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"""
Outlier detection via leave-one-out
===================================
Outliers can sometimes be identified by assessing the influence of each
datapoint. To assess the influence of one point, we fit the dataset while the
point and compare the result with the fit of the full dataset. The code below
shows how to do this with lmfit. Note that the presented method is very basic.
"""
from collections import defaultdict
import matplotlib.pyplot as plt
import numpy as np
import lmfit
plt.rcParams['figure.dpi'] = 130
plt.rcParams['figure.autolayout'] = True
###############################################################################
# Generate test data and model. Apply the model to the data
x = np.linspace(0.3, 10, 100)
np.random.seed(1)
y = 1.0 / (0.1 * x) + 2.0 + 3 * np.random.randn(x.size)
params = lmfit.Parameters()
params.add_many(('a', 0.1), ('b', 1))
def func(x, a, b):
return 1.0 / (a * x) + b
# Make 5 points outliers
idx = np.random.randint(0, x.size, 5)
y[idx] += 10 * np.random.randn(idx.size)
# Fit the data
model = lmfit.Model(func, independent_vars=['x'])
fit_result = model.fit(y, x=x, a=0.1, b=2)
###############################################################################
# and gives the plot and fitting results below:
fit_result.plot_fit()
plt.plot(x[idx], y[idx], 'o', color='r', label='outliers')
plt.show()
print(fit_result.fit_report())
###############################################################################
# Fit the dataset while omitting one data point
best_vals = defaultdict(lambda: np.zeros(x.size))
stderrs = defaultdict(lambda: np.zeros(x.size))
chi_sq = np.zeros_like(x)
for i in range(x.size):
idx2 = np.arange(0, x.size)
idx2 = np.delete(idx2, i)
tmp_x = x[idx2]
tmp = model.fit(y[idx2],
x=tmp_x,
a=fit_result.params['a'],
b=fit_result.params['b'])
chi_sq[i] = tmp.chisqr
for p in tmp.params:
tpar = tmp.params[p]
best_vals[p][i] = tpar.value
stderrs[p][i] = (tpar.stderr / fit_result.params[p].stderr)
###############################################################################
# Plot the influence on the red. chisqr of each point
fig, ax = plt.subplots()
ax.plot(x, (fit_result.chisqr - chi_sq) / chi_sq)
ax.scatter(x[idx],
fit_result.chisqr / chi_sq[idx] - 1,
color='r',
label='outlier')
ax.set_ylabel(r'Relative red. $\chi^2$ change')
ax.set_xlabel('x')
ax.legend()
###############################################################################
# Plot the influence on the parameter value and error of each point
fig, axs = plt.subplots(4, figsize=(4, 7), sharex='col')
axs[0].plot(x, best_vals['a'])
axs[0].scatter(x[idx], best_vals['a'][idx], color='r', label='outlier')
axs[0].set_ylabel('best a')
axs[1].plot(x, best_vals['b'])
axs[1].scatter(x[idx], best_vals['b'][idx], color='r', label='outlier')
axs[1].set_ylabel('best b')
axs[2].plot(x, stderrs['a'])
axs[2].scatter(x[idx], stderrs['a'][idx], color='r', label='outlier')
axs[2].set_ylabel('err a change')
axs[3].plot(x, stderrs['b'])
axs[3].scatter(x[idx], stderrs['b'][idx], color='r', label='outlier')
axs[3].set_ylabel('err b change')
axs[3].set_xlabel('x')
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