sweights

sWeights implementation.

sWeights are used to statistically separate signal and background contributions in a data sample when the distributions overlap.

cal_sweights(pdfs, N=None)

Calculate sWeights for a set of PDF components.

Given normalized PDFs for each component (signal, background, etc.), computes weights such that the weighted sum of events gives the yield for each component.

Parameters:

• pdfs (list of array-like) –

  List of PDF values evaluated at each data point. Each element should be an array of shape (n_events,).

  Two calling conventions are supported:

  1. Normalized PDFs with N parameter:

     >>> import numpy as np
     >>> from scipy.stats import norm
     >>> np.random.seed(42)
     >>> x_sig = np.random.normal(5.0, 0.5, 50)
     >>> x_bkg = np.random.uniform(0, 10, 100)
     >>> x = np.concatenate([x_sig, x_bkg])
     >>> f_sig = norm.pdf(x, 5.0, 0.5)
     >>> f_bkg = np.ones_like(x) / 10.0
     >>> sw = cal_sweights([f_sig, f_bkg], N=[50, 100])
     >>> bool(np.isclose(np.sum(sw[0]), 50))
     True
     

  2. Unnormalized PDFs without N parameter:

     >>> sw2 = cal_sweights([f_sig * 50, f_bkg * 100])
     >>> bool(np.allclose(np.sum(sw2[0]), 50, rtol=0.1))
     True
     

• N (array-like, optional) – Expected yields for each component. If None, yields are estimated from the PDFs assuming the PDFs are already scaled by yields.

Returns:

sweights – Array of shape (n_components, n_events) containing sWeights for each component at each event.

Return type:

ndarray

Raises:

LinAlgError – If the covariance matrix is singular and cannot be inverted.

Notes

The sWeights are computed using Cowan’s formula:

\[p(x) = \sum_i N_i f_i(x)\]

\[V_{ij}^{-1} = \sum_k \frac{f_i(x_k) f_j(x_k)}{p(x_k)^2}\]

\[w_i(x_k) = \frac{\sum_j V_{ij} f_j(x_k)}{p(x_k)}\]

where:

• \(N_i\) is the yield for component i

• \(f_i(x)\) is the normalized PDF for component i

• \(p(x)\) is the total PDF

• \(V\) is the covariance matrix of the yields

Reference: sPlot paper