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Fix typos and prettify code in ar1_turningpts.md #251

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101 changes: 49 additions & 52 deletions lectures/ar1_turningpts.md
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Original file line number Diff line number Diff line change
Expand Up @@ -31,9 +31,9 @@ We consider two sorts of statistics:

- prospective values $y_{t+j}$ of a random process $\{y_t\}$ that is governed by the AR(1) process

- sample path properties that are defined as non-linear functions of future values $\{y_{t+j}\}_{j \geq 1}$ at time $t$.
- sample path properties that are defined as non-linear functions of future values $\{y_{t+j}\}_{j \geq 1}$ at time $t$

**Sample path properties** are things like "time to next turning point" or "time to next recession"
**Sample path properties** are things like "time to next turning point" or "time to next recession".

To investigate sample path properties we'll use a simulation procedure recommended by Wecker {cite}`wecker1979predicting`.

Expand All @@ -43,17 +43,14 @@ Let's start with some imports.

```{code-cell} ipython3
import numpy as np

import arviz as az
import pymc as pmc

import matplotlib.pyplot as plt
import seaborn as sns

sns.set_style('white')
colors = sns.color_palette()


import logging
logging.basicConfig()
logger = logging.getLogger('pymc')
Expand Down Expand Up @@ -106,9 +103,9 @@ We also want to compute some predictive distributions of "sample path statistic
- the time until the next "recession",
- the minimum value of $Y$ over the next 8 periods,
- "severe recession", and
- the time until the next turning point (positive or negative)
- the time until the next turning point (positive or negative).

To accomplish that for situations in which we are uncertain about parameter values, we shall extend the approach Wecker {cite}`wecker1979predicting` in the following way.
To accomplish that for situations in which we are uncertain about parameter values, we shall extend Wecker's {cite}`wecker1979predicting` approach in the following way.

- first simulate an initial path of length $T_0$;
- for a given prior, draw a sample of size $N$ from the posterior joint distribution of parameters $\left(\rho,\sigma\right)$ after observing the initial path;
Expand All @@ -130,12 +127,12 @@ def AR1_simulate(rho, sigma, y0, T):

# Allocate space and draw epsilons
y = np.empty(T)
eps = np.random.normal(0,sigma,T)
eps = np.random.normal(0, sigma, T)

# Initial condition and step forward
y[0] = y0
for t in range(1, T):
y[t] = rho*y[t-1] + eps[t]
y[t] = rho * y[t-1] + eps[t]

return y

Expand All @@ -144,26 +141,26 @@ def plot_initial_path(initial_path):
"""
Plot the initial path and the preceding predictive densities
"""
# compute .9 confidence interval]
# Compute .9 confidence interval]
y0 = initial_path[-1]
center = np.array([rho**j*y0 for j in range(T1)])
vars = np.array([sigma**2*(1-rho**(2*j))/(1-rho**2) for j in range(T1)])
center = np.array([rho**j * y0 for j in range(T1)])
vars = np.array([sigma**2 * (1 - rho**(2 * j)) / (1 - rho**2) for j in range(T1)])
y_bounds1_c95, y_bounds2_c95 = center + 1.96 * np.sqrt(vars), center - 1.96 * np.sqrt(vars)
y_bounds1_c90, y_bounds2_c90 = center + 1.65 * np.sqrt(vars), center - 1.65 * np.sqrt(vars)

# plot
fig, ax = plt.subplots(1,1, figsize = (12, 6))
# Plot
fig, ax = plt.subplots(1, 1, figsize=(12, 6))
ax.set_title("Initial Path and Predictive Densities", fontsize=15)
ax.plot(np.arange(-T0+1, 1), initial_path)
ax.plot(np.arange(-T0 + 1, 1), initial_path)
ax.set_xlim([-T0, T1])
ax.axvline(0, linestyle='--', alpha=.4, color='k', lw=1)

# simulate future paths
# Simulate future paths
for i in range(10):
y_future = AR1_simulate(rho, sigma, y0, T1)
ax.plot(np.arange(T1), y_future, color='grey', alpha=.5)

# plot 90% CI
# Plot 90% CI
ax.fill_between(np.arange(T1), y_bounds1_c95, y_bounds2_c95, alpha=.3, label='95% CI')
ax.fill_between(np.arange(T1), y_bounds1_c90, y_bounds2_c90, alpha=.35, label='90% CI')
ax.plot(np.arange(T1), center, color='red', alpha=.7, label='expected mean')
Expand All @@ -176,11 +173,11 @@ rho = 0.9
T0, T1 = 100, 100
y0 = 10

# simulate
# Simulate
np.random.seed(145)
initial_path = AR1_simulate(rho, sigma, y0, T0)

# plot
# Plot
plot_initial_path(initial_path)
```

Expand All @@ -196,7 +193,7 @@ He called these functions "path properties" to contrast them with properties of

He studied two special prospective path properties of a given series $\{y_t\}$.

The first was **time until the next turning point**
The first was **time until the next turning point**.

* he defined a **"turning point"** to be the date of the second of two successive declines in $y$.

Expand Down Expand Up @@ -227,7 +224,7 @@ W_t(\omega):= \inf \{ k\geq 1 \mid Z_{t+k}(\omega) = 1\}
$$

Wecker {cite}`wecker1979predicting` also studied **the minimum value of $Y$ over the next 8 quarters**
which can be defined as the random variable
which can be defined as the random variable.

$$
M_t(\omega) := \min \{ Y_{t+1}(\omega); Y_{t+2}(\omega); \dots; Y_{t+8}(\omega)\}
Expand Down Expand Up @@ -316,22 +313,22 @@ def draw_from_posterior(sample):

with AR1_model:

#start with priors
rho = pmc.Uniform('rho',lower=-1.,upper=1.) #assume stable rho
# Start with priors
rho = pmc.Uniform('rho',lower=-1.,upper=1.) # Assume stable rho
sigma = pmc.HalfNormal('sigma', sigma = np.sqrt(10))

# Expected value of y at the next period (rho * y)
yhat = rho*sample[:-1]
yhat = rho * sample[:-1]

# Likelihood of the actual realization.
y_like = pmc.Normal('y_obs', mu = yhat, sigma=sigma, observed=sample[1:])
y_like = pmc.Normal('y_obs', mu=yhat, sigma=sigma, observed=sample[1:])

with AR1_model:
trace = pmc.sample(10000, tune=5000)

# check condition
with AR1_model:
az.plot_trace(trace, figsize=(17,6))
az.plot_trace(trace, figsize=(17, 6))

rhos = trace.posterior.rho.values.flatten()
sigmas = trace.posterior.sigma.values.flatten()
Expand All @@ -350,7 +347,7 @@ The graphs on the left portray posterior marginal distributions.

## Calculating Sample Path Statistics

Our next step is to prepare Python codeto compute our sample path statistics.
Our next step is to prepare Python code to compute our sample path statistics.

```{code-cell} ipython3
# define statistics
Expand All @@ -374,9 +371,9 @@ def severe_recession(omega):
n = z.shape[0]

sr = (z < -.02).astype(int)
indices = np.where(sr==1)[0]
indices = np.where(sr == 1)[0]

if len(indices)==0:
if len(indices) == 0:
return T1
else:
return indices[0] + 1
Expand All @@ -401,8 +398,8 @@ def next_turning_point(omega):
(omega[i+2] > omega[i+3]) and (omega[i+3] > omega[i+4])):
T[i] = -1

up_turn = np.where(T==1)[0][0] + 1 if (1 in T) == True else T1
down_turn = np.where(T==-1)[0][0] + 1 if (-1 in T) == True else T1
up_turn = np.where(T == 1)[0][0] + 1 if (1 in T) == True else T1
down_turn = np.where(T == -1)[0][0] + 1 if (-1 in T) == True else T1

return up_turn, down_turn
```
Expand All @@ -417,31 +414,31 @@ def plot_Wecker(initial_path, N, ax):
"""
Plot the predictive distributions from "pure" Wecker's method.
"""
# store outcomes
# Store outcomes
next_reces = np.zeros(N)
severe_rec = np.zeros(N)
min_vals = np.zeros(N)
next_up_turn, next_down_turn = np.zeros(N), np.zeros(N)

# compute .9 confidence interval]
# Compute .9 confidence interval]
y0 = initial_path[-1]
center = np.array([rho**j*y0 for j in range(T1)])
vars = np.array([sigma**2*(1-rho**(2*j))/(1-rho**2) for j in range(T1)])
center = np.array([rho**j * y0 for j in range(T1)])
vars = np.array([sigma**2 * (1 - rho**(2 * j)) / (1 - rho**2) for j in range(T1)])
y_bounds1_c95, y_bounds2_c95 = center + 1.96 * np.sqrt(vars), center - 1.96 * np.sqrt(vars)
y_bounds1_c90, y_bounds2_c90 = center + 1.65 * np.sqrt(vars), center - 1.65 * np.sqrt(vars)

# plot
# Plot
ax[0, 0].set_title("Initial path and predictive densities", fontsize=15)
ax[0, 0].plot(np.arange(-T0+1, 1), initial_path)
ax[0, 0].plot(np.arange(-T0 + 1, 1), initial_path)
ax[0, 0].set_xlim([-T0, T1])
ax[0, 0].axvline(0, linestyle='--', alpha=.4, color='k', lw=1)

# plot 90% CI
# Plot 90% CI
ax[0, 0].fill_between(np.arange(T1), y_bounds1_c95, y_bounds2_c95, alpha=.3)
ax[0, 0].fill_between(np.arange(T1), y_bounds1_c90, y_bounds2_c90, alpha=.35)
ax[0, 0].plot(np.arange(T1), center, color='red', alpha=.7)

# simulate future paths
# Simulate future paths
for n in range(N):
sim_path = AR1_simulate(rho, sigma, initial_path[-1], T1)
next_reces[n] = next_recession(np.hstack([initial_path[-3:-1], sim_path]))
Expand All @@ -452,7 +449,7 @@ def plot_Wecker(initial_path, N, ax):
if n%(N/10) == 0:
ax[0, 0].plot(np.arange(T1), sim_path, color='gray', alpha=.3, lw=1)

# return next_up_turn, next_down_turn
# Return next_up_turn, next_down_turn
sns.histplot(next_reces, kde=True, stat='density', ax=ax[0, 1], alpha=.8, label='True parameters')
ax[0, 1].set_title("Predictive distribution of time until the next recession", fontsize=13)

Expand All @@ -478,42 +475,42 @@ plt.show()
Now we apply we apply our "extended" Wecker method based on predictive densities of $y$ defined by
{eq}`ar1-tp-eq4` that acknowledge posterior uncertainty in the parameters $\rho, \sigma$.

To approximate the intergration on the right side of {eq}`ar1-tp-eq4`, we repeately draw parameters from the joint posterior distribution each time we simulate a sequence of future values from model {eq}`ar1-tp-eq1`.
To approximate the intergration on the right side of {eq}`ar1-tp-eq4`, we repeatedly draw parameters from the joint posterior distribution each time we simulate a sequence of future values from model {eq}`ar1-tp-eq1`.

```{code-cell} ipython3
def plot_extended_Wecker(post_samples, initial_path, N, ax):
"""
Plot the extended Wecker's predictive distribution
"""
# select a sample
index = np.random.choice(np.arange(len(post_samples['rho'])), N+1, replace=False)
# Select a sample
index = np.random.choice(np.arange(len(post_samples['rho'])), N + 1, replace=False)
rho_sample = post_samples['rho'][index]
sigma_sample = post_samples['sigma'][index]

# store outcomes
# Store outcomes
next_reces = np.zeros(N)
severe_rec = np.zeros(N)
min_vals = np.zeros(N)
next_up_turn, next_down_turn = np.zeros(N), np.zeros(N)

# plot
# Plot
ax[0, 0].set_title("Initial path and future paths simulated from posterior draws", fontsize=15)
ax[0, 0].plot(np.arange(-T0+1, 1), initial_path)
ax[0, 0].plot(np.arange(-T0 + 1, 1), initial_path)
ax[0, 0].set_xlim([-T0, T1])
ax[0, 0].axvline(0, linestyle='--', alpha=.4, color='k', lw=1)

# simulate future paths
# Simulate future paths
for n in range(N):
sim_path = AR1_simulate(rho_sample[n], sigma_sample[n], initial_path[-1], T1)
next_reces[n] = next_recession(np.hstack([initial_path[-3:-1], sim_path]))
severe_rec[n] = severe_recession(sim_path)
min_vals[n] = minimum_value(sim_path)
next_up_turn[n], next_down_turn[n] = next_turning_point(sim_path)

if n%(N/10) == 0:
if n % (N / 10) == 0:
ax[0, 0].plot(np.arange(T1), sim_path, color='gray', alpha=.3, lw=1)

# return next_up_turn, next_down_turn
# Return next_up_turn, next_down_turn
sns.histplot(next_reces, kde=True, stat='density', ax=ax[0, 1], alpha=.6, color=colors[1], label='Sampling from posterior')
ax[0, 1].set_title("Predictive distribution of time until the next recession", fontsize=13)

Expand All @@ -529,19 +526,19 @@ def plot_extended_Wecker(post_samples, initial_path, N, ax):
sns.histplot(next_down_turn, kde=True, stat='density', ax=ax[2, 1], alpha=.6, color=colors[1], label='Sampling from posterior')
ax[2, 1].set_title("Predictive distribution of time until the next negative turn", fontsize=13)

fig, ax = plt.subplots(3, 2, figsize=(15,12))
fig, ax = plt.subplots(3, 2, figsize=(15, 12))
plot_extended_Wecker(post_samples, initial_path, 1000, ax)
plt.show()
```

## Comparison

Finally, we plot both the original Wecker method and the extended method with parameter values drawn from the posterior together to compare the differnces that emerge from pretending to know parameter values when they are actually uncertain.
Finally, we plot both the original Wecker method and the extended method with parameter values drawn from the posterior together to compare the differences that emerge from pretending to know parameter values when they are actually uncertain.

```{code-cell} ipython3
fig, ax = plt.subplots(3, 2, figsize=(15,12))
plot_Wecker(initial_path, 1000, ax)
ax[0,0].clear()
ax[0, 0].clear()
plot_extended_Wecker(post_samples, initial_path, 1000, ax)
plt.legend()
plt.show()
Expand Down