Stratified resampling

Stratified resampling#

The Jolly and Hampton [1990] algorithm for estimating the mean and variance of stratified random transects is implemented using a bootstrapping method. This enables the calculation of confidence intervals for stratum-specific and and overall survey population estimates, as well as characterizing the overall variance via the coefficient of variation (\(\textit{CV}\), Eq. 2.21).

Import necessary modules#

This algorithm is implemented via the JollyHampton class from the survey sub-package.

from echopop.survey import JollyHampton

The JollyHampton class can be initialized with two arguments:

  • model_parameters: A dictionary that can be configured to include the following keys:

    • num_replicates: The number of bootstrap replicates.

    • strata_transect_proportion: The proportion of transects to sample per stratum.

    • transects_per_latitude: The number of transects per degree latitude. However, this key is only necessary for virtual transect generation which will be discussed later.

  • resample_seed: An optional argument that sets the random seed for reproducible bootstrapping. If no seed is provided, then this defaults to being set to None.

Transects#

For the transect data, we only need strata_transect_proportion and num_replicates for the JollyHampton-class:

TRANSECT_JOLLYHAMPTON_PARAMETERS = {
    "strata_transect_proportion": 0.75,
    "num_replicates": 1000,
}

jh_transect = JollyHampton(TRANSECT_JOLLYHAMPTON_PARAMETERS)

The next step involves performing the bootstrap itself, which executes the built-in analysis pipeline. This is done by the stratified_bootstrap method, which has three required inputs:

  • data: Input pandas.DataFrame containing transect data.

  • stratum_dim: Stratification column name (e.g., "geostratum_inpfc").

  • variable: Name of the response variable column in data_df (e.g., "biomass").

jh_transect.stratified_bootstrap(
    data=df_nasc_no_age1_prt, stratum_dim="geostratum_inpfc", variable="biomass"
)

This method populates several attributes that can be accessed. The first is transect_summary, which details the summed biomass (kg), distance, area coverage, and areal/line biomass densities (kg nmi-2) for each transect (note that only the top ten rows are displayed below).

biomass distance area biomass_areal_density biomass_distance_density
geostratum_inpfc transect_num
1 1.0 4.010244e+06 34.809236 348.092365 11520.632646 115206.326459
2.0 3.101135e+05 35.341287 353.532537 877.185052 8774.820805
3.0 0.000000e+00 50.660997 507.671065 0.000000 0.000000
4.0 1.312977e+07 48.246504 483.030217 27182.089235 272139.315187
5.0 0.000000e+00 48.119031 480.726902 0.000000 0.000000
6.0 1.209238e+06 42.895494 428.615484 2821.266345 28190.337432
7.0 1.541750e+06 45.527123 455.378065 3385.648627 33864.430843
8.0 2.774609e+06 44.055394 440.435515 6299.694199 62980.008285
9.0 3.618213e+06 40.192165 402.164809 8996.842110 90022.850305
10.0 1.377956e+07 34.756911 347.653526 39635.892922 396455.200679

The next attribute is strata_summary, which provides the transect count (and resampling counts) total distance, area, biomass, and mean areal/line biomass densities for each stratum.

transect_counts num_transects_to_sample distance area biomass biomass_distance_density biomass_density
geostratum_inpfc
1 10 8 424.604141 4247.300484 4.037350e+07 95085.028859 9505.684182
2 24 18 899.718778 10057.202226 4.257742e+08 473230.298614 42335.251530
3 13 10 504.277720 5773.969771 3.274201e+08 649285.338462 56706.242571
4 15 11 619.190882 7060.629519 4.466629e+08 721365.418732 63261.057510
5 15 11 666.979139 7068.645037 2.607504e+08 390942.338685 36888.312143
6 36 27 1392.286678 20673.571163 1.758493e+08 126302.502696 8505.994951

Next is survey_summary, which provides the overall total population estimate and density for each stratum and for the entire survey. These can be navigated between the two keys "strata" and "survey".

jh_transect.survey_summary["strata"]
biomass biomass_density
geostratum_inpfc 1 2 3 4 5 6 1 2 3 4 5 6
0 4.037350e+07 4.257742e+08 3.274201e+08 4.466629e+08 2.607504e+08 1.758493e+08 9505.684182 42335.25153 56706.242571 63261.05751 36888.312143 8505.994951 
jh_transect.survey_summary["survey"]
biomass biomass_density cv
0 1.677048e+09 30553.755539 0.109971

The last attribute to note is bootstrap_replicates, which is a DataFrame comprising the computed estimators (e.g., total coverage area, biomass, weighted biomass density) for each of the bootstrap replicates. This can be used to gain insights from the distributions of different metrics within each strata and for the entire survey across replicates.

area biomass ... distance_weighted_biomass_density distance_weighted_variance
geostratum_inpfc 1 2 3 4 5 6 1 2 3 4 ... 3 4 5 6 1 2 3 4 5 6
replicate
0 3491.603138 7509.820151 4511.928114 5513.006534 5457.315275 14318.621027 4.555646e+07 4.346331e+08 3.410848e+08 5.305976e+08 ... 2.828432e+07 3.448754e+07 1.998586e+07 7.926901e+06 2.551602e+12 2.434465e+13 8.731225e+13 1.150339e+14 4.055778e+13 3.275461e+12
1 3362.105458 7484.795160 4437.237772 5335.221508 5092.035493 15363.064304 2.953189e+07 3.314311e+08 3.443247e+08 4.483489e+08 ... 2.876824e+07 2.998217e+07 2.131988e+07 2.120924e+06 9.404633e+11 1.112345e+13 9.044058e+13 8.693969e+13 4.602187e+13 2.386535e+11
2 3335.654784 7954.843829 4213.283876 4795.224655 5210.857907 17234.327973 3.254311e+07 4.857275e+08 2.918134e+08 3.245640e+08 ... 2.242866e+07 2.210942e+07 2.090768e+07 7.328970e+06 1.171072e+12 2.893518e+13 5.310228e+13 4.710995e+13 4.461204e+13 2.786318e+12
3 3545.675582 7623.011593 4658.918907 4770.894617 5377.238005 16149.719545 4.506642e+07 3.828920e+08 3.510499e+08 3.718640e+08 ... 2.898080e+07 2.488468e+07 2.314116e+07 6.932840e+06 2.483088e+12 1.485016e+13 9.179948e+13 5.978116e+13 5.421045e+13 2.539439e+12
4 3404.700160 7536.521289 4522.591358 4976.292893 4742.204911 13191.110861 4.247584e+07 4.267015e+08 3.251184e+08 5.221232e+08 ... 2.706904e+07 3.381357e+07 1.462609e+07 8.508163e+06 2.308959e+12 2.332454e+13 7.996204e+13 1.101892e+14 2.058707e+13 3.865258e+12
5 3545.675582 7361.792693 4681.977201 5329.516331 5015.109211 13626.957423 4.506642e+07 3.431288e+08 3.219473e+08 4.529638e+08 ... 2.698873e+07 2.966433e+07 1.789599e+07 8.546989e+06 2.483088e+12 1.168723e+13 7.952485e+13 8.536872e+13 3.160122e+13 3.876241e+12
6 3453.332432 7778.876291 4661.664549 5358.256039 5228.580027 14272.287058 4.661097e+07 4.222089e+08 3.239509e+08 3.985688e+08 ... 2.712935e+07 2.631528e+07 2.187929e+07 3.124508e+06 2.691887e+12 2.275028e+13 8.043238e+13 6.687868e+13 4.860928e+13 5.164220e+11
7 3459.211443 7467.118724 4480.570122 5018.232907 5338.222629 17603.312432 2.977416e+07 3.009117e+08 3.543826e+08 4.984857e+08 ... 2.924806e+07 3.228357e+07 2.179290e+07 7.216606e+06 1.284094e+12 9.251804e+12 9.346134e+13 1.002539e+14 4.795979e+13 2.701337e+12
8 3363.306935 7710.787861 4452.249166 5246.909016 5202.153736 16506.412792 4.702814e+07 4.543071e+08 2.978632e+08 4.019570e+08 ... 2.507716e+07 2.611321e+07 2.035400e+07 7.055204e+06 2.888872e+12 2.574308e+13 6.877361e+13 6.616927e+13 4.219341e+13 2.687097e+12
9 3418.481217 7684.859244 4684.746919 5377.169006 5397.603790 15359.356385 4.545407e+07 4.379739e+08 3.341958e+08 4.049182e+08 ... 2.782286e+07 2.646311e+07 1.977879e+07 3.290864e+06 2.717833e+12 2.429742e+13 8.449816e+13 6.792275e+13 3.977680e+13 5.543117e+11

10 rows × 36 columns

These bootstrap replicates can then be processed into summary statistics computed for each stratum as well as the entire survey. This uses the summarize method, which as the arguments:

  • ci_percentile: Confidence level for estimating the uncertainty interval. This defaults to 0.95.

  • ci_method: The method used for computing the confidence interval (\(\textit{CI}\)). Currently, valid entries for ci_method are:

    • "student_jackknife": This is the default method that computes the studentized \(\textit{CI}\) using jackknife (or leave-one-out) resampling.

    • "bc": Bias-corrected \(\textit{CI}\) that adjusts for bias in the bootstrap distribution using the empirical cumulative distribution function.

    • "bca": Bias-corrected and accelerated \(\textit{CI}\) that not only accounts for bias in the bootstrap sample but also corrects for skewness using finite-sample jackknife resampling.

    • "empirical": Sometimes known as the “delta method”, this selection uses the distribution of bootstrapped deviations between the replicate means and population statistic.

    • "normal": This assumes that the bootstrap replicates are normally distributed.

    • "percentile": The \(\textit{CI}\) are constructed directly from the bootstrap distribution quantiles.

    • "student_standard": This assumes that the bootstrap replicates are approximately \(t\)-distributed.

jh_transect.summarize()
biomass biomass_density cv
metric low mean high bias low mean high bias low mean high bias
1 2.820906e+07 4.040993e+07 5.010102e+07 3.643418e+04 6641.644748 9514.262378 11795.966714 8.578196 NaN NaN NaN NaN
2 3.070271e+08 4.225634e+08 5.103502e+08 -3.210779e+06 30528.078222 42015.999774 50744.749903 -319.251756 NaN NaN NaN NaN
3 2.437218e+08 3.256368e+08 3.793792e+08 -1.783342e+06 42210.443573 56397.383683 65705.088947 -308.858888 NaN NaN NaN NaN
4 3.242482e+08 4.452046e+08 5.418683e+08 -1.458327e+06 45923.418822 63054.514082 76745.039140 -206.543428 NaN NaN NaN NaN
5 1.935299e+08 2.630395e+08 3.194195e+08 2.289147e+06 27378.645785 37212.157417 45188.219506 323.845274 NaN NaN NaN NaN
6 8.319200e+07 1.750265e+08 2.291056e+08 -8.227529e+05 4024.075188 8466.197624 11082.052377 -39.797327 NaN NaN NaN NaN
survey 1.462387e+09 1.671881e+09 1.865804e+09 -5.166822e+06 26646.356142 30463.567847 33997.061607 -90.187691 0.123976 0.130357 0.141477 0.020386

This yields a pandas.DataFrame indexed by each stratum and overall survey, with columns organized by their respective metrics. In this case, we have "biomass", "biomass_density", and "cv". The columns "low", "mean", and "high" correspond to the lower bound of the \(\textit{CI}\), the distribution mean, and the upper bound of the \(\textit{CI}\), respectively. The "bias" represents the deviation between the bootstrapped means and the original estimates. The metric "cv" is only calculated for the entire survey, so there are no valid values to report for each stratum (i.e., NaN).

Kriged mesh#

We can also run this analysis for the kriged mesh estimates, although we have to incorporate a few modifications to make the gridded points compatible with the expected transect sampling design. So we now initialize the JollyHampton-class object with the dictionary key "transects_per_latitude", which defines the number of virtual transects that will be generated per degree latitude.

KRIGED_JOLLYHAMPTON_PARAMETERS = {
    "transects_per_latitude": 5,
    "strata_transect_proportion": 0.75,
    "num_replicates": 1000,
}

jh_kriged = JollyHampton(KRIGED_JOLLYHAMPTON_PARAMETERS)

Before using the stratified_bootstrap method, the create_virtual_transects method is required. This method creates virtual transects from the gridded data that are then subsequently assigned to individual strata. This has four arguments:

  • mesh_data: A pandas.DataFrame containing gridded kriged data.

  • geostrata: A pandas.DataFrame containing geographical stratum boundaries and definitions (e.g., "geostratum_inpfc").

  • stratum_dim: Stratification column name (e.g., "geostratum_inpfc").

  • variable: Name of the response variable column in data_df (e.g., "biomass").

This returns a pandas.DataFrame that can then be fed into the other JollyHampton-class methods as if they were standard transect data.

kriged_transects = jh_kriged.create_virtual_transects(
    mesh_data=unextrapolated_results,
    geostrata=df_dict_geostrata["inpfc"],
    stratum_dim="geostratum_inpfc",
    variable="biomass",
)
transect_num latitude transect_distance transect_area biomass geostratum_inpfc
0 1 34.6 48.675265 292.051592 1.278193e+06 1
1 2 34.8 54.211796 650.541558 1.061705e+06 1
2 3 35.0 54.655882 655.870584 1.071039e+07 1
3 4 35.2 56.585071 679.020851 6.468483e+05 1
4 5 35.4 51.338622 616.063464 1.791419e+06 1
5 6 35.6 57.280884 687.370604 3.334136e+06 1
6 7 35.8 51.768742 621.224904 3.610589e+06 1
7 8 36.0 48.788137 585.457646 1.494046e+07 2
8 9 36.2 47.058672 564.704058 8.543800e+06 2
9 10 36.4 44.176795 530.121544 3.189396e+06 2

So now we can take these virtual transects to compute the bootstrap replicates and subsequent statistics.

# Generate replicates
jh_kriged.stratified_bootstrap(
    data=kriged_transects, stratum_dim="geostratum_inpfc", variable="biomass"
)

# Summarize
jh_kriged.summarize()
biomass biomass_density cv
metric low mean high bias low mean high bias low mean high bias
1 1.078635e+07 2.232470e+07 2.905677e+07 -1.085791e+05 2540.437518 5257.987864 6843.547876 -25.572912 NaN NaN NaN NaN
2 3.082708e+08 3.940729e+08 4.648138e+08 2.132049e+05 23037.811113 29450.005131 34736.640264 15.933312 NaN NaN NaN NaN
3 2.565377e+08 3.439625e+08 4.101752e+08 -9.524314e+05 36675.415948 49173.923503 58639.899534 -136.162483 NaN NaN NaN NaN
4 3.903927e+08 4.926153e+08 5.581010e+08 2.772988e+06 49920.542282 62992.023841 71365.848143 354.589325 NaN NaN NaN NaN
5 1.880855e+08 2.663654e+08 3.106342e+08 1.098833e+06 20822.068401 29488.068006 34388.863662 121.646641 NaN NaN NaN NaN
6 1.191697e+08 1.966672e+08 2.516261e+08 8.902240e+04 3320.180188 5479.333932 7010.538926 2.480248 NaN NaN NaN NaN
survey 1.517318e+09 1.716008e+09 1.876902e+09 2.931529e+06 19611.808670 22179.938517 24259.542433 40.236983 0.128841 0.134447 0.141746 0.018865