Gaussian similarity metric for pitch coordinates

[InakiRaba91]

Hey everyone. I was just looking through the implementation of the similarity metric in pitch domain, and it seems to differ from the one provided in Eq. 4 in the paper.

Paper: exp(ln(0.05)(gt-pred)^2/t^2) = 0.05^((gt-pred)^2/t^2)
Code: exp(-0.5(gt-pred)^2/t^2)

Is there any reason for this?

Thanks in advance!

[VlSomers]

Hi @InakiRaba91, the two formulas are actually not the same as you wrote—one uses σ (sigma) and the other uses τ (tau). In the paper, τ is the distance tolerance (e.g. 5 meters), while in the code, σ is used in the standard Gaussian formula.

In sn-trackeval, given a tau, the corresponding sigma is computed using the "calculate_sigma" function (see bottom of trackeval/datasets/soccernet_gs.py). This ensures both formulas are mathematically equivalent when tau = 5 m.

You can check out the attached script that compares both and confirms they match perfectly when σ is computed from τ. Does this make sense now? Let me know if you spot anything that still seems off!

import numpy as np
import matplotlib.pyplot as plt

# Function from sn-trackeval to calculate sigma based on the distance tolerance parameter tau
# This function calculates sigma such that the Gaussian similarity function matches the desired similarity score (i.e. 0.05) at a distance of tau
# (From Sect 5.1 of the paper) The desired similarity score is set to 0.05 because the GS-HOTA is integrated over the discrete interval α ∈ [0.05, 0.95]
# This means that (P, G) pairs with a similarity below or equal to 0.05 are never matched.
def calculate_sigma(x):
    return np.sqrt(x**2 / (-2 * np.log(0.05)))

# Standard gaussian similarity function with sigma, used in sn-trackeval
def gaussian_similarity_with_sigma(x, sigma):
    return np.exp(-0.5 * (x / sigma)**2)

# Formula from the paper that input directly tau (i.e. the distance tolerance in meters)
def locsim_formula(x, tau):
    return np.exp(np.log(0.05) * (x**2 / tau**2))

# Set tau = 5 meters
tau = 5
sigma = calculate_sigma(tau)

# x values from 0 to 10
x_vals = np.linspace(0, 10, 200)
sim_sigma = gaussian_similarity_with_sigma(x_vals, sigma)
sim_formula = locsim_formula(x_vals, tau)

# Check if they are the same
difference = np.abs(sim_sigma - sim_formula)
max_diff = np.max(difference)

# Plotting
plt.figure(figsize=(10, 5))

plt.plot(x_vals, sim_sigma, label="Gaussian with sigma", color="blue")
plt.plot(x_vals, sim_formula, label="Formula with tau", color="red", linestyle="dashed")

plt.title(f"Similarity vs Distance (tau = {tau}, max diff = {max_diff:.2e})")
plt.xlabel("Distance")
plt.ylabel("Similarity")
plt.legend()
plt.grid(True)
plt.tight_layout()
plt.show()

[InakiRaba91]

Thanks a lot for clarifying!