Full Rank Markov and Geographic Rank Markov

Author: Wei Kang weikang9009@gmail.com

import libpysal as ps
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
import seaborn as sns
import pandas as pd
import geopandas as gpd

Full Rank Markov

from giddy.markov import FullRank_Markov
income_table = pd.read_csv(ps.examples.get_path("usjoin.csv"))
income_table.head()
Name STATE_FIPS 1929 1930 1931 1932 1933 1934 1935 1936 ... 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009
0 Alabama 1 323 267 224 162 166 211 217 251 ... 23471 24467 25161 26065 27665 29097 30634 31988 32819 32274
1 Arizona 4 600 520 429 321 308 362 416 462 ... 25578 26232 26469 27106 28753 30671 32552 33470 33445 32077
2 Arkansas 5 310 228 215 157 157 187 207 247 ... 22257 23532 23929 25074 26465 27512 29041 31070 31800 31493
3 California 6 991 887 749 580 546 603 660 771 ... 32275 32750 32900 33801 35663 37463 40169 41943 42377 40902
4 Colorado 8 634 578 471 354 353 368 444 542 ... 32949 34228 33963 34092 35543 37388 39662 41165 41719 40093

5 rows × 83 columns

pci = income_table[list(map(str,range(1929,2010)))].values
pci
array([[  323,   267,   224, ..., 31988, 32819, 32274],
       [  600,   520,   429, ..., 33470, 33445, 32077],
       [  310,   228,   215, ..., 31070, 31800, 31493],
       ...,
       [  460,   408,   356, ..., 29769, 31265, 31843],
       [  673,   588,   469, ..., 35839, 36594, 35676],
       [  675,   585,   476, ..., 43453, 45177, 42504]])
m = FullRank_Markov(pci)
m.ranks
array([[45, 45, 44, ..., 41, 40, 39],
       [24, 25, 25, ..., 36, 38, 41],
       [46, 47, 45, ..., 43, 43, 43],
       ...,
       [34, 34, 34, ..., 47, 46, 42],
       [17, 17, 22, ..., 25, 26, 25],
       [16, 18, 19, ...,  6,  6,  7]])
m.transitions
array([[66.,  5.,  5., ...,  0.,  0.,  0.],
       [ 8., 51.,  9., ...,  0.,  0.,  0.],
       [ 2., 13., 44., ...,  0.,  0.,  0.],
       ...,
       [ 0.,  0.,  0., ..., 40., 17.,  0.],
       [ 0.,  0.,  0., ..., 15., 54.,  2.],
       [ 0.,  0.,  0., ...,  2.,  1., 77.]])

Full rank Markov transition probability matrix

m.p
array([[0.825 , 0.0625, 0.0625, ..., 0.    , 0.    , 0.    ],
       [0.1   , 0.6375, 0.1125, ..., 0.    , 0.    , 0.    ],
       [0.025 , 0.1625, 0.55  , ..., 0.    , 0.    , 0.    ],
       ...,
       [0.    , 0.    , 0.    , ..., 0.5   , 0.2125, 0.    ],
       [0.    , 0.    , 0.    , ..., 0.1875, 0.675 , 0.025 ],
       [0.    , 0.    , 0.    , ..., 0.025 , 0.0125, 0.9625]])

Full rank first mean passage times

m.fmpt
array([[  48.        ,   87.96280048,   68.1089084 , ...,  443.76689275,
         518.31000749, 1628.59025557],
       [ 225.92564594,   48.        ,   78.75804364, ...,  440.0173313 ,
         514.56045127, 1624.84070661],
       [ 271.55443692,  102.484092  ,   48.        , ...,  438.93288204,
         513.47599512, 1623.75624059],
       ...,
       [ 727.11189921,  570.15910508,  546.61934646, ...,   48.        ,
         117.41906375, 1278.96860316],
       [ 730.40467469,  573.45179415,  549.91216045, ...,   49.70722573,
          48.        , 1202.06279368],
       [ 754.8761577 ,  597.92333477,  574.38361779, ...,   43.23574191,
         104.9460425 ,   48.        ]])
m.sojourn_time
array([ 5.71428571,  2.75862069,  2.22222222,  1.77777778,  1.66666667,
        1.73913043,  1.53846154,  1.53846154,  1.53846154,  1.42857143,
        1.42857143,  1.56862745,  1.53846154,  1.40350877,  1.29032258,
        1.21212121,  1.31147541,  1.37931034,  1.29032258,  1.25      ,
        1.15942029,  1.12676056,  1.25      ,  1.17647059,  1.19402985,
        1.08108108,  1.19402985,  1.25      ,  1.25      ,  1.14285714,
        1.33333333,  1.26984127,  1.25      ,  1.37931034,  1.42857143,
        1.31147541,  1.26984127,  1.25      ,  1.31147541,  1.25      ,
        1.19402985,  1.25      ,  1.53846154,  1.6       ,  1.86046512,
        2.        ,  3.07692308, 26.66666667])
df_fullrank = pd.DataFrame(np.c_[m.p.diagonal(),m.sojourn_time], columns=["Staying Probability","Sojourn Time"], index = np.arange(m.p.shape[0])+1)
df_fullrank.head()
Staying Probability Sojourn Time
1 0.8250 5.714286
2 0.6375 2.758621
3 0.5500 2.222222
4 0.4375 1.777778
5 0.4000 1.666667
df_fullrank.plot(subplots=True, layout=(1,2), figsize=(15,5))
array([[<matplotlib.axes._subplots.AxesSubplot object at 0x1a2ad213c8>,
        <matplotlib.axes._subplots.AxesSubplot object at 0x1a2ad64668>]],
      dtype=object)

png

sns.distplot(m.fmpt.flatten(),kde=False)
<matplotlib.axes._subplots.AxesSubplot at 0x1a2ae70908>

png

Geographic Rank Markov

from giddy.markov import GeoRank_Markov, Markov, sojourn_time
gm = GeoRank_Markov(pci)
gm.transitions
array([[38.,  0.,  8., ...,  0.,  0.,  0.],
       [ 0., 15.,  0., ...,  0.,  1.,  0.],
       [ 6.,  0., 44., ...,  5.,  0.,  0.],
       ...,
       [ 2.,  0.,  5., ..., 34.,  0.,  0.],
       [ 0.,  0.,  0., ...,  0., 18.,  2.],
       [ 0.,  0.,  0., ...,  0.,  3., 14.]])
gm.p
array([[0.475 , 0.    , 0.1   , ..., 0.    , 0.    , 0.    ],
       [0.    , 0.1875, 0.    , ..., 0.    , 0.0125, 0.    ],
       [0.075 , 0.    , 0.55  , ..., 0.0625, 0.    , 0.    ],
       ...,
       [0.025 , 0.    , 0.0625, ..., 0.425 , 0.    , 0.    ],
       [0.    , 0.    , 0.    , ..., 0.    , 0.225 , 0.025 ],
       [0.    , 0.    , 0.    , ..., 0.    , 0.0375, 0.175 ]])
gm.sojourn_time[:10]
array([1.9047619 , 1.23076923, 2.22222222, 1.73913043, 1.15942029,
       3.80952381, 1.70212766, 1.25      , 1.31147541, 1.11111111])
gm.sojourn_time
array([ 1.9047619 ,  1.23076923,  2.22222222,  1.73913043,  1.15942029,
        3.80952381,  1.70212766,  1.25      ,  1.31147541,  1.11111111,
        1.73913043,  1.37931034,  1.17647059,  1.21212121,  1.33333333,
        1.37931034,  1.09589041,  2.10526316,  2.        ,  1.45454545,
        1.26984127, 26.66666667,  1.19402985,  1.23076923,  1.09589041,
        1.56862745,  1.26984127,  2.42424242,  1.50943396,  2.        ,
        1.29032258,  1.09589041,  1.6       ,  1.42857143,  1.25      ,
        1.45454545,  1.29032258,  1.6       ,  1.17647059,  1.56862745,
        1.25      ,  1.37931034,  1.45454545,  1.42857143,  1.29032258,
        1.73913043,  1.29032258,  1.21212121])
gm.fmpt
array([[ 48.        ,  63.35532038,  92.75274652, ...,  82.47515731,
         71.01114491,  68.65737127],
       [108.25928005,  48.        , 127.99032986, ...,  92.03098299,
         63.36652935,  61.82733039],
       [ 76.96801786,  64.7713783 ,  48.        , ...,  73.84595169,
         72.24682723,  69.77497173],
       ...,
       [ 93.3107474 ,  62.47670463, 105.80634118, ...,  48.        ,
         69.30121319,  67.08838421],
       [113.65278078,  61.1987031 , 133.57991745, ...,  96.0103924 ,
         48.        ,  56.74165107],
       [114.71894813,  63.4019776 , 134.73381719, ...,  97.287895  ,
         61.45565054,  48.        ]])
income_table["geo_sojourn_time"] = gm.sojourn_time
i = 0
for state in income_table["Name"]:
    income_table["geo_fmpt_to_" + state] = gm.fmpt[:,i]
    income_table["geo_fmpt_from_" + state] = gm.fmpt[i,:]
    i = i + 1
income_table.head()
Name STATE_FIPS 1929 1930 1931 1932 1933 1934 1935 1936 ... geo_fmpt_to_Virginia geo_fmpt_from_Virginia geo_fmpt_to_Washington geo_fmpt_from_Washington geo_fmpt_to_West Virginia geo_fmpt_from_West Virginia geo_fmpt_to_Wisconsin geo_fmpt_from_Wisconsin geo_fmpt_to_Wyoming geo_fmpt_from_Wyoming
0 Alabama 1 323 267 224 162 166 211 217 251 ... 72.186055 109.828532 82.994754 118.769984 82.475157 93.310747 71.011145 113.652781 68.657371 114.718948
1 Arizona 4 600 520 429 321 308 362 416 462 ... 67.544447 60.838807 76.090895 66.729262 92.030983 62.476705 63.366529 61.198703 61.827330 63.401978
2 Arkansas 5 310 228 215 157 157 187 207 247 ... 73.650943 129.533691 84.071211 138.692513 73.845952 105.806341 72.246827 133.579917 69.774972 134.733817
3 California 6 991 887 749 580 546 603 660 771 ... 71.377700 111.644884 62.230417 97.908341 104.922271 121.670243 69.368408 110.668388 59.998457 105.965215
4 Colorado 8 634 578 471 354 353 368 444 542 ... 69.627179 57.106339 66.353930 52.229230 98.797636 66.464398 60.762589 52.324565 55.559020 53.872702

5 rows × 180 columns

geo_table = gpd.read_file(ps.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()
AREA PERIMETER STATE_ STATE_ID STATE_NAME STATE_FIPS_x SUB_REGION STATE_ABBR geometry Name ... geo_fmpt_to_Virginia geo_fmpt_from_Virginia geo_fmpt_to_Washington geo_fmpt_from_Washington geo_fmpt_to_West Virginia geo_fmpt_from_West Virginia geo_fmpt_to_Wisconsin geo_fmpt_from_Wisconsin geo_fmpt_to_Wyoming geo_fmpt_from_Wyoming
0 20.750 34.956 1 1 Washington 53 Pacific WA (POLYGON ((-122.400749206543 48.22539520263672... Washington ... 71.663055 73.756804 48.000000 48.000000 101.592400 81.692586 65.219124 70.701226 53.126177 64.476985
1 45.132 34.527 2 2 Montana 30 Mtn MT POLYGON ((-111.4746322631836 44.70223999023438... Montana ... 69.918931 59.067897 76.184088 64.710823 90.781850 58.795201 63.455248 58.975522 60.881954 60.553000
2 9.571 18.899 3 3 Maine 23 N Eng ME (POLYGON ((-69.77778625488281 44.0740737915039... Maine ... 69.431862 53.872836 77.512381 62.862378 87.734760 54.244823 66.257807 56.905741 61.978506 58.336426
3 21.874 21.353 4 4 North Dakota 38 W N Cen ND POLYGON ((-98.73005676269531 45.93829727172852... North Dakota ... 69.441690 56.526347 76.659646 62.823668 85.031218 49.511240 67.362718 58.717458 64.386382 59.728719
4 22.598 22.746 5 5 South Dakota 46 W N Cen SD POLYGON ((-102.7879333496094 42.99532318115234... South Dakota ... 68.229894 61.548209 78.886304 68.794083 88.192659 55.754109 66.187694 63.802359 64.336311 65.070022

5 rows × 189 columns

complete_table.columns
Index(['AREA', 'PERIMETER', 'STATE_', 'STATE_ID', 'STATE_NAME', 'STATE_FIPS_x',
       'SUB_REGION', 'STATE_ABBR', 'geometry', 'Name',
       ...
       'geo_fmpt_to_Virginia', 'geo_fmpt_from_Virginia',
       'geo_fmpt_to_Washington', 'geo_fmpt_from_Washington',
       'geo_fmpt_to_West Virginia', 'geo_fmpt_from_West Virginia',
       'geo_fmpt_to_Wisconsin', 'geo_fmpt_from_Wisconsin',
       'geo_fmpt_to_Wyoming', 'geo_fmpt_from_Wyoming'],
      dtype='object', length=189)

Visualizing first mean passage time from/to California/Mississippi:

fig, axes = plt.subplots(nrows=2, ncols=2,figsize = (15,7))
target_states = ["California","Mississippi"]
directions = ["from","to"]
for i, direction in enumerate(directions):
    for j, target in enumerate(target_states):
        ax = axes[i,j]
        col = direction+"_"+target
        complete_table.plot(ax=ax,column = "geo_fmpt_"+ col,cmap='OrRd', 
                    scheme='quantiles', legend=True)
        ax.set_title("First Mean Passage Time "+direction+" "+target)
        ax.axis('off')
        leg = ax.get_legend()
        leg.set_bbox_to_anchor((0.8, 0.15, 0.16, 0.2))
plt.tight_layout()

png

Visualizing sojourn time for each US state:

fig, axes = plt.subplots(nrows=1, ncols=2,figsize = (15,7))
schemes = ["Quantiles","Equal_Interval"]
for i, scheme in enumerate(schemes):
    ax = axes[i]
    complete_table.plot(ax=ax,column = "geo_sojourn_time",cmap='OrRd', 
                scheme=scheme, legend=True)
    ax.set_title("Rank Sojourn Time ("+scheme+")")
    ax.axis('off')
    leg = ax.get_legend()
    leg.set_bbox_to_anchor((0.8, 0.15, 0.16, 0.2))
plt.tight_layout()

png