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Computing the b-value

The b-value is a common metric for the seismicity of a region. You can imagine you would like to calculate it often when working with earthquake data. For tasks like this that you will do often, it is best to write a function! So, write a function with signature b_value(mags, mt, perc=[2.5, 97.5], n_reps=None) that returns the b-value and (optionally, if n_reps is not None) its confidence interval for a set of magnitudes, mags. The completeness threshold is given by mt. The perc keyword argument gives the percentiles for the lower and upper bounds of the confidence interval, and n_reps is the number of bootstrap replicates to use in computing the confidence interval.

This exercise is part of the course

Case Studies in Statistical Thinking

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Exercise instructions

  • Define a function with signature b_value(mags, mt, perc=[2.5, 97.5], n_reps=None) that does the following:
    • Slice magnitudes out of mags at and above the completeness threshold mt using Boolean indexing. Store the result in the variable m.
    • Compute the best estimate of the b-value. Remember, the best estimate for the b-value is b = (m - mt)·ln(10). Store the result in the variable b.
    • if n_reps is not None, do the following.
      • Draw n_reps bootstrap replicates of the mean of m. Store the result in the variable m_bs_reps.
      • Convert the bootstrap replicates of the mean of m to replicates of the b-value. Store the result in b_bs_reps.
      • Compute the confidence interval from the bootstrap replicates of the b-value. Store the result in conf_int.
    • Return b and conf_int, or just b if n_reps is None.

Hands-on interactive exercise

Have a go at this exercise by completing this sample code.

____ ____:
    """Compute the b-value and optionally its confidence interval."""
    # Extract magnitudes above completeness threshold: m
    m = ____[____ >= ____]

    # Compute b-value: b
    b = ____

    # Draw bootstrap replicates
    if n_reps is None:
        return b
    else:
        m_bs_reps = ____

        # Compute b-value from replicates: b_bs_reps
        b_bs_reps = (____ - ____) * ____

        # Compute confidence interval: conf_int
        conf_int = ____
    
        return b, conf_int
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