MarginalCDFExtrapolation

class axtreme.qoi.marginal_cdf_extrapolation.MarginalCDFExtrapolation(env_iterable: Iterable[Tensor] | Iterable[list[Tensor]], period_len: int, quantile: Tensor | None = None, input_transform: InputTransform | None = None, outcome_transform: OutcomeTransform | None = None, posterior_sampler: PosteriorSampler | None = None, response_distribution: type[Distribution] = Gumbel, quantile_accuracy: Tensor | None = None, dtype: dtype = torch.float64, *, device: device | None = None, no_grad: bool = True)

Bases: MeanVarPosteriorSampledEstimates, QoIEstimator

Estimate an ERD QoI using the marginal CDF for a single timestep.

Basic Idea:

• Get the marginal CDF of the response for a single timestep (e.g. 1 hour).

  Marginal CDF: The CDF after marginalising out all sources of randomness within a timestep. E.g if you considered all the variablity in a single timestep (e.g all the different weather conditions), collected the different CDFs they produce, and then averaged them, you would have the marginal CDF.

• Using the average CDF you can then calculate:

  • “The probablity that a response of size y won’t be exceeded in 1 hour” as follows:

    • \(CDF(y = .8) = .5\)

  • “The probablity that a response of size y won’t be exceeded in 2 hours” as follows:

    • [Prb not exceeded in hour one] AND [Prb not exceeded in hour two]

    • \(CDF(y = .8) * CDF(y = .8) = .5 * .5 = .25\)

  • “The probablity that a response of size y won’t be exceeded in N hours” as follows:

    • \(CDF(y = .8)^N = (.5)^N\)

This is possible because we are using the ‘average’ timestep, which is the same for all timesteps.

Strengths:

• Once the ‘average’ CDF has been obtained, very large values of N can be calculated quickly.

Challenge:

• Need to obtain the ‘average’ CDF, and it must be very accurate.

See [#TODO(sw 2024_11_4) put in link to pre-print] for details.

__init__(env_iterable: Iterable[Tensor] | Iterable[list[Tensor]], period_len: int, quantile: Tensor | None = None, input_transform: InputTransform | None = None, outcome_transform: OutcomeTransform | None = None, posterior_sampler: PosteriorSampler | None = None, response_distribution: type[Distribution] = Gumbel, quantile_accuracy: Tensor | None = None, dtype: dtype = torch.float64, *, device: device | None = None, no_grad: bool = True) → None

Initialise the QOI estimator.

Parameters:

• env_iterable –

  An interable that produces the env data to be used. Typically this is a DataLoader. Supports standard env data, and weighted env data (e.g Importance sampled).

  • Standards env data: Expects tensor of shape (batch_size, d)

  • Weighted data: Expectes list of tensors.

    • The first: the env data of shape (batch_size,d).

    • The second item: the weight, of shape (batch_size,).

• period_len – The number of env samples in the period of interest.

• quantile – Shape (,). The quantile of the ERD to be estimate (the QoI). Should be within range (0,1). default .5.

• input_transform – transforms that should be applied to the env_samples before being passed to the model

• outcome_transform – transforms that should be applied to the output of the model before they are used

• posterior_sampler –

  The sampler to use to draw samples from the posterior of the GP.

  • n_posterior_samples: is set in the PosteriorSampler

  • NOTE: if env_iterable contains batches, a batch compatible sampler such as NormalIndependentSampler or “ut_sampler” should be selected.

• response_distribution – The distribution which models the stochastic response of the simulation at a single input point.

• quantile_accuracy – shape (,). Default value .01. Internally, optimisation is used to find the quantile. The optimiser is allowed to terminate once in the region quantile +/- quantile_accuracy. The greater the accuracy required, the longer the optimisation will take. Typically other sources of uncertainty produce far greater uncertainty.

• dtype – The dtype used for the distribution. This has implications for the numerical accuracy possible.

• device – The device QoI should be performed on.

• no_grad – If gradient requires tracking through this QOI calculation.

Methods

__init__(env_iterable, period_len[, ...])	Initialise the QOI estimator.
mean(x)	Function that computes the mean of the estimates produced by using self.posterior_sampler.
var(x)	Function that computes the variance of the estimates produced by using self.posterior_sampler.

Attributes

posterior_sampler