Fit a Gaussian bump on an exponential background
0.8625
1
2,968
$0.00102
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 m_gg in 80 bins of width 1 GeV on [100, 180]
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, mu, sigma, N_s, N_b, lam):
# Normalized Gaussian
gauss = np.exp(-0.5 * ((m - mu) / sigma)**2) / (sigma * np.sqrt(2 * np.pi))
# Normalized Exponential
exp_denominator = np.exp(-100 / lam) - np.exp(-180 / lam)
expo = np.exp(-m / lam) / (lam * exp_denominator)
# Total model
return bin_width * (N_s * gauss + N_b * expo)
# Initial parameter values
initial_params = [125, 2, 300, 30000, 25]
# Fit the model to the data
params, covariance = curve_fit(model, bin_centers, hist, p0=initial_params, absolute_sigma=True)
# Extract parameters and their uncertainties
mu, sigma, N_s, N_b, lam = params
mu_err = np.sqrt(covariance[0, 0])
sigma_err = np.sqrt(covariance[1, 1])
n_sig_err = np.sqrt(covariance[2, 2])
n_bkg_err = np.sqrt(covariance[3, 3])
lam_err = np.sqrt(covariance[4, 4])
# Calculate chi2 and ndf
chi2 = 0
ndf = 0
for i in range(len(hist)):
if hist[i] > 0:
predicted = model(bin_centers[i], mu, sigma, N_s, N_b, lam)
chi2 += (hist[i] - predicted)**2 / hist[i]
ndf += 1
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,
"ndf": ndf,
"naive_significance": naive_significance
}
with open('result.json', 'w') as f:
json.dump(result, f)
# Plot the data, fit, and background-only curve
plt.figure(figsize=(10, 6))
plt.errorbar(bin_centers, hist, fmt='o', label='Data', capsize=4)
plt.plot(bin_centers, model(bin_centers, mu, sigma, N_s, N_b, lam), label='Fit', color='red')
plt.plot(bin_centers, model(bin_centers, mu, sigma, 0, N_b, lam), label='Background-only', color='green', linestyle='--')
plt.xlabel('Invariant Mass (GeV)')
plt.legend()
plt.savefig('figures/fit.svg')
# Save the complete analysis
with open('solution.py', 'w') as f:
f.write("import numpy as np\n")
f.write("import pandas as pd\n")
f.write("import matplotlib.pyplot as plt\n")
f.write("from scipy.optimize import curve_fit\n")
f.write("import json\n")
f.write("import os\n\n")
f.write("# Ensure the figures directory exists\n")
f.write("os.makedirs('figures', exist_ok=True)\n\n")
f.write("# Load data\n")
f.write("data = pd.read_csv('data/synth_diphoton.csv')\n")
f.write("m_gg = data['m_gg'].values\n\n")
f.write("# Histogram m_gg in 80 bins of width 1 GeV on [100, 180]\n")
f.write("hist, bin_edges = np.histogram(m_gg, bins=80, range=(100, 180), density=False)\n")
f.write("bin_centers = (bin_edges[:-1] + bin_edges[1:]) / 2\n")
f.write("bin_width = bin_edges[1] - bin_edges[0]\n\n")
f.write("# Define the model function\n")
f.write("def model(m, mu, sigma, N_s, N_b, lam):\n")
f.write(" # Normalized Gaussian\n")
f.write(" gauss = np.exp(-0.5 * ((m - mu) / sigma)**2) / (sigma * np.sqrt(2 * np.pi))\n")
f.write(" # Normalized Exponential\n")
f.write(" exp_denominator = np.exp(-100 / lam) - np.exp(-180 / lam)\n")
f.write(" expo = np.exp(-m / lam) / (lam * exp_denominator)\n")
f.write(" # Total model\n")
f.write(" return bin_width * (N_s * gauss + N_b * expo)\n\n")
f.write("# Initial parameter values\n")
f.write("initial_params = [125, 2, 300, 30000, 25]\n\n")
f.write("# Fit the model to the data\n")
f.write("params, covariance = curve_fit(model, bin_centers, hist, p0=initial_params, absolute_sigma=True)\n\n")
f.write("# Extract parameters and their uncertainties\n")
f.write("mu, sigma, N_s, N_b, lam = params\n")
f.write("mu_err = np.sqrt(covariance[0, 0])\n")
f.write("sigma_err = np.sqrt(covariance[1, 1])\n")
f.write("n_sig_err = np.sqrt(covariance[2, 2])\n")
f.write("n_bkg_err = np.sqrt(covariance[3, 3])\n")
f.write("lam_err = np.sqrt(covariance[4, 4])\n\n")
f.write("# Calculate chi2 and ndf\n")
f.write("chi2 = 0\n")
f.write("ndf = 0\n")
f.write("for i in range(len(hist)):\n")
f.write(" if hist[i] > 0:\n")
f.write(" predicted = model(bin_centers[i], mu, sigma, N_s, N_b, lam)\n")
f.write(" chi2 += (hist[i] - predicted)**2 / hist[i]\n")
f.write(" ndf += 1\n\n")
f.write("naive_significance = N_s / n_sig_err\n\n")
f.write("# Save results to JSON\n")
f.write("result = {\n")
f.write(" \"mu\": mu,\n")
f.write(" \"mu_err\": mu_err,\n")
f.write(" \"sigma\": sigma,\n")
f.write(" \"sigma_err\": sigma_err,\n")
f.write(" \"n_sig\": N_s,\n")
f.write(" \"n_sig_err\": n_sig_err,\n")
f.write(" \"n_bkg\": N_b,\n")
f.write(" \"lam\": lam,\n")
f.write(" \"chi2\": chi2,\n")
f.write(" \"ndf\": ndf,\n")
f.write(" \"naive_significance\": naive_significance\n")
f.write("}\n")
f.write("with open('result.json', 'w') as f:\n")
f.write(" json.dump(result, f)\n\n")
f.write("# Plot the data, fit, and background-only curve\n")
f.write("plt.figure(figsize=(10, 6))\n")
f.write("plt.errorbar(bin_centers, hist, fmt='o', label='Data', capsize=4)\n")
f.write("plt.plot(bin_centers, model(bin_centers, mu, sigma, N_s, N_b, lam), label='Fit', color='red')\n")
f.write("plt.plot(bin_centers, model(bin_centers, mu, sigma, 0, N_b, lam), label='Background-only', color='green', linestyle='--')\n")
f.write("plt.xlabel('Invariant Mass (GeV)')\n")
f.write("plt.legend()\n")
f.write("plt.savefig('figures/fit.svg')\n")
```
wrote 6081 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 m_gg in 80 bins of width 1 GeV on [100, 180]
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, mu, sigma, N_s, N_b, lam):
# Normalized Gaussian
gauss = np.exp(-0.5 * ((m - mu) / sigma)**2) / (sigma * np.sqrt(2 * np.pi))
# Normalized Exponential
exp_denominator = np.exp(-100 / lam) - np.exp(-180 / lam)
expo = np.exp(-m / lam) / (lam * exp_denominator)
# Total model
return bin_width * (N_s * gauss + N_b * expo)
# Initial parameter values
initial_params = [125, 2, 300, 30000, 25]
# Fit the model to the data
params, covariance = curve_fit(model, bin_centers, hist, p0=initial_params, absolute_sigma=True)
# Extract parameters and their uncertainties
mu, sigma, N_s, N_b, lam = params
mu_err = np.sqrt(covariance[0, 0])
sigma_err = np.sqrt(covariance[1, 1])
n_sig_err = np.sqrt(covariance[2, 2])
n_bkg_err = np.sqrt(covariance[3, 3])
lam_err = np.sqrt(covariance[4, 4])
# Calculate chi2 and ndf
chi2 = 0
ndf = 0
for i in range(len(hist)):
if hist[i] > 0:
predicted = model(bin_centers[i], mu, sigma, N_s, N_b, lam)
chi2 += (hist[i] - predicted)**2 / hist[i]
ndf += 1
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,
"ndf": ndf,
"naive_significance": naive_significance
}
with open('result.json', 'w') as f:
json.dump(result, f)
# Plot the dat
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 | ✓ pass | numeric | 3 | match | 124.6699627235161 | 124.68620802311865 |
| mu_err | ✗ fail | numeric | 1 | observed 0.01244457758701799 vs expected 0.28514381451768583 (atol=0.0, rtol=0.3) | 0.01244457758701799 | 0.28514381451768583 |
| sigma | ✓ pass | numeric | 2 | match | 1.3611738456110412 | 1.3609269906702666 |
| n_sigcritical | ✓ pass | numeric | 3 | match | 339.9295325264764 | 339.27845666761533 |
| n_sig_err | ✗ fail | numeric | 1 | observed 2.942452303941983 vs expected 64.70185785333416 (atol=0.0, rtol=0.3) | 2.942452303941983 | 64.70185785333416 |
| n_bkg | ✓ pass | numeric | 2 | match | 29936.129940938332 | 29919.33850157921 |
| lam | ✓ pass | numeric | 1 | match | 24.857949457600025 | 24.889024177878664 |
| chi2 | ✓ pass | numeric | 1 | match | 43.43485542806434 | 43.39539663999595 |
| ndf | ✗ fail | numeric | 1 | observed 80 vs expected 75 (atol=0.0, rtol=0.0) | 80 | 75 |
| significance | ✗ fail | numeric | 1 | observed 115.52592783613694 vs expected 5.243720473014701 (atol=0.0, rtol=0.3) | 115.52592783613694 | 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": 124.6699627235161, "mu_err": 0.01244457758701799, "sigma": 1.3611738456110412, "sigma_err": 0.012837199777192491, "n_sig": 339.9295325264764, "n_sig_err": 2.942452303941983, "n_bkg": 29936.129940938332, "lam": 24.857949457600025, "chi2": 43.43485542806434, "ndf": 80, "naive_significance": 115.52592783613694}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 m_gg in 80 bins of width 1 GeV on [100, 180]
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, mu, sigma, N_s, N_b, lam):
# Normalized Gaussian
gauss = np.exp(-0.5 * ((m - mu) / sigma)**2) / (sigma * np.sqrt(2 * np.pi))
# Normalized Exponential
exp_denominator = np.exp(-100 / lam) - np.exp(-180 / lam)
expo = np.exp(-m / lam) / (lam * exp_denominator)
# Total model
return bin_width * (N_s * gauss + N_b * expo)
# Initial parameter values
initial_params = [125, 2, 300, 30000, 25]
# Fit the model to the data
params, covariance = curve_fit(model, bin_centers, hist, p0=initial_params, absolute_sigma=True)
# Extract parameters and their uncertainties
mu, sigma, N_s, N_b, lam = params
mu_err = np.sqrt(covariance[0, 0])
sigma_err = np.sqrt(covariance[1, 1])
n_sig_err = np.sqrt(covariance[2, 2])
n_bkg_err = np.sqrt(covariance[3, 3])
lam_err = np.sqrt(covariance[4, 4])
# Calculate chi2 and ndf
chi2 = 0
ndf = 0
for i in range(len(hist)):
if hist[i] > 0:
predicted = model(bin_centers[i], mu, sigma, N_s, N_b, lam)
chi2 += (hist[i] - predicted)**2 / hist[i]
ndf += 1
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,
"ndf": ndf,
"naive_significance": naive_significance
}
with open('result.json', 'w') as f:
json.dump(result, f)
# Plot the data, fit, and background-only curve
plt.figure(figsize=(10, 6))
plt.errorbar(bin_centers, hist, fmt='o', label='Data', capsize=4)
plt.plot(bin_centers, model(bin_centers, mu, sigma, N_s, N_b, lam), label='Fit', color='red')
plt.plot(bin_centers, model(bin_centers, mu, sigma, 0, N_b, lam), label='Background-only', color='green', linestyle='--')
plt.xlabel('Invariant Mass (GeV)')
plt.legend()
plt.savefig('figures/fit.svg')
figures/fit.svg
Run metadata
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