Removed bivariate example; updated formatting

examples/time_series/Euler-Maruyama_and_SDEs.myst.md

@@ -32,12 +32,20 @@ run_control:
 slideshow:
   slide_type: '-'
 ---
+import warnings
+
 import arviz as az
 import matplotlib.pyplot as plt
 import numpy as np
 import pymc as pm
 import pytensor.tensor as pt
 import scipy as sp
+
+# Ignore UserWarnings
+warnings.filterwarnings("ignore", category=UserWarning)
+
+RANDOM_SEED = 8927
+np.random.seed(RANDOM_SEED)
 ```
 
 ```{code-cell} ipython3
@@ -96,16 +104,19 @@ run_control:
 slideshow:
   slide_type: subslide
 ---
-plt.figure(figsize=(10, 3))
-plt.plot(x_t[:30], "k", label="$x(t)$", alpha=0.5)
-plt.plot(z_t[:30], "r", label="$z(t)$", alpha=0.5)
-plt.title("Transient")
-plt.legend()
-plt.subplot(122)
-plt.plot(x_t[30:], "k", label="$x(t)$", alpha=0.5)
-plt.plot(z_t[30:], "r", label="$z(t)$", alpha=0.5)
-plt.title("All time")
-plt.legend();
+fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 3))
+
+ax1.plot(x_t[:30], "k", label="$x(t)$", alpha=0.5)
+ax1.plot(z_t[:30], "r", label="$z(t)$", alpha=0.5)
+ax1.set_title("Transient")
+ax1.legend()
+
+ax2.plot(x_t[30:], "k", label="$x(t)$", alpha=0.5)
+ax2.plot(z_t[30:], "r", label="$z(t)$", alpha=0.5)
+ax2.set_title("All time")
+ax2.legend()
+
+plt.tight_layout()
 ```
 
 +++ {"button": false, "new_sheet": false, "run_control": {"read_only": false}}
@@ -123,7 +134,7 @@ new_sheet: false
 run_control:
   read_only: false
 ---
-def lin_sde(x, lam):
+def lin_sde(x, lam, s2):
     return lam * x, s2
 ```
 
@@ -144,11 +155,12 @@ slideshow:
 ---
 with pm.Model() as model:
     # uniform prior, but we know it must be negative
-    l = pm.Flat("l")
+    l = pm.HalfCauchy("l", beta=1)
+    s = pm.Uniform("s", 0.005, 0.5)
 
     # "hidden states" following a linear SDE distribution
     # parametrized by time step (det. variable) and lam (random variable)
-    xh = pm.EulerMaruyama("xh", dt=dt, sde_fn=lin_sde, sde_pars=(l,), shape=N)
+    xh = pm.EulerMaruyama("xh", dt=dt, sde_fn=lin_sde, sde_pars=(-l, s**2), shape=N, initval=x_t)
 
     # predicted observation
     zh = pm.Normal("zh", mu=xh, sigma=5e-3, observed=z_t)
@@ -166,7 +178,7 @@ run_control:
   read_only: false
 ---
 with model:
-    trace = pm.sample()
+    trace = pm.sample(nuts_sampler="nutpie", random_seed=RANDOM_SEED, target_accept=0.99)
 ```
 
 +++ {"button": false, "new_sheet": false, "run_control": {"read_only": false}, "slideshow": {"slide_type": "subslide"}}
@@ -185,7 +197,7 @@ plt.plot(x_t, "r", label="$x(t)$")
 plt.legend()
 
 plt.subplot(122)
-plt.hist(az.extract(trace.posterior)["l"], 30, label=r"$\hat{\lambda}$", alpha=0.5)
+plt.hist(-1 * az.extract(trace.posterior)["l"], 30, label=r"$\hat{\lambda}$", alpha=0.5)
 plt.axvline(lam, color="r", label=r"$\lambda$", alpha=0.5)
 plt.legend();
 ```
@@ -218,119 +230,6 @@ plt.legend();
 
 +++ {"button": false, "new_sheet": false, "run_control": {"read_only": false}}
 
-Note that 
-
-- inference also estimates the initial conditions
-- the observed data $z(t)$ lies fully within the 95% interval of the PPC.
-- there are many other ways of evaluating fit
-
-+++ {"button": false, "new_sheet": false, "run_control": {"read_only": false}, "slideshow": {"slide_type": "slide"}}
-
-### Toy model 2
-
-As the next model, let's use a 2D deterministic oscillator, 
-\begin{align}
-\dot{x} &= \tau (x - x^3/3 + y) \\
-\dot{y} &= \frac{1}{\tau} (a - x)
-\end{align}
-
-with noisy observation $z(t) = m x + (1 - m) y + N(0, 0.05)$.
-
-```{code-cell} ipython3
-N, tau, a, m, s2 = 200, 3.0, 1.05, 0.2, 1e-1
-xs, ys = [0.0], [1.0]
-for i in range(N):
-    x, y = xs[-1], ys[-1]
-    dx = tau * (x - x**3.0 / 3.0 + y)
-    dy = (1.0 / tau) * (a - x)
-    xs.append(x + dt * dx + np.sqrt(dt) * s2 * np.random.randn())
-    ys.append(y + dt * dy + np.sqrt(dt) * s2 * np.random.randn())
-xs, ys = np.array(xs), np.array(ys)
-zs = m * xs + (1 - m) * ys + np.random.randn(xs.size) * 0.1
-
-plt.figure(figsize=(10, 2))
-plt.plot(xs, label="$x(t)$")
-plt.plot(ys, label="$y(t)$")
-plt.plot(zs, label="$z(t)$")
-plt.legend()
-```
-
-+++ {"button": false, "new_sheet": false, "run_control": {"read_only": false}, "slideshow": {"slide_type": "subslide"}}
-
-Now, estimate the hidden states $x(t)$ and $y(t)$, as well as parameters $\tau$, $a$ and $m$.
-
-As before, we rewrite our SDE as a function returned drift & diffusion coefficients:
-
-```{code-cell} ipython3
----
-button: false
-new_sheet: false
-run_control:
-  read_only: false
----
-def osc_sde(xy, tau, a):
-    x, y = xy[:, 0], xy[:, 1]
-    dx = tau * (x - x**3.0 / 3.0 + y)
-    dy = (1.0 / tau) * (a - x)
-    dxy = pt.stack([dx, dy], axis=0).T
-    return dxy, s2
-```
-
-+++ {"button": false, "new_sheet": false, "run_control": {"read_only": false}}
-
-As before, the Euler-Maruyama discretization of the SDE is written as a prediction of the state at step $i+1$ based on the state at step $i$.
-
-+++ {"button": false, "new_sheet": false, "run_control": {"read_only": false}, "slideshow": {"slide_type": "subslide"}}
-
-We can now write our statistical model as before, with uninformative priors on $\tau$, $a$ and $m$:
-
-```{code-cell} ipython3
----
-button: false
-new_sheet: false
-run_control:
-  read_only: false
----
-xys = np.c_[xs, ys]
-
-with pm.Model() as model:
-    tau_h = pm.Uniform("tau_h", lower=0.1, upper=5.0)
-    a_h = pm.Uniform("a_h", lower=0.5, upper=1.5)
-    m_h = pm.Uniform("m_h", lower=0.0, upper=1.0)
-    xy_h = pm.EulerMaruyama(
-        "xy_h", dt=dt, sde_fn=osc_sde, sde_pars=(tau_h, a_h), shape=xys.shape, initval=xys
-    )
-    zh = pm.Normal("zh", mu=m_h * xy_h[:, 0] + (1 - m_h) * xy_h[:, 1], sigma=0.1, observed=zs)
-```
-
-```{code-cell} ipython3
-pm.__version__
-```
-
-```{code-cell} ipython3
----
-button: false
-new_sheet: false
-run_control:
-  read_only: false
----
-with model:
-    pm.sample_posterior_predictive(trace, extend_inferencedata=True)
-```
-
-```{code-cell} ipython3
-plt.figure(figsize=(10, 3))
-plt.plot(
-    trace.posterior_predictive.quantile((0.025, 0.975), dim=("chain", "draw"))["zh"].values.T,
-    "k",
-    label=r"$z_{95\% PP}(t)$",
-)
-plt.plot(z_t, "r", label="$z(t)$")
-plt.legend();
-```
-
-+++ {"button": false, "new_sheet": false, "run_control": {"read_only": false}}
-
 Note that the initial conditions are also estimated, and that most of the observed data $z(t)$ lies within the 95% interval of the PPC.
 
 Another approach is to look at draws from the sampling distribution of the data relative to the observed data. This too shows a good fit across the range of observations -- the posterior predictive mean almost perfectly tracks the data.
@@ -350,6 +249,17 @@ az.plot_ppc(trace)
 :filter: docname in docnames
 :::
 
+## Authors
+- Authored by @maedoc in July 2016
+- Updated to PyMC v5 by @fonnesbeck in September 2024
+
++++
+
+## References
+:::{bibliography}
+:filter: docname in docnames
+:::
+
 ```{code-cell} ipython3
 %load_ext watermark
 %watermark -n -u -v -iv -w