Multivariate Normal CDF in Python using scipy

Multivariate Normal CDF in Python using scipy

In Python, you can compute the multivariate normal cumulative distribution function (CDF) using the scipy.stats module, specifically multivariate_normal from scipy.stats. Here's how you can do it:

Example: Computing Multivariate Normal CDF

import numpy as np from scipy.stats import multivariate_normal # Define mean vector and covariance matrix mean = np.array([0, 0]) covariance = np.array([[1, 0.5], [0.5, 1]]) # Create a multivariate normal distribution object mvn = multivariate_normal(mean=mean, cov=covariance) # Define the lower and upper bounds of integration lower_bound = np.array([-np.inf, -np.inf]) # Lower bounds (negative infinity for -∞) upper_bound = np.array([1, 2]) # Upper bounds (1 for x1, 2 for x2) # Calculate the multivariate normal CDF from lower_bound to upper_bound cdf_value = mvn.cdf(upper_bound) - mvn.cdf(lower_bound) print(f"Multivariate Normal CDF from {lower_bound} to {upper_bound}: {cdf_value}")

Explanation:

  1. Imports:

    • import numpy as np: Imports NumPy for numerical operations.
    • from scipy.stats import multivariate_normal: Imports multivariate_normal from scipy.stats for handling multivariate normal distributions.

  2. Mean and Covariance Matrix:

    • mean: Mean vector of the multivariate normal distribution.
    • covariance: Covariance matrix of the multivariate normal distribution.

  3. Create Multivariate Normal Distribution:

    • mvn = multivariate_normal(mean=mean, cov=covariance): Creates an instance of multivariate_normal with the specified mean vector and covariance matrix.

  4. Define Integration Bounds:

    • lower_bound and upper_bound: Arrays defining the lower and upper bounds of integration. Here, lower_bound is set to negative infinity (-np.inf) for both dimensions, and upper_bound is set to [1, 2].

  5. Compute Multivariate Normal CDF:

    • cdf_value = mvn.cdf(upper_bound) - mvn.cdf(lower_bound): Calculates the CDF of the multivariate normal distribution from lower_bound to upper_bound.

  6. Output:

    • Prints the computed CDF value for the given range.

Notes:

  • Ensure that your covariance matrix (covariance) is positive semidefinite for valid multivariate normal distribution calculations.
  • Adjust mean, covariance, lower_bound, and upper_bound according to your specific application requirements.

This example demonstrates how to use scipy.stats.multivariate_normal to compute the multivariate normal CDF efficiently in Python. Adjust parameters and integration bounds as necessary for your specific use case.

Examples

  1. "python scipy multivariate normal cdf example"

    • Description: Calculate the cumulative distribution function (CDF) of a multivariate normal distribution using scipy in Python.
    • Code:
      import numpy as np from scipy.stats import mvn def multivariate_normal_cdf(mean, cov, lower, upper): lower_bound = np.array(lower) upper_bound = np.array(upper) mean = np.array(mean) cov = np.array(cov) result, error = mvn.mvnun(lower_bound, upper_bound, mean, cov) return result # Example usage mean = [0, 0] cov = [[1, 0.5], [0.5, 1]] lower_bound = [-np.inf, -np.inf] upper_bound = [0.5, 0.5] result = multivariate_normal_cdf(mean, cov, lower_bound, upper_bound) print("Multivariate Normal CDF:", result)

    Explanation: Defines a function multivariate_normal_cdf to compute the CDF of a 2-dimensional multivariate normal distribution using scipy's mvn.mvnun.

  2. "python scipy multivariate normal cumulative distribution function"

    • Description: Using scipy to compute the cumulative distribution function (CDF) of a multivariate normal distribution in Python.
    • Code:
      import numpy as np from scipy.stats import multivariate_normal # Parameters mean = np.array([0, 0]) cov = np.array([[1, 0.5], [0.5, 1]]) # Create a multivariate normal distribution mvn_dist = multivariate_normal(mean=mean, cov=cov) # Calculate CDF lower_bound = [-np.inf, -np.inf] upper_bound = [0.5, 0.5] result = mvn_dist.cdf(upper_bound) - mvn_dist.cdf(lower_bound) print("Multivariate Normal CDF:", result)

    Explanation: Creates a multivariate normal distribution using scipy.stats.multivariate_normal and computes the CDF within a specified range.

  3. "python mvn CDF scipy example"

    • Description: Example of using scipy to compute the cumulative distribution function (CDF) of a multivariate normal distribution in Python.
    • Code:
      import numpy as np from scipy.stats import mvn def multivariate_normal_cdf(mean, cov, lower, upper): lower_bound = np.array(lower) upper_bound = np.array(upper) mean = np.array(mean) cov = np.array(cov) result, error = mvn.mvnun(lower_bound, upper_bound, mean, cov) return result # Example usage mean = [0, 0] cov = [[1, 0.5], [0.5, 1]] lower_bound = [-np.inf, -np.inf] upper_bound = [0.5, 0.5] result = multivariate_normal_cdf(mean, cov, lower_bound, upper_bound) print("Multivariate Normal CDF:", result)

    Explanation: Defines a function multivariate_normal_cdf to compute the CDF of a 2-dimensional multivariate normal distribution using scipy's mvn.mvnun.

  4. "python scipy mvn CDF with covariance matrix"

    • Description: Compute the cumulative distribution function (CDF) of a multivariate normal distribution using scipy with a covariance matrix.
    • Code:
      import numpy as np from scipy.stats import mvn def multivariate_normal_cdf(mean, cov, lower, upper): lower_bound = np.array(lower) upper_bound = np.array(upper) mean = np.array(mean) cov = np.array(cov) result, error = mvn.mvnun(lower_bound, upper_bound, mean, cov) return result # Example usage mean = [0, 0] cov = [[1, 0.5], [0.5, 1]] lower_bound = [-np.inf, -np.inf] upper_bound = [0.5, 0.5] result = multivariate_normal_cdf(mean, cov, lower_bound, upper_bound) print("Multivariate Normal CDF:", result)

    Explanation: Implements a function multivariate_normal_cdf to compute the CDF of a multivariate normal distribution with a specified covariance matrix using mvn.mvnun.

  5. "python scipy multivariate normal cdf 2d"

    • Description: Calculate the cumulative distribution function (CDF) of a 2-dimensional multivariate normal distribution using scipy in Python.
    • Code:
      import numpy as np from scipy.stats import mvn def multivariate_normal_cdf(mean, cov, lower, upper): lower_bound = np.array(lower) upper_bound = np.array(upper) mean = np.array(mean) cov = np.array(cov) result, error = mvn.mvnun(lower_bound, upper_bound, mean, cov) return result # Example usage mean = [0, 0] cov = [[1, 0.5], [0.5, 1]] lower_bound = [-np.inf, -np.inf] upper_bound = [0.5, 0.5] result = multivariate_normal_cdf(mean, cov, lower_bound, upper_bound) print("Multivariate Normal CDF:", result)

    Explanation: Defines a function multivariate_normal_cdf to compute the CDF of a 2-dimensional multivariate normal distribution using scipy's mvn.mvnun.

  6. "python scipy mvn CDF example"

    • Description: Example of computing the cumulative distribution function (CDF) of a multivariate normal distribution using scipy in Python.
    • Code:
      import numpy as np from scipy.stats import mvn def multivariate_normal_cdf(mean, cov, lower, upper): lower_bound = np.array(lower) upper_bound = np.array(upper) mean = np.array(mean) cov = np.array(cov) result, error = mvn.mvnun(lower_bound, upper_bound, mean, cov) return result # Example usage mean = [0, 0] cov = [[1, 0.5], [0.5, 1]] lower_bound = [-np.inf, -np.inf] upper_bound = [0.5, 0.5] result = multivariate_normal_cdf(mean, cov, lower_bound, upper_bound) print("Multivariate Normal CDF:", result)

    Explanation: Implements a function multivariate_normal_cdf to compute the CDF of a multivariate normal distribution using scipy's mvn.mvnun.

  7. "python scipy mvn cumulative distribution function example"

    • Description: Example of using scipy to compute the cumulative distribution function (CDF) of a multivariate normal distribution in Python.
    • Code:
      import numpy as np from scipy.stats import mvn def multivariate_normal_cdf(mean, cov, lower, upper): lower_bound = np.array(lower) upper_bound = np.array(upper) mean = np.array(mean) cov = np.array(cov) result, error = mvn.mvnun(lower_bound, upper_bound, mean, cov) return result # Example usage mean = [0, 0] cov = [[1, 0.5], [0.5, 1]] lower_bound = [-np.inf, -np.inf] upper_bound = [0.5, 0.5] result = multivariate_normal_cdf(mean, cov, lower_bound, upper_bound) print("Multivariate Normal CDF:", result)

    Explanation: Defines a function multivariate_normal_cdf to compute the CDF of a multivariate normal distribution using scipy's mvn.mvnun.

  8. "python scipy multivariate normal distribution cdf"

    • Description: Calculate the cumulative distribution function (CDF) of a multivariate normal distribution using scipy in Python.
    • Code:
      import numpy as np from scipy.stats import mvn def multivariate_normal_cdf(mean, cov, lower, upper): lower_bound = np.array(lower) upper_bound = np.array(upper) mean = np.array(mean) cov = np.array(cov) result, error = mvn.mvnun(lower_bound, upper_bound, mean, cov) return result # Example usage mean = [0, 0] cov = [[1, 0.5], [0.5, 1]] lower_bound = [-np.inf, -np.inf] upper_bound = [0.5, 0.5] result = multivariate_normal_cdf(mean, cov, lower_bound, upper_bound) print("Multivariate Normal CDF:", result)

    Explanation: Defines a function multivariate_normal_cdf to compute the CDF of a multivariate normal distribution using scipy's mvn.mvnun.

  9. "python multivariate normal cumulative distribution function scipy"

    • Description: Using scipy to compute the cumulative distribution function (CDF) of a multivariate normal distribution in Python.
    • Code:
      import numpy as np from scipy.stats import mvn def multivariate_normal_cdf(mean, cov, lower, upper): lower_bound = np.array(lower) upper_bound = np.array(upper) mean = np.array(mean) cov = np.array(cov) result, error = mvn.mvnun(lower_bound, upper_bound, mean, cov) return result # Example usage mean = [0, 0] cov = [[1, 0.5], [0.5, 1]] lower_bound = [-np.inf, -np.inf] upper_bound = [0.5, 0.5] result = multivariate_normal_cdf(mean, cov, lower_bound, upper_bound) print("Multivariate Normal CDF:", result)

    Explanation: Defines a function multivariate_normal_cdf to compute the CDF of a multivariate normal distribution using scipy's mvn.mvnun.

  10. "python scipy mvn CDF covariance matrix"

    • Description: Compute the cumulative distribution function (CDF) of a multivariate normal distribution using scipy with a covariance matrix.
    • Code:
      import numpy as np from scipy.stats import mvn def multivariate_normal_cdf(mean, cov, lower, upper): lower_bound = np.array(lower) upper_bound = np.array(upper) mean = np.array(mean) cov = np.array(cov) result, error = mvn.mvnun(lower_bound, upper_bound, mean, cov) return result # Example usage mean = [0, 0] cov = [[1, 0.5], [0.5, 1]] lower_bound = [-np.inf, -np.inf] upper_bound = [0.5, 0.5] result = multivariate_normal_cdf(mean, cov, lower_bound, upper_bound) print("Multivariate Normal CDF:", result)

    Explanation: Implements a function multivariate_normal_cdf to compute the CDF of a multivariate normal distribution with a specified covariance matrix using mvn.mvnun.

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