**Author: Serge Rey [email protected], Wei Kang [email protected]**

This notebook introduces Discrete Markov Chains (DMC) model and its two variants which explicitly incorporate spatial effects. We will demonstrate the usage of these methods by an empirical study for understanding regional income dynamics in the US. The dataset is the per capita incomes observed annually from 1929 to 2009 for the lower 48 US states.

Note that a full execution of this notebook requires **pandas**, **matplotlib** and light-weight geovisualization package pysal-**splot**.

`giddy.markov.Markov(self, class_ids, classes=None)`

We start with a look at a simple example of classic DMC methods implemented in PySAL-giddy. A Markov chain may be in one of $k$ different states/classes at any point in time. These states are exhaustive and mutually exclusive. If one had a time series of remote sensing images used to develop land use classifications, then the states could be defined as the specific land use classes and interest would center on the transitions in and out of different classes for each pixel.

For example, suppose there are 5 pixels, each of which takes on one of 3 states (a,b,c) at 3 consecutive periods:

In [1]:

```
import numpy as np
c = np.array([['b','a','c'],['c','c','a'],['c','b','c'],['a','a','b'],['a','b','c']])
```

So the first pixel was in state ‘b’ in period 1, state ‘a’ in period 2, and state ‘c’ in period 3. Each pixel's trajectory (row) owns Markov property, meaning that which state a pixel takes on today is only dependent on its immediate past.

Let's suppose that all the 5 pixels are governed by the same transition dynamics rule. That is, each trajectory is a realization of a Discrete Markov Chain process. We could pool all the 5 trajectories from which to estimate a transition probability matrix. To do that, we utlize the **Markov** class in **giddy**:

In [2]:

```
import giddy
m = giddy.markov.Markov(c)
```

You may turn off the summary for the Markov chain by assigning `summary=False`

when initializing the Markov Chain.

In [3]:

```
m = giddy.markov.Markov(c, summary=False)
```

In this way, we create a **Markov** instance - $m$. Its attribute $classes$ gives 3 unique classes these pixels can take on, which are 'a','b' and 'c'.

In [4]:

```
print(m.classes)
```

In [5]:

```
print(len(m.classes))
```

In addition to extracting the unique states as an attribute, our **Markov** instance will also have the attribute *transitions* which is a transition matrix counting the number of transitions from one state to another. Since there are 3 unique states, we will have a $(3,3)$ transtion matrix:

In [6]:

```
print(m.transitions)
```

The above transition matrix indicates that of the four pixels that began a transition interval in state ‘a’, 1 remained in that state, 2 transitioned to state ‘b’ and 1 transitioned to state ‘c’. Another attribute $p$ gives the transtion probability matrix which is the transition dynamics rule ubiquitous to all the 5 pixels across the 3 periods. The maximum likehood estimator for each element $p_{i,j}$ is shown below where $n_{i,j}$ is the number of transitions from state $i$ to state $j$ and $k$ is the number of states (here $k=3$):

\begin{equation} \hat{p}_{i,j} = \frac{n_{i,j}}{\sum_{q=1}^k n_{i,q} } \end{equation}In [7]:

```
print(m.p)
```

This means that if any of the 5 pixels was in state 'c', the probability of staying at 'c' or transitioning to any other states ('a', 'b') in the next period is the same (0.333). If a pixel was in state 'b', there is a high possibility that it would take on state 'c' in the next period because $\hat{p}_{2,3}=0.667$.

In [8]:

```
m.steady_state # steady state distribution
```

Out[8]:

This simple example illustrates the basic creation of a Markov instance, but the small sample size makes it unrealistic for the more advanced features of this approach. For a larger example, we will look at an application of Markov methods to understanding regional income dynamics in the US. Here we will load in data on per capita incomes observed annually from 1929 to 2010 for the lower 48 US states:

Firstly, we load in data on per capita incomes observed annually from 1929 to 2009 for the lower 48 US states. We use the example dataset in **libpysal** which was downloaded from US Bureau of Economic Analysis.

In [9]:

```
import libpysal
f = libpysal.io.open(libpysal.examples.get_path("usjoin.csv"))
pci = np.array([f.by_col[str(y)] for y in range(1929,2010)])
print(pci.shape)
```

The first row of the array is the per capita incomes for the 48 US states for the year 1929:

In [10]:

```
print(pci[0, :])
```

In order to apply the classic Markov approach to this series, we first have to discretize the distribution by defining our classes. There are many ways to do this including quantiles classification scheme, equal interval classification scheme, Fisher Jenks classification scheme, etc. For a list of classification methods, please refer to the pysal package **mapclassify**.

Here we will use the quintiles for each annual income distribution to define the classes. It should be noted that using quintiles for the pooled income distribution to define the classes will result in a different interpretation of the income dynamics. Quintiles for each annual income distribution (the former) will reveal more of relative income dynamics while those for the pooled income distribution (the latter) will provide insights in absolute dynamics.

In [11]:

```
import matplotlib.pyplot as plt
%matplotlib inline
years = range(1929,2010)
names = np.array(f.by_col("Name"))
order1929 = np.argsort(pci[0,:])
order2009 = np.argsort(pci[-1,:])
names1929 = names[order1929[::-1]]
names2009 = names[order2009[::-1]]
first_last = np.vstack((names1929,names2009))
from pylab import rcParams
rcParams['figure.figsize'] = 15,10
plt.plot(years,pci)
for i in range(48):
plt.text(1915,54530-(i*1159), first_last[0][i],fontsize=12)
plt.text(2010.5,54530-(i*1159), first_last[1][i],fontsize=12)
plt.xlim((years[0], years[-1]))
plt.ylim((0, 54530))
plt.ylabel(r"$y_{i,t}$",fontsize=14)
plt.xlabel('Years',fontsize=12)
plt.title('Absolute Dynamics',fontsize=18)
```

Out[11]:

In [12]:

```
years = range(1929,2010)
rpci= (pci.T / pci.mean(axis=1)).T
names = np.array(f.by_col("Name"))
order1929 = np.argsort(rpci[0,:])
order2009 = np.argsort(rpci[-1,:])
names1929 = names[order1929[::-1]]
names2009 = names[order2009[::-1]]
first_last = np.vstack((names1929,names2009))
from pylab import rcParams
rcParams['figure.figsize'] = 15,10
plt.plot(years,rpci)
for i in range(48):
plt.text(1915,1.91-(i*0.041), first_last[0][i],fontsize=12)
plt.text(2010.5,1.91-(i*0.041), first_last[1][i],fontsize=12)
plt.xlim((years[0], years[-1]))
plt.ylim((0, 1.94))
plt.ylabel(r"$y_{i,t}/\bar{y}_t$",fontsize=14)
plt.xlabel('Years',fontsize=12)
plt.title('Relative Dynamics',fontsize=18)
```

Out[12]:

In [13]:

```
import mapclassify as mc
q5 = np.array([mc.Quantiles(y,k=5).yb for y in pci]).transpose()
print(q5[:, 0])
```

In [14]:

```
print(f.by_col("Name"))
```

A number of things need to be noted here. First, we are relying on the classification methods in **mapclassify** for defining our quintiles. The class *Quantiles* uses quintiles ($k=5$) as the default and will create an instance of this class that has multiple attributes, the one we are extracting in the first line is $yb$ - the class id for each observation. The second thing to note is the transpose operator which gets our resulting array $q5$ in the proper structure required for use of Markov. Thus we see that the first spatial unit (Alabama with an income of 323) fell in the first quintile in 1929, while the last unit (Wyoming with an income of 675) fell in the fourth quintile.

So now we have a time series for each state of its quintile membership. For example, Colorado’s quintile time series is:

In [15]:

```
print(q5[4, :])
```

indicating that it has occupied the 3rd, 4th and 5th quintiles in the distribution at the first 3 periods. To summarize the transition dynamics for all units, we instantiate a Markov object:

In [16]:

```
m5 = giddy.markov.Markov(q5)
```

The number of transitions between any two quintile classes could be counted:

In [17]:

```
print(m5.transitions)
```

By assuming the first-order Markov property, time homogeneity, spatial homogeneity and spatial independence, a transition probability matrix could be estimated which holds for all the 48 US states across 1929-2010:

In [18]:

```
print(m5.p)
```

The fact that each of the 5 diagonal elements is larger than $0.78$ indicates a high stability of US regional income dynamics system.

Another very important feature of DMC model is the steady state distribution $\pi$ (also called limiting distribution) defined as $\pi p = \pi$. The attribute $steady\_state$ gives $\pi$ as follows:

In [19]:

```
print(m5.steady_state)
```

If the distribution at $t$ is a steady state distribution as shown above, then any distribution afterwards is the same distribution.

With the transition probability matrix in hand, we can estimate the first mean passage time which is the average number of steps to go from a state/class to another state for the first time:

In [20]:

```
print(giddy.ergodic.fmpt(m5.p))
```

Thus, for a state with income in the first quintile, it takes on average 11.5 years for it to first enter the second quintile, 29.6 to get to the third quintile, 53.4 years to enter the fourth, and 103.6 years to reach the richest quintile.

Thus far we have treated all the spatial units as independent to estimate the transition probabilities. This hides an implicit assumption: the movement of a spatial unit in the income distribution is independent of the movement of its neighbors or the position of the neighbors in the distribution. But what if spatial context matters??

We could plot the choropleth maps of per capita incomes of US states to get a first impression of the spatial distribution.

In [21]:

```
import geopandas as gpd
import pandas as pd
```

In [22]:

```
geo_table = gpd.read_file(libpysal.examples.get_path('us48.shp'))
income_table = pd.read_csv(libpysal.examples.get_path("usjoin.csv"))
complete_table = geo_table.merge(income_table,left_on='STATE_NAME',right_on='Name')
complete_table.head()
```

Out[22]:

In [23]:

```
index_year = range(1929,2010,15)
fig, axes = plt.subplots(nrows=2, ncols=3,figsize = (15,7))
for i in range(2):
for j in range(3):
ax = axes[i,j]
complete_table.plot(ax=ax, column=str(index_year[i*3+j]), cmap='OrRd', scheme='quantiles', legend=True)
ax.set_title('Per Capita Income %s Quintiles'%str(index_year[i*3+j]))
ax.axis('off')
leg = ax.get_legend()
leg.set_bbox_to_anchor((0.8, 0.15, 0.16, 0.2))
plt.tight_layout()
```