TST: (styleguide) qe-writing-001 (sonnet 4.5)

Author: mmcky

lectures/additive_functionals.md

@@ -41,7 +41,9 @@ For example, outputs, prices, and dividends typically display  irregular but per
 
 Asymptotic stationarity and ergodicity are key assumptions needed to make it possible to learn by applying statistical methods.
 
-But  there are good ways to model time series that have persistent growth that still enable statistical learning based on a law of large numbers for an asymptotically stationary and ergodic process.
+But  there are good ways to model time series that have persistent growth.
+
+These models still enable statistical learning based on a law of large numbers for an asymptotically stationary and ergodic process.
 
 Thus, {cite}`Hansen_2012_Eca`  described  two classes of time series models that accommodate growth.
 
@@ -83,8 +85,9 @@ from scipy.stats import norm, lognorm
 
 This lecture focuses on a subclass of these: a scalar process $\{y_t\}_{t=0}^\infty$ whose increments are driven by a Gaussian vector autoregression.
 
-Our special additive functional displays interesting time series behavior while also being easy to construct, simulate, and analyze
-by using linear state-space tools.
+Our special additive functional displays interesting time series behavior.
+
+It is also easy to construct, simulate, and analyze by using linear state-space tools.
 
 We construct our  additive functional from two pieces, the first of which is a **first-order vector autoregression** (VAR)
 
@@ -186,7 +189,9 @@ which is a standard linear state space system.
 
 To study it, we could map it into an instance of [LinearStateSpace](https://github.com/QuantEcon/QuantEcon.py/blob/master/quantecon/lss.py) from [QuantEcon.py](http://quantecon.org/quantecon-py).
 
-But here we will use a different set of code for simulation, for reasons described below.
+But here we will use a different set of code for simulation.
+
+The reasons for this are described below.
 
 ## Dynamics
 
@@ -225,9 +230,13 @@ While {eq}`ftaf` is not a first order system like {eq}`old1_additive_functionals
 
 * For an example of such a mapping, see [this example](https://python.quantecon.org/linear_models.html#second-order-difference-equation).
 
-In fact, this whole model can be mapped into the additive functional system definition in {eq}`old1_additive_functionals` -- {eq}`old2_additive_functionals`  by appropriate selection of the matrices $A, B, D, F$.
+In fact, this whole model can be mapped into the additive functional system definition in {eq}`old1_additive_functionals` -- {eq}`old2_additive_functionals`.
 
-You can try writing these matrices down now as an exercise --- correct expressions appear in the code below.
+This requires appropriate selection of the matrices $A, B, D, F$.
+
+You can try writing these matrices down now as an exercise.
+
+Correct expressions appear in the code below.
 
 ### Simulation
 
@@ -755,12 +764,13 @@ It is convenient for us to introduce the following notation:
 
 We want to characterize and simulate components $\tau_t, m_t, s_t$ of the decomposition.
 
-A convenient way to do this is to construct an appropriate instance of a [linear state space system](https://python-intro.quantecon.org/linear_models.html) by using [LinearStateSpace](https://github.com/QuantEcon/QuantEcon.py/blob/master/quantecon/lss.py) from [QuantEcon.py](http://quantecon.org/quantecon-py).
+A convenient way to do this is to construct an appropriate instance of a [linear state space system](https://python-intro.quantecon.org/linear_models.html).
+
+We use [LinearStateSpace](https://github.com/QuantEcon/QuantEcon.py/blob/master/quantecon/lss.py) from [QuantEcon.py](http://quantecon.org/quantecon-py).
 
 This will allow us to use the routines in [LinearStateSpace](https://github.com/QuantEcon/QuantEcon.py/blob/master/quantecon/lss.py) to study dynamics.
 
-To start, observe that, under the dynamics in {eq}`old1_additive_functionals` and {eq}`old2_additive_functionals` and with the
-definitions just given,
+To start, observe that, under the dynamics in {eq}`old1_additive_functionals` and {eq}`old2_additive_functionals` and with the definitions just given, we have the following.
 
 $$
 \begin{bmatrix}
@@ -837,24 +847,21 @@ $$
 \end{aligned}
 $$
 
-By picking out components of $\tilde y_t$, we can track all variables of
-interest.
+By picking out components of $\tilde y_t$, we can track all variables of interest.
 
 ## Code
 
 The class `AMF_LSS_VAR` mentioned {ref}`above <amf_lss>` does all that we want to study our additive functional.
 
-In fact, `AMF_LSS_VAR` does more
-because it allows us to study  an associated multiplicative functional as well.
+In fact, `AMF_LSS_VAR` does more because it allows us to study  an associated multiplicative functional as well.
 
-(A hint that it does more is the name of the class -- here AMF stands for
-"additive and multiplicative functional" -- the code computes and displays objects associated with
-multiplicative functionals too.)
+(A hint that it does more is the name of the class -- here AMF stands for "additive and multiplicative functional".)
+
+The code computes and displays objects associated with multiplicative functionals too.
 
 Let's use this code (embedded above) to explore the {ref}`example process described above <addfunc_eg1>`.
 
-If you run {ref}`the code that first simulated that example <addfunc_egcode>` again and then the method call
-you will generate (modulo randomness) the plot
+If you run {ref}`the code that first simulated that example <addfunc_egcode>` again and then the method call you will generate (modulo randomness) the plot.
 
 ```{code-cell} ipython3
 plot_additive(amf, T)
@@ -863,14 +870,14 @@ plt.show()
 
 When we plot multiple realizations of a component in the 2nd, 3rd, and 4th panels, we also plot the population 95% probability coverage sets computed using the LinearStateSpace class.
 
-We have chosen to simulate many paths, all starting from the *same* non-random initial conditions $x_0, y_0$ (you can tell this from the shape of the 95% probability coverage shaded areas).
+We have chosen to simulate many paths, all starting from the *same* non-random initial conditions $x_0, y_0$.
+
+You can tell this from the shape of the 95% probability coverage shaded areas.
 
-Notice tell-tale signs of these probability coverage shaded areas
+Notice tell-tale signs of these probability coverage shaded areas:
 
-* the purple one for the martingale component $m_t$ grows with
-  $\sqrt{t}$
-* the green one for the stationary component $s_t$ converges to a
-  constant band
+* the purple one for the martingale component $m_t$ grows with $\sqrt{t}$
+* the green one for the stationary component $s_t$ converges to a constant band
 
 ### Associated multiplicative functional
 
@@ -909,8 +916,7 @@ An instance of class `AMF_LSS_VAR` ({ref}`above <amf_lss>`)  includes this assoc
 
 Let's plot this multiplicative functional for our example.
 
-If you run {ref}`the code that first simulated that example <addfunc_egcode>` again and then the method call in the cell below you'll
-obtain the graph in the next cell.
+If you run {ref}`the code that first simulated that example <addfunc_egcode>` again and then the method call in the cell below you'll obtain the graph in the next cell.
 
 ```{code-cell} ipython3
 plot_multiplicative(amf, T)
@@ -919,28 +925,25 @@ plt.show()
 
 As before, when we plotted multiple realizations of a component in the 2nd, 3rd, and 4th panels, we also plotted population 95% confidence bands computed using the LinearStateSpace class.
 
-Comparing this figure and the last also helps show how geometric growth differs from
-arithmetic growth.
+Comparing this figure and the last also helps show how geometric growth differs from arithmetic growth.
+
+The top right panel of the above graph shows a panel of martingales associated with the panel of $M_t = \exp(y_t)$.
 
-The top right panel of the above graph shows a panel of martingales associated with the panel of $M_t = \exp(y_t)$ that we have generated
-for a limited horizon $T$.
+We have generated these for a limited horizon $T$.
 
 It is interesting to how the martingale behaves as $T \rightarrow +\infty$.
 
 Let's see what happens when we set $T = 12000$ instead of $150$.
 
 ### Peculiar large sample property
 
-Hansen and Sargent {cite}`Hans_Sarg_book` (ch. 8) describe the following two properties of the  martingale component
-$\widetilde M_t$ of the multiplicative decomposition
+Hansen and Sargent {cite}`Hans_Sarg_book` (ch. 8) describe the following two properties of the  martingale component $\widetilde M_t$ of the multiplicative decomposition:
 
-* while $E_0 \widetilde M_t = 1$ for all $t \geq 0$,
-  nevertheless $\ldots$
+* while $E_0 \widetilde M_t = 1$ for all $t \geq 0$, nevertheless $\ldots$
 * as $t \rightarrow +\infty$, $\widetilde M_t$ converges to
   zero almost surely
 
-The first property follows from the fact that $\widetilde M_t$ is a multiplicative martingale with initial condition
-$\widetilde M_0 = 1$.
+The first property follows from the fact that $\widetilde M_t$ is a multiplicative martingale with initial condition $\widetilde M_0 = 1$.
 
 The second is a **peculiar property** noted and proved by Hansen and Sargent {cite}`Hans_Sarg_book`.
 
@@ -960,8 +963,7 @@ The purple 95 percent frequency coverage interval collapses around zero, illustr
 
 ## More about the multiplicative martingale
 
-Let's drill down and study probability distribution of the multiplicative martingale  $\{\widetilde M_t\}_{t=0}^\infty$  in
-more detail.
+Let's drill down and study probability distribution of the multiplicative martingale  $\{\widetilde M_t\}_{t=0}^\infty$  in more detail.
 
 As we  have seen, it has representation
 
@@ -977,8 +979,9 @@ It follows that $\log {\widetilde M}_t \sim {\mathcal N} ( -\frac{t H \cdot H}{2
 
 Next, we want a program to simulate the likelihood ratio process $\{ \tilde{M}_t \}_{t=0}^\infty$.
 
-In particular, we want to simulate 5000 sample paths of length $T$ for the case in which $x$ is a scalar and
-$[A, B, D, F] = [0.8, 0.001, 1.0, 0.01]$ and $\nu = 0.005$.
+In particular, we want to simulate 5000 sample paths of length $T$ for the case in which $x$ is a scalar.
+
+We use $[A, B, D, F] = [0.8, 0.001, 1.0, 0.01]$ and $\nu = 0.005$.
 
 After accomplishing this, we want to display and study histograms of $\tilde{M}_T^i$ for various values of  $T$.
 
@@ -988,7 +991,9 @@ Here is code that accomplishes these tasks.
 
 Let's write a program to simulate sample paths of $\{ x_t, y_{t} \}_{t=0}^{\infty}$.
 
-We'll do this by formulating the additive functional as a linear state space model and putting the [LinearStateSpace](https://github.com/QuantEcon/QuantEcon.py/blob/master/quantecon/lss.py) class to work.
+We'll do this by formulating the additive functional as a linear state space model.
+
+We put the [LinearStateSpace](https://github.com/QuantEcon/QuantEcon.py/blob/master/quantecon/lss.py) class to work.
 
 ```{code-cell} ipython3
 class AMF_LSS_VAR:
@@ -1142,8 +1147,7 @@ def population_means(amf, T=150):
     return xmean, ymean
 ```
 
-Now that we have these functions in our toolkit, let's apply them to run some
-simulations.
+Now that we have these functions in our toolkit, let's apply them to run some simulations.
 
 ```{code-cell} ipython3
 def simulate_martingale_components(amf, T=1000, I=5000):
@@ -1189,9 +1193,7 @@ Then let's use the plots to  investigate how these densities evolve through time
 We will plot the densities of $\log {\widetilde M}_t$ for different values of $t$.
 
 ```{note}
-`scipy.stats.lognorm` expects you to pass the standard deviation
-first $(tH \cdot H)$ and then the exponent of the mean as a
-keyword argument `scale` (`scale=np.exp(-t * H2 / 2)`).
+`scipy.stats.lognorm` expects you to pass the standard deviation first $(tH \cdot H)$ and then the exponent of the mean as a keyword argument `scale` (`scale=np.exp(-t * H2 / 2)`).
 
 * See the documentation [here](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.lognorm.html#scipy.stats.lognorm).
 
@@ -1248,21 +1250,17 @@ plt.tight_layout()
 plt.show()
 ```
 
-These probability density functions help us understand mechanics underlying the  **peculiar property** of our multiplicative martingale
+These probability density functions help us understand mechanics underlying the  **peculiar property** of our multiplicative martingale:
 
 * As $T$ grows, most of the probability mass shifts leftward toward zero.
-* For example, note that most  mass is near $1$ for $T =10$ or $T = 100$ but
-  most of it is near $0$ for $T = 5000$.
+* For example, note that most  mass is near $1$ for $T =10$ or $T = 100$ but most of it is near $0$ for $T = 5000$.
 * As $T$ grows, the tail of the density of $\widetilde M_T$ lengthens toward the right.
-* Enough mass moves toward the right tail to keep $E \widetilde M_T = 1$
-  even as most mass in the distribution of $\widetilde M_T$ collapses around $0$.
+* Enough mass moves toward the right tail to keep $E \widetilde M_T = 1$ even as most mass in the distribution of $\widetilde M_T$ collapses around $0$.
 
 ### Multiplicative martingale as likelihood ratio process
 
-[This lecture](https://python.quantecon.org/likelihood_ratio_process.html) studies **likelihood processes**
-and **likelihood ratio processes**.
+[This lecture](https://python.quantecon.org/likelihood_ratio_process.html) studies **likelihood processes** and **likelihood ratio processes**.
 
 A **likelihood ratio process** is  a  multiplicative  martingale with mean unity.
 
-Likelihood ratio processes exhibit the peculiar property that naturally also appears
-[here](https://python.quantecon.org/likelihood_ratio_process.html).
+Likelihood ratio processes exhibit the peculiar property that naturally also appears [here](https://python.quantecon.org/likelihood_ratio_process.html).