Uncertainty quantification#
R.A. Collenteur
In this notebook it is shown how to compute the uncertainty of the model simulation using the built-in uncertainty quantification options of Pastas.
Confidence interval of simulation
Prediction interval of simulation
Confidence interval of step response
Confidence interval of block response
Confidence interval of contribution
Custom confidence intervals
The compute the confidence intervals, parameters sets are drawn from a multivariate normal distribution based on the jacobian matrix obtained during parameter optimization. This method to quantify uncertainties has some underlying assumptions on the model residuals (or noise) that should be checked. This notebook only deals with parameter uncertainties and not with model structure uncertainties.
import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
from scipy.stats import norm
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
Create a model#
We first create a toy model to simulate the groundwater levels in southeastern Austria. We will use this model to illustrate how the different methods for uncertainty quantification can be used.
gwl = (
pd.read_csv("data_wagna/head_wagna.csv", index_col=0, parse_dates=True, skiprows=2)
.squeeze()
.loc["2006":]
.iloc[0::10]
)
gwl = gwl.loc[~gwl.index.duplicated(keep="first")]
evap = pd.read_csv(
"data_wagna/evap_wagna.csv", index_col=0, parse_dates=True, skiprows=2
).squeeze()
prec = pd.read_csv(
"data_wagna/rain_wagna.csv", index_col=0, parse_dates=True, skiprows=2
).squeeze()
# Model settings
tmin = pd.Timestamp("2007-01-01") # Needs warmup
tmax = pd.Timestamp("2016-12-31")
ml = ps.Model(gwl)
sm = ps.RechargeModel(
prec, evap, recharge=ps.rch.FlexModel(), rfunc=ps.Exponential(), name="rch"
)
ml.add_stressmodel(sm)
# Add the ARMA(1,1) noise model and solve the Pastas model
ml.add_noisemodel(ps.ArmaNoiseModel())
ml.solve(tmin=tmin, tmax=tmax, report="full")
Fit report GWL Fit Statistics
======================================================
nfev 34 EVP 77.36
nobs 365 R2 0.77
noise True RMSE 0.18
tmin 2007-01-01 00:00:00 AICc -2053.02
tmax 2016-12-31 00:00:00 BIC -2018.42
freq D Obj 0.63
freq_obs None ___
warmup 3650 days 00:00:00 Interp. No
Parameters (9 optimized)
======================================================
optimal initial vary stderr
rch_A 0.638119 0.529381 True ±13.10%
rch_a 74.655762 10.000000 True ±15.26%
rch_srmax 364.057456 250.000000 True ±41.22%
rch_lp 0.250000 0.250000 False nan
rch_ks 353.703440 100.000000 True ±153.63%
rch_gamma 3.804077 2.000000 True ±15.06%
rch_kv 1.283578 1.000000 True ±18.34%
rch_simax 2.000000 2.000000 False nan
constant_d 262.598597 263.166264 True ±3.25e-02%
noise_alpha 111.959374 10.000000 True ±28.47%
noise_beta 8.261044 1.000000 True ±14.93%
Parameter correlations |rho| > 0.5
======================================================
rch_A rch_a 0.80
rch_A constant_d -0.74
rch_a constant_d -0.70
rch_srmax rch_ks 0.97
rch_srmax rch_gamma 0.58
rch_srmax rch_kv 0.66
rch_ks rch_gamma 0.73
rch_ks rch_kv 0.58
Diagnostic Checks#
Before we perform the uncertainty quantification, we should check if the underlying statistical assumptions are met. We refer to the notebook on Diagnostic checking for more details on this.
ml.plots.diagnostics();
Confidence intervals#
After the model is calibrated, a solver attribute is added to the Pastas Model object (ml.solver). This object contains information about the optimizations (e.g., the jacobian matrix) and a number of methods that can be used to quantify uncertainties.
ci = ml.solver.ci_simulation(alpha=0.05, n=1000)
ax = ml.plot(figsize=(10, 3))
ax.fill_between(ci.index, ci.iloc[:, 0], ci.iloc[:, 1], color="lightgray")
ax.legend(["Observations", "Simulation", "95% Confidence interval"], ncol=3, loc=2)
<matplotlib.legend.Legend at 0x7d4cbd7f1d30>
Prediction interval#
ci = ml.solver.prediction_interval(n=1000)
ax = ml.plot(figsize=(10, 3))
ax.fill_between(ci.index, ci.iloc[:, 0], ci.iloc[:, 1], color="lightgray")
ax.legend(["Observations", "Simulation", "95% Prediction interval"], ncol=3, loc=2)
<matplotlib.legend.Legend at 0x7d4cc42596a0>
Checking the quality of the prediction interval#
We can compute the PICP to see what percentage of the measurements are within the 95% prediction interval, which should theoretically be around 0.95.
ps.stats.picp(gwl[ci.index[0] : ci.index[-1]], ci)
np.float64(0.9780821917808219)
Uncertainty of step response#
ci = ml.solver.ci_step_response("rch")
ax = ml.plots.step_response(figsize=(6, 2))
ax.fill_between(ci.index, ci.iloc[:, 0], ci.iloc[:, 1], color="lightgray")
ax.legend(["Simulation", "95% Confidence interval"], ncol=3, loc=4)
<matplotlib.legend.Legend at 0x7d4cc444cec0>
Uncertainty of block response#
ci = ml.solver.ci_block_response("rch")
ax = ml.plots.block_response(figsize=(6, 2))
ax.fill_between(ci.index, ci.iloc[:, 0], ci.iloc[:, 1], color="lightgray")
ax.legend(["Simulation", "95% Confidence interval"], ncol=3, loc=1)
<matplotlib.legend.Legend at 0x7d4cc42156a0>
Uncertainty of the contributions#
ci = ml.solver.ci_contribution("rch")
r = ml.get_contribution("rch")
ax = r.plot(figsize=(10, 3))
ax.fill_between(ci.index, ci.iloc[:, 0], ci.iloc[:, 1], color="lightgray")
ax.legend(["Simulation", "95% Confidence interval"], ncol=3, loc=1)
plt.tight_layout()
Custom Confidence intervals#
It is also possible to compute the confidence intervals manually, for example to estimate the uncertainty in the recharge or statistics (e.g., SGI, NSE). We can call ml.solver.get_parameter_sample to obtain random parameter samples from a multivariate normal distribution using the optimal parameters and the covariance matrix. Next, we use the parameter sets to obtain multiple simulations of ‘something’, here the recharge.
params = ml.solver.get_parameter_sample(n=1000, name="rch")
data = {}
# Here we run the model n times with different parameter samples
for i, param in enumerate(params):
data[i] = ml.stressmodels["rch"].get_stress(p=param)
df = pd.DataFrame.from_dict(data, orient="columns").loc[tmin:tmax].resample("YE").sum()
ci = df.quantile([0.025, 0.975], axis=1).transpose()
r = ml.get_stress("rch").resample("YE").sum()
ax = r.plot.bar(figsize=(10, 2), width=0.5, yerr=[r - ci.iloc[:, 0], ci.iloc[:, 1] - r])
ax.set_xticklabels(labels=r.index.year, rotation=0, ha="center")
ax.set_ylabel("Recharge [mm a$^{-1}$]")
ax.legend(ncol=3);
Uncertainty of the NSE#
The code pattern shown above can be used for many types of uncertainty analyses. Another example is provided below, where we compute the uncertainty of the Nash-Sutcliffe efficacy.
params = ml.solver.get_parameter_sample(n=1000)
data = []
# Here we run the model n times with different parameter samples
for i, param in enumerate(params):
sim = ml.simulate(p=param)
data.append(ps.stats.nse(obs=ml.observations(), sim=sim))
fig, ax = plt.subplots(1, 1, figsize=(4, 3))
plt.hist(data, bins=50, density=True)
ax.axvline(ml.stats.nse(), linestyle="--", color="k")
ax.set_xlabel("NSE [-]")
ax.set_ylabel("frequency [-]")
mu, std = norm.fit(data)
# Plot the PDF.
xmin, xmax = ax.set_xlim()
x = np.linspace(xmin, xmax, 100)
p = norm.pdf(x, mu, std)
ax.plot(x, p, "k", linewidth=2)
[<matplotlib.lines.Line2D at 0x7d4cbff9dfd0>]
