chap05soln


Modeling and Simulation in Python

Chapter 5

Copyright 2017 Allen Downey

License: Creative Commons Attribution 4.0 International


In [1]:
# Configure Jupyter so figures appear in the notebook
%matplotlib inline

# Configure Jupyter to display the assigned value after an assignment
%config InteractiveShell.ast_node_interactivity='last_expr_or_assign'

# import functions from the modsim.py module
from modsim import *

Reading data

Pandas is a library that provides tools for reading and processing data. read_html reads a web page from a file or the Internet and creates one DataFrame for each table on the page.


In [2]:
from pandas import read_html

The data directory contains a downloaded copy of https://en.wikipedia.org/wiki/World_population_estimates

The arguments of read_html specify the file to read and how to interpret the tables in the file. The result, tables, is a sequence of DataFrame objects; len(tables) reports the length of the sequence.


In [3]:
filename = 'data/World_population_estimates.html'
tables = read_html(filename, header=0, index_col=0, decimal='M')
len(tables)


Out[3]:
6

We can select the DataFrame we want using the bracket operator. The tables are numbered from 0, so tables[2] is actually the third table on the page.

head selects the header and the first five rows.


In [4]:
table2 = tables[2]
table2.head()


Out[4]:
United States Census Bureau (2017)[28] Population Reference Bureau (1973–2016)[15] United Nations Department of Economic and Social Affairs (2015)[16] Maddison (2008)[17] HYDE (2007)[24] Tanton (1994)[18] Biraben (1980)[19] McEvedy & Jones (1978)[20] Thomlinson (1975)[21] Durand (1974)[22] Clark (1967)[23]
Year
1950 2557628654 2.516000e+09 2.525149e+09 2.544000e+09 2.527960e+09 2.400000e+09 2.527000e+09 2.500000e+09 2.400000e+09 NaN 2.486000e+09
1951 2594939877 NaN 2.572851e+09 2.571663e+09 NaN NaN NaN NaN NaN NaN NaN
1952 2636772306 NaN 2.619292e+09 2.617949e+09 NaN NaN NaN NaN NaN NaN NaN
1953 2682053389 NaN 2.665865e+09 2.665959e+09 NaN NaN NaN NaN NaN NaN NaN
1954 2730228104 NaN 2.713172e+09 2.716927e+09 NaN NaN NaN NaN NaN NaN NaN

tail selects the last five rows.


In [5]:
table2.tail()


Out[5]:
United States Census Bureau (2017)[28] Population Reference Bureau (1973–2016)[15] United Nations Department of Economic and Social Affairs (2015)[16] Maddison (2008)[17] HYDE (2007)[24] Tanton (1994)[18] Biraben (1980)[19] McEvedy & Jones (1978)[20] Thomlinson (1975)[21] Durand (1974)[22] Clark (1967)[23]
Year
2012 7013871313 7.057075e+09 7.080072e+09 NaN NaN NaN NaN NaN NaN NaN NaN
2013 7092128094 7.136796e+09 7.162119e+09 NaN NaN NaN NaN NaN NaN NaN NaN
2014 7169968185 7.238184e+09 7.243784e+09 NaN NaN NaN NaN NaN NaN NaN NaN
2015 7247892788 7.336435e+09 7.349472e+09 NaN NaN NaN NaN NaN NaN NaN NaN
2016 7325996709 7.418152e+09 NaN NaN NaN NaN NaN NaN NaN NaN NaN

Long column names are awkard to work with, but we can replace them with abbreviated names.


In [6]:
table2.columns = ['census', 'prb', 'un', 'maddison', 
                  'hyde', 'tanton', 'biraben', 'mj', 
                  'thomlinson', 'durand', 'clark']

Here's what the DataFrame looks like now.


In [7]:
table2.head()


Out[7]:
census prb un maddison hyde tanton biraben mj thomlinson durand clark
Year
1950 2557628654 2.516000e+09 2.525149e+09 2.544000e+09 2.527960e+09 2.400000e+09 2.527000e+09 2.500000e+09 2.400000e+09 NaN 2.486000e+09
1951 2594939877 NaN 2.572851e+09 2.571663e+09 NaN NaN NaN NaN NaN NaN NaN
1952 2636772306 NaN 2.619292e+09 2.617949e+09 NaN NaN NaN NaN NaN NaN NaN
1953 2682053389 NaN 2.665865e+09 2.665959e+09 NaN NaN NaN NaN NaN NaN NaN
1954 2730228104 NaN 2.713172e+09 2.716927e+09 NaN NaN NaN NaN NaN NaN NaN

The first column, which is labeled Year, is special. It is the index for this DataFrame, which means it contains the labels for the rows.

Some of the values use scientific notation; for example, 2.544000e+09 is shorthand for $2.544 \cdot 10^9$ or 2.544 billion.

NaN is a special value that indicates missing data.

Series

We can use dot notation to select a column from a DataFrame. The result is a Series, which is like a DataFrame with a single column.


In [8]:
census = table2.census
census.head()


Out[8]:
Year
1950    2557628654
1951    2594939877
1952    2636772306
1953    2682053389
1954    2730228104
Name: census, dtype: int64

In [9]:
census.tail()


Out[9]:
Year
2012    7013871313
2013    7092128094
2014    7169968185
2015    7247892788
2016    7325996709
Name: census, dtype: int64

Like a DataFrame, a Series contains an index, which labels the rows.

1e9 is scientific notation for $1 \cdot 10^9$ or 1 billion.

From here on, we will work in units of billions.


In [10]:
un = table2.un / 1e9
un.head()


Out[10]:
Year
1950    2.525149
1951    2.572851
1952    2.619292
1953    2.665865
1954    2.713172
Name: un, dtype: float64

In [11]:
census = table2.census / 1e9
census.head()


Out[11]:
Year
1950    2.557629
1951    2.594940
1952    2.636772
1953    2.682053
1954    2.730228
Name: census, dtype: float64

Here's what these estimates look like.


In [12]:
plot(census, ':', label='US Census')
plot(un, '--', label='UN DESA')
    
decorate(xlabel='Year',
         ylabel='World population (billion)')

savefig('figs/chap05-fig01.pdf')


Saving figure to file figs/chap05-fig01.pdf

The following expression computes the elementwise differences between the two series, then divides through by the UN value to produce relative errors, then finds the largest element.

So the largest relative error between the estimates is about 1.3%.


In [13]:
max(abs(census - un) / un) * 100


Out[13]:
1.3821293828998855

Exercise: Break down that expression into smaller steps and display the intermediate results, to make sure you understand how it works.

  1. Compute the elementwise differences, census - un
  2. Compute the absolute differences, abs(census - un)
  3. Compute the relative differences, abs(census - un) / un
  4. Compute the percent differences, abs(census - un) / un * 100

In [14]:
# Solution

census - un


Out[14]:
Year
1950    0.032480
1951    0.022089
1952    0.017480
1953    0.016188
1954    0.017056
1955    0.020448
1956    0.023728
1957    0.028307
1958    0.032107
1959    0.030321
1960    0.016999
1961    0.001137
1962   -0.000978
1963    0.008650
1964    0.017462
1965    0.021303
1966    0.023203
1967    0.021812
1968    0.020639
1969    0.021050
1970    0.021525
1971    0.023573
1972    0.023695
1973    0.022914
1974    0.021304
1975    0.018063
1976    0.014049
1977    0.011268
1978    0.008441
1979    0.007486
          ...   
1987   -0.018115
1988   -0.023658
1989   -0.028560
1990   -0.031861
1991   -0.037323
1992   -0.038763
1993   -0.040597
1994   -0.042404
1995   -0.042619
1996   -0.041576
1997   -0.040716
1998   -0.040090
1999   -0.039403
2000   -0.039129
2001   -0.038928
2002   -0.038837
2003   -0.039401
2004   -0.040006
2005   -0.041050
2006   -0.041964
2007   -0.043192
2008   -0.044599
2009   -0.046508
2010   -0.057599
2011   -0.061999
2012   -0.066201
2013   -0.069991
2014   -0.073816
2015   -0.101579
2016         NaN
Length: 67, dtype: float64

In [15]:
# Solution

abs(census - un)


Out[15]:
Year
1950    0.032480
1951    0.022089
1952    0.017480
1953    0.016188
1954    0.017056
1955    0.020448
1956    0.023728
1957    0.028307
1958    0.032107
1959    0.030321
1960    0.016999
1961    0.001137
1962    0.000978
1963    0.008650
1964    0.017462
1965    0.021303
1966    0.023203
1967    0.021812
1968    0.020639
1969    0.021050
1970    0.021525
1971    0.023573
1972    0.023695
1973    0.022914
1974    0.021304
1975    0.018063
1976    0.014049
1977    0.011268
1978    0.008441
1979    0.007486
          ...   
1987    0.018115
1988    0.023658
1989    0.028560
1990    0.031861
1991    0.037323
1992    0.038763
1993    0.040597
1994    0.042404
1995    0.042619
1996    0.041576
1997    0.040716
1998    0.040090
1999    0.039403
2000    0.039129
2001    0.038928
2002    0.038837
2003    0.039401
2004    0.040006
2005    0.041050
2006    0.041964
2007    0.043192
2008    0.044599
2009    0.046508
2010    0.057599
2011    0.061999
2012    0.066201
2013    0.069991
2014    0.073816
2015    0.101579
2016         NaN
Length: 67, dtype: float64

In [16]:
# Solution

abs(census - un) / un


Out[16]:
Year
1950    0.012862
1951    0.008585
1952    0.006674
1953    0.006072
1954    0.006286
1955    0.007404
1956    0.008439
1957    0.009887
1958    0.011011
1959    0.010208
1960    0.005617
1961    0.000369
1962    0.000311
1963    0.002702
1964    0.005350
1965    0.006399
1966    0.006829
1967    0.006289
1968    0.005827
1969    0.005821
1970    0.005832
1971    0.006258
1972    0.006166
1973    0.005847
1974    0.005332
1975    0.004437
1976    0.003388
1977    0.002670
1978    0.001965
1979    0.001712
          ...   
1987    0.003591
1988    0.004604
1989    0.005461
1990    0.005988
1991    0.006900
1992    0.007054
1993    0.007277
1994    0.007490
1995    0.007423
1996    0.007142
1997    0.006903
1998    0.006709
1999    0.006511
2000    0.006386
2001    0.006274
2002    0.006183
2003    0.006197
2004    0.006216
2005    0.006302
2006    0.006365
2007    0.006473
2008    0.006604
2009    0.006805
2010    0.008328
2011    0.008860
2012    0.009350
2013    0.009772
2014    0.010190
2015    0.013821
2016         NaN
Length: 67, dtype: float64

In [17]:
# Solution

max(abs(census - un) / census) * 100


Out[17]:
1.4014999251669376

max and abs are built-in functions provided by Python, but NumPy also provides version that are a little more general. When you import modsim, you get the NumPy versions of these functions.

Constant growth

We can select a value from a Series using bracket notation. Here's the first element:


In [18]:
census[1950]


Out[18]:
2.557628654

And the last value.


In [19]:
census[2016]


Out[19]:
7.325996709

But rather than "hard code" those dates, we can get the first and last labels from the Series:


In [20]:
t_0 = get_first_label(census)


Out[20]:
1950

In [21]:
t_end = get_last_label(census)


Out[21]:
2016

In [22]:
elapsed_time = t_end - t_0


Out[22]:
66

And we can get the first and last values:


In [23]:
p_0 = get_first_value(census)


Out[23]:
2.557628654

In [24]:
p_end = get_last_value(census)


Out[24]:
7.325996709

Then we can compute the average annual growth in billions of people per year.


In [25]:
total_growth = p_end - p_0


Out[25]:
4.768368055

In [26]:
annual_growth = total_growth / elapsed_time


Out[26]:
0.07224800083333333

TimeSeries

Now let's create a TimeSeries to contain values generated by a linear growth model.


In [27]:
results = TimeSeries()


Out[27]:
values

Initially the TimeSeries is empty, but we can initialize it so the starting value, in 1950, is the 1950 population estimated by the US Census.


In [28]:
results[t_0] = census[t_0]
results


Out[28]:
values
1950 2.557629

After that, the population in the model grows by a constant amount each year.


In [29]:
for t in linrange(t_0, t_end):
    results[t+1] = results[t] + annual_growth

Here's what the results looks like, compared to the actual data.


In [30]:
plot(census, ':', label='US Census')
plot(un, '--', label='UN DESA')
plot(results, color='gray', label='model')

decorate(xlabel='Year', 
         ylabel='World population (billion)',
         title='Constant growth')

savefig('figs/chap05-fig02.pdf')


Saving figure to file figs/chap05-fig02.pdf

The model fits the data pretty well after 1990, but not so well before.

Exercises

Optional Exercise: Try fitting the model using data from 1970 to the present, and see if that does a better job.

Hint:

  1. Copy the code from above and make a few changes. Test your code after each small change.

  2. Make sure your TimeSeries starts in 1950, even though the estimated annual growth is based on later data.

  3. You might want to add a constant to the starting value to match the data better.


In [31]:
# Solution

def compute_annual_growth(t_0, t_end):
    """Average annual growth over given period.
    
    t_0: start date
    t_end: end_date
    
    returns: average annual growth
    """
    elapsed_time = t_end - t_0
    p_0 = census[t_0]
    p_end = census[t_end]
    total_growth = p_end - p_0
    annual_growth = total_growth / elapsed_time
    return annual_growth

# compute annual growth using data from 1970 to the end
t_0 = 1970
t_end = get_last_label(census)
annual_growth = compute_annual_growth(t_0, t_end)

# Run the simulation over the whole time range.
# I subtract 0.45 from the initial value to shift
# the fitted curve down so it fits the data better.
t_0 = get_first_label(census)
t_end = get_last_label(census)
p_0 = get_first_value(census) - 0.45

# initialize the result
results = TimeSeries()
results[t_0] = p_0

# run the simulation
for t in linrange(t_0, t_end):
    results[t+1] = results[t] + annual_growth
    
# plot the results
plot(census, ':', label='US Census')
plot(un, '--', label='UN DESA')
plot(results, '--', color='gray', label='model')

decorate(xlabel='Year', 
         ylabel='World population (billion)',
         title='Constant growth')



In [32]:
census.loc[1960:1970]


Out[32]:
Year
1960    3.043002
1961    3.083967
1962    3.140093
1963    3.209828
1964    3.281201
1965    3.350426
1966    3.420678
1967    3.490334
1968    3.562314
1969    3.637159
1970    3.712698
Name: census, dtype: float64

In [ ]: