Fit a Gaussian bump on an exponential background
0.4844
1
2,587
$0.00085
32 s
0
Task given to the agent
`data/synth_diphoton.csv` (data card in README.md) contains a diphoton invariant mass `m_gg` per event. Fit the spectrum with the following fully specified procedure: - Histogram `m_gg` in 80 bins of width 1 GeV on [100, 180]. - Model for the expected count in a bin centred at m (bin width w = 1 GeV): f(m) = w * [ N_s * Gauss(m; mu, sigma) + N_b * Expo(m; lam) ] where Gauss is the normalised Gaussian density and Expo(m; lam) = exp(-m/lam) / (lam * (exp(-100/lam) - exp(-180/lam))) is the exponential density normalised on [100, 180]. - Minimise chi2 = sum over bins with n > 0 of (n - f)^2 / n (i.e. least squares with sigma = sqrt(n); bins with zero counts are excluded), starting from mu = 125, sigma = 2, N_s = 300, N_b = 30000, lam = 25. - Parameter uncertainties from the covariance matrix of the least-squares fit (e.g. `scipy.optimize.curve_fit` with `absolute_sigma=True`). Deliverables: - `result.json` with `mu`, `mu_err`, `sigma`, `sigma_err`, `n_sig`, `n_sig_err`, `n_bkg`, `lam` (floats), `chi2` (float), `ndf` (int, number of fitted bins minus number of parameters) and `naive_significance` = n_sig / n_sig_err (float). - `figures/fit.svg`: data points with error bars, the total fit curve and the background-only curve, legend entries containing "data" and "fit", x-axis label including "GeV". - `solution.py`: the complete analysis; `python solution.py` from a clean copy of this directory must regenerate every deliverable.
Trajectory
```python
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
import json
import os
# Ensure the figures directory exists
os.makedirs('figures', exist_ok=True)
# Load data
data = pd.read_csv('data/synth_diphoton.csv')
m_gg = data['m_gg'].values
# Histogram the data
hist, bin_edges = np.histogram(m_gg, bins=80, range=(100, 180), density=False)
bin_centers = (bin_edges[:-1] + bin_edges[1:]) / 2
bin_width = bin_edges[1] - bin_edges[0]
# Define the model function
def model(m, N_s, mu, sigma, N_b, lam):
# Calculate the Gaussian component
gauss = np.exp(-((m - mu) ** 2) / (2 * sigma ** 2)) / (sigma * np.sqrt(2 * np.pi))
# Calculate the exponential component
exp_term = np.exp(-m / lam) / (lam * (np.exp(-100 / lam) - np.exp(-180 / lam)))
# Combine the components and multiply by bin width
return bin_width * (N_s * gauss + N_b * exp_term)
# Define the chi2 function
def chi2(params, m, n):
N_s, mu, sigma, N_b, lam = params
f = model(m, N_s, mu, sigma, N_b, lam)
return np.sum((n - f) ** 2 / n)
# Initial parameter values
initial_params = [300, 125, 2, 30000, 25]
# Perform the fit
params, covariance = curve_fit(
lambda x, *p: chi2(p, x, hist),
bin_centers,
hist,
p0=initial_params,
absolute_sigma=True
)
# Extract parameters and their uncertainties
N_s, mu, sigma, N_b, lam = params
mu_err = np.sqrt(covariance[0, 0])
sigma_err = np.sqrt(covariance[2, 2])
n_sig_err = np.sqrt(covariance[0, 0]) * N_s
n_bkg_err = np.sqrt(covariance[3, 3]) * N_b
chi2_value = chi2(params, bin_centers, hist)
ndf = len(hist) - len(params)
# Calculate naive significance
naive_significance = N_s / n_sig_err
# Save results to JSON
result = {
"mu": mu,
"mu_err": mu_err,
"sigma": sigma,
"sigma_err": sigma_err,
"n_sig": N_s,
"n_sig_err": n_sig_err,
"n_bkg": N_b,
"lam": lam,
"chi2": chi2_value,
"ndf": ndf,
"naive_significance": naive_significance
}
with open('result.json', 'w') as f:
json.dump(result, f)
# Plot the data and fit
plt.figure(figsize=(10, 6))
plt.errorbar(bin_centers, hist, fmt='o', label='Data', capsize=4)
plt.plot(bin_centers, model(bin_centers, *params), label='Fit', color='red')
plt.plot(bin_centers, model(bin_centers, *initial_params), label='Background-only', color='green', linestyle='--')
plt.xlabel('Invariant Mass (GeV)')
plt.ylabel('Counts')
plt.legend()
plt.title('Diphoton Invariant Mass Spectrum')
plt.savefig('figures/fit.svg')
# Save the complete analysis
with open('solution.py', 'w') as f:
f.write('''import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
import json
import os
# Ensure the figures directory exists
os.makedirs('figures', exist_ok=True)
# Load data
data = pd.read_csv('data/synth_diphoton.csv')
m_gg = data['m_gg'].values
# Histogram the data
hist, bin_edges = np.histogram(m_gg, bins=80, range=(100, 180), density=False)
bin_centers = (bin_edges[:-1] + bin_edges[1:]) / 2
bin_width = bin_edges[1] - bin_edges[0]
# Define the model function
def model(m, N_s, mu, sigma, N_b, lam):
# Calculate the Gaussian component
gauss = np.exp(-((m - mu) ** 2) / (2 * sigma ** 2)) / (sigma * np.sqrt(2 * np.pi))
# Calculate the exponential component
exp_term = np.exp(-m / lam) / (lam * (np.exp(-100 / lam) - np.exp(-180 / lam)))
# Combine the components and multiply by bin width
return bin_width * (N_s * gauss + N_b * exp_term)
# Define the chi2 function
def chi2(params, m, n):
N_s, mu, sigma, N_b, lam = params
f = model(m, N_s, mu, sigma, N_b, lam)
return np.sum((n - f) ** 2 / n)
# Initial parameter values
initial_params = [300, 125, 2, 30000, 25]
# Perform the fit
params, covariance = curve_fit(
lambda x, *p: chi2(p, x, hist),
bin_centers,
hist,
p0=initial_params,
absolute_sigma=True
)
# Extract parameters and their uncertainties
N_s, mu, sigma, N_b, lam = params
mu_err = np.sqrt(covariance[0, 0])
sigma_err = np.sqrt(covariance[2, 2])
n_sig_err = np.sqrt(covariance[0, 0]) * N_s
n_bkg_err = np.sqrt(covariance[3, 3]) * N_b
chi2_value = chi2(params, bin_centers, hist)
ndf = len(hist) - len(params)
# Calculate naive significance
naive_significance = N_s / n_sig_err
# Save results to JSON
result = {
"mu": mu,
"mu_err": mu_err,
"sigma": sigma,
"sigma_err": sigma_err,
"n_sig": N_s,
"n_sig_err": n_sig_err,
"n_bkg": N_b,
"lam": lam,
"chi2": chi2_value,
"ndf": ndf,
"naive_significance": naive_significance
}
with open('result.json', 'w') as f:
json.dump(result, f)
# Plot the data and fit
plt.figure(figsize=(10, 6))
plt.errorbar(bin_centers, hist, fmt='o', label='Data', capsize=4)
plt.plot(bin_centers, model(bin_centers, *params), label='Fit', color='red')
plt.plot(bin_centers, model(bin_centers, *initial_params), label='Background-only', color='green', linestyle='--')
plt.xlabel('Invariant Mass (GeV)')
plt.ylabel('Counts')
plt.legend()
plt.title('Diphoton Invariant Mass Spectrum')
plt.savefig('figures/fit.svg')
''')
```
wrote 5104 chars to solution.py
exit code: 0 files created: figures/fit.svg, result.json files modified: solution.py
Final answer
```python
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
import json
import os
# Ensure the figures directory exists
os.makedirs('figures', exist_ok=True)
# Load data
data = pd.read_csv('data/synth_diphoton.csv')
m_gg = data['m_gg'].values
# Histogram the data
hist, bin_edges = np.histogram(m_gg, bins=80, range=(100, 180), density=False)
bin_centers = (bin_edges[:-1] + bin_edges[1:]) / 2
bin_width = bin_edges[1] - bin_edges[0]
# Define the model function
def model(m, N_s, mu, sigma, N_b, lam):
# Calculate the Gaussian component
gauss = np.exp(-((m - mu) ** 2) / (2 * sigma ** 2)) / (sigma * np.sqrt(2 * np.pi))
# Calculate the exponential component
exp_term = np.exp(-m / lam) / (lam * (np.exp(-100 / lam) - np.exp(-180 / lam)))
# Combine the components and multiply by bin width
return bin_width * (N_s * gauss + N_b * exp_term)
# Define the chi2 function
def chi2(params, m, n):
N_s, mu, sigma, N_b, lam = params
f = model(m, N_s, mu, sigma, N_b, lam)
return np.sum((n - f) ** 2 / n)
# Initial parameter values
initial_params = [300, 125, 2, 30000, 25]
# Perform the fit
params, covariance = curve_fit(
lambda x, *p: chi2(p, x, hist),
bin_centers,
hist,
p0=initial_params,
absolute_sigma=True
)
# Extract parameters and their uncertainties
N_s, mu, sigma, N_b, lam = params
mu_err = np.sqrt(covariance[0, 0])
sigma_err = np.sqrt(covariance[2, 2])
n_sig_err = np.sqrt(covariance[0, 0]) * N_s
n_bkg_err = np.sqrt(covariance[3, 3]) * N_b
chi2_value = chi2(params, bin_centers, hist)
ndf = len(hist) - len(params)
# Calculate naive significance
naive_significance = N_s / n_sig_err
# Save results to JSON
result = {
"mu": mu,
"mu_err": mu_err,
"sigma": sigma,
"sigma_err": sigma_err,
"n_sig": N_s,
"n_sig_err": n_sig_err,
"n_bkg": N_b,
"lam": lam,
"chi2": chi2_value,
"ndf": ndf,
"naive_significance": naive_significance
}
Grader verdict
| Check | Result | Category | Weight | Detail | Observed | Expected |
|---|---|---|---|---|---|---|
| result_existscritical | ✓ pass | artifact | 1 | ok | ||
| figure_exists | ✓ pass | artifact | 1 | ok | ||
| solution_exists | ✓ pass | artifact | 1 | ok | ||
| mucritical | ✗ fail | numeric | 3 | observed 130.06747515704183 vs expected 124.68620802311865 (atol=0.1, rtol=0.0) | 130.06747515704183 | 124.68620802311865 |
| mu_err | ✗ fail | numeric | 1 | observed 1.749297464797609e+29 vs expected 0.28514381451768583 (atol=0.0, rtol=0.3) | 1.749297464797609e+29 | 0.28514381451768583 |
| sigma | ✗ fail | numeric | 2 | observed 1.9893451839994816 vs expected 1.3609269906702666 (atol=0.2, rtol=0.0) | 1.9893451839994816 | 1.3609269906702666 |
| n_sigcritical | ✗ fail | numeric | 3 | observed 530.6472469329591 vs expected 339.27845666761533 (atol=0.0, rtol=0.15) | 530.6472469329591 | 339.27845666761533 |
| n_sig_err | ✗ fail | numeric | 1 | observed 9.282598837616562e+31 vs expected 64.70185785333416 (atol=0.0, rtol=0.3) | 9.282598837616562e+31 | 64.70185785333416 |
| n_bkg | ✗ fail | numeric | 2 | observed 31356.347794731253 vs expected 29919.33850157921 (atol=0.0, rtol=0.02) | 31356.347794731253 | 29919.33850157921 |
| lam | ✗ fail | numeric | 1 | observed 26.428477465349747 vs expected 24.889024177878664 (atol=0.0, rtol=0.05) | 26.428477465349747 | 24.889024177878664 |
| chi2 | ✗ fail | numeric | 1 | observed 378.7576696544784 vs expected 43.39539663999595 (atol=0.0, rtol=0.1) | 378.7576696544784 | 43.39539663999595 |
| ndf | ✓ pass | numeric | 1 | match | 75 | 75 |
| significance | ✗ fail | numeric | 1 | observed 5.716580628073444e-30 vs expected 5.243720473014701 (atol=0.0, rtol=0.3) | 5.716580628073444e-30 | 5.243720473014701 |
| svg_legend | ✓ pass | plot | 1 | labels present | ["Matplotlib v3.9.2, https://matplotlib.org/","100","110","120","130","140","150 | ["data","fit","GeV"] |
| svg_drawn | ✓ pass | plot | 1 | drawn | 112 | ">=10" |
| reruns | ✓ pass | reproducibility | 1 | reproduced exactly | {"ok":true,"artifacts_reproduced":true,"output_matches":true,"exit_code":0} | |
| not_hardcoded | ✓ pass | compliance | 1 | no expected values hard-coded | [] |
Reproducibility rerun
✓ pass
✓ pass
✓ exact
✓ pass
no randomness used
reproduced exactly
Artifacts
result.json
{"mu": 130.06747515704183, "mu_err": 1.749297464797609e+29, "sigma": 1.9893451839994816, "sigma_err": 1.4130345480323933e+27, "n_sig": 530.6472469329591, "n_sig_err": 9.282598837616562e+31, "n_bkg": 31356.347794731253, "lam": 26.428477465349747, "chi2": 378.7576696544784, "ndf": 75, "naive_significance": 5.716580628073444e-30}solution.py
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
import json
import os
# Ensure the figures directory exists
os.makedirs('figures', exist_ok=True)
# Load data
data = pd.read_csv('data/synth_diphoton.csv')
m_gg = data['m_gg'].values
# Histogram the data
hist, bin_edges = np.histogram(m_gg, bins=80, range=(100, 180), density=False)
bin_centers = (bin_edges[:-1] + bin_edges[1:]) / 2
bin_width = bin_edges[1] - bin_edges[0]
# Define the model function
def model(m, N_s, mu, sigma, N_b, lam):
# Calculate the Gaussian component
gauss = np.exp(-((m - mu) ** 2) / (2 * sigma ** 2)) / (sigma * np.sqrt(2 * np.pi))
# Calculate the exponential component
exp_term = np.exp(-m / lam) / (lam * (np.exp(-100 / lam) - np.exp(-180 / lam)))
# Combine the components and multiply by bin width
return bin_width * (N_s * gauss + N_b * exp_term)
# Define the chi2 function
def chi2(params, m, n):
N_s, mu, sigma, N_b, lam = params
f = model(m, N_s, mu, sigma, N_b, lam)
return np.sum((n - f) ** 2 / n)
# Initial parameter values
initial_params = [300, 125, 2, 30000, 25]
# Perform the fit
params, covariance = curve_fit(
lambda x, *p: chi2(p, x, hist),
bin_centers,
hist,
p0=initial_params,
absolute_sigma=True
)
# Extract parameters and their uncertainties
N_s, mu, sigma, N_b, lam = params
mu_err = np.sqrt(covariance[0, 0])
sigma_err = np.sqrt(covariance[2, 2])
n_sig_err = np.sqrt(covariance[0, 0]) * N_s
n_bkg_err = np.sqrt(covariance[3, 3]) * N_b
chi2_value = chi2(params, bin_centers, hist)
ndf = len(hist) - len(params)
# Calculate naive significance
naive_significance = N_s / n_sig_err
# Save results to JSON
result = {
"mu": mu,
"mu_err": mu_err,
"sigma": sigma,
"sigma_err": sigma_err,
"n_sig": N_s,
"n_sig_err": n_sig_err,
"n_bkg": N_b,
"lam": lam,
"chi2": chi2_value,
"ndf": ndf,
"naive_significance": naive_significance
}
with open('result.json', 'w') as f:
json.dump(result, f)
# Plot the data and fit
plt.figure(figsize=(10, 6))
plt.errorbar(bin_centers, hist, fmt='o', label='Data', capsize=4)
plt.plot(bin_centers, model(bin_centers, *params), label='Fit', color='red')
plt.plot(bin_centers, model(bin_centers, *initial_params), label='Background-only', color='green', linestyle='--')
plt.xlabel('Invariant Mass (GeV)')
plt.ylabel('Counts')
plt.legend()
plt.title('Diphoton Invariant Mass Spectrum')
plt.savefig('figures/fit.svg')
figures/fit.svg
Run metadata
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"temperature": 0,
"max_tokens": 2500,
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"config": {
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"preset": "gariyuu-qwen3-8b",
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"agent": {
"name": "single_shot",
"max_steps": 25,
"max_debug_rounds": 3
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"environment": {
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"started_at": "2026-09-06T17:19:38.066206+00:00",
"finished_at": "2026-09-06T17:20:11.548995+00:00"
}