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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 Comparing Results from Kernel Density Estimates of Parameters
Step1:  Load data for three different approaches to estimating stellar parameters from MCMC simulations.
Step2:  Compare KDE and MLE methods. MLE is a kernel density estimate with a constant bandwidth and should provide more detail about the underlying posterior distribution, whereas the KDE is a kernel density estimate with a "rule-of-thumb" determination of the bandwidth (Silverman's rule). They provide a modal estimate for the distributions.
Step3:  Analysis.
Now, comparing KDE to estimate of distribution median.
Step4:  Analysis.
How does this choice affect the inferred relationship between Teff, [M/H], log(g) and the mixing length parameter?
Step5:  To permit a better comparison with 3D RHD models, we should isolate stars that have a roughly solar metallicity. Since 3D RHD simluations are only performed at solar metallicity, it may bias the comparison given that we a significantly larger spread in metallicity.
Step6:  Now try only a limited range of effective temperatures to mitigate effects related to the temperature sensitivity of various opacity sources.
Step7:  This subset is also more comparable to results from Bonaca et al. (2012), who used asteroseismic data to constrain how the convective mixing length parameter changes with stellar properties.
Step8:  Calculate correlation coefficients from Spearman r test.
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Python Code:
%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
Explanation: Comparing Results from Kernel Density Estimates of Parameters
End of explanation
kde = np.genfromtxt('data/run08_kde_props.txt')
mle = np.genfromtxt('data/run08_mle_props.txt')
med = np.genfromtxt('data/run08_median_props.txt')
Explanation: Load data for three different approaches to estimating stellar parameters from MCMC simulations.
End of explanation
fig, ax = plt.subplots(3, 3, figsize=(12., 12.))
for i in range(9):
    row  = i/3
    col  = i%3
    axis = ax[row, col]
    
    # set axis labels and ranges
    axis.set_xlabel('KDE', fontsize=14.)
    axis.set_ylabel('MLE', fontsize=14.)
    axis.grid(True)
    axis.tick_params(which='major', axis='both', labelsize=14., length=12.)
    
    # 1-to-1 correlation
    axis.plot([min(kde[:, i]), max(kde[:, i])], [min(kde[:, i]), max(kde[:, i])], lw=2, dashes=(20., 5.), c='#b22222')
    axis.plot(kde[:, i], mle[:, i], 'o', markersize=7., c='#555555', alpha=0.7)
fig.tight_layout()
Explanation: Compare KDE and MLE methods. MLE is a kernel density estimate with a constant bandwidth and should provide more detail about the underlying posterior distribution, whereas the KDE is a kernel density estimate with a "rule-of-thumb" determination of the bandwidth (Silverman's rule). They provide a modal estimate for the distributions.
End of explanation
fig, ax = plt.subplots(3, 3, figsize=(12., 12.))
for i in range(9):
    row  = i/3
    col  = i%3
    axis = ax[row, col]
    
    # set axis labels and ranges
    axis.set_xlabel('KDE', fontsize=14.)
    axis.set_ylabel('Median', fontsize=14.)
    axis.grid(True)
    axis.tick_params(which='major', axis='both', labelsize=14., length=12.)
    
    # 1-to-1 correlation
    axis.plot([min(kde[:, i]), max(kde[:, i])], [min(kde[:, i]), max(kde[:, i])], lw=2, dashes=(20., 5.), c='#b22222')
    axis.plot(kde[:, i], med[:, i], 'o', markersize=7., c='#555555', alpha=0.7)
fig.tight_layout()
Explanation: Analysis.
Now, comparing KDE to estimate of distribution median.
End of explanation
fig, ax = plt.subplots(2, 3, figsize=(12., 8.))
# KDE
ax[0, 0].plot(10**kde[:, 6], kde[:, 5], 'o', markersize=7., c='#555555', alpha=0.7)
ax[0, 1].plot(kde[:, 1], kde[:, 5], 'o', markersize=7., c='#555555', alpha=0.7)
ax[0, 2].plot(kde[:, 0], kde[:, 5], 'o', markersize=7., c='#555555', alpha=0.7)
# Median
ax[1, 0].plot(10**med[:, 6], med[:, 5], 'o', markersize=7., c='#555555', alpha=0.7)
ax[1, 1].plot(med[:, 1], med[:, 5], 'o', markersize=7., c='#555555', alpha=0.7)
ax[1, 2].plot(med[:, 0], med[:, 5], 'o', markersize=7., c='#555555', alpha=0.7)
fig.tight_layout()
Explanation: Analysis.
How does this choice affect the inferred relationship between Teff, [M/H], log(g) and the mixing length parameter?
End of explanation
solar_kde = np.array([star for star in kde if -0.1 < star[1] < 0.1])
fig, ax = plt.subplots(1, 3, figsize=(12., 4.))
# KDE
ax[0].plot(10**solar_kde[:, 6], solar_kde[:, 5], 'o', markersize=7., c='#555555', alpha=0.7)
ax[1].plot(solar_kde[:, 1], solar_kde[:, 5], 'o', markersize=7., c='#555555', alpha=0.7)
ax[2].plot(solar_kde[:, 0], solar_kde[:, 5], 'o', markersize=7., c='#555555', alpha=0.7)
fig.tight_layout()
Explanation: To permit a better comparison with 3D RHD models, we should isolate stars that have a roughly solar metallicity. Since 3D RHD simluations are only performed at solar metallicity, it may bias the comparison given that we a significantly larger spread in metallicity.
End of explanation
warm_kde = np.array([star for star in kde if 5000. <= 10**star[6] <= 6000. and star[1] > -0.35])
fig, ax = plt.subplots(1, 3, figsize=(12., 4.))
# KDE
ax[0].plot(10**warm_kde[:, 6], warm_kde[:, 5], 'o', markersize=7., c='#555555', alpha=0.7)
ax[1].plot(warm_kde[:, 1], warm_kde[:, 5], 'o', markersize=7., c='#555555', alpha=0.7)
ax[2].plot(warm_kde[:, 0], warm_kde[:, 5], 'o', markersize=7., c='#555555', alpha=0.7)
fig.tight_layout()
Explanation: Now try only a limited range of effective temperatures to mitigate effects related to the temperature sensitivity of various opacity sources.
End of explanation
import scipy.stats as stat
Explanation: This subset is also more comparable to results from Bonaca et al. (2012), who used asteroseismic data to constrain how the convective mixing length parameter changes with stellar properties.
End of explanation
stat.spearmanr(10**warm_kde[:, 6], warm_kde[:, 5]), stat.spearmanr(warm_kde[:, 1], warm_kde[:, 5]), \
    stat.spearmanr(warm_kde[:, 0], warm_kde[:, 5])
Explanation: Calculate correlation coefficients from Spearman r test.
End of explanation
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