.. SPDX-FileCopyrightText: 2026 Koen van Greevenbroek
..
.. SPDX-License-Identifier: CC-BY-4.0

.. _sensitivity-analysis:

Sensitivity Analysis
====================

The food systems model involves many uncertain inputs: emission factors for
different greenhouse gases, crop yield potentials, health dose-response
parameters, and policy-level choices like carbon prices and health valuations.
Understanding which of these uncertainties matter most for model outcomes is
essential for directing research priorities and communicating result robustness.

Global sensitivity analysis answers the question: *which input uncertainties
drive the most variation in model outcomes?* This is quantified through
**Sobol indices**, which decompose the variance of each output into
contributions from individual inputs and their interactions.

- **First-order index** (:math:`S_1`): The fraction of output variance
  attributable to a single input, acting alone.
- **Total-order index** (:math:`S_T`): The fraction of output variance
  attributable to a single input, including all its interactions with other
  inputs.

A parameter with :math:`S_1 \approx S_T` influences the output mainly through
its direct effect. A parameter with :math:`S_T \gg S_1` is involved in
significant interactions with other parameters.

This implementation supports four surrogate modelling approaches:

- **Polynomial Chaos Expansion (PCE)**: Computes Sobol indices analytically
  from a polynomial approximation to the model response.
- **Random Forest (RF)**, **XGBoost (XGB)**, and
  **Multivariate Adaptive Regression Splines (MARS)**: Tree- and
  spline-based regressors; Sobol indices are computed via Saltelli
  pick-freeze Monte Carlo on the fitted surrogate.

All four share a single fitting pipeline (the ``build_surrogate`` rule)
that persists the trained model as a pickled :class:`SurrogateBundle`
alongside a validation parquet.  The bundle is the single artifact
consumed by downstream rules and notebooks: Sobol-index computation,
uncertainty-band plots, and policy sweeps all load the bundle instead of
refitting.  Which surrogate is used for a given downstream consumer is
controlled by ``sensitivity_analysis.default_surrogate`` in the config.


Methodology
-----------

Polynomial Chaos Expansion
~~~~~~~~~~~~~~~~~~~~~~~~~~

The central idea is to approximate the model's input-output mapping as a
polynomial in the uncertain inputs. If :math:`\mathbf{X} = (X_1, \ldots, X_d)`
are the uncertain parameters and :math:`Y` is a scalar output, the PCE
representation is:

.. math::

   Y \approx \sum_{\boldsymbol{\alpha}} c_{\boldsymbol{\alpha}}\,
   \Psi_{\boldsymbol{\alpha}}(\mathbf{X})

where :math:`\boldsymbol{\alpha} = (\alpha_1, \ldots, \alpha_d)` is a
multi-index, :math:`c_{\boldsymbol{\alpha}}` are scalar coefficients, and
:math:`\Psi_{\boldsymbol{\alpha}}` are multivariate orthonormal polynomials
with respect to the joint input distribution. Each
:math:`\Psi_{\boldsymbol{\alpha}}` is a product of univariate orthonormal
polynomials:

.. math::

   \Psi_{\boldsymbol{\alpha}}(\mathbf{X}) =
   \prod_{i=1}^d \psi_{\alpha_i}^{(i)}(X_i)

where :math:`\psi_k^{(i)}` is the degree-:math:`k` orthonormal polynomial for
the marginal distribution of :math:`X_i`. For uniform inputs these are
(normalised) Legendre polynomials; for normal inputs, Hermite polynomials.

The orthonormality property
:math:`\mathbb{E}[\Psi_{\boldsymbol{\alpha}} \Psi_{\boldsymbol{\beta}}] =
\delta_{\boldsymbol{\alpha}\boldsymbol{\beta}}` is what makes the variance
decomposition exact: the variance of the expansion is simply the sum of
squared coefficients. Once the coefficients are known, Sobol indices follow
directly without further model evaluations.

The polynomial basis is generated using `chaospy
<https://chaospy.readthedocs.io/>`_. A **cross-truncation** parameter
:math:`q \in (0, 1]` controls the multi-index set: lower values favour
lower-order interaction terms, keeping the basis compact. The default is
:math:`q = 0.5`.


Sparse Fitting via LARS
~~~~~~~~~~~~~~~~~~~~~~~~

With many parameters and moderate polynomial degree, the number of candidate
basis terms can exceed the number of model samples. Rather than requiring a
full tensor-product design, the implementation uses **Least Angle Regression
(LARS)** with cross-validation to select a sparse subset of active terms.

LARS incrementally adds basis terms that are most correlated with the current
residual, using cross-validation to choose the optimal number of active terms.
This produces a parsimonious expansion that captures the dominant polynomial
structure without overfitting.

The fitting uses `sklearn.linear_model.LarsCV
<https://scikit-learn.org/stable/modules/generated/sklearn.linear_model.LarsCV.html>`_
with 5-fold cross-validation.

**Validation**: Surrogate quality is assessed through multiple metrics:

- **Holdout error**: When ``holdout_fraction > 0`` (recommended), the tail of
  the Sobol sequence is reserved as held-out test data. The surrogate is fitted
  on the remaining training samples and evaluated on the holdout set. This gives
  an honest, out-of-sample error estimate. Using the tail preserves the
  space-filling quality of the training design (Sobol sequences front-load
  coverage).
- **Leave-one-out error** (PCE only): A relative error computed via the hat
  matrix without refitting. Reported alongside holdout error for comparison.
- **Out-of-bag R²** (RF only): The OOB score from bootstrap aggregation.
  Reported alongside holdout error for comparison.
- **R-squared** (:math:`R^2`): Coefficient of determination on training data.

The primary ``validation_error`` field in output files is the holdout error when
available, falling back to LOO (PCE) or OOB (RF) error otherwise.

Random Forest
~~~~~~~~~~~~~

As an alternative to PCE, a **Random Forest** ensemble can be fitted to the
same model samples. Sobol indices are computed via Monte Carlo integration:
for each parameter, the marginal effect is estimated by permuting that
parameter's values while holding others fixed, then comparing the variance
reduction.

Random Forests are non-parametric and can capture non-smooth or discontinuous
model responses that polynomials may miss. However, they are more expensive to
evaluate conditionally (requiring Monte Carlo samples at each grid point) and
produce noisier sensitivity estimates.

The implementation uses `sklearn.ensemble.RandomForestRegressor
<https://scikit-learn.org/stable/modules/generated/sklearn.ensemble.RandomForestRegressor.html>`_
with OOB scoring enabled.


Sobol Indices from PCE Coefficients
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Thanks to the orthonormality of the basis, the total variance of the expansion
is:

.. math::

   D = \sum_{\boldsymbol{\alpha} \neq \mathbf{0}}
   c_{\boldsymbol{\alpha}}^2

The first-order Sobol index for parameter :math:`i` sums the squared
coefficients of terms where *only* :math:`\alpha_i > 0` (no other parameter
is active):

.. math::

   S_{1,i} = \frac{1}{D}
   \sum_{\substack{\boldsymbol{\alpha}:\, \alpha_i > 0 \\
   \alpha_j = 0\; \forall\, j \neq i}}
   c_{\boldsymbol{\alpha}}^2

The total-order Sobol index sums all terms where :math:`\alpha_i > 0`,
regardless of other active parameters:

.. math::

   S_{T,i} = \frac{1}{D}
   \sum_{\boldsymbol{\alpha}:\, \alpha_i > 0}
   c_{\boldsymbol{\alpha}}^2

These indices satisfy :math:`0 \le S_{1,i} \le S_{T,i} \le 1` and
:math:`\sum_i S_{1,i} \le 1` (with equality when there are no interactions).


Conditional Sensitivity Analysis
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Some parameters are **policy choices** (e.g., GHG price, value per YLL) rather
than epistemic uncertainties. It is often useful to ask: *given a specific
policy setting, how sensitive are outcomes to the remaining uncertain
parameters?*

Designated **slice parameters** are fixed at specific conditioning values.
For **PCE**, the conditioning is performed analytically: for each slice
parameter :math:`j` with conditioning value :math:`x_j^*`, the univariate
basis is evaluated at that value, and the resulting factors are absorbed into
the PCE coefficients:

.. math::

   c'_{\boldsymbol{\alpha}} = c_{\boldsymbol{\alpha}} \cdot
   \prod_{j \in \text{slice}} \psi_{\alpha_j}^{(j)}(x_j^*)

Sobol indices are then computed from the transformed coefficients
:math:`c'_{\boldsymbol{\alpha}}`, considering only terms where at least one
non-slice parameter is active. The result is a set of Sobol indices for the
remaining parameters, conditional on the policy choices — showing how
sensitivity patterns shift as policy values change.

For **RF**, conditioning is done via Monte Carlo: slice parameters are held
fixed while remaining parameters are sampled from their marginal distributions,
and Sobol indices are computed from the resulting predictions.


Experimental Design
-------------------

Sobol Quasi-Random Sequences
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The model is sampled using a **scrambled Sobol sequence** — a quasi-random,
low-discrepancy design that provides better coverage of the parameter space
than simple random sampling, especially in high dimensions. The Sobol sequence
fills the :math:`[0, 1]^d` hypercube with balanced coverage, and an inverse
CDF transform maps each dimension to the corresponding parameter distribution.

The implementation uses `scipy.stats.qmc.Sobol
<https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.qmc.Sobol.html>`_
with scrambling enabled for improved uniformity. Scrambling is seeded for
deterministic reproducibility (default seed: 42).

.. note::

   The number of samples should be a **power of 2** for optimal balance
   of the Sobol sequence (e.g., 64, 128, 256, 512, 1024).


.. _supported-distributions:

Supported Distributions
~~~~~~~~~~~~~~~~~~~~~~~~

Each uncertain parameter is assigned an independent marginal distribution.
The joint distribution is the product of the marginals (i.e., parameters are
assumed independent).

.. csv-table::
   :header: Distribution, Required fields, Optional fields, Description

   ``uniform`` (default), "``lower``, ``upper``", , "Flat distribution over [lower, upper]"
   ``log_uniform``, "``lower``, ``upper``", , "Log-uniform: uniform on log scale over [lower, upper] (both > 0)"
   ``normal``, "``mean``, ``std``", ``bounds``, "Gaussian with given mean and standard deviation"
   ``normal_ci``, "``lower``, ``upper``", "``confidence``, ``bounds``", "Normal distribution where [lower, upper] defines a confidence interval"
   ``lognormal``, "``mu``, ``sigma``", , "Log-normal with log-scale mean and std"

When the ``distribution`` field is omitted, ``uniform`` is assumed (requiring
only ``lower`` and ``upper``).

The ``normal_ci`` distribution derives its mean as ``(lower + upper) / 2`` and
its standard deviation from the confidence level (default: 0.9, i.e. 90% CI).
This is useful when the literature reports uncertainty as a confidence interval
around a central value rather than as explicit standard deviations.

The optional ``bounds`` field (a two-element list ``[lo, hi]``) truncates
``normal`` and ``normal_ci`` distributions to the given range using a truncated
normal. Use ``null`` for an unbounded side (e.g., ``bounds: [0, null]`` enforces
non-negativity). This prevents physically meaningless values (e.g., negative
multiplicative factors) in the tails of the distribution.


Configuration
-------------

Sensitivity analysis is configured through the ``_generators`` DSL in the
scenarios file (see :doc:`configuration` for the full generator syntax). A
sensitivity generator uses ``mode: sensitivity`` and specifies parameter
distributions rather than fixed value lists.

**Full example** (from the production configuration):

.. code-block:: yaml

   # config/gsa.yaml
   name: "gsa"

   scenarios:
     default: {}

     _generators:
       - name: gsa_{sample_id}
         mode: sensitivity
         samples: 4096
         slice_parameters: [value_per_yll, ghg_price]
         parameters:
           yield_factor:
             lower: 0.8
             upper: 1.2
             bounds: [0, null]
           ch4_factor:
             distribution: normal_ci
             lower: 0.5
             upper: 1.5
             confidence: 0.9
             bounds: [0, null]
           # ... (remaining parameters)
         template:
           sensitivity:
             crop_yields:
               all: "{yield_factor}"
             emission_factors:
               ch4: "{ch4_factor}"
               # ...
           health:
             value_per_yll: "{value_per_yll}"
           emissions:
             ghg_price: "{ghg_price}"

   # Surrogate fitting methods (applied independently to the same scenarios)
   sensitivity_analysis:
     holdout_fraction: 0.15
     methods:
       pce:
         grid_resolution: 50
         method_options:
           n_mc_conditional: 4096
           cross_truncation: 0.8
       rf:
         grid_resolution: 50
         method_options:
           n_estimators: 500

The generator defines only the scenario sampling design (parameters,
distributions, sample count). Surrogate method configuration lives in
the separate ``sensitivity_analysis`` section, allowing multiple methods
to be applied to the same solved scenarios without duplication.

Health relative risk parameters use a **quantile parameterization**: each
``rr_<risk_factor>`` value is a quantile :math:`q \in [0, 1]` that interpolates
between the GBD confidence bounds [#gbd]_ at every dose-response breakpoint:

.. math::

   \log(\text{RR}(q)) = (1 - q) \cdot \log(\text{RR}_{\text{low}})
   + q \cdot \log(\text{RR}_{\text{high}})

- :math:`q = 0`: GBD lower bound (strongest protective effect for beneficial foods)
- :math:`q = 1`: GBD upper bound (weakest protective effect for beneficial foods)

This is applied per (risk factor, cause, exposure) point, so a single quantile
parameter per risk factor produces cause-specific adjustments automatically.

**Generator field reference**:

- ``name``: Scenario name pattern. Use ``{sample_id}`` as a placeholder for the
  zero-indexed sample number (e.g., ``gsa_{sample_id}`` produces ``gsa_0``,
  ``gsa_1``, ..., ``gsa_4095``).
- ``mode: sensitivity``: Activates space-filling Sobol sampling with
  distribution-based parameter specifications.
- ``samples``: Number of samples to draw. Should be a power of 2.
- ``seed`` (optional): Random seed for the scrambled Sobol sequence. Default: 42.
- ``slice_parameters`` (optional): List of parameter names to use as
  conditioning variables in the conditional Sobol analysis.
- ``parameter_groups`` (optional): Mapping of group names to parameter lists,
  used for plot organisation and colour grouping.
- ``parameters``: Mapping of parameter names to distribution specifications (see
  :ref:`supported-distributions` above).
- ``template``: Configuration template with ``{param_name}`` placeholders that
  are substituted with sampled values. Type is preserved when the placeholder
  is the entire value.

**``sensitivity_analysis`` field reference**:

- ``holdout_fraction``: Fraction of samples reserved for out-of-sample
  validation (e.g., 0.15 for 15%). Set to 0 to disable holdout.
- ``methods``: Mapping of method names (``pce``, ``rf``) to method-specific
  config. Each method entry supports:

  - ``grid_resolution`` (default: 100): Number of grid points for conditional
    Sobol evaluation along each slice parameter axis.
  - ``method_options``: Method-specific hyperparameters:

    - **PCE**: ``max_degree`` (default: 3), ``cross_truncation`` (default: 0.5),
      ``n_mc_conditional`` (default: 4096).
    - **RF**: ``n_estimators`` (default: 500), ``n_mc_global`` (default: 16384),
      ``n_mc_conditional`` (default: 8192).


.. _sensitivity-parameter-ranges:

Parameter Range Justification
-----------------------------

This section documents the uncertainty ranges assigned to each sensitivity
parameter, with references to the scientific literature. The range represents
the multiplicative factor applied to the model's default values.

**Distribution choice.** Parameters whose ranges are derived from formal
confidence intervals reported in the literature use ``normal_ci`` distributions,
where ``lower`` and ``upper`` define the 90% confidence interval of a normal
distribution (truncated at zero to prevent physically meaningless negative
factors). This applies to the three emission-related factors (CH\ :sub:`4`,
N\ :sub:`2`\ O, and LUC), whose ranges are grounded in IPCC confidence
intervals. The remaining parameters (crop yields, food loss & waste, feed
conversion ratios) use ``uniform`` distributions because their ranges are
derived from informal expert assessments, inter-estimate disagreement, or error
metrics that do not correspond to a well-defined confidence level.

The CH\ :sub:`4` and N\ :sub:`2`\ O sensitivity factors represent combined
uncertainty in both the underlying **emission factors** (measurement and
methodology uncertainty) and the **100-year global warming potentials** (GWP100)
used to convert physical emissions to CO\ :sub:`2`-equivalents. The IPCC AR6
reports GWP100 90% confidence intervals of ±40% for CH\ :sub:`4` and ±47% for
N\ :sub:`2`\ O [#gwp]_. Since EF and GWP uncertainties are independent (the
former is an agricultural measurement question, the latter a climate science
question), they combine approximately in quadrature.

CH\ :sub:`4` factor (``ch4_factor``: 0.5–1.5)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

This factor scales all CH\ :sub:`4` emissions in the model: enteric
fermentation, manure management (from animal production links), and rice paddy
emissions (from wetland rice crop production links).

**Emission factor uncertainty.** The IPCC 2019 Refinement reports a Tier 1
uncertainty of ±30–50% (95% CI) for livestock CH\ :sub:`4` emission factors
[#ipcc_ch10]_. Enteric fermentation dominates total livestock CH\ :sub:`4`
(roughly 80–90% of the total). Species-specific standard deviations in the 2019
Refinement (Tables 10.12–10.13) translate to 95% CIs of ±26% (sheep) to ±39%
(cattle/buffalo). Manure management uncertainty is considerably larger: Hristov
et al. report a 95% CI of ±63–65% for US manure CH\ :sub:`4` using gridded
Monte Carlo analysis [#hristov]_. A GLEAM one-at-a-time sensitivity analysis
found approximately ±39% total variation in ruminant CH\ :sub:`4` when
perturbing all 92 input parameters [#gleam]_. The midpoint of the IPCC Tier 1
range, ±40%, is a reasonable central estimate for the emission factor alone.

**Combined uncertainty.** Adding GWP100 uncertainty (±40%, 90% CI [#gwp]_) in
quadrature with the emission factor uncertainty (±40%, 95% CI) gives a combined
uncertainty of approximately ±57%. The ±50% range is a conservative rounding of
this combined estimate.

**Distribution.** Since both the IPCC GWP100 (90% CI) and Tier 1 emission factor
(95% CI) uncertainties are reported as formal confidence intervals, the combined
range is treated as a 90% CI of a ``normal_ci`` distribution (truncated at
zero). This is slightly conservative relative to the quadrature result.

N\ :sub:`2`\ O factor (``n2o_factor``: 0.3–1.7)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

This factor scales all N\ :sub:`2`\ O emissions in the model: manure management
and manure applied to soils (from animal production links), and direct and
indirect emissions from synthetic fertiliser application (from fertiliser
distribution links). N\ :sub:`2`\ O emission factors are among the most
uncertain parameters in agricultural GHG inventories.

**Emission factor uncertainty.** The IPCC 2019 Refinement [#ipcc_ch11]_ reports
the aggregated Tier 1 direct emission factor EF\ :sub:`1` = 0.01 with a 95% CI
of 0.001–0.018, corresponding to multiplicative factors of ×0.1–1.8. When
disaggregated by climate, the 95% CIs are tighter (e.g., wet-climate synthetic
fertiliser: 0.013–0.019 around 0.016, i.e., ±19%) but differ substantially
between wet and dry climates.

Since the model uses the aggregated EF\ :sub:`1` = 0.01 uniformly across
countries, it mixes climate zones where the true EF differs by a factor of ~3
(wet 0.016 vs. dry 0.005). An emission-factor-only range of approximately ±50%
captures:

- The **global aggregate** N\ :sub:`2`\ O **uncertainty** from Tian et al., who
  synthesise bottom-up inventories to estimate total agricultural N\ :sub:`2`\ O
  at 3.8 Tg N/yr with a min–max range across methodologies of 2.5–5.8 Tg N/yr,
  i.e., factors of ×0.66–1.53 relative to the central estimate [#tian]_.
- The **structural uncertainty** from using aggregated rather than
  climate-disaggregated emission factors. Hergoualc'h et al. showed that
  propagating the 95% CIs of disaggregated EFs gives a global estimate of
  883–1,285 Gg N\ :sub:`2`\ O-N/yr (±19% around the midpoint), whereas the
  aggregated method spans 539–2,713 Gg/yr [#hergoualch]_.
- **Indirect** N\ :sub:`2`\ O **pathways** with very wide 95% CIs
  (EF\ :sub:`4` for volatilisation: 0.002–0.05; Frac\ :sub:`LEACH`:
  0.10–0.80) [#ipcc_ch11]_.

The range is narrower than the full IPCC aggregated 95% CI [×0.1, ×1.8] because
global aggregation across countries and N sources averages out regional
extremes. Evidence of possible systematic underestimation (legacy effects
suggesting a true global mean EF of ~1.9% [#legacy_n2o]_) supports including
factors well above 1.0 in the range.

**Combined uncertainty.** Adding GWP100 uncertainty (±47%, 90% CI [#gwp]_) in
quadrature with the emission factor uncertainty (±50%) gives a combined
uncertainty of approximately ±69%, rounded to ±70%.

**Distribution.** As with CH\ :sub:`4`, both contributing uncertainties are
formal confidence intervals, so the combined range is treated as a 90% CI of a
``normal_ci`` distribution (truncated at zero).

Land-use change emissions (``luc_factor``: 0.3–1.7)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

This factor scales CO\ :sub:`2` emissions from land conversion (both cropland
and pasture expansion). LUC emissions are among the most uncertain components of
the global carbon budget, driven by uncertainty in carbon stocks, the spatial
pattern of conversion, and methodological choices.

The ±70% range aligns with the IPCC AR6 WGIII assessment, which reports a 90%
CI of ±70% for CO\ :sub:`2`-LULUCF emissions — the largest fractional
uncertainty of any major emission category [#ipcc_ar6_luc]_. This is consistent
with:

- The **Global Carbon Budget**: E\ :sub:`LUC` for 2022 is 1.2 ± 0.7 GtC/yr
  (semi-quantitative 1σ / 68% CI), with biogeochemical parameterisation as the
  dominant uncertainty component [#gcb2023]_.
- **Above-ground biomass** carbon stock estimates: the IPCC 2019 Refinement
  default AGB values for tropical forests have standard deviations averaging
  ~55% of the mean across ecological zones and continents [#rozendaal]_. The
  GlobBiomass dataset used in global maps has a global mean error of ~50%
  [#spawn]_.
- **Soil organic carbon** stock change factors (F\ :sub:`LU`, F\ :sub:`MG`,
  F\ :sub:`I`): 95% CIs of ±11–16% per factor, propagating to a combined
  SOC-change uncertainty of ~20–25% for well-characterised climate zones,
  reaching ±50% in poorly sampled regions [#ipcc_ch5]_.
- **Amortisation period** choices: switching from the conventional 20-year to a
  30-year period changes annualised emissions by ~33% [#maciel]_.

**Distribution.** The IPCC AR6 WGIII directly reports ±70% as a 90% CI, making
this the most straightforward case for a ``normal_ci`` distribution (truncated
at zero).

Crop yield factor (``yield_factor``: 0.8–1.2)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

This factor scales all crop production yields (efficiency on ``crop_production``
links), representing uncertainty in attainable yield estimates. The model uses
GAEZ v5 attainable yield data, which are subject to climate model uncertainty,
agronomic model limitations, and spatial aggregation error.

The ±20% range is supported by:

- **Grid-cell prediction error**: Normalised root-mean-square error (NRMSE)
  between GAEZ attainable yields and observed yields is 18–28% across major
  crop groups [#mueller]_.
- **Inter-annual variability**: Coefficients of variation in national crop
  yields are 13–22% for major cereals, reflecting weather-driven fluctuations
  that the model's single-year snapshot does not capture [#ray]_.
- **Climate model spread**: GAEZ v5 provides yield projections under five
  GCMs (GFDL-ESM4, IPSL-CM6A-LR, MPI-ESM1-2-HR, MRI-ESM2-0, UKESM1-0-LL);
  inter-model yield spread is typically 10–25% for a given crop and region.

Yield uncertainty propagates strongly to land use (more yield means less land
required) and GHG emissions (through reduced land-use change pressure).

**Distribution.** A ``uniform`` distribution is used because the range is
synthesised from heterogeneous error metrics (NRMSE, CV, model spread) that do
not correspond to a formal confidence interval at any specific level.

Food loss and waste factor (``flw_factor``: 0.7–1.3)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

This factor scales the efficiency of ``food_processing`` links, which
incorporate food loss and waste fractions from SDG 12.3.1 data (food loss index
from FAO + food waste index from UNEP). A factor below 1.0 means more food is
lost than the baseline estimates suggest (higher FLW); above 1.0 means less food
is lost (lower FLW).

The ±30% range is supported by:

- **Inter-estimate disagreement**: Global FLW estimates range from ~24% of food
  supply on a caloric basis [#kummu]_ to ~33% on a mass basis [#gustavsson]_ to
  ~40% when on-farm harvest losses are included [#wwf]_, giving multiplicative
  factors of ×0.73–1.21 relative to the FAO central estimate.
- **Measurement bias**: Self-reported consumer waste data (which feed into the
  UNEP Food Waste Index) systematically underestimate actual waste by 20–40%
  compared to direct waste compositional analysis [#quested]_.
- **Country-level data gaps**: The SDG 12.3.1 Food Loss Index has a stated
  random error of ~25% at the country level [#fao_sofa]_. Only ~12% of the
  global population lives in countries that directly track FLW. Much
  post-harvest loss data for developing countries was collected over 30 years
  ago [#parfitt]_.

The dominant bias direction in the literature is underestimation, which would
support an asymmetric range skewed higher. The symmetric ±30% range is a
conservative simplification.

**Distribution.** A ``uniform`` distribution is used because the range is derived
from inter-estimate disagreement and data gap assessments rather than formal
confidence intervals.

Feed conversion ratio factor (``fcr_factor``: 0.8–1.2)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

This factor scales feed conversion efficiencies (efficiency on
``animal_production`` links), representing uncertainty in how much feed is
required per unit of animal product. Higher values mean better conversion (more
product per unit feed). The model uses Wirsenius (2000) regional feed energy
requirements [#wirsenius]_ converted via NRC/NASEM net-energy-to-metabolisable-energy
factors [#nrc_beef]_ [#nasem_beef]_ [#nrc_dairy]_.

The ±20% range is supported by:

- **Inter-source disagreement**: The model applies calibration multipliers (up
  to 2.0×) to reconcile Wirsenius-based efficiencies with GLEAM feed baselines.
  After calibration, residual disagreement between Wirsenius and GLEAM / Herrero
  et al. [#herrero]_ is typically 10–30% for a given region–product pair.
- **Energy conversion uncertainty**: The NE-to-ME conversion factors (k_m=0.65,
  k_g=0.43, k_l=0.64) are central NRC/NASEM values for a typical mixed diet
  [#nasem_beef]_ [#nrc_dairy]_. Published ranges span k_m: 0.55–0.70 and
  k_g: 0.35–0.50 (depending on diet metabolizability q), introducing ~10–15%
  uncertainty.
- **Temporal lag**: The Wirsenius data reflects ~1994–1998 conditions.
  Monogastric FCRs have improved ~10–20% since then through genetic progress
  (~0.5–1%/year for poultry and pork); ruminant improvement is minimal.
- **Precedent**: Alexander et al. [#alexander]_ and Springmann et al.
  [#springmann]_ both used ±20% for FCR uncertainty in Monte Carlo global food
  system analyses. The IPCC 2019 Refinement reports ±20% uncertainty for Tier 2
  feed intake estimates [#ipcc_ch10]_.

The ±20% range captures data source disagreement, conversion factor
uncertainty, and temporal lag without bleeding into inter-system variation
(which is already represented by the model's regionalized FCR assignment).

**Distribution.** A ``uniform`` distribution is used because the range is based
on inter-source disagreement and precedent from other studies that also used
uniform distributions for FCR uncertainty [#alexander]_ [#springmann]_.

Health relative risk parameters (``rr_protective``, ``rr_harmful``: 0–1)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Relative risk uncertainty is parameterized as quantiles :math:`q \in [0, 1]`
that interpolate between the GBD lower and upper confidence bounds for
dose-response relative risks. At :math:`q = 0` the strongest effect estimate is
used; at :math:`q = 1` the weakest.

Rather than specifying a separate quantile per risk factor, two **grouped**
parameters reduce dimensionality:

- ``rr_protective``: applies to all risk factors whose RR *decreases* with
  intake (e.g., fruits, vegetables, whole grains, legumes, nuts & seeds).
- ``rr_harmful``: applies to all risk factors whose RR *increases* with
  intake (e.g., red meat, processed meat).

Direction is inferred automatically from the dose-response data: for each risk
factor, log_rr at the lowest intake is compared with log_rr at the highest
intake. This grouping is justified because protective food groups share a common
uncertainty mechanism (GBD confidence interval bounds).

Individual risk factor keys (e.g., ``whole_grains: 0.5``) remain supported and
take precedence over group keys when both are specified. However, specifying
both a group key and an individual key for the same risk factor raises an error.

Reforestation cap fraction (``reforest_fraction``: 0.05--1.0)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

A per-country cap on the fraction of spareable agricultural land
(baseline cropland + existing pasture) that may be spared for carbon
sequestration, wired to the solve-time constraint
``land.reforestation_cap.max_fraction`` (see :ref:`reforestation-cap`).
At ``1.0`` the cap is inactive; under heavy emission weighting the
unconstrained model reforests 80--90% of some countries' agricultural
area, so this parameter tests how strongly results depend on permitting
such concentrated reforestation. Unlike the other parameters it is a
policy/plausibility lever rather than an empirical uncertainty, so it is
sampled uniformly over a wide range rather than from a fitted
distribution. The ``sequestration`` output (net CO\ :sub:`2` from
spared-land sequestration credits) is included as a Sobol target to
capture its most direct effect.

.. _sensitivity-prod-stability-cost:

Deviation-penalty regime (companion ``gsa_l1.yaml`` config)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The L1 deviation-penalty regime is a structural modelling choice rather than
an empirical uncertainty, so it is not sampled as a Sobol parameter.
``config/gsa.yaml`` runs the single calibrated central regime
(``deviation_penalty.{land.crops,land.grassland,feed}.l1_cost:
"calibrated"``). The lower/higher regimes
(``l1_cost_factor = 1/sqrt(10)`` and ``sqrt(10)``, i.e. half an order of
magnitude below/above the calibrated centre on cropland, grassland and
feed) live in a companion config, ``gsa_l1.yaml``, maintained alongside
the paper. Each regime has its own consumer-values baseline so that the
piecewise food utility blocks are calibrated against the matching L1
regime; the regimes share the rest of the GSA design (parameters,
sampling, slice parameters). Comparing Sobol indices across regimes
shows how sensitivity structure shifts with the stability regime,
without spending Sobol degrees of freedom on a non-empirical axis.

Policy slice parameters
~~~~~~~~~~~~~~~~~~~~~~~

The **value per YLL** (``value_per_yll``: 0–20,000 USD) and **GHG price**
(``ghg_price``: 0–300 USD/tCO\ :sub:`2`-eq) are policy-choice parameters rather
than epistemic uncertainties. They are designated as **slice parameters**:
included in the PCE fit but analytically conditioned on specific values to show
how sensitivity patterns shift across the policy space. Their ranges are chosen
to span a wide policy-relevant domain without claiming to represent a specific
probability distribution.


.. rubric:: References

.. [#gwp] IPCC, 2021: *Climate Change 2021: The Physical Science Basis*,
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   GWP100 (N\ :sub:`2`\ O): 273 ± 130 (90% CI).
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.. [#ipcc_ch10] IPCC, 2019: *2019 Refinement to the 2006 IPCC Guidelines for
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.. [#hristov] Hristov, A. N. et al., 2017: Discrepancies and uncertainties in
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.. [#gleam] Rivera Moncada, A. et al., 2025: Sensitivity analysis of
   parameters, emission factors, and coefficients for estimating animal
   emissions of ruminant species in GLEAM. *Int. J. Life Cycle Assess.*
   One-at-a-time perturbation of 92 input parameters.
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.. [#ipcc_ch11] IPCC, 2019: *2019 Refinement to the 2006 IPCC Guidelines for
   National Greenhouse Gas Inventories*, Vol. 4, Ch. 11: N\ :sub:`2`\ O
   Emissions from Managed Soils, and CO\ :sub:`2` Emissions from Lime and Urea
   Application. EF\ :sub:`1` = 0.01 (95% CI: 0.001–0.018).
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.. [#tian] Tian, H. et al., 2020: A comprehensive quantification of global
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.. [#hergoualch] Hergoualc'h, K. et al., 2021: Improved accuracy and reduced
   uncertainty in greenhouse gas inventories by refining the IPCC emission
   factor for direct N\ :sub:`2`\ O emissions from nitrogen inputs to managed
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   by propagating IPCC 95% CIs of disaggregated emission factors.
   https://doi.org/10.1111/gcb.15884

.. [#legacy_n2o] Qian, H. et al., 2025: Legacy effects cause systematic
   underestimation of N\ :sub:`2`\ O emission factors. *Nat. Commun.*, 16.
   https://doi.org/10.1038/s41467-025-58090-0

.. [#ipcc_ar6_luc] IPCC, 2022: *Climate Change 2022: Mitigation of Climate
   Change*, WG III, Chapter 7: Agriculture, Forestry, and Other Land Uses.
   CO\ :sub:`2`-LULUCF uncertainty: ±70% (90% CI).
   https://www.ipcc.ch/report/ar6/wg3/chapter/chapter-7/

.. [#gcb2023] Friedlingstein, P. et al., 2023: Global Carbon Budget 2023.
   *Earth Syst. Sci. Data*, 15, 5301–5369. E\ :sub:`LUC` uncertainty described
   as a "semi-quantitative" 1σ (68% CI).
   https://doi.org/10.5194/essd-15-5301-2023

.. [#rozendaal] Rozendaal, D. M. A. et al., 2022: Aboveground forest biomass
   varies across continents, ecological zones and successional stages: refined
   IPCC default values for tropical and subtropical forests. *Environ. Res.
   Lett.*, 17, 014047.
   https://doi.org/10.1088/1748-9326/ac45b3

.. [#spawn] Spawn, S. A. et al., 2020: Harmonized global maps of above and
   belowground biomass carbon density in the year 2010. *Sci. Data*, 7, 112.
   https://doi.org/10.1038/s41597-020-0444-4

.. [#ipcc_ch5] IPCC, 2019: *2019 Refinement to the 2006 IPCC Guidelines for
   National Greenhouse Gas Inventories*, Vol. 4, Ch. 5: Cropland. SOC stock
   change factor uncertainties: ±11–50%.
   https://www.ipcc-nggip.iges.or.jp/public/2019rf/pdf/4_Volume4/19R_V4_Ch05_Cropland.pdf

.. [#maciel] Maciel, V. G. et al., 2022: Towards a non-ambiguous view of the
   amortization period for quantifying direct land-use change in LCA. *Int. J.
   Life Cycle Assess.*, 27, 1299–1315.
   https://doi.org/10.1007/s11367-022-02103-3

.. [#gbd] GBD 2017 Diet Collaborators, 2019: Health effects of dietary risks
   in 195 countries, 1990–2017. *Lancet*, 393, 1958–1972.
   https://doi.org/10.1016/S0140-6736(19)30041-8

.. [#mueller] Mueller, N. D. et al., 2012: Closing yield gaps through nutrient
   and water management. *Nature*, 490, 254–257.
   https://doi.org/10.1038/nature11420

.. [#ray] Ray, D. K. et al., 2015: Climate variation explains a third of
   global crop yield variability. *Nat. Commun.*, 6, 5989.
   https://doi.org/10.1038/ncomms6989

.. [#kummu] Kummu, M. et al., 2012: Lost food, wasted resources: Global food
   supply chain losses and their impacts on freshwater, cropland, and fertiliser
   use. *Sci. Total Environ.*, 438, 477–489.
   https://doi.org/10.1016/j.scitotenv.2012.08.092

.. [#gustavsson] Gustavsson, J. et al., 2011: *Global food losses and food
   waste: extent, causes and prevention*. FAO, Rome.
   https://www.fao.org/4/mb060e/mb060e00.htm

.. [#wwf] WWF-UK, 2021: *Driven to Waste: The Global Impact of Food Loss and
   Waste on Farms*. WWF-UK, Woking. Estimates total FLW at ~40% of production
   when on-farm losses are included.
   https://wwfint.awsassets.panda.org/downloads/driven_to_waste___the_global_impact_of_food_loss_and_waste_on_farms.pdf

.. [#quested] Quested, T. E. et al., 2020: Comparing diaries and waste
   compositional analysis for measuring food waste in the home. *J. Clean.
   Prod.*, 279, 123635. Self-reported estimates 7–40% lower than compositional
   analysis.
   https://doi.org/10.1016/j.jclepro.2020.123635

.. [#fao_sofa] FAO, 2019: *The State of Food and Agriculture 2019: Moving
   forward on food loss and waste reduction*. FAO, Rome. SDG 12.3.1 Food Loss
   Index country-level random error: ~25%.
   https://www.fao.org/3/ca6030en/ca6030en.pdf

.. [#parfitt] Parfitt, J. et al., 2010: Food waste within food supply chains:
   quantification and potential for change to 2050. *Phil. Trans. R. Soc. B*,
   365, 3065–3081.
   https://doi.org/10.1098/rstb.2010.0126

.. [#wirsenius] Wirsenius, S., 2000: *Human Use of Land and Organic Materials:
   Modeling the Turnover of Biomass in the Global Food System*. PhD thesis,
   Chalmers University of Technology.

.. [#nrc_beef] NRC, 2000: *Nutrient Requirements of Beef Cattle*, 7th
   revised ed., update 2000. Washington, DC: The National Academies Press.
   Source of the California Net Energy System used for ruminant ME→NE
   conversion (k_m, k_g). https://doi.org/10.17226/9791

.. [#nasem_beef] NASEM, 2016: *Nutrient Requirements of Beef Cattle*, 8th
   revised ed. Washington, DC: The National Academies Press. Provides the
   updated cubic-in-ME equations for k_m and k_g; evaluated at typical
   mixed-diet metabolizability q ≈ 0.60 these give k_m ≈ 0.65 and
   k_g ≈ 0.43. https://doi.org/10.17226/19014

.. [#nrc_dairy] NRC, 2001: *Nutrient Requirements of Dairy Cattle*, 7th
   revised ed. Washington, DC: The National Academies Press. Specifies
   the fixed ME-to-NEL efficiency k_l = 0.64 used for dairy.
   https://doi.org/10.17226/9825

.. [#herrero] Herrero, M. et al., 2013: Biomass use, production, feed
   efficiencies, and greenhouse gas emissions from global livestock systems.
   *PNAS*, 110, 20888–20893.
   https://doi.org/10.1073/pnas.1308149110

.. [#alexander] Alexander, P. et al., 2016: Human appropriation of land for
   food: The role of diet. *Global Environ. Change*, 41, 88–98. Used ±20% FCR
   uncertainty in Monte Carlo sensitivity analysis.
   https://doi.org/10.1016/j.gloenvcha.2016.09.005

.. [#springmann] Springmann, M. et al., 2018: Options for keeping the food
   system within environmental limits. *Nature*, 562, 519–525. Used ±20% FCR
   uncertainty in Monte Carlo analysis (500 samples, uniform distribution).
   https://doi.org/10.1038/s41586-018-0594-0


Running the Analysis
--------------------

The sensitivity analysis has four stages: build sampled scenarios, solve
them, fit a surrogate over the scalar outputs, and compute Sobol indices
(or other diagnostics) from the surrogate. Snakemake handles all
dependencies automatically.

**Run the full pipeline** (build + solve + analyze for all samples):

.. code-block:: bash

   tools/smk -j4 --configfile config/gsa.yaml

**Fit a surrogate only** (after scenarios are already solved):

.. code-block:: bash

   # XGBoost surrogate for the default scenario group
   tools/smk -j4 --configfile config/gsa.yaml -- \
       results/gsa/surrogates/surrogate_gsa_xgb.pkl

The ``build_surrogate`` rule writes a pickled :class:`SurrogateBundle` (the
trained model, generator spec, and parameter metadata) plus a flat
validation parquet.  The bundle is consumed by downstream rules and
notebooks (policy sweeps, uncertainty-band plots).

.. note::

   When a sweep solves outside Snakemake (typically the cluster path
   driven by ``tools/batch-solve``; see :doc:`cluster_execution`), set
   ``sensitivity_analysis.discover_scenarios_on_disk: true`` in the
   config (already set in ``config/gsa.yaml``).  With the flag on, the surrogate-fit
   rule scans the analysis directory and fits over scenarios with
   complete outputs on disk, dropping the small fraction that may have
   hit per-solve TimeLimit.  The rule errors out if more than 50 % of
   scenarios are missing, as a guardrail against running it before the
   solve+analyse phase has finished.  When the flag is left at its
   default ``false``, ``build_surrogate`` declares every Sobol scenario
   as an input so a single ``tools/smk`` call drives the whole
   solve→analyse→surrogate chain — the canonical Snakemake idiom and
   the right choice for small / test runs.

**Compute Sobol indices from a surrogate**:

.. code-block:: bash

   tools/smk -j4 --configfile config/gsa.yaml -- \
       results/gsa/analysis/sobol_global_indices_gsa_xgb.parquet

Output paths use two wildcards: ``{group}`` identifies the scenario sampling
group (e.g., ``gsa``, ``gsa-l1-low``) and ``{method}`` selects the surrogate
type (``pce``, ``rf``, ``mars``, ``xgb``).  All methods consume the same
solved scenarios, and ``sensitivity_analysis.default_surrogate`` selects the
surrogate downstream consumers (notebooks, uncertainty plots) load by
default.

.. note::

   Each sample requires a full model build and solve. Start with a small sample
   count (32--64) for testing, then increase (1024--4096) for production. A
   coarser spatial resolution also reduces per-sample cost.


Output Files
------------

Per (group, method) combination, the workflow writes:

- A pickled surrogate bundle at
  ``results/{name}/surrogates/surrogate_{group}_{method}.pkl``.  Loaded via
  :func:`workflow.scripts.analysis.surrogate.load_bundle`.
- A flat validation parquet at
  ``results/{name}/surrogates/surrogate_validation_{group}_{method}.parquet``
  (surrogate fit quality per output target).
- Three Sobol parquets at ``results/{name}/analysis/``:

**sobol_global_indices_{group}_{method}.parquet** — Global Sobol indices

.. csv-table::
   :header: Column, Type, Description

   ``output``, string, "Output metric (``total_cost``, ``co2``, ``ch4``, ``n2o``, ``land_use``, ``yll``; per-gas emissions are in MtCO\u2082eq)"
   ``parameter``, string, "Parameter name from generator spec"
   ``S1``, float, "First-order Sobol index"
   ``ST``, float, "Total-order Sobol index"

One row per (output, parameter) pair.

**sobol_conditional_indices_{group}_{method}.parquet** — Conditional Sobol indices (1D slices)

.. csv-table::
   :header: Column, Type, Description

   ``output``, string, "Output metric"
   ``parameter``, string, "Parameter name (non-slice parameters only)"
   ``S1_cond``, float, "Conditional first-order Sobol index"
   ``ST_cond``, float, "Conditional total-order Sobol index"
   ``conditional_variance``, float, "Output variance when slice parameters are fixed"
   *slice columns*, float, "One column per slice parameter with the conditioning value"

One row per (output, parameter, conditioning-value combination).

**sobol_conditional_joint_indices_{group}_{method}.parquet** — Joint conditional Sobol indices (2D grid)

Same schema as above, but conditioned on *all* slice parameters simultaneously
over a 2D grid. Used by the dominant factor phase diagram plot.

**surrogate_validation_{group}_{method}.parquet** — Surrogate quality metrics
(lives under ``surrogates/``, not ``analysis/``, because it describes the
surrogate fit rather than the Sobol indices).

.. csv-table::
   :header: Column, Type, Description

   ``output``, string, "Output metric"
   ``validation_error``, float, "Primary error metric (holdout error when available)"
   ``r2_train``, float, "R² on training data"
   ``r2_test``, float, "R² on holdout data (null if holdout disabled)"
   ``n_train``, int, "Number of training samples"
   ``n_test``, int, "Number of holdout samples"
   ``method``, string, "Surrogate method (``pce``, ``rf``, ``mars``, ``xgb``)"

Additional method-specific columns: ``loo_error``, ``n_terms``,
``n_active_terms``, ``max_degree`` (PCE); ``oob_error``, ``n_estimators``
(RF); ``n_estimators`` (XGB); ``gcv``, ``n_basis`` (MARS).

**Plots**

Three types of sensitivity plots are generated per (group, method):

.. code-block:: bash

   tools/smk -j4 --configfile config/gsa.yaml -- \
       results/gsa/plots/sobol_conditional_s1_vs_ghg_price_gsa_pce.pdf

- **Stacked area charts** (``sobol_conditional_s1_vs_{slice}_{group}_{method}.pdf``):
  Conditional first-order Sobol shares vs each slice parameter. One panel per
  model output.
- **Dominant factor phase diagrams** (``sobol_conditional_dominant_factor_{group}_{method}.pdf``):
  2D policy space coloured by which parameter has the highest conditional S1.
- **Contour surfaces** (``sobol_conditional_s1_surface_{parameter}_{group}_{method}.pdf``):
  Per-parameter conditional S1 surface over the 2D policy space.


Interpreting Results
--------------------

**Reading Sobol indices**:

- :math:`S_1 \approx 1`: This parameter is the dominant driver; reducing its
  uncertainty would substantially reduce output uncertainty.
- :math:`S_T \gg S_1`: This parameter is involved in significant interactions
  with other parameters.
- :math:`S_1 \approx S_T \approx 0`: This parameter has negligible influence
  on the output.

**Example interpretation**: If ``yield_factor`` has :math:`S_1 = 0.6` for
``co2``, then 60% of the variance in net CO\u2082 emissions is explained by
crop yield uncertainty alone.  Total GHG emissions are the sum of the
``co2``, ``ch4``, and ``n2o`` outputs (all in MtCO\u2082eq); for tree methods
(RF, XGBoost) fit with a shared multi-output structure, that sum is also the
direct surrogate prediction for total emissions, as linear relations between
outputs are preserved exactly under MSE loss.

**Validation quality**:

- Validation error < 0.1 indicates a reliable surrogate.
- Validation error > 0.1 suggests the surrogate approximation is insufficient.
  For PCE, consider increasing samples or polynomial degree. For RF, consider
  increasing the number of estimators. Comparing PCE and RF errors can reveal
  whether the issue is model non-smoothness (where RF may outperform PCE) or
  insufficient data.

**Conditional indices**: These show how sensitivity patterns shift with policy
choices. For instance, at low GHG prices, yield uncertainty may dominate
emissions variance, while at high GHG prices, land-use-change factors may
become more important as the model restructures production patterns.