confidence_interval

Module that is used to calculate the confidence interval given the estimated parameters

pygom.loss.confidence_interval.asymptotic(obj, alpha=0.05, theta=None, lb=None, ub=None)

Finds the confidence interval at the \(\alpha\) level under the \(\mathcal{X}^{2}\) assumption for the likelihood

Parameters:	obj: ode object an object initialized from `BaseLoss` alpha: numeric, optional confidence level, \(0 < \alpha < 1\). Defaults to 0.05. theta: array like, optional the MLE parameters. Defaults to None which then theta will be inferred from the input obj lb: array like, optional expected lower bound ub: array like, optional expected upper bound
Returns:	l: array like lower confidence interval u: array like upper confidence interval

pygom.loss.confidence_interval.profile(obj, alpha, theta=None, lb=None, ub=None, full_output=False)

Finds the profile confidence interval at the \(\alpha\) level under the \(\mathcal{X}^{2}\) assumption for the likelihood

Parameters:

obj: ode object: an object initialized from BaseLoss
alpha: numeric: confidence level, \(0 < \alpha < 1\)
theta: array like, optional: the MLE parameters. When None given, it tries to estimate the optimal using methods provided by obj
lb: array like, optional: expected lower bound
ub: array like, optional: expected upper bound
full_output: bool, optional: if more output is desired

Returns:

pygom.loss.confidence_interval.bootstrap(obj, alpha=0.05, theta=None, lb=None, ub=None, iteration=0, full_output=False)

Finds the confidence interval at the \(\alpha\) level via bootstrap

Parameters:

obj: ode object: an object initialized from BaseLoss
alpha: numeric, optional: confidence level, \(0 < \alpha < 1\). Defaults to 0.05.
theta: array like, optional: the MLE parameters
lb: array like, optional: upper bound for the parameters
ub: array like, optional: lower bound for the parameters
iteration: int, optional: number of bootstrap samples, defaults to 0 which is interpreted as \(min(2n, 100)\) where \(n\) is the number of data points.
full_output: bool: if the full set of estimates is required.

Returns:

pygom.loss.confidence_interval.geometric(obj, alpha=0.05, theta=None, method='jtj', geometry='o', full_output=False)

Finds the geometric confidence interval under profiling at the \(\alpha\) level the normal approximation

Parameters:

obj: ode object: an object initialized from BaseLoss
alpha: numeric: confidence level, \(0 < \alpha < 1\)
theta: array like, optional: the MLE parameters. When None given, it tries to estimate the optimal using methods provided by obj
method: string: construction of the covariance matrix. jtj is the \(J^{\top}\) where \(J\) is the Jacobian of the ode. ‘hessian’ is the hessian of the ode while ‘fisher’ is the fisher information found by \(cov(\nabla_{\theta}\mathcal{L})\).
geometry: string: the two types of geometry defined in [Moolgavkar1987]. c geometry uses the covariance at the maximum likelihood estimate \(\hat{\theta}\), while the ‘o’ geometry is the covariance defined at point \(\theta\).
full_output: bool, optional: If True then both the l_path and u_path will be outputted, else only the point estimates of l and u

Returns:

l: array like: lower confidence interval
u: array like: upper confidence interval
l_path: list: path from \(\hat{\theta}\) to the lower \(1 - \alpha/2\) point for all parameters
u_path: list: same as l_path but for the upper confidence interval