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---

In addition to what's in Anaconda, this lecture will need the following libraries:

```{code-cell} ipython

---

tags: [hide-output]

---

```{code-cell} ipython3

:tags: [hide-output]

!pip install quantecon

```

We'll need the following imports:

```{code-cell} ipython

```{code-cell} ipython3

import matplotlib.pyplot as plt

plt.rcParams["figure.figsize"] = (11, 5) #set default figure size

from scipy import linalg

import numpy as np

import matplotlib.cm as cm

This density $p(x)$ is shown below as a contour map, with the center of the red ellipse being equal to $\hat x$.

```{code-cell} python3

---

tags: [output_scroll]

---

```{code-cell} ipython3

:tags: [output_scroll]

# Set up the Gaussian prior density p

Σ = [[0.4, 0.3], [0.3, 0.45]]

Σ = np.matrix(Σ)

def gen_gaussian_plot_vals(μ, C):

"Z values for plotting the bivariate Gaussian N(μ, C)"

m_x, m_y = float(μ[0]), float(μ[1])

m_x, m_y = float(μ[0,0]), float(μ[1,0])

s_x, s_y = np.sqrt(C[0, 0]), np.sqrt(C[1, 1])

s_xy = C[0, 1]

return bivariate_normal(X, Y, s_x, s_y, m_x, m_y, s_xy)

The next figure shows the original prior $p(x)$ and the new reported

location $y$

```{code-cell} python3

```{code-cell} ipython3

fig, ax = plt.subplots(figsize=(10, 8))

ax.grid()

Z = gen_gaussian_plot_vals(x_hat, Σ)

ax.contourf(X, Y, Z, 6, alpha=0.6, cmap=cm.jet)

cs = ax.contour(X, Y, Z, 6, colors="black")

ax.clabel(cs, inline=1, fontsize=10)

ax.text(float(y[0]), float(y[1]), "$y$", fontsize=20, color="black")

ax.text(float(y[0].item()), float(y[1].item()), "$y$", fontsize=20, color="black")

plt.show()

```

The original density is left in as contour lines for comparison

```{code-cell} python3

```{code-cell} ipython3

fig, ax = plt.subplots(figsize=(10, 8))

ax.grid()

cs2 = ax.contour(X, Y, new_Z, 6, colors="black")

ax.clabel(cs2, inline=1, fontsize=10)

ax.contourf(X, Y, new_Z, 6, alpha=0.6, cmap=cm.jet)

ax.text(float(y[0]), float(y[1]), "$y$", fontsize=20, color="black")

ax.text(float(y[0].item()), float(y[1].item()), "$y$", fontsize=20, color="black")

plt.show()

```

Q = 0.3 * \Sigma

$$

```{code-cell} python3

```{code-cell} ipython3

fig, ax = plt.subplots(figsize=(10, 8))

ax.grid()

cs3 = ax.contour(X, Y, new_Z, 6, colors="black")

ax.clabel(cs3, inline=1, fontsize=10)

ax.contourf(X, Y, new_Z, 6, alpha=0.6, cmap=cm.jet)

ax.text(float(y[0]), float(y[1]), "$y$", fontsize=20, color="black")

ax.text(float(y[0].item()), float(y[1].item()), "$y$", fontsize=20, color="black")

plt.show()

```

:class: dropdown

```

```{code-cell} python3

```{code-cell} ipython3

# Parameters

θ = 10 # Constant value of state x_t

A, C, G, H = 1, 0, 1, 1

for i in range(N):

# Record the current predicted mean and variance

m, v = [float(z) for z in (kalman.x_hat, kalman.Sigma)]

m, v = [float(z) for z in (kalman.x_hat.item(), kalman.Sigma.item())]

# Plot, update filter

ax.plot(xgrid, norm.pdf(xgrid, loc=m, scale=np.sqrt(v)), label=f'$t={i}$')

kalman.update(y[i])

:class: dropdown

```

```{code-cell} python3

```{code-cell} ipython3

ϵ = 0.1

θ = 10 # Constant value of state x_t

A, C, G, H = 1, 0, 1, 1

for t in range(T):

# Record the current predicted mean and variance and plot their densities

m, v = [float(temp) for temp in (kalman.x_hat, kalman.Sigma)]

m, v = [float(temp) for temp in (kalman.x_hat.item(), kalman.Sigma.item())]

f = lambda x: norm.pdf(x, loc=m, scale=np.sqrt(v))

integral, error = quad(f, θ - ϵ, θ + ϵ)

:class: dropdown

```

```{code-cell} python3

```{code-cell} ipython3

# Define A, C, G, H

G = np.identity(2)

H = np.sqrt(0.5) * np.identity(2)