Additional edits

examples/time_series/Euler-Maruyama_and_SDEs.myst.md

@@ -32,20 +32,12 @@ 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
@@ -104,19 +96,16 @@ run_control:
 slideshow:
   slide_type: subslide
 ---
-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()
+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();
 ```
 
 +++ {"button": false, "new_sheet": false, "run_control": {"read_only": false}}
@@ -134,7 +123,7 @@ new_sheet: false
 run_control:
   read_only: false
 ---
-def lin_sde(x, lam, s2):
+def lin_sde(x, lam):
     return lam * x, s2
 ```
 
@@ -155,12 +144,11 @@ slideshow:
 ---
 with pm.Model() as model:
     # uniform prior, but we know it must be negative
-    l = pm.HalfCauchy("l", beta=1)
-    s = pm.Uniform("s", 0.005, 0.5)
+    l = pm.Flat("l")
 
     # "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, s**2), shape=N, initval=x_t)
+    xh = pm.EulerMaruyama("xh", dt=dt, sde_fn=lin_sde, sde_pars=(l,), shape=N)
 
     # predicted observation
     zh = pm.Normal("zh", mu=xh, sigma=5e-3, observed=z_t)
@@ -178,7 +166,7 @@ run_control:
   read_only: false
 ---
 with model:
-    trace = pm.sample(nuts_sampler="nutpie", random_seed=RANDOM_SEED, target_accept=0.99)
+    trace = pm.sample()
 ```
 
 +++ {"button": false, "new_sheet": false, "run_control": {"read_only": false}, "slideshow": {"slide_type": "subslide"}}
@@ -197,7 +185,7 @@ plt.plot(x_t, "r", label="$x(t)$")
 plt.legend()
 
 plt.subplot(122)
-plt.hist(-1 * az.extract(trace.posterior)["l"], 30, label=r"$\hat{\lambda}$", alpha=0.5)
+plt.hist(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();
 ```
@@ -230,6 +218,119 @@ 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.