Newer
Older
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""
Created on Tue Dec 10 2019
@author: Martin Renoult
correspondence: martin.renoult@misu.su.se
"""
## Library
from pymc3 import *
import numpy as np
import matplotlib.pyplot as plt
import scipy.stats as stat
from scipy.optimize import curve_fit
from adjustText import adjust_text
import math
#------------------------------------------------------------------------------
## Lists to save the data while computing
list_predict_t = list()
list_predict_t_stats_66 = list()
list_predict_t_stats_90 = list()
lb_90 = list()
ub_90 = list()
lb_66 = list()
ub_66 = list()
#------------------------------------------------------------------------------
## Model data
# x = ECS
# y = Tropical temperature
# Pliocene from PlioMIP1
# In the following order:
# [CCSM4, IPSLCM5A, MIROC4m, GISS-E2-R, COSMOS, MRI-CGCM2.3, HadCM3,NorESM-L,
# FGOALS-g2, GISS-E2-1-G, IPSL-CM6A-LR, NorESM1-F,CESM2,EC-EARTH3.3]
# FGOALS has not been recomputed: Taken from Hargreaves et Annan, 2016
# ECS of FGOALS and HadCM3 are taken from LGM ECS for consistency
#x = [3.2, 3.4, 4.05, 2.8, 4.1, 3.2, 3.3, 3.1,
#
#y = [1.0256042, 1.3337059, 1.9900227, 1.1576538, 2.1806774, 1.151104, 1.9331722, 1.445343,
# 2.14, 0.9211941,2.1174774, 1.3736095,3.4950447,2.94250999999997]
# Latest model versions approach
x = [3.2, 4.05, 4.1, 3.2, 3.3,
y = [1.0256042, 1.9900227, 2.1806774, 1.151104, 1.9331722,
2.14, 0.9211941,2.1174774, 1.3736095,3.4950447,2.94250999999997]
# PlioMIP1 only
#x = [3.2, 3.4, 4.05, 2.8, 4.1, 3.2, 3.1, 3.1,3.7]
# Modified ECS, new HadCM3 and FGOALS-g2 to be consistent with LGM
#x = [3.2, 3.4, 4.05, 2.8, 4.1, 3.2, 3.3, 3.1,3.37]
#y = [1.0256042, 1.3337059, 1.9900227, 1.1576538, 2.1806774, 1.151104, 1.9331722, 1.445343,2.14]
# Full PlioMIP1 plot
#pliomip1x = [3.2, 3.4, 4.05, 2.8, 4.1, 3.2, 3.3, 3.1,3.37]
#pliomip1y = [1.0256042, 1.3337059, 1.9900227, 1.1576538, 2.1806774, 1.151104, 1.9331722, 1.445343,2.14]
# Latest version plot
pliomip1x = [3.2, 4.05, 4.1, 3.2, 3.3, 3.37]
pliomip1y = [1.0256042, 1.9900227, 2.1806774, 1.151104, 1.9331722 ,2.14]
pliomip2x = [2.6, 4.50, 2.29]
pliomip2y = [0.9211941,2.1174774, 1.3736095]
#------------------------------------------------------------------------------
### Uncomment to see the data distribution
#
#fig, ax = plt.subplots(figsize=(7, 7))
#
# #plt.plot(pliomip1x, pliomip1y, '.', label='PlioMIP1', ms=17, color='#009999', mec='#006666')
# #plt.plot(pliomip2x, pliomip2y, '.', label='PlioMIP2', ms=17, color='#cc0066',mec='#800040')
#
#plt.xlim(-1, 6)
#plt.ylim(-0.5, 3)
#plt.legend(loc='upper left', bbox_to_anchor=(0.2, 0.35), fancybox=True)
#plt.xlabel('Climate Sensitivity (K)', labelpad=-40, weight='bold')
#plt.ylabel('Mid-Pliocene Tropical Ocean SST anomaly (K)', position=(0,0.8), weight='bold')
#ax.spines['top'].set_visible(False)
#ax.spines['right'].set_visible(False)
#ax.spines['bottom'].set_position(('data', 0))
#ax.spines['left'].set_position(('data', 0))
#ax.spines['left'].set_linewidth(2)
#ax.spines['bottom'].set_linewidth(2)
#ax.tick_params('x', direction='out', pad=20, width=2)
#ax.tick_params('y', width=2)
#plt.yticks(ticks=[0.5, 1, 1.5, 2, 2.5, 3,3.5], labels=['0.5', '1','1.5', '2','2.5', '3','3.5'],
# weight='bold')
#plt.xticks(ticks=[1, 2, 3, 4, 5, 6], labels=['1', '2','3', '4','5', '6'],
# weight='bold')
#------------------------------------------------------------------------------
## MCMC model
with Model() as model: # model specifications in PyMC3 are wrapped in a with-statement
# Define prior means, sd or precision (see conjugate for precision)
mn_intercept = 0.0
mn_slope = 1.0
sd_reg = 1.0
precis_intercept = np.sqrt(0.5)
precis_slope = np.sqrt(0.5)
# Define priors: Non-conjugate approach
intercept = Normal('Intercept', mn_intercept, sd=sd_reg)
x_coeff = Normal('x', mn_slope, sd=sd_reg)
# Priors for a Normal-Inverse Gamma conjugate approach.
# In conjugate approach, priors on intercept and slope depends on a scaled sigma
# and need to be defined that way to match mathematical equations.
# This part should be mainly used for comparison / check with the conjugate approach code
# sigma = InverseGamma('Sigma',alpha=0.5,beta=0.5)
# intercept = Normal('Intercept', mn_intercept, sd=sigma/precis_intercept)
# x_coeff = Normal('x', mn_slope, sd=sigma/precis_slope)
likelihood = Normal('y', mu=intercept + x_coeff * x,
sd=sigma, observed=y)
# Inference! 4 jobs in parallel (convergence check)
trace = sample(progressbar=True, draws=100000, cores=4, tune=5000)
values_intercept = trace['Intercept']
values_sigma = trace['Sigma']
# Gelman-Rubin test for convergence of the model
# If BFMI = Gelman-Rubin, then you have convergence
# It compares the variance between the chains to the variance inside a chain
# and both variances should be equal if all the chains (the model) converged
#bfmi = bfmi(trace)
#max_gr = max(np.max(gr_stats) for gr_stats in gelman_rubin(trace).values())
#
#(energyplot(trace, legend=True, figsize=(6, 4))
# .set_title("BFMI = {}\nGelman-Rubin = {}".format(bfmi, max_gr)));
#------------------------------------------------------------------------------
## Predicted temperature calculation
## Create predicted ensemble for 5-95% estimate
# Discrete sample of sensitivity
ran = np.linspace(0, 10, 500)
# Loop for predicted temperature based on trace and line above
for j in ran:
predicted_t = values_x * j + values_intercept + np.random.normal(loc=0, scale=values_sigma)
# Calculate and save the 5-95% interval of the prediction
stats_predict_t_90 = np.percentile(predicted_t, q=(5,95))
# Calculate and save the 17-83% interval of the prediction
stats_predict_t_66 = np.percentile(predicted_t, q=(17,83))
# Save in a list the intervals for every sample of sensitivity "ran"
list_predict_t_stats_66.append(stats_predict_t_66)
list_predict_t_stats_90.append(stats_predict_t_90)
#------------------------------------------------------------------------------
## Bayesian framework
# Priors on sensitivity
#prior_S = np.random.uniform(0, 20, size=400000)
#prior_S = stat.gamma.rvs(a=1, loc=0, scale=5, size=400000)
def truncated_cauchy_rvs(loc=0, scale=1, a=-1, b=1, size=None):
"""
Generate random samples from a truncated Cauchy distribution.
`loc` and `scale` are the location and scale parameters of the distribution.
`a` and `b` define the interval [a, b] to which the distribution is to be
limited.
With the default values of the parameters, the samples are generated
from the standard Cauchy distribution limited to the interval [-1, 1].
"""
ua = np.arctan((a - loc)/scale)/np.pi + 0.5
ub = np.arctan((b - loc)/scale)/np.pi + 0.5
U = np.random.uniform(ua, ub, size=size)
rvs = loc + scale * np.tan(np.pi*(U - 0.5))
return rvs
# Truncated-at-zero Cauchy distribution
prior_S = truncated_cauchy_rvs(loc=2.5, scale=3, a=1/math.inf, b=math.inf, size=400000)
prior_stats_90 = np.percentile(prior_S, q=(5, 95))
prior_stats_66 = np.percentile(prior_S, q=(17, 83))
# Model to generate a single point based on the prior on S
def gen_mod(alpha, s, beta, error):
return alpha * s + beta + np.random.normal(loc=0, scale=error)
def likelihood(sim, obs, std):
return stat.norm.pdf(x=sim, loc=obs, scale=std)
# Generate temperatures
model_T = gen_mod(values_x, prior_S, values_intercept, values_sigma)
# "Real" observed data
# Tropical Pliocene T
T = 0.8
stdT = 1
gauss_obs = np.random.normal(loc=T, scale=stdT, size=800000)
obs_stats_90 = np.percentile(gauss_obs, q=(5, 95))
# Create weights through importance sampling
weight = likelihood(model_T, T, stdT)
weight = weight/weight.sum()
# Bayesian updating of the prior with importance sampling
posterior = np.random.choice(prior_S, size=100000, p=weight)
post_median = np.median(posterior)
# Compute 5-95% and 17-83% posterior intervals
post_stats_90 = np.percentile(posterior, q=(5, 95))
post_stats_66 = np.percentile(posterior, q=(17, 83))
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
#------------------------------------------------------------------------------
## Plot part
## 1st plot: Trace plot
plt.figure(figsize=(7, 7))
traceplot(trace)
plt.tight_layout()
#plt.savefig('Trace_PlioMIP.pdf')
#-------------------------------
## 2nd plot: BLR
# Plot the data
fig, ax = plt.subplots(figsize=(7, 7))
# Range of the plotted MCMC lines and plot of the lines
range_eval = np.linspace(-1, 10, 100)
plots.plot_posterior_predictive_glm(trace, samples=100, eval=range_eval,
label='Predictive regression lines', alpha=0.35)
# Plot the 5-95% interval
for h in list_predict_t_stats_90:
lb_90.append(h[0])
ub_90.append(h[1])
for g in list_predict_t_stats_66:
lb_66.append(g[0])
ub_66.append(g[1])
# Compute running mean to smooth the confidence interval
rand_rm = np.convolve(ran, np.ones((50,))/50, mode='valid')
low_rm_90 = np.convolve(lb_90, np.ones((50,))/50, mode='valid')
up_rm_90 = np.convolve(ub_90, np.ones((50,))/50, mode='valid')
low_rm_66 = np.convolve(lb_66, np.ones((50,))/50, mode='valid')
up_rm_66 = np.convolve(ub_66, np.ones((50,))/50, mode='valid')
plt.plot(rand_rm, low_rm_90, linestyle='-', color='red', label='5-95% interval', alpha=0.75, linewidth=2)
plt.plot(rand_rm, up_rm_90, linestyle='-', color='red', alpha=0.75, linewidth=2)
plt.plot(rand_rm, low_rm_66, linestyle='--', color='red', label='17-83% interval', alpha=0.75, linewidth=2)
plt.plot(rand_rm, up_rm_66, linestyle='--', color='red', alpha=0.75, linewidth=2)
ylim = plt.ylim(-0.5, 3)
xlim = plt.xlim(-1,8)
# Plot 2 std on the figure instead of one (esthetic change)
stdT = 1.6
# Line for observed value, 2 standard deviations
plt.axvline(x=0.2, ymin=(-0.5/(ylim[0]-ylim[1]))-((T-stdT)/(ylim[0]-ylim[1])), ymax=(-0.5/(ylim[0]-ylim[1]))-((T+stdT)/(ylim[0]-ylim[1])),
color='#009900', label='5-95% observed value', linewidth=2)
plt.axvline(x=0.2, ymin=(-0.5/(ylim[0]-ylim[1]))-((T-stdT)/(ylim[0]-ylim[1])), ymax=(-0.5/(ylim[0]-ylim[1]))-((T-stdT)/(ylim[0]-ylim[1])),
color='#009900', marker='v')
plt.axvline(x=0.2, ymin=(-0.5/(ylim[0]-ylim[1]))-((T+stdT)/(ylim[0]-ylim[1])), ymax=(-0.5/(ylim[0]-ylim[1]))-((T+stdT)/(ylim[0]-ylim[1])),
color='#009900', marker='^')
plt.axvline(x=0.2, ymin=(-0.5/(ylim[0]-ylim[1]))-((T)/(ylim[0]-ylim[1])), ymax=(-0.5/(ylim[0]-ylim[1]))-((T)/(ylim[0]-ylim[1])),
color='#009900', marker='.', ms=12)
plt.axhline(y=0.08, xmin=(-1/(xlim[0]-xlim[1]))-(post_stats_90[0]/(xlim[0]-xlim[1])),
xmax=(-1/(xlim[0]-xlim[1]))-(post_stats_90[1]/(xlim[0]-xlim[1])), c='#9933ff', label='5-95% posterior', linewidth=2)
plt.axhline(y=0.08, xmin=(-1/(xlim[0]-xlim[1]))-(post_stats_90[0]/(xlim[0]-xlim[1])),
xmax=(-1/(xlim[0]-xlim[1]))-(post_stats_90[0]/(xlim[0]-xlim[1])), marker='<', c='#9933ff')
plt.axhline(y=0.08, xmin=(-1/(xlim[0]-xlim[1]))-(post_stats_90[1]/(xlim[0]-xlim[1])),
xmax=(-1/(xlim[0]-xlim[1]))-(post_stats_90[1]/(xlim[0]-xlim[1])), marker='>', c='#9933ff')
plt.axhline(y=0.08, xmin=(-1/(xlim[0]-xlim[1]))-(post_median/(xlim[0]-xlim[1])),
xmax=(-1/(xlim[0]-xlim[1]))-(post_median/(xlim[0]-xlim[1])), c='#9933ff', marker='.', ms=12)
plt.plot(pliomip1x, pliomip1y, '.', label='PlioMIP1',markersize=17, color='#009999', mec='#006666')
plt.plot(pliomip2x, pliomip2y, '.', label='PlioMIP2',markersize=17, color='#cc0066',mec='#800040')
# Adjust text function. Computes location of model numbers based on all points (esthetic)
texts = [plt.text(x[i], y[i], '%s' %(i+15), ha='center', va='center', fontsize=15) for i in range(0, 1, 1)]
adjust_text(texts)
#texts2 = [plt.text(x[i], y[i], '%s' %(i+16), ha='center', va='center', fontsize=15) for i in range(1, 2, 1)]
#adjust_text(texts2)
#texts3 = [plt.text(x[i], y[i], '%s' %(i+17), ha='center', va='center', fontsize=15) for i in range(2, 4, 1)]
#adjust_text(texts3)
#texts4 = [plt.text(x[i], y[i], '%s' %(i+17), ha='center', va='center', fontsize=15) for i in range(4, 5, 1)]
#adjust_text(texts4)
#texts6 = [plt.text(x[i], y[i], '%s' %(i+18), ha='center', va='center', fontsize=15) for i in range(5, 6, 1)]
#adjust_text(texts6)
#texts5 = [plt.text(x[i], y[i], '%s' %(i+20), ha='center', va='center', fontsize=15) for i in range(6, 9, 1)]
#adjust_text(texts5)
# Make it pretty
plt.legend(loc='best', bbox_to_anchor=(0.6, 0.45), fancybox=True, edgecolor='k')
plt.xlabel('Climate sensitivity (K)', labelpad=10, position=(0.6,0),fontsize=16)
plt.ylabel('Mid-Pliocene tropical (30° S - 30° N) \nocean SST anomaly (K)',position=(0,0.6),fontsize=16)
ax.spines['top'].set_alpha(0)
ax.spines['right'].set_visible(False)
ax.spines['bottom'].set_position(('data', 0))
ax.spines['left'].set_position(('data', 0))
ax.spines['top'].set_position(('data', 0))
ax.spines['left'].set_linewidth(2)
ax.spines['bottom'].set_linewidth(2)
ax.tick_params('x', direction='out', pad=5, width=2)
ax.tick_params('y', width=2)
plt.yticks(ticks=[0.5, 1, 1.5, 2, 2.5, 3], labels=['0.5', '1','1.5', '2','2.5', '3'], fontsize=14)
plt.xticks(ticks=np.arange(1,9,1), fontsize=14)
plt.tight_layout()
#plt.savefig('Bayes_PlioMIP.pdf', dpi=300)
#-------------------------------
## 3rd plot: Posterior
fig, ax = plt.subplots(figsize=(7,7))
## Prior distribution line plot
scale = 5
loc = 2.5
a = 0
b = 10
k = 3
cauchy_scale = 3
x = np.linspace(0, 15, 1000)
# /!\ *1.3 account for the truncation at zero for the Cauchy distribution (esthetic approximation for the Figure)
cauchy = (1/((np.pi*cauchy_scale)*(1+((x-loc)/cauchy_scale)**2)))*1.3
#uniform = np.linspace(1/(b-a), 1/(b-a), 1000)
#gamma = (x**(k-1)*np.exp(-x/scale))/(scale**k)
plt.plot(x, cauchy, '-', color='darkorange', label='Prior',linewidth=4, linestyle='--')
#plt.plot(x, uniform, '-', c='darkorange', label='Prior',linewidth=4, linestyle='--')
#plt.plot(x, gamma, '-', c='darkorange', label='Prior',linewidth=4, linestyle='--')
# Fit function. Doesn't work well sometimes...
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
# Changing the value of alpha plot the true histogram behind
def fit_function(x, A, beta, B, mu, sigma):
return (A * np.exp(-x/beta) + B * np.exp(-1.0 * (x - mu)**2 / (2 * sigma**2)))
n, bins, patches = plt.hist(posterior, density=True, bins=500, alpha=0)
xspace = np.linspace(0, 8, 1000000)
binscenters = np.array([0.5 * (bins[ijk] + bins[ijk+1]) for ijk in range(len(bins)-1)])
popt, pcov = curve_fit(fit_function, xdata=binscenters, ydata=n)
plt.plot(xspace, fit_function(xspace, *popt), color='darkorange', linewidth=4, label='Posterior')
ylim = plt.ylim(-0.02, 0.5)
xlim = plt.xlim(-1,8)
plt.xlabel('Climate sensitivity (K)',fontsize=16)
plt.ylabel('Probability',fontsize=16)
plt.axhline(y=0.008, xmin=(-1/(xlim[0]-xlim[1]))-(post_stats_90[0]/(xlim[0]-xlim[1])),
xmax=(-1/(xlim[0]-xlim[1]))-(post_stats_90[1]/(xlim[0]-xlim[1])), c='#9933ff', label='5-95% estimate', linewidth=2)
plt.axhline(y=0.008, xmin=(-1/(xlim[0]-xlim[1]))-(post_stats_90[0]/(xlim[0]-xlim[1])),
xmax=(-1/(xlim[0]-xlim[1]))-(post_stats_90[0]/(xlim[0]-xlim[1])), marker='<', c='#9933ff')
plt.axhline(y=0.008, xmin=(-1/(xlim[0]-xlim[1]))-(post_stats_90[1]/(xlim[0]-xlim[1])),
xmax=(-1/(xlim[0]-xlim[1]))-(post_stats_90[1]/(xlim[0]-xlim[1])), marker='>', c='#9933ff')
plt.axhline(y=0.008, xmin=(-1/(xlim[0]-xlim[1]))-(post_median/(xlim[0]-xlim[1])),
xmax=(-1/(xlim[0]-xlim[1]))-(post_median/(xlim[0]-xlim[1])), c='#9933ff', marker='.', ms=12)
plt.axhline(y=0.02, xmin=(-1/(xlim[0]-xlim[1]))-(post_stats_66[0]/(xlim[0]-xlim[1])),
xmax=(-1/(xlim[0]-xlim[1]))-(post_stats_66[1]/(xlim[0]-xlim[1])), linestyle=':', c='#9933ff', label='17-83% estimate', linewidth=2)
plt.axhline(y=0.02, xmin=(-1/(xlim[0]-xlim[1]))-(post_stats_66[0]/(xlim[0]-xlim[1])),
xmax=(-1/(xlim[0]-xlim[1]))-(post_stats_66[0]/(xlim[0]-xlim[1])), marker='<', c='#9933ff')
plt.axhline(y=0.02, xmin=(-1/(xlim[0]-xlim[1]))-(post_stats_66[1]/(xlim[0]-xlim[1])),
xmax=(-1/(xlim[0]-xlim[1]))-(post_stats_66[1]/(xlim[0]-xlim[1])), marker='>', c='#9933ff')
plt.axhline(y=0.02, xmin=(-1/(xlim[0]-xlim[1]))-(post_median/(xlim[0]-xlim[1])),
xmax=(-1/(xlim[0]-xlim[1]))-(post_median/(xlim[0]-xlim[1])), c='#9933ff', marker='.', ms=12)
ax.spines['top'].set_visible(False)
ax.spines['right'].set_visible(False)
ax.spines['bottom'].set_position(('data', 0))
ax.spines['left'].set_position(('data', 0))
ax.spines['left'].set_linewidth(2)
ax.spines['bottom'].set_linewidth(2)
ax.tick_params('x', width=2)
ax.tick_params('y', width=2)
plt.xticks(ticks=np.arange(1,9,1),fontsize=14)
plt.yticks([0.1, 0.2, 0.3, 0.4, 0.5],fontsize=14)
plt.legend(loc='upper right', edgecolor='k')
plt.tight_layout()
#plt.savefig('Posterior_PlioMIP.pdf', dpi=300)