Bayesian uncertainty analysis#

R.A. Collenteur, Eawag, June, 2023

In this notebook it is shown how the MCMC-algorithm can be used to estimate the model parameters and quantify the (parameter) uncertainties for a Pastas model using a Bayesian approach. For this the EmceeSolver is introduced, based on the emcee Python package.

Besides Pastas the following Python Packages have to be installed to run this notebook:

emcee
corner

Note: The EmceeSolver is still an experimental feature and some of the arguments might be changed in the near future (2023/06/22). We welcome testing and feedback on this new feature!.

import corner
import emcee
import matplotlib.pyplot as plt
import numpy as np
import pandas as pd

import pastas as ps

ps.set_log_level("ERROR")
ps.show_versions()

Pastas     : 2.0.0
Python     : 3.14.0
Numpy      : 2.4.6
Pandas     : 3.0.3
Scipy      : 1.17.1
Matplotlib : 3.10.9
Numba      : 0.65.1

1. Create a Pastas Model#

The first step is to create a Pastas Model, including the RechargeModel to simulate the effect of precipitation and evaporation on the heads. Here, we first estimate the model parameters using the standard least-squares approach.

head = pd.read_csv(
    "data/B32C0639001.csv", parse_dates=["date"], index_col="date"
).squeeze()

evap = pd.read_csv("data/evap_260.csv", index_col=0, parse_dates=[0]).squeeze()
rain = pd.read_csv("data/rain_260.csv", index_col=0, parse_dates=[0]).squeeze()

ml = ps.Model(head)
ml.add_noisemodel(ps.ArNoiseModel())

# Select a recharge model
rch = ps.rch.FlexModel()

rm = ps.RechargeModel(rain, evap, recharge=rch, rfunc=ps.Gamma(), name="rch")
ml.add_stressmodel(rm)

ml.solve(tmin="1990")

ax = ml.plot(figsize=(10, 3))

Fit report head                     Fit Statistics
==================================================
nfev     35                     EVP          87.37
nobs     351                    R2            0.87
noise    True                   RMSE          0.07
tmin     1990-01-01 00:00:00    AICc      -2048.61
tmax     2005-10-14 00:00:00    BIC       -2014.39
freq     D                      Obj           0.49
freq_obs None                   ___               
warmup   3650 days 00:00:00     Interp.         No

Parameters (9 optimized)
==================================================
                optimal     initial   vary
rch_A          0.473048    0.630436   True
rch_n          0.674025    1.000000   True
rch_a        311.234331   10.000000   True
rch_srmax     75.363926  250.000000   True
rch_lp         0.250000    0.250000  False
rch_ks        50.997527  100.000000   True
rch_gamma      2.203450    2.000000   True
rch_kv         1.999647    1.000000   True
rch_simax      2.000000    2.000000  False
constant_d     0.790329    1.359779   True
noise_alpha   42.580834   15.000000   True

../_images/14c8d1919f6e8e2928e6ffb44ac66cd8ed00f92400e03df3e52f770198feaf9b.png

2. Use the EmceeSolver#

We will now use the EmceeSolve solver to estimate the model parameters and their uncertainties. This solver wraps the Emcee package, which implements different versions of MCMC. A good understanding of Emcee helps when using this solver, so it comes recommended to check out their documentation as well.

To set up the solver, a number of decisions need to be made:

Determine the priors of the parameters
Choose a (log) likelihood function
Choose the number of steps and thinning

2a. Choose and set the priors#

The first step is to choose and set the priors of the parameters. This is done by using the ml.set_parameter method and the dist argument (from distribution). Any distribution from the scipy.stats can be chosen (https://docs.scipy.org/doc/scipy/tutorial/stats/continuous.html), for example uniform, norm, or lognorm. Here, for the sake of the example, we set all prior distributions to a normal distribution.

# Set the initial parameters to a normal distribution
for name in ml.parameters.index:
    ml.set_parameter(name, dist="norm")

ml.parameters

	initial	pmin	pmax	vary	name	dist	stderr	optimal
rch_A	0.630436	0.00001	63.043598	True	rch	norm	0.047620	0.473048
rch_n	1.000000	0.10000	5.000000	True	rch	norm	0.031943	0.674025
rch_a	10.000000	0.01000	10000.000000	True	rch	norm	64.881837	311.234331
rch_srmax	250.000000	0.00001	1000.000000	True	rch	norm	42.277454	75.363926
rch_lp	0.250000	0.00001	1.000000	False	rch	norm	NaN	0.250000
rch_ks	100.000000	0.00001	10000.000000	True	rch	norm	59.061896	50.997527
rch_gamma	2.000000	0.00001	20.000000	True	rch	norm	0.201305	2.203450
rch_kv	1.000000	0.25000	2.000000	True	rch	norm	0.424225	1.999647
rch_simax	2.000000	0.00000	10.000000	False	rch	norm	NaN	2.000000
constant_d	1.359779	NaN	NaN	True	constant	norm	0.052656	0.790329
noise_alpha	15.000000	0.00001	5000.000000	True	noise	norm	6.544576	42.580834

Pastas will use the initial value of the parameter for the loc argument of the distribution (e.g., the mean of a normal distribution), and the stderr as the scale argument (e.g., the standard deviation of a normal distribution). Only for the parameters with a uniform distribution, the pmin and pmax values are used to determine a uniform prior. By default, all parameters are assigned a uniform prior.

Note: This means that either the `pmin` and `pmax` should be set for uniform distributions, or the `stderr` for any other distribution. That is why in this example model was first solved using LeastSquares, in order to obtain estimates for the `stderr`. In practice, these could also be set based on expert judgement or information about the parameters.

2b. Create the solver instance#

The next step is to create an instance of the EmceeSolve solver class. At this stage all the settings need to be provided on how the Ensemble Sampler is created (https://emcee.readthedocs.io/en/stable/user/sampler/). Important settings are the nwalkers, the moves, the objective_function. More advanced options are to parallelize the MCMC algorithm (parallel=True), and to set a backend to store the results. Here’s an example:

# Choose the objective function
ln_prob = ps.objfunc.GaussianLikelihoodAr1()

# Create the EmceeSolver with some settings
s = ps.EmceeSolve(
    nwalkers=20,
    moves=emcee.moves.DEMove(),
    objective_function=ln_prob,
    progress_bar=True,
    parallel=False,
)

In the above code we created an EmceeSolve instance with 20 walkers, which take steps according to the DEMove move algorithm (see Emcee docs), and a Gaussian likelihood function that assumes AR1 correlated errors. Different objective functions are available, see the Pastas documentation on the different options.

Depending on the likelihood function, a number of additional parameters need to be inferred. These parameters are not added to the Pastas Model instance, but are available from the solver object. Using the set_parameter method of the solver, these parameters can be changed. In this example where we use the GaussianLikelihoodAr1 function the sigma and theta are estimated; the unknown standard deviation of the errors and the autoregressive parameter.

s.parameters

	initial	pmin	pmax	vary	stderr	name	dist
ln_var	0.05	1.000000e-10	1.00000	True	0.01	ln	uniform
ln_phi	0.50	1.000000e-10	0.99999	True	0.20	ln	uniform

s.set_parameter("ln_var", initial=0.0028, vary=False, dist="norm")
s.parameters

	initial	pmin	pmax	vary	stderr	name	dist
ln_var	0.0028	1.000000e-10	1.00000	False	0.01	ln	norm
ln_phi	0.5000	1.000000e-10	0.99999	True	0.20	ln	uniform

2c. Run the solver and solve the model#

After setting the parameters and creating a EmceeSolve solver instance we are now ready to run the MCMC analysis. We can do this by running ml.solve. We can pass the same parameters that we normally provide to this method (e.g., tmin or fit_constant). Here we use the initial parameters from our least-square solve, and do not fit a noise model, because we take autocorrelated errors into account through the likelihood function.

All the arguments that are not used by ml.solve, for example steps and tune, are passed on to the run_mcmc method from the sampler (see Emcee docs). The most important is the steps argument, that determines how many steps each of the walkers takes.

# Use the solver to run MCMC
ml.del_noisemodel()
ml.solve(
    solver=s,
    initial=False,
    fit_constant=False,
    tmin="1990",
    steps=100,
    tune=True,
)

Fit report head                     Fit Statistics
==================================================
nfev     nan                    EVP          88.10
nobs     351                    R2            0.88
noise    False                  RMSE          0.07
tmin     1990-01-01 00:00:00    AICc      -1835.06
tmax     2005-10-14 00:00:00    BIC       -1808.36
freq     D                      Obj            nan
freq_obs None                   ___               
warmup   3650 days 00:00:00     Interp.         No

Parameters (7 optimized)
==================================================
               optimal     initial   vary
rch_A         0.476766    0.473048   True
rch_n         0.678931    0.674025   True
rch_a       304.906105  311.234331   True
rch_srmax    52.436214   75.363926   True
rch_lp        0.250000    0.250000  False
rch_ks       29.583133   50.997527   True
rch_gamma     2.473316    2.203450   True
rch_kv        1.979099    1.999647   True
rch_simax     2.000000    2.000000  False
constant_d    0.815720    0.000000  False

  0%|          | 0/100 [00:00<?, ?it/s]
/home/docs/checkouts/readthedocs.org/user_builds/pastas/envs/latest/lib/python3.14/site-packages/emcee/moves/red_blue.py:99: RuntimeWarning: invalid value encountered in scalar subtract
  lnpdiff = f + nlp - state.log_prob[j]
  2%|▏         | 2/100 [00:00<00:05, 18.54it/s]
  4%|▍         | 4/100 [00:00<00:05, 16.08it/s]
  6%|▌         | 6/100 [00:00<00:06, 14.09it/s]
  8%|▊         | 8/100 [00:00<00:06, 13.45it/s]
 10%|█         | 10/100 [00:00<00:07, 12.12it/s]
 12%|█▏        | 12/100 [00:00<00:07, 11.58it/s]
 14%|█▍        | 14/100 [00:01<00:07, 11.04it/s]
 16%|█▌        | 16/100 [00:01<00:08, 10.43it/s]
 18%|█▊        | 18/100 [00:01<00:08, 10.09it/s]
 20%|██        | 20/100 [00:01<00:07, 10.58it/s]
 22%|██▏       | 22/100 [00:01<00:07, 10.45it/s]
 24%|██▍       | 24/100 [00:02<00:07, 10.34it/s]
 26%|██▌       | 26/100 [00:02<00:07, 10.20it/s]
 28%|██▊       | 28/100 [00:02<00:06, 10.36it/s]
 30%|███       | 30/100 [00:02<00:06, 10.33it/s]
 32%|███▏      | 32/100 [00:02<00:06, 10.27it/s]
 34%|███▍      | 34/100 [00:03<00:06, 10.60it/s]
 36%|███▌      | 36/100 [00:03<00:06, 10.11it/s]
 38%|███▊      | 38/100 [00:03<00:06, 10.03it/s]
 40%|████      | 40/100 [00:03<00:05, 10.81it/s]
 42%|████▏     | 42/100 [00:03<00:05, 10.98it/s]
 44%|████▍     | 44/100 [00:04<00:05, 10.89it/s]
 46%|████▌     | 46/100 [00:04<00:05, 10.31it/s]
 48%|████▊     | 48/100 [00:04<00:04, 10.43it/s]
 50%|█████     | 50/100 [00:04<00:04, 10.47it/s]
 52%|█████▏    | 52/100 [00:04<00:04, 10.45it/s]
 54%|█████▍    | 54/100 [00:05<00:04, 10.07it/s]
 56%|█████▌    | 56/100 [00:05<00:04,  9.64it/s]
 57%|█████▋    | 57/100 [00:05<00:04,  9.58it/s]
 58%|█████▊    | 58/100 [00:05<00:04,  9.55it/s]
 59%|█████▉    | 59/100 [00:05<00:04,  9.42it/s]
 61%|██████    | 61/100 [00:05<00:04,  9.54it/s]
 62%|██████▏   | 62/100 [00:05<00:04,  9.47it/s]
 63%|██████▎   | 63/100 [00:06<00:03,  9.44it/s]
 64%|██████▍   | 64/100 [00:06<00:03,  9.33it/s]
 65%|██████▌   | 65/100 [00:06<00:03,  9.28it/s]
 66%|██████▌   | 66/100 [00:06<00:03,  9.24it/s]
 67%|██████▋   | 67/100 [00:06<00:03,  9.42it/s]
 69%|██████▉   | 69/100 [00:06<00:03,  9.78it/s]
 71%|███████   | 71/100 [00:06<00:02, 10.31it/s]
 73%|███████▎  | 73/100 [00:07<00:02,  9.96it/s]
 74%|███████▍  | 74/100 [00:07<00:02,  9.84it/s]
 76%|███████▌  | 76/100 [00:07<00:02, 10.04it/s]
 78%|███████▊  | 78/100 [00:07<00:02, 10.07it/s]
 80%|████████  | 80/100 [00:07<00:02,  9.97it/s]
 81%|████████  | 81/100 [00:07<00:01,  9.96it/s]
 82%|████████▏ | 82/100 [00:07<00:01,  9.95it/s]
 83%|████████▎ | 83/100 [00:08<00:01,  9.79it/s]
 84%|████████▍ | 84/100 [00:08<00:01,  9.79it/s]
 85%|████████▌ | 85/100 [00:08<00:01,  9.70it/s]
 87%|████████▋ | 87/100 [00:08<00:01,  9.73it/s]
 89%|████████▉ | 89/100 [00:08<00:01,  9.74it/s]
 90%|█████████ | 90/100 [00:08<00:01,  9.75it/s]
 91%|█████████ | 91/100 [00:08<00:00,  9.53it/s]
 92%|█████████▏| 92/100 [00:08<00:00,  9.45it/s]
 94%|█████████▍| 94/100 [00:09<00:00,  9.65it/s]
 95%|█████████▌| 95/100 [00:09<00:00,  9.63it/s]
 97%|█████████▋| 97/100 [00:09<00:00, 10.03it/s]
 99%|█████████▉| 99/100 [00:09<00:00, 10.16it/s]
100%|██████████| 100/100 [00:09<00:00, 10.25it/s]

3. Posterior parameter distributions#

The results from the MCMC analysis are stored in the sampler object, accessible through ml.solver.sampler variable. The object ml.solver.sampler.flatchain contains a Pandas DataFrame with \(n\) the parameter samples, where \(n\) is calculated as follows:

\(n = \frac{\left(\text{steps}-\text{burn}\right)\cdot\text{nwalkers}}{\text{thin}} \)

Corner.py#

Corner is a simple but great python package that makes creating corner graphs easy. A couple of lines of code suffice to create a plot of the parameter distributions and the covariances between the parameters.

# Corner plot of the results
fig = plt.figure(figsize=(8, 8))

labels = list(ml.parameters.index[ml.parameters.vary]) + list(
    ml.solver.parameters.index[ml.solver.parameters.vary]
)
labels = [label.split("_")[1] for label in labels]

best = list(ml.parameters[ml.parameters.vary].optimal) + list(
    ml.solver.parameters[ml.solver.parameters.vary].optimal
)

axes = corner.corner(
    ml.solver.sampler.get_chain(flat=True, discard=50),
    quantiles=[0.025, 0.5, 0.975],
    labelpad=0.1,
    show_titles=True,
    title_kwargs=dict(fontsize=10),
    label_kwargs=dict(fontsize=10),
    max_n_ticks=3,
    fig=fig,
    labels=labels,
    truths=best,
)

plt.show()

../_images/f579716f17be0606b802ef06b3f484d502fb593f8ddbb7ae3fcc4d532d9246a1.png

4. What happens to the walkers at each step?#

The walkers take steps in different directions for each step. It is expected that after a number of steps, the direction of the step becomes random, as a sign that an optimum has been found. This can be checked by looking at the autocorrelation, which should be insignificant after a number of steps. Below we just show how to obtain the different chains, the interpretation of which is outside the scope of this notebook.

fig, axes = plt.subplots(len(labels), figsize=(10, 7), sharex=True)

samples = ml.solver.sampler.get_chain(flat=True)
for i in range(len(labels)):
    ax = axes[i]
    ax.plot(samples[:, i], "k", alpha=0.5)
    ax.set_xlim(0, len(samples))
    ax.set_ylabel(labels[i])
    ax.yaxis.set_label_coords(-0.1, 0.5)

axes[-1].set_xlabel("step number")

Text(0.5, 0, 'step number')

../_images/83e022ee7035c4ac30353863a6dacf3bdd2e24fbf98db0f6fbc502d20f9c6c1b.png

5. Plot some simulated time series to display uncertainty?#

We can now draw parameter sets from the chain and simulate the uncertainty in the head simulation.

# Plot results and uncertainty
ax = ml.plot(figsize=(10, 3))
plt.title(None)

chain = ml.solver.sampler.get_chain(flat=True, discard=50)
inds = np.random.randint(len(chain), size=100)
for ind in inds:
    params = chain[ind]
    p = ml.parameters.optimal.copy().to_numpy(copy=True)
    p[ml.parameters.vary] = params[: ml.parameters.vary.sum()]
    _ = ml.simulate(p, tmin="1990").plot(c="gray", alpha=0.1, zorder=-1)

plt.legend(["Measurements", "Simulation", "Ensemble members"], numpoints=3);

../_images/97a33cef28ce00d67696c5cf3fb3682339e1df7045e334023ca400942da32c83.png