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Update driver.rst
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adowling2 authored Jun 1, 2020
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Expand Up @@ -148,9 +148,9 @@ in parameter estimation.
Covariance matrix estimation
----------------------------

If the optional argument ``calc_cov=True`` is specified for ``theta_est()``, parmest will calculate the covariance matrix of the fitted parameters as follows:
If the optional argument ``calc_cov=True`` is specified for ``theta_est()``, parmest will calculate the covariance matrix :math:`V_{\theta}` as follows:

.. math::
V_{\theta} = 2 \sigma^2 H^{-1}
This formula assumes all measurement errors are independent and identically distributed with variance :math:`\sigma^2`. `H^{-1}` is the inverse of the Hessian matrix for a unweighted sum of least squares problem. Currently, the covariance approximation is only valid if the objective given to parmest is the sum of squared error. Moreover, parmest approximates the variance of the measurement errors as :math:`\sigma^2 = \frac{1}{n-l} \sum e_i^2` where :math:`n` is the number of data points, :math:`l` is the number of fitted parameters, and :math:`e_i` is the residual for experiment :math:`i`.
This formula assumes all measurement errors are independent and identically distributed with variance :math:`\sigma^2`. :math:`H^{-1}` is the inverse of the Hessian matrix for a unweighted sum of least squares problem. Currently, the covariance approximation is only valid if the objective given to parmest is the sum of squared error. Moreover, parmest approximates the variance of the measurement errors as :math:`\sigma^2 = \frac{1}{n-l} \sum e_i^2` where :math:`n` is the number of data points, :math:`l` is the number of fitted parameters, and :math:`e_i` is the residual for experiment :math:`i`.

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