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Commit 87e4e1b8 authored by Frisinghelli Daniel's avatar Frisinghelli Daniel
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Preliminary validation metrics for temperature.

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%% Cell type:markdown id:f15afea1-9ea4-4201-bdd7-32ae377db6a9 tags:
# Evaluate ERA-5 downscaling
%% Cell type:markdown id:7d2c4c86-18e4-44c4-bdf7-0e8249614749 tags:
We used **1981-1991 as training** period and **1991-2010 as reference** period. The results shown in this notebook are based on the model predictions on the reference period.
%% Cell type:markdown id:4186c89c-b55b-4559-a818-8b712baaf44e tags:
Define the predictand and the model to evaluate:
%% Cell type:code id:bb24b6ed-2d0a-44e0-b9a9-abdcb2a8294d tags:
``` python
# define the model parameters
PREDICTAND = 'tasmin'
PREDICTAND = 'tasmax'
MODEL = 'USegNet'
PPREDICTORS = 'ztuvq'
PLEVELS = ['500', '850']
SPREDICTORS = 'p'
DEM = 'dem'
DOY = 'doy'
```
%% Cell type:code id:686a7c1d-e71d-4954-8d79-26b9a84f648e tags:
``` python
# mapping from predictands to variable names
NAMES = {'tasmin': 'minimum temperature', 'tasmax': 'maximum temperature', 'pr': 'precipitation'}
```
%% Cell type:markdown id:4d5d12c5-50fd-4c5c-9240-c3df78e49b44 tags:
### Imports
%% Cell type:code id:1afb0fab-5d2a-4875-9032-29b99c6dec89 tags:
``` python
# builtins
import datetime
import warnings
import calendar
# externals
import xarray as xr
import numpy as np
import matplotlib.pyplot as plt
import scipy.stats as stats
from IPython.display import Image
from sklearn.metrics import r2_score
# locals
from climax.main.io import ERA5_PATH, OBS_PATH, TARGET_PATH
from pysegcnn.core.utils import search_files
```
%% Cell type:markdown id:2c2d0b1d-9630-4cfe-ae9d-35b05b94a1a5 tags:
### Model architecture
%% Cell type:code id:f98e1c9f-0581-43ea-a569-fd05fdaf36c1 tags:
``` python
Image("./Figures/architecture.png", width=900, height=400)
```
%% Cell type:markdown id:f47caa41-9380-4c02-8785-4febcf2cb2d0 tags:
### Load datasets
%% Cell type:code id:020dfe33-ce3c-467f-ad0a-295cc338b1a9 tags:
``` python
# model predictions and observations NetCDF
y_pred = TARGET_PATH.joinpath(PREDICTAND, '_'.join([MODEL, PREDICTAND, PPREDICTORS, *PLEVELS, SPREDICTORS, DEM, DOY]) + '.nc')
if PREDICTAND == 'tas':
# read both tasmax and tasmin
tasmax = xr.open_dataset(search_files(OBS_PATH.joinpath('tasmax'), '.nc$').pop())
tasmin = xr.open_dataset(search_files(OBS_PATH.joinpath('tasmin'), '.nc$').pop())
y_true = xr.merge([tasmax, tasmin])
else:
y_true = xr.open_dataset(search_files(OBS_PATH.joinpath(PREDICTAND), '.nc$').pop())
```
%% Cell type:code id:1dc2a386-d63b-4c6a-8e63-00365927559d tags:
``` python
# load datasets
y_pred = xr.open_dataset(y_pred)
y_true = y_true.sel(time=y_pred.time) # subset to time period covered by predictions
```
%% Cell type:code id:966d85fb-9185-408f-ac2b-1e4ca829ccd1 tags:
``` python
# align datasets and mask missing values in model predictions
y_true, y_pred = xr.align(y_true, y_pred, join='override')
y_pred = y_pred.where(~np.isnan(y_true), other=np.nan)
```
%% Cell type:markdown id:ddebdf9f-862c-461e-aa57-cd344d54eee9 tags:
## Model validation: temperature
%% Cell type:markdown id:ab15d557-c7ea-40c0-9977-a3d410fea784 tags:
### Coefficient of determination
%% Cell type:code id:619d5dc9-4d36-43a3-b23c-a4ea51229c78 tags:
``` python
# get predicted and observed values over entire time series and grid points
y_pred_values = y_pred[PREDICTAND].values.flatten()
y_true_values = y_true[PREDICTAND].values.flatten()
```
%% Cell type:code id:49dff6ce-a629-460b-a43b-d1a0ef447351 tags:
``` python
# apply mask of valid pixels
mask = (~np.isnan(y_pred_values) & ~np.isnan(y_true_values))
y_pred_values = y_pred_values[mask]
y_true_values = y_true_values[mask]
```
%% Cell type:code id:e9e46770-4176-4257-8ea2-7050d3325e98 tags:
``` python
# calculate coefficient of determination
r2 = r2_score(y_true_values, y_pred_values)
```
%% Cell type:code id:703ab604-5193-4032-92eb-80f2cff9fc2c tags:
``` python
# scatter plot of observations vs. predictions
fig, ax = plt.subplots(1, 1, figsize=(10, 10))
# plot only a subset of data: otherwise plot is overloaded ...
subset = np.random.choice(np.arange(0, len(y_pred_values)), size=int(1e7), replace=False)
ax.plot(y_true_values[subset], y_pred_values[subset], 'o', alpha=.5, markeredgecolor='grey', markerfacecolor='none', markersize=3);
# plot entire dataset
# ax.plot(y_true_values, y_pred_values, 'o', alpha=.5, markeredgecolor='grey', markerfacecolor='none', markersize=3);
# plot 1:1 mapping line
interval = np.arange(-40, 45, 5)
ax.plot(interval, interval, color='k', lw=2, ls='--')
# add coefficient of determination: calculated on entire dataset!
ax.text(interval[-1] - 1, interval[0] + 1, s='Coefficient of determination R$^2$ = {:.2f}'.format(r2), ha='right', fontsize=14)
# format axes
ax.set_ylim(-40, 40)
ax.set_xlim(-40, 40)
ax.set_xticks(interval)
ax.set_xticklabels(interval, fontsize=14)
ax.set_yticks(interval)
ax.set_yticklabels(interval, fontsize=14)
ax.set_xlabel('Observed', fontsize=14)
ax.set_ylabel('Predicted', fontsize=14)
ax.set_title('{} (°C): 1991 - 2010'.format(NAMES[PREDICTAND].capitalize()), fontsize=16, pad=10);
# save figure
fig.savefig('../Notebooks/Figures/{}_r2.png'.format(PREDICTAND), dpi=300, bbox_inches='tight')
```
%% Cell type:markdown id:5e8be24e-8ca2-4582-98c0-b56c6db289d2 tags:
### Bias
%% Cell type:markdown id:a4f7177c-7d09-401f-957b-0e493b9ef5d0 tags:
Calculate yearly average bias over entire reference period:
%% Cell type:code id:746bf95f-a78b-4da8-a063-1fa48e3c5da8 tags:
``` python
# yearly average bias over reference period
y_pred_yearly_avg = y_pred.groupby('time.year').mean(dim='time')
y_true_yearly_avg = y_true.groupby('time.year').mean(dim='time')
bias_yearly_avg = y_pred_yearly_avg - y_true_yearly_avg
for var in bias:
print('Yearly average bias {}: {:.2f}'.format(var, bias_yearly_avg[var].mean().item()))
for var in bias_yearly_avg:
print('Yearly average bias of {}: {:.2f}'.format(var, bias_yearly_avg[var].mean().item()))
```
%% Cell type:code id:2ef19dce-29ac-4f6f-9999-e67745a4afd1 tags:
``` python
# mean absolute error over reference period
mae_avg = np.abs(y_pred_yearly_avg - y_true_yearly_avg).mean()
for var in mae:
print('Yearly average mean absolute error {}: {:.2f}'.format(var, mae[var].item()))
for var in mae_avg:
print('Yearly average MAE of {}: {:.2f}'.format(var, mae_avg[var].item()))
```
%% Cell type:code id:4c4dd156-f763-482c-84e6-c4329bfd3fe4 tags:
``` python
# root mean squared error over reference period
rmse_avg = ((y_pred_yearly_avg - y_true_yearly_avg) ** 2).mean()
for var in rmse:
print('Root mean squared error {}: {:.2f}'.format(var, rmse[var].item()))
for var in rmse_avg:
print('Yearly average RMSE of {}: {:.2f}'.format(var, rmse_avg[var].item()))
```
%% Cell type:code id:ed54a50e-41c2-4a7f-9839-05357996f0c4 tags:
``` python
# Pearson's correlation coefficient over reference period
for var in y_pred_yearly_avg:
correlations = []
for year in y_pred_yearly_avg.year:
y_p = y_pred_yearly_avg[var].sel(year=year).values
y_t = y_true_yearly_avg[var].sel(year=year).values
r, _ = stats.pearsonr(y_p[~np.isnan(y_p)], y_t[~np.isnan(y_t)])
correlations.append(r)
print('Yearly average Pearson correlation coefficient for {}: {:.2f}'.format(var, np.asarray(r).mean()))
```
%% Cell type:code id:760f86ce-9e04-4938-b24f-d2819fbf622e tags:
``` python
# plot average of observation, prediction, and bias
vmin, vmax = (-15, 15) if PREDICTAND == 'tasmin' else (0, 25)
fig, axes = plt.subplots(len(y_pred_yearly_avg.data_vars), 3, figsize=(24, len(y_pred_yearly_avg.data_vars) * 6),
sharex=True, sharey=True)
axes = axes.reshape(len(y_pred_yearly_avg.data_vars), -1)
for i, var in enumerate(y_pred_yearly_avg):
for ds, ax in zip([y_true_yearly_avg, y_pred_yearly_avg, bias_yearly_avg], axes[i, ...]):
if ds is bias_yearly_avg:
ds = ds[var].mean(dim='year')
im2 = ax.imshow(ds.values, origin='lower', cmap='RdBu_r', vmin=-2, vmax=2)
ax.text(x=ds.shape[0] - 2, y=2, s='Average: {:.2f}°C'.format(ds.mean().item()), fontsize=14, ha='right')
else:
im1 = ax.imshow(ds[var].mean(dim='year').values, origin='lower', cmap='RdYlBu_r', vmin=-15, vmax=15)
im1 = ax.imshow(ds[var].mean(dim='year').values, origin='lower', cmap='RdYlBu_r', vmin=vmin, vmax=vmax)
# set titles
axes[0, 0].set_title('Observed', fontsize=16);
axes[0, 1].set_title('Predicted', fontsize=16);
axes[0, 2].set_title('Bias', fontsize=16);
axes[0, 0].set_title('Observed', fontsize=16, pad=10);
axes[0, 1].set_title('Predicted', fontsize=16, pad=10);
axes[0, 2].set_title('Bias', fontsize=16, pad=10);
# adjust axes
for ax in axes.flat:
ax.axes.get_xaxis().set_ticklabels([])
ax.axes.get_xaxis().set_ticks([])
ax.axes.get_yaxis().set_ticklabels([])
ax.axes.get_yaxis().set_ticks([])
ax.axes.axis('tight')
ax.set_xlabel('')
ax.set_ylabel('')
# adjust figure
fig.suptitle('Average {}: 1991 - 2010'.format(NAMES[PREDICTAND]), fontsize=20);
fig.subplots_adjust(hspace=0, wspace=0, top=0.85)
# add colorbar for bias
axes = axes.flatten()
cbar_ax = fig.add_axes([axes[-1].get_position().x1 + 0.01, axes[-1].get_position().y0,
0.01, axes[-1].get_position().y1 - axes[-1].get_position().y0])
cbar = fig.colorbar(im2, cax=cbar_ax)
cbar.set_label(label='Bias / (°C)', fontsize=16)
cbar.ax.tick_params(labelsize=14)
cbar_ax_bias = fig.add_axes([axes[-1].get_position().x1 + 0.01, axes[-1].get_position().y0,
0.01, axes[-1].get_position().y1 - axes[-1].get_position().y0])
cbar_bias = fig.colorbar(im2, cax=cbar_ax_bias)
cbar_bias.set_label(label='Bias / (°C)', fontsize=16)
cbar_bias.ax.tick_params(labelsize=14)
# add colorbar for predictand
cbar_ax_predictand = fig.add_axes([axes[0].get_position().x0, axes[0].get_position().y0 - 0.1,
axes[-1].get_position().x0 - axes[0].get_position().x0,
0.05])
cbar_predictand = fig.colorbar(im1, cax=cbar_ax_predictand, orientation='horizontal')
cbar_predictand.set_label(label='{} / (°C)'.format(NAMES[PREDICTAND].capitalize()), fontsize=16)
cbar_predictand.ax.tick_params(labelsize=14)
# add metrics: MAE and RMSE
axes[1].text(x=ds.shape[0] - 2, y=2, s='MAE = {:.2f}°C'.format(mae_avg[PREDICTAND].item()), fontsize=14, ha='right')
axes[1].text(x=ds.shape[0] - 2, y=12, s='RMSE = {:.2f}°C$^2$'.format(rmse_avg[PREDICTAND].item()), fontsize=14, ha='right')
# save figure
fig.savefig('../Notebooks/Figures/{}_average_bias.png'.format(PREDICTAND), dpi=300, bbox_inches='tight')
```
%% Cell type:markdown id:aa4ef730-5a9d-40dc-a318-2f43a4cf1cd2 tags:
### Seasonal bias
%% Cell type:markdown id:eda455a2-e8ee-4644-bb85-b0cf76acd11a tags:
Calculate seasonal bias:
%% Cell type:code id:24aadff5-b19d-4f4b-a4c5-32ee656e64cd tags:
``` python
# group data by season: (DJF, MAM, JJA, SON)
y_true_snl = y_true.groupby('time.season').mean(dim='time')
y_pred_snl = y_pred.groupby('time.season').mean(dim='time')
bias_snl = y_pred_snl - y_true_snl
```
%% Cell type:code id:2d0d5a46-0652-4289-9e1c-6d47aafcfef0 tags:
``` python
# print average bias per season
for var in bias_snl.data_vars:
for season in bias_snl.tasmin.season:
for season in bias_snl[PREDICTAND].season:
print('Average bias of {} for season {}: {:.2f}'.format(var, season.values.item(), bias_snl[var].sel(season=season).mean().item()))
```
%% Cell type:markdown id:c4232ae9-a557-4d61-8ab3-d7eda6201f98 tags:
Plot seasonal differences, taken from the [xarray documentation](xarray.pydata.org/en/stable/examples/monthly-means.html).
%% Cell type:code id:b39a6cc0-614c-452d-bb85-bc10e5179948 tags:
``` python
# plot seasonal differences
seasons = ('DJF', 'JJA')
fig, axes = plt.subplots(nrows=1, ncols=len(seasons) + 1, figsize=(24,8), sharex=True, sharey=True)
axes = axes.flatten()
# plot annual average bias
ds = bias_yearly_avg[PREDICTAND].mean(dim='year')
axes[0].imshow(ds.values, origin='lower', cmap='RdBu_r', vmin=-2, vmax=2)
axes[0].set_title('Annual', fontsize=16);
axes[0].text(x=ds.shape[0] - 2, y=2, s='Average: {:.2f}°C'.format(ds.mean().item()), fontsize=14, ha='right')
# plot seasonal average bias
for ax, season in zip(axes[1:], seasons):
ds = bias_snl[PREDICTAND].sel(season=season)
ax.imshow(ds.values, origin='lower', cmap='RdBu_r', vmin=-2, vmax=2)
ax.set_title(season, fontsize=16);
ax.text(x=ds.shape[0] - 2, y=2, s='Average: {:.2f}°C'.format(ds.mean().item()), fontsize=14, ha='right')
# adjust axes
for ax in axes.flat:
ax.axes.get_xaxis().set_ticklabels([])
ax.axes.get_xaxis().set_ticks([])
ax.axes.get_yaxis().set_ticklabels([])
ax.axes.get_yaxis().set_ticks([])
ax.axes.axis('tight')
ax.set_xlabel('')
ax.set_ylabel('')
# adjust figure
fig.suptitle('Average bias of {}: 1991 - 2010'.format(NAMES[PREDICTAND]), fontsize=20);
fig.subplots_adjust(hspace=0, wspace=0, top=0.85)
# add colorbar for bias
axes = axes.flatten()
cbar_ax = fig.add_axes([axes[-1].get_position().x1 + 0.01, axes[-1].get_position().y0,
0.01, axes[-1].get_position().y1 - axes[-1].get_position().y0])
cbar = fig.colorbar(im2, cax=cbar_ax)
cbar.set_label(label='Bias / (°C)', fontsize=16)
cbar.ax.tick_params(labelsize=14)
# save figure
fig.savefig('../Notebooks/Figures/{}_average_bias_seasonal.png'.format(PREDICTAND), dpi=300, bbox_inches='tight')
```
%% Cell type:markdown id:c70b369d-2d16-42e3-9300-4a18757ad1b2 tags:
%% Cell type:markdown id:41600269-2f8c-4717-8f74-b3dfaef60359 tags:
### Bias of extreme values
Calculate the mean annual cycle:
%% Cell type:code id:4acfc3f2-20ed-498c-ab35-f392ae0e64f9 tags:
%% Cell type:code id:c9f27c01-4dfc-4d16-8d29-00e69b7794cd tags:
``` python
# TODO: smooth quantiles
# group timeseries by month and calculate mean over time and space
y_pred_ac = y_pred.groupby('time.month').mean(dim=('time', 'y', 'x'))
y_true_ac = y_true.groupby('time.month').mean(dim=('time', 'y', 'x'))
```
%% Cell type:code id:bcc63e73-2636-49c1-a66b-241eb5407e2e tags:
``` python
# plot mean annual cycle
fig, ax = plt.subplots(1, 1, figsize=(10, 10))
ax.plot(y_pred_ac[PREDICTAND].values, ls='--', color='k', label='Predicted')
ax.plot(y_true_ac[PREDICTAND].values, ls='-', color='k', label='Observed')
ax.legend(frameon=False, fontsize=14);
ax.set_yticks(np.arange(np.floor(y_true_ac[PREDICTAND].min().item()), np.ceil(y_true_ac[PREDICTAND].max().item()) + 1, 1))
ax.set_yticklabels(np.arange(np.floor(y_true_ac[PREDICTAND].min().item()), np.ceil(y_true_ac[PREDICTAND].max().item()) + 1, 1), fontsize=12)
ax.set_xticks(np.arange(0, 12))
ax.set_xticklabels([calendar.month_name[i + 1] for i in np.arange(0, 12)], rotation=90, fontsize=12)
ax.set_title('Mean annual cycle of {}: 1991 - 2010'.format(NAMES[PREDICTAND]), pad=20, fontsize=16);
ax.set_ylabel('{} / (°C)'.format(NAMES[PREDICTAND].capitalize()), fontsize=14)
ax.set_xlabel('Month', fontsize=14);
# save figure
fig.savefig('../Notebooks/Figures/{}_mean_annual_cycle.png'.format(PREDICTAND), dpi=300, bbox_inches='tight')
```
%% Cell type:markdown id:c70b369d-2d16-42e3-9300-4a18757ad1b2 tags:
### Bias of extreme values
%% Cell type:code id:33198267-8eb6-4b84-bb6d-c903b27ccbc5 tags:
``` python
# percentiles of interest
percentiles = [0.01, 0.02, 0.98, 0.99]
# extreme quantile of interest
quantile = 0.02 if PREDICTAND == 'tasmin' else 0.98
```
%% Cell type:code id:51137d23-a380-4d48-a005-fd1edaf554eb tags:
%% Cell type:code id:c3da76d3-7261-4084-b5ec-f65682fd6596 tags:
``` python
# calculate percentiles over reference period
# calculate extreme quantile for each year
with warnings.catch_warnings():
warnings.simplefilter('ignore')
y_pred_dist = y_pred.quantile(q=percentiles, dim='time')
y_true_dist = y_true.quantile(q=percentiles, dim='time')
warnings.simplefilter('ignore', category=RuntimeWarning)
y_pred_ex = y_pred.groupby('time.year').quantile(quantile, dim='time')
y_true_ex = y_true.groupby('time.year').quantile(quantile, dim='time')
```
%% Cell type:code id:20db96c8-e04f-4acb-886b-abb740863fbb tags:
``` python
# calculate bias in extreme quantile for each year
bias_ex = y_pred_ex - y_true_ex
for var in bias_ex:
print('Yearly average bias for P{:.0f} of {}: {:.2f}'.format(quantile * 100, var, bias_ex[var].mean().item()))
```
%% Cell type:code id:b461c774-c5fc-4a50-8609-34a7e7674a34 tags:
%% Cell type:code id:c7aeff55-deef-4d3c-a251-eac877c9afd9 tags:
``` python
# calculate bias in each percentile over entire reference period
bias_dist = y_pred_dist - y_true_dist
# mean absolute error in extreme quantile
mae_ex = np.abs(y_pred_ex - y_true_ex).mean()
for var in mae_ex:
print('Yearly average MAE for P{:.0f} of {}: {:.2f}'.format(quantile * 100, var, mae_avg[var].item()))
```
%% Cell type:code id:bd2ec314-87ed-407f-af8b-5e2c6785e9cc tags:
%% Cell type:code id:2dc6b7cf-7ae1-4b0e-8483-376eab59f5dd tags:
``` python
# calculate correlation coefficient for extreme values
for var in y_pred_dist:
for q in percentiles:
y_p = y_pred_dist[var].sel(quantile=q).values[~np.isnan(y_pred_dist[var].sel(quantile=q))]
y_t = y_true_dist[var].sel(quantile=q).values[~np.isnan(y_true_dist[var].sel(quantile=q))]
r, _ = stats.pearsonr(y_p, y_t)
print('Pearson correlation for {}, q={:.2f}: R={:.2f}'.format(var, q, r))
# root mean squared error over reference period
rmse_ex = ((y_pred_ex - y_true_ex) ** 2).mean()
for var in rmse_ex:
print('Yearly average RMSE for P{:.0f} of {}: {:.2f}'.format(quantile * 100, var, rmse_ex[var].item()))
```
%% Cell type:code id:04a1b4fc-8fc0-4e1e-adbe-bf1eaafeac5d tags:
%% Cell type:code id:cfc196fa-0999-4603-959c-2c82f038c8fa tags:
``` python
# plot bias in each percentile
fig, axes = plt.subplots(len(y_pred_dist.data_vars), len(percentiles), sharex=True, sharey=True, figsize=(32, 6))
axes = axes.reshape(len(y_pred_dist.data_vars), -1)
for ax, var in zip(axes, y_pred_dist):
# iterate over percentiles
for axis, q in zip(ax, percentiles):
ds = bias_dist.sel(quantile=q).to_array()
ds.plot(ax=axis, vmin=-2, vmax=2, cmap='RdBu_r')
axis.text(x=bias_dist.x[-1], y=bias_dist.y[0], s='Avg: {:.2f}'.format(ds.mean().item()), ha='right', va='bottom')
# plot extremes of observation, prediction, and bias
vmin, vmax = (-20, 0) if PREDICTAND == 'tasmin' else (10, 40)
fig, axes = plt.subplots(len(y_pred_ex.data_vars), 3, figsize=(24, len(y_pred_ex.data_vars) * 6),
sharex=True, sharey=True)
axes = axes.reshape(len(y_pred_ex.data_vars), -1)
for i, var in enumerate(y_pred_ex):
for ds, ax in zip([y_true_ex, y_pred_ex, bias_ex], axes[i, ...]):
if ds is bias_ex:
ds = ds[var].mean(dim='year')
im2 = ax.imshow(ds.values, origin='lower', cmap='RdBu_r', vmin=-2, vmax=2)
ax.text(x=ds.shape[0] - 2, y=2, s='Average: {:.2f}°C'.format(ds.mean().item()), fontsize=14, ha='right')
else:
im1 = ax.imshow(ds[var].mean(dim='year').values, origin='lower', cmap='Blues_r' if PREDICTAND == 'tasmin' else 'Reds',
vmin=vmin, vmax=vmax)
# set titles
axes[0, 0].set_title('Observed', fontsize=16, pad=10);
axes[0, 1].set_title('Predicted', fontsize=16, pad=10);
axes[0, 2].set_title('Bias', fontsize=16, pad=10);
# adjust axes
for ax in axes.flat:
ax.axes.get_xaxis().set_ticklabels([])
ax.axes.get_xaxis().set_ticks([])
ax.axes.get_yaxis().set_ticklabels([])
ax.axes.get_yaxis().set_ticks([])
ax.axes.axis('tight')
ax.set_xlabel('')
ax.set_ylabel('')
# adjust figure
fig.suptitle('Average P{:.0f} of {}: 1991 - 2010'.format(quantile * 100, NAMES[PREDICTAND]), fontsize=20);
fig.subplots_adjust(hspace=0, wspace=0, top=0.85)
# add colorbar for bias
axes = axes.flatten()
cbar_ax = fig.add_axes([axes[-1].get_position().x1 + 0.01, axes[-1].get_position().y0,
0.01, axes[-1].get_position().y1 - axes[-1].get_position().y0])
cbar = fig.colorbar(im2, cax=cbar_ax)
cbar.set_label(label='Bias / (°C)', fontsize=16)
cbar.ax.tick_params(labelsize=14)
# add colorbar for predictand
cbar_ax_predictand = fig.add_axes([axes[0].get_position().x0, axes[0].get_position().y0 - 0.1,
axes[-1].get_position().x0 - axes[0].get_position().x0,
0.05])
cbar_predictand = fig.colorbar(im1, cax=cbar_ax_predictand, orientation='horizontal')
cbar_predictand.set_label(label='{} / (°C)'.format(NAMES[PREDICTAND].capitalize()), fontsize=16)
cbar_predictand.ax.tick_params(labelsize=14)
# add metrics: MAE and RMSE
axes[1].text(x=ds.shape[0] - 2, y=2, s='MAE = {:.2f}°C'.format(mae_ex[PREDICTAND].item()), fontsize=14, ha='right')
axes[1].text(x=ds.shape[0] - 2, y=12, s='RMSE = {:.2f}°C$^2$'.format(rmse_ex[PREDICTAND].item()), fontsize=14, ha='right')
# save figure
fig.savefig('../Notebooks/Figures/{}_average_bias_p{:.0f}.png'.format(PREDICTAND, quantile * 100), dpi=300, bbox_inches='tight')
```
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