Parameter Identifiability

This module provides a function to compute confidence intervals (CIs) and significance statistics for model parameters using linear approximation based on the covariance matrix from a nonlinear least squares fit, commonly known as also Wald Intervals.

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Purpose

Given:

• Best-fit parameter estimates (popt)
• Their covariance matrix (pcov)
• The observed data (target)

This function estimates:

• Standard errors
• t-statistics
• Two-sided p-values
• 95% confidence intervals (or another level via alpha_val)

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Mathematical Background

1. Standard Error

The standard error of each parameter is estimated as:

$$ \text{SE}(\beta_i) = \sqrt{ \text{Var}(\beta_i) } = \sqrt{ \text{diag}(\text{pcov})_i } $$

Where pcov is the covariance matrix from the curve fitting routine (typically from scipy.optimize.curve_fit).

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2. Degrees of Freedom

$$ \text{df} = n_{\text{obs}} - n_{\text{params}} $$

Used to select the correct t-distribution for the confidence interval and p-value computation.

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3. t-Statistic

For each parameter:

$$ t_i = \frac{\hat{\beta}_i}{\text{SE}(\hat{\beta}_i)} $$

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4. p-Value

Two-sided p-value from the Student’s t-distribution:

$$ p_i = 2 \cdot P(T > |t_i|) = 2 \cdot \text{sf}(|t_i|, \text{df}) $$

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5. Confidence Interval

Using the t-critical value $t^*$:

$$ t^* = t_{1 - \alpha/2, \text{df}} $$

Confidence bounds:

$$ \text{CI}_i = \left[ \max\left(0, \hat{\beta}_i - t^* \cdot \text{SE}(\hat{\beta}_i)\right),\ \hat{\beta}_i + t^* \cdot \text{SE}(\hat{\beta}_i) \right] $$

Lower bound is clipped to zero for non-negative parameters.

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When to Use

• To assess parameter certainty
• To report statistical significance and error bars