaturationBaselined, MichaelisMentenSaturation, HillSaturation, + HillSaturationSigmoid, RootSaturation, ] } diff --git a/pymc_marketing/mmm/transformers.py b/pymc_marketing/mmm/transformers.py index 83f4e6d2..2ae28d86 100644 --- a/pymc_marketing/mmm/transformers.py +++ b/pymc_marketing/mmm/transformers.py @@ -899,13 +899,86 @@ def michaelis_menten( return alpha * x / (lam + x) -def hill_saturation( +def hill_function( + x: pt.TensorLike, slope: pt.TensorLike, kappa: pt.TensorLike +) -> pt.TensorVariable: + r"""Hill Function + + .. math:: + f(x) = 1 - \frac{\kappa^s}{\kappa^s + x^s} + + where: + - :math:`s` is the slope of the hill. + - :math:`\kappa` is the half-saturation point as :math:`f(\kappa) = 0.5` for any value of :math:`s` and :math:`\kappa`. + - :math:`x` is the independent variable and must be non-negative. + + Hill function from Equation (5) in the paper [1]_. + + .. plot:: + :context: close-figs + + import numpy as np + import matplotlib.pyplot as plt + from pymc_marketing.mmm.transformers import hill_function + x = np.linspace(0, 10, 100) + # Varying slope + slopes = [0.3, 0.7, 1.2] + fig, axes = plt.subplots(1, 3, figsize=(12, 4), sharey=True) + for i, slope in enumerate(slopes): + plt.subplot(1, 3, i+1) + y = hill_function(x, slope, 2).eval() + plt.plot(x, y) + plt.xlabel('x') + plt.title(f'Slope = {slope}') + plt.subplot(1,3,1) + plt.ylabel('Hill Saturation Sigmoid') + plt.tight_layout() + plt.show() + # Varying kappa + kappas = [1, 5, 10] + fig, axes = plt.subplots(1, 3, figsize=(12, 4), sharey=True) + for i, kappa in enumerate(kappas): + plt.subplot(1, 3, i+1) + y = hill_function(x, 1, kappa).eval() + plt.plot(x, y) + plt.xlabel('x') + plt.title(f'Kappa = {kappa}') + plt.subplot(1,3,1) + plt.ylabel('Hill Saturation Sigmoid') + plt.tight_layout() + plt.show() + + Parameters + ---------- + x : float or array-like + The independent variable, typically representing the concentration of a + substrate or the intensity of a stimulus. + slope : float + The slope of the hill. Must pe non-positive. + kappa : float + The half-saturation point as :math:`f(\kappa) = 0.5` for any value of :math:`s` and :math:`\kappa`. + + Returns + ------- + float + The value of the Hill function given the parameters. + + References + ---------- + .. [1] Jin, Yuxue, et al. “Bayesian methods for media mix modeling with carryover and shape effects.” (2017). + """ # noqa: E501 + return pt.as_tensor_variable( + 1 - pt.power(kappa, slope) / (pt.power(kappa, slope) + pt.power(x, slope)) + ) + + +def hill_saturation_sigmoid( x: pt.TensorLike, sigma: pt.TensorLike, beta: pt.TensorLike, lam: pt.TensorLike, ) -> pt.TensorVariable: - r"""Hill Saturation Function + r"""Hill Saturation Sigmoid Function .. math:: f(x) = \frac{\sigma}{1 + e^{-\beta(x - \lambda)}} - \frac{\sigma}{1 + e^{\beta\lambda}} @@ -929,19 +1002,19 @@ def hill_saturation( import numpy as np import matplotlib.pyplot as plt - from pymc_marketing.mmm.transformers import hill_saturation + from pymc_marketing.mmm.transformers import hill_saturation_sigmoid x = np.linspace(0, 10, 100) # Varying sigma sigmas = [0.5, 1, 1.5] fig, axes = plt.subplots(1, 3, figsize=(12, 4), sharey=True) for i, sigma in enumerate(sigmas): plt.subplot(1, 3, i+1) - y = hill_saturation(x, sigma, 2, 5).eval() + y = hill_saturation_sigmoid(x, sigma, 2, 5).eval() plt.plot(x, y) plt.xlabel('x') plt.title(f'Sigma = {sigma}') plt.subplot(1,3,1) - plt.ylabel('Hill Saturation') + plt.ylabel('Hill Saturation Sigmoid') plt.tight_layout() plt.show() # Varying beta @@ -949,12 +1022,12 @@ def hill_saturation( fig, axes = plt.subplots(1, 3, figsize=(12, 4), sharey=True) for i, beta in enumerate(betas): plt.subplot(1, 3, i+1) - y = hill_saturation(x, 1, beta, 5).eval() + y = hill_saturation_sigmoid(x, 1, beta, 5).eval() plt.plot(x, y) plt.xlabel('x') plt.title(f'Beta = {beta}') plt.subplot(1,3,1) - plt.ylabel('Hill Saturation') + plt.ylabel('Hill Saturation Sigmoid') plt.tight_layout() plt.show() # Varying lam @@ -962,12 +1035,12 @@ def hill_saturation( fig, axes = plt.subplots(1, 3, figsize=(12, 4), sharey=True) for i, lam in enumerate(lams): plt.subplot(1, 3, i+1) - y = hill_saturation(x, 1, 2, lam).eval() + y = hill_saturation_sigmoid(x, 1, 2, lam).eval() plt.plot(x, y) plt.xlabel('x') plt.title(f'Lambda = {lam}') plt.subplot(1,3,1) - plt.ylabel('Hill Saturation') + plt.ylabel('Hill Saturation Sigmoid') plt.tight_layout() plt.show() @@ -977,8 +1050,8 @@ def hill_saturation( The independent variable, typically representing the concentration of a substrate or the intensity of a stimulus. sigma : float - The upper asymptote of the curve, representing the maximum value the - function will approach as x grows large. + The upper asymptote of the curve, representing the approximate maximum value the + function will approach as x grows large. The true maximum value is at `sigma * (1 - 1 / (1 + exp(beta * lam)))` beta : float The slope parameter, determining the steepness of the curve. lam : float @@ -988,7 +1061,7 @@ def hill_saturation( Returns ------- float or array-like - The value of the Hill function for each input value of x. + The value of the Hill saturation sigmoid function for each input value of x. """ return sigma / (1 + pt.exp(-beta * (x - lam))) - sigma / (1 + pt.exp(beta * lam)) diff --git a/tests/mmm/components/test_saturation.py b/tests/mmm/components/test_saturation.py index 43245f9b..cebd9cd6 100644 --- a/tests/mmm/components/test_saturation.py +++ b/tests/mmm/components/test_saturation.py @@ -22,6 +22,7 @@ from pymc_marketing.mmm import ( HillSaturation, + HillSaturationSigmoid, InverseScaledLogisticSaturation, LogisticSaturation, MichaelisMentenSaturation, @@ -48,6 +49,7 @@ def saturation_functions(): TanhSaturationBaselined(), MichaelisMentenSaturation(), HillSaturation(), + HillSaturationSigmoid(), RootSaturation(), ] @@ -104,6 +106,7 @@ def test_support_for_lift_test_integrations(saturation) -> None: ("tanh_baselined", TanhSaturationBaselined), ("michaelis_menten", MichaelisMentenSaturation), ("hill", HillSaturation), + ("hill_sigmoid", HillSaturationSigmoid), ("root", RootSaturation), ], ) diff --git a/tests/mmm/test_transformers.py b/tests/mmm/test_transformers.py index 98179c39..c1454c32 100644 --- a/tests/mmm/test_transformers.py +++ b/tests/mmm/test_transformers.py @@ -27,7 +27,8 @@ batched_convolution, delayed_adstock, geometric_adstock, - hill_saturation, + hill_function, + hill_saturation_sigmoid, inverse_scaled_logistic_saturation, logistic_saturation, michaelis_menten, @@ -466,9 +467,9 @@ def test_michaelis_menten(self, x, alpha, lam, expected): (3, 2, -1), ], ) - def test_hill_monotonicity(self, sigma, beta, lam): + def test_hill_sigmoid_monotonicity(self, sigma, beta, lam): x = np.linspace(-10, 10, 100) - y = hill_saturation(x, sigma, beta, lam).eval() + y = hill_saturation_sigmoid(x, sigma, beta, lam).eval() assert np.all(np.diff(y) >= 0), "The function is not monotonic." @pytest.mark.parametrize( @@ -479,8 +480,8 @@ def test_hill_monotonicity(self, sigma, beta, lam): (3, 2, -1), ], ) - def test_hill_zero(self, sigma, beta, lam): - y = hill_saturation(0, sigma, beta, lam).eval() + def test_hill_sigmoid_zero(self, sigma, beta, lam): + y = hill_saturation_sigmoid(0, sigma, beta, lam).eval() assert y == pytest.approx(0.0) @pytest.mark.parametrize( @@ -491,8 +492,8 @@ def test_hill_zero(self, sigma, beta, lam): (-3, 3, 2, -1), ], ) - def test_hill_sigma_upper_bound(self, x, sigma, beta, lam): - y = hill_saturation(x, sigma, beta, lam).eval() + def test_hill_sigmoid_sigma_upper_bound(self, x, sigma, beta, lam): + y = hill_saturation_sigmoid(x, sigma, beta, lam).eval() assert y <= sigma, f"The output {y} exceeds the upper bound sigma {sigma}." @pytest.mark.parametrize( @@ -503,8 +504,8 @@ def test_hill_sigma_upper_bound(self, x, sigma, beta, lam): (-1, 3, 2, -1, 1.5), ], ) - def test_hill_behavior_at_lambda(self, x, sigma, beta, lam, expected): - y = hill_saturation(x, sigma, beta, lam).eval() + def test_hill_sigmoid_behavior_at_lambda(self, x, sigma, beta, lam, expected): + y = hill_saturation_sigmoid(x, sigma, beta, lam).eval() offset = sigma / (1 + np.exp(beta * lam)) expected_with_offset = expected - offset np.testing.assert_almost_equal( @@ -522,8 +523,8 @@ def test_hill_behavior_at_lambda(self, x, sigma, beta, lam, expected): (np.array([1, 2, 3]), 3, 2, 2), ], ) - def test_hill_vectorized_input(self, x, sigma, beta, lam): - y = hill_saturation(x, sigma, beta, lam).eval() + def test_hill_sigmoid_vectorized_input(self, x, sigma, beta, lam): + y = hill_saturation_sigmoid(x, sigma, beta, lam).eval() assert ( y.shape == x.shape ), "The function did not return the correct shape for vectorized input." @@ -536,9 +537,9 @@ def test_hill_vectorized_input(self, x, sigma, beta, lam): (3, 2, -1), ], ) - def test_hill_asymptotic_behavior(self, sigma, beta, lam): + def test_hill_sigmoid_asymptotic_behavior(self, sigma, beta, lam): x = 1e6 # A very large value to approximate infinity - y = hill_saturation(x, sigma, beta, lam).eval() + y = hill_saturation_sigmoid(x, sigma, beta, lam).eval() offset = sigma / (1 + np.exp(beta * lam)) expected = sigma - offset np.testing.assert_almost_equal( @@ -548,6 +549,102 @@ def test_hill_asymptotic_behavior(self, sigma, beta, lam): err_msg="The function does not approach sigma as x approaches infinity.", ) + @pytest.mark.parametrize( + argnames=["slope", "kappa"], + argvalues=[ + (1, 1), + (2, 0.5), + (3, 2), + ], + ids=["slope=1, kappa=1", "slope=2, kappa=0.5", "slope=3, kappa=2"], + ) + def test_hill_monotonicity(self, slope, kappa): + x = np.linspace(0, 10, 100) + y = hill_function(x, slope, kappa).eval() + assert np.all(np.diff(y) >= 0), "The function is not monotonic." + + @pytest.mark.parametrize( + argnames=["slope", "kappa"], + argvalues=[ + (1, 1), + (2, 0.5), + (3, 2), + ], + ids=["slope=1, kappa=1", "slope=2, kappa=0.5", "slope=3, kappa=2"], + ) + def test_hill_zero(self, slope, kappa): + y = hill_function(0, slope, kappa).eval() + assert y == pytest.approx(0.0) + + @pytest.mark.parametrize( + argnames=["x", "slope", "kappa"], + argvalues=[ + (1, 1, 1), + (2, 0.5, 0.5), + (3, 2, 2), + ], + ids=[ + "x=1, slope=1, kappa=1", + "x=2, slope=0.5, kappa=0.5", + "x=3, slope=2, kappa=2", + ], + ) + def test_hill_upper_bound(self, x, slope, kappa): + y = hill_function(x, slope, kappa).eval() + assert y <= 1, f"The output {y} exceeds the upper bound 1." + + @pytest.mark.parametrize( + argnames=["slope", "kappa"], + argvalues=[ + (1, 1), + (2, 0.5), + (3, 2), + ], + ids=["slope=1, kappa=1", "slope=2, kappa=0.5", "slope=3, kappa=2"], + ) + def test_hill_behavior_at_midpoint(self, slope, kappa): + y = hill_function(kappa, slope, kappa).eval() + expected = 0.5 + np.testing.assert_almost_equal( + y, + expected, + decimal=5, + err_msg="The function does not behave as expected at the midpoint.", + ) + + @pytest.mark.parametrize( + "x, slope, kappa", + [ + (np.array([0, 1, 2]), 1, 1), + (np.array([-1, 0, 1]), 2, 0.5), + (np.array([1, 2, 3]), 3, 2), + ], + ) + def test_hill_vectorized_input(self, x, slope, kappa): + y = hill_function(x, slope, kappa).eval() + assert ( + y.shape == x.shape + ), "The function did not return the correct shape for vectorized input." + + @pytest.mark.parametrize( + "slope, kappa", + [ + (1, 1), + (2, 0.5), + (3, 2), + ], + ) + def test_hill_asymptotic_behavior(self, slope, kappa): + x = 1e6 # A very large value to approximate infinity + y = hill_function(x, slope, kappa).eval() + expected = 1 + np.testing.assert_almost_equal( + y, + expected, + decimal=5, + err_msg="The function does not approach sigma as x approaches infinity.", + ) + class TestTransformersComposition: @pytest.mark.parametrize(

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