Expected Number of Detections and Sensitivity Estimation

In order to calculate the expected number of detections \(\mu(\Lambda)\), sensitivity of the detectors is also required. Mathematically, it is expressed as:

\[ \mu(\Lambda) = T_{\mathrm{obs}} \int \int \rho(\theta,z | \Lambda) \frac{dV_c}{dz} \frac{1}{1+z}p_{\mathrm{det}}(\theta;z) d\theta dz \]

If redshift is not a parameter in the population model or can be factored out, then

\[ \mu(\Lambda) = T_{\mathrm{obs}} \int \rho(\theta | \Lambda) \underbrace{\int \frac{dV_c}{dz} \frac{1}{1+z}p_{\mathrm{det}}(\theta;z) dz}_{\mathrm{VT}(\theta)} d\theta \]

where, \(\mathrm{VT}(\theta)\) is called Sensitive Spacetime Volume of a network with probability of detection \(p_{\mathrm{det}}(\theta;z)\) for a source with parameters \(\theta\) at redshift \(z\). \(T_{\mathrm{obs}}\) is the total observation time.

GWKokab has three major methods to estimate the number of detections.

Probability of Detection

To avoid interpolation, GWKokab trains a small Multilayer Perceptron (MLP) model to estimate the probability of detection \(p_{\mathrm{det}}(\theta;z)\). These trained models serves as a fast surrogate to estimate the probability of detection for a given source parameters \(\theta\) and redshift \(z\). For a multi-source population model, the overall probability of detection is calculated by averaging over the individual source probabilities. If \(\hat{p}_{\mathrm{det}}(\theta;z)\) is the neural network estimate of the probability of detection, then for \(K\) sources, the overall probability of detection is given by:

\[ \hat{\mu}(\Lambda) = T_{\mathrm{obs}} \sum_{k=1}^{K}\mathcal{R}_{k}\left<Z_k\hat{p}_{\mathrm{det}}(\theta_i;z)\right>_{\theta_i,z\sim \rho^*_k(\theta,z|\Lambda)} \]

where, \(Z_k\) is any factor required to normalize \(\rho_k\), and \(\rho^*_k\) is the normalized \(\rho_k\).

An example configuration for probability of detection estimator is shown below:

{
    "estimator_type": "neural_pdet",
    "filename": "neural_pdet_with_TaylorF2Ecc_uniform_injections.hdf5",
    "time_scale": 1.0,
    "num_samples": 1500,
    "batch_size": 150
}

Here, the configuration specifies the use of a neural network to estimate the probability of detection. "estimator_type" indicates the type of estimator, "filename" is the path to the trained neural network model, "time_scale" is the observation time (in appropriate units), "num_samples" is the number of samples to draw from the population model, and "batch_size" is the size of a batch to evaluate the neural network.

Sensitive Spacetime Volume

If population model is independent of redshift or redshift can be factored out, then one can integrate out the redshift dependence to calculate the sensitive spacetime volume \(\mathrm{VT}(\theta)\), and train a MLP model to estimate \(\mathrm{VT}(\theta)\) directly.

\[ \hat{\mu}(\Lambda) = T_{\mathrm{obs}} \int \rho(\theta | \Lambda) \hat{\mathrm{VT}}(\theta) d\theta = T_{\mathrm{obs}} \sum_{k=1}^{K}\mathcal{R}_{k}\left<Z_k\hat{\mathrm{VT}}(\theta_i)\right>_{\theta_i\sim \rho^*_k(\theta|\Lambda)} \]

where, \(Z_k\) is any factor required to normalize \(\rho_k\), and \(\rho^*_k\) is the normalized \(\rho_k\).

An example configuration for sensitive spacetime volume estimator is shown below:

{
    "estimator_type": "neural_vt",
    "filename": "neural_vt_1_200_1000_ecc_matters.hdf5",
    "time_scale": 248.0,
    "num_samples": 2000,
    "batch_size": 1000
}

Here, the configuration specifies the use of a neural network to estimate the sensitive spacetime volume. "estimator_type" indicates the type of estimator, "filename" is the path to the trained neural network model, "time_scale" is the observation time (in appropriate units), "num_samples" is the number of samples to draw from the population model, and "batch_size" is the size of a batch to evaluate the neural network.

Injection Based Method

Expected number of detections can also be estimated given a set of injections and their sampling probability density. If we have \(N_{\mathrm{inj}}\) injections with parameters \(\{\theta_i,z_i\}_{i=1}^{N_{\mathrm{inj}}}\), and out of these \(N_{\mathrm{det}}\) injections are detected by the search pipeline, then the expected number of detections is given by:

\[ \hat{\mu}(\Lambda) = \frac{T_{\mathrm{obs}}}{N_{\mathrm{inj}}} \sum_{i=1}^{N_{\mathrm{det}}} \frac{\rho(\theta_i,z_i|\Lambda)}{p_{\mathrm{inj}}(\theta_i,z_i)} \]

An example configuration for o1o2o3o4a injection based estimator is shown below:

{
    "estimator_type": "injection",
    "filename": "mixture-semi_o1_o2-real_o3_o4a-cartesian_spins_20250503134659UTC.hdf",
    "batch_size": 20000,
    "snr_cut": 10.0,
    "far_cut": 1.0
}

Here, the configuration specifies the use of an injection-based method to estimate the expected number of detections. "estimator_type" indicates the type of estimator, "filename" is the path to the injection data file, "batch_size" is the size of a batch to process injections, "snr_cut" is the signal-to-noise ratio threshold for detection, and "far_cut" is the false alarm rate threshold for detection.

Custom Estimator

If none of the above estimators fit your needs, you can pass a custom estimator by implementing a function named custom_poisson_mean_estimator in a python module and that has a signature like below:

from typing import Callable, Sequence, Optional
from jaxtyping import PRNGKeyArray, ArrayLike
from gwkokab.models.utils import ScaleMixture

def custom_poisson_mean_estimator(
    key: PRNGKeyArray,
    parameters: Sequence[str],
    filename: str,
    batch_size: Optional[int],
    **kwargs) -> Callable[[ScaleMixture], ArrayLike]: ...

Its configuration would look like:

{
    "estimator_type": "custom",
    "module": "my_custom_module.py",
    "filename": "my_custom_file.hdf5",
    "batch_size": 1000,
    "other_param_1": "value_1",
    "other_param_2": 10
}