The Generalized Logistic Model¶

In [1]:

import numpy as np
import matplotlib.pyplot as plt
import ipywidgets as widgets
from ipywidgets import interact
import plotly.graph_objects as go
import pandas as pd
import patsy as pt

from yeastdnnexplorer.utils.sigmoid import sigmoid
from yeastdnnexplorer.ml_models.GeneralizedLogisticModel import GeneralizedLogisticModel

import numpy as np import matplotlib.pyplot as plt import ipywidgets as widgets from ipywidgets import interact import plotly.graph_objects as go import pandas as pd import patsy as pt from yeastdnnexplorer.utils.sigmoid import sigmoid from yeastdnnexplorer.ml_models.GeneralizedLogisticModel import GeneralizedLogisticModel

The Sigmoid Function¶

The sigmoid() function takes the following arguments:

• The observations as a model matrix, where the first column is a constant vector
• Upper Asymptote: The upper bound of the sigmoid curve.
• Lower Asymptote: The lower bound of the sigmoid curve.
• Coefficients (B): The coefficients of the linear combination of input variables that will control the shape of the sigmoid curve.

$$ Y(X) = \frac{right\_asymptote - left\_asymptote}{1 + e^{- X \cdot B }} + left\_asymptote $$

Fitting the Sigmoid Model to Data¶

The GeneralizedLogisticModel is used to fit the sigmoid parameters to data. The following two examples are with simulated data

Single Variable Example¶

In [4]:

glm_single_variable = GeneralizedLogisticModel()

# Generate sample data
np.random.seed(42)
X1 = np.linspace(-10, 10, 100)
X0 = np.ones_like(X1)
X = np.column_stack([X0, X1])
# True sigmoid parameters
true_left_asymptote = 1.2
true_right_asymptote = 6.0
true_slope = [-9, 5]

# Generate Y data using the true sigmoid function
Y_true = sigmoid(
    X,
    true_left_asymptote,
    true_right_asymptote,
    np.array(true_slope))


# Add some noise to the Y values
noise = 0.05 * np.random.randn(len(X1))
Y_noisy = Y_true.ravel() + noise

glm_single_variable.model(Y_noisy, X)

glm_single_variable.fit(method="lm") # , max_nfev=10000

Y_fitted = glm_single_variable.predict(X)

# Plot the results
plt.figure(figsize=(10, 6))
plt.scatter(X1, Y_noisy, color="gray", alpha=0.5, label="Noisy Data")
plt.plot(X1, Y_fitted, color="blue", label="Fitted Sigmoid Curve")
plt.title("Sigmoid Curve Fit for Single Predictor Variable")
plt.xlabel("X1")
plt.ylabel("Y")
plt.legend()
plt.grid(True)
plt.show()

glm_single_variable = GeneralizedLogisticModel() # Generate sample data np.random.seed(42) X1 = np.linspace(-10, 10, 100) X0 = np.ones_like(X1) X = np.column_stack([X0, X1]) # True sigmoid parameters true_left_asymptote = 1.2 true_right_asymptote = 6.0 true_slope = [-9, 5] # Generate Y data using the true sigmoid function Y_true = sigmoid( X, true_left_asymptote, true_right_asymptote, np.array(true_slope)) # Add some noise to the Y values noise = 0.05 * np.random.randn(len(X1)) Y_noisy = Y_true.ravel() + noise glm_single_variable.model(Y_noisy, X) glm_single_variable.fit(method="lm") # , max_nfev=10000 Y_fitted = glm_single_variable.predict(X) # Plot the results plt.figure(figsize=(10, 6)) plt.scatter(X1, Y_noisy, color="gray", alpha=0.5, label="Noisy Data") plt.plot(X1, Y_fitted, color="blue", label="Fitted Sigmoid Curve") plt.title("Sigmoid Curve Fit for Single Predictor Variable") plt.xlabel("X1") plt.ylabel("Y") plt.legend() plt.grid(True) plt.show()

In [5]:

glm_single_variable.plot()

glm_single_variable.summary()

glm_single_variable.plot() glm_single_variable.summary()

Generalized Logistic Model Summary

| Parameter       |   Estimate |
|:----------------|-----------:|
| left_asymptote  |     1.1904 |
| right_asymptote |     5.9996 |
| coef_0          |    -8.9032 |
| coef_1          |     4.9538 |

Model Diagnostics Comparison
| Metric                               |   Sigmoid Model |   Linear Model |
|:-------------------------------------|----------------:|---------------:|
| Variance Explained (R-squared)       |          0.9996 |         0.7537 |
| Akaike Information Criterion (AIC)   |       -329.001  |       315.751  |
| Bayesian Information Criterion (BIC) |       -318.58   |       320.962  |

Likelihood Ratio Test (LRT) vs Linear Model
|:-----------------------------|---------:|
| Linear Model Log-Likelihood  | -155.876 |
| Sigmoid Model Log-Likelihood |  168.5   |
| LRT Statistic                |  648.752 |
| p-value                      |    0     |

LRT Comparisons with Reduced Models
| Model                                |   Log-Likelihood |   LRT Statistic |   p-value |
|:-------------------------------------|-----------------:|----------------:|----------:|
| Reduced Model (with first 1 columns) |         -225.936 |         788.873 |         0 |

Multivariate Example¶

In [6]:

glm_two_variable = GeneralizedLogisticModel()

# Generate sample data
np.random.seed(42)
X1 = np.linspace(-10, 10, 100)
X2 = np.linspace(-10, 10, 100)
X0 = np.ones_like(X1)

# True sigmoid parameters
true_left_asymptote = 1.3
true_right_asymptote = 9.5
true_slope = np.array([-7, 1.6, 1.6])  # Slopes for both variables

# Stack X1 and X2 to form a design matrix with two variables
X_two_vars = np.column_stack([X0, X1, X2])

# Generate Y data using the true sigmoid function
Y_true = sigmoid(
    X_two_vars,
    true_left_asymptote,
    true_right_asymptote,
    true_slope
)

# Add some noise to the Y values
noise = 0.75 * np.random.randn(len(X1))
Y_noisy = Y_true.ravel() + noise

# Fit the model to the noisy data
glm_two_variable.model(Y_noisy, X_two_vars)

# note that bounds and other initial parameters can be passed to the fit method
# there is one additional method, dogbox, which is available
# from scipy.optimize.curve_fit
glm_two_variable.fit(method="trf", max_nfev=100000)

# Predict using the fitted model
Y_fitted = glm_two_variable.predict(X_two_vars)

# glm_two_variable.plot()

glm_two_variable.summary()

glm_two_variable = GeneralizedLogisticModel() # Generate sample data np.random.seed(42) X1 = np.linspace(-10, 10, 100) X2 = np.linspace(-10, 10, 100) X0 = np.ones_like(X1) # True sigmoid parameters true_left_asymptote = 1.3 true_right_asymptote = 9.5 true_slope = np.array([-7, 1.6, 1.6]) # Slopes for both variables # Stack X1 and X2 to form a design matrix with two variables X_two_vars = np.column_stack([X0, X1, X2]) # Generate Y data using the true sigmoid function Y_true = sigmoid( X_two_vars, true_left_asymptote, true_right_asymptote, true_slope ) # Add some noise to the Y values noise = 0.75 * np.random.randn(len(X1)) Y_noisy = Y_true.ravel() + noise # Fit the model to the noisy data glm_two_variable.model(Y_noisy, X_two_vars) # note that bounds and other initial parameters can be passed to the fit method # there is one additional method, dogbox, which is available # from scipy.optimize.curve_fit glm_two_variable.fit(method="trf", max_nfev=100000) # Predict using the fitted model Y_fitted = glm_two_variable.predict(X_two_vars) # glm_two_variable.plot() glm_two_variable.summary()

Generalized Logistic Model Summary

| Parameter       |   Estimate |
|:----------------|-----------:|
| left_asymptote  |     1.1573 |
| right_asymptote |     9.5234 |
| coef_0          |    -6.2911 |
| coef_1          |    -2.2532 |
| coef_2          |     5.1284 |

Model Diagnostics Comparison
| Metric                               |   Sigmoid Model |   Linear Model |
|:-------------------------------------|----------------:|---------------:|
| Variance Explained (R-squared)       |          0.9716 |         0.7273 |
| Akaike Information Criterion (AIC)   |        214.452  |       434.633  |
| Bayesian Information Criterion (BIC) |        227.478  |       439.844  |

Likelihood Ratio Test (LRT) vs Linear Model
|:-----------------------------|---------:|
| Linear Model Log-Likelihood  | -215.317 |
| Sigmoid Model Log-Likelihood | -102.226 |
| LRT Statistic                |  226.181 |
| p-value                      |    0     |

LRT Comparisons with Reduced Models
| Model                                |   Log-Likelihood |   LRT Statistic |   p-value |
|:-------------------------------------|-----------------:|----------------:|----------:|
| Reduced Model (with first 2 columns) |         -102.226 |           0     |    0.9999 |
| Reduced Model (with first 1 columns) |         -280.286 |         356.121 |    0      |

In [8]:

glm_two_variable.plot()

glm_two_variable.plot()

In [9]:

glm_two_variable.summary()

glm_two_variable.summary()

Generalized Logistic Model Summary

| Parameter       |   Estimate |
|:----------------|-----------:|
| left_asymptote  |     1.1573 |
| right_asymptote |     9.5234 |
| coef_0          |    -6.2911 |
| coef_1          |    -2.2532 |
| coef_2          |     5.1284 |

Model Diagnostics Comparison
| Metric                               |   Sigmoid Model |   Linear Model |
|:-------------------------------------|----------------:|---------------:|
| Variance Explained (R-squared)       |          0.9716 |         0.7273 |
| Akaike Information Criterion (AIC)   |        214.452  |       434.633  |
| Bayesian Information Criterion (BIC) |        227.478  |       439.844  |

Likelihood Ratio Test (LRT) vs Linear Model
|:-----------------------------|---------:|
| Linear Model Log-Likelihood  | -215.317 |
| Sigmoid Model Log-Likelihood | -102.226 |
| LRT Statistic                |  226.181 |
| p-value                      |    0     |

LRT Comparisons with Reduced Models
| Model                                |   Log-Likelihood |   LRT Statistic |   p-value |
|:-------------------------------------|-----------------:|----------------:|----------:|
| Reduced Model (with first 2 columns) |         -102.226 |           0     |    0.9999 |
| Reduced Model (with first 1 columns) |         -280.286 |         356.121 |    0      |

Fitting the Sigmoid model to real data¶

In [10]:

# Note that for this to work in your environment, this path must point to a real file
met28_met31 = pd.read_csv("../../tmp/met28_met31.csv")
model_string = "met28_shifted_lrr ~ met28_shifted_lrb + met28_shifted_lrb:met31_shifted_lrb"
y_met28_met31, X_met28_met31_interaction = pt.dmatrices(model_string, data=met28_met31)

y_met28_met31 = np.asanyarray(y_met28_met31).ravel()
X_met28_met31_interaction = np.asarray(X_met28_met31_interaction)


met28_met31_model_interaction = GeneralizedLogisticModel()
met28_met31_model_interaction.model(y_met28_met31, X_met28_met31_interaction)
met28_met31_model_interaction.fit(method="lm")

met28_met31_model_interaction.summary()
met28_met31_model_interaction.plot(interactor_diagnostic=True)

# Note that for this to work in your environment, this path must point to a real file met28_met31 = pd.read_csv("../../tmp/met28_met31.csv") model_string = "met28_shifted_lrr ~ met28_shifted_lrb + met28_shifted_lrb:met31_shifted_lrb" y_met28_met31, X_met28_met31_interaction = pt.dmatrices(model_string, data=met28_met31) y_met28_met31 = np.asanyarray(y_met28_met31).ravel() X_met28_met31_interaction = np.asarray(X_met28_met31_interaction) met28_met31_model_interaction = GeneralizedLogisticModel() met28_met31_model_interaction.model(y_met28_met31, X_met28_met31_interaction) met28_met31_model_interaction.fit(method="lm") met28_met31_model_interaction.summary() met28_met31_model_interaction.plot(interactor_diagnostic=True)

Generalized Logistic Model Summary

| Parameter       |   Estimate |
|:----------------|-----------:|
| left_asymptote  |     0.3136 |
| right_asymptote |     1.4538 |
| coef_0          |    -5.2551 |
| coef_1          |     0.6793 |
| coef_2          |     1.0907 |

Model Diagnostics Comparison
| Metric                               |   Sigmoid Model |   Linear Model |
|:-------------------------------------|----------------:|---------------:|
| Variance Explained (R-squared)       |          0.1155 |         0.1028 |
| Akaike Information Criterion (AIC)   |       1510.23   |      1594.22   |
| Bayesian Information Criterion (BIC) |       1543.85   |      1614.39   |

Likelihood Ratio Test (LRT) vs Linear Model
|:-----------------------------|----------:|
| Linear Model Log-Likelihood  | -794.11   |
| Sigmoid Model Log-Likelihood | -750.115  |
| LRT Statistic                |   87.9894 |
| p-value                      |    0      |

LRT Comparisons with Reduced Models
| Model                                |   Log-Likelihood |   LRT Statistic |   p-value |
|:-------------------------------------|-----------------:|----------------:|----------:|
| Reduced Model (with first 2 columns) |         -775.808 |         51.3855 | 7.589e-13 |
| Reduced Model (with first 1 columns) |        -1127.68  |        755.134  | 0         |

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