appletmagic.com Thayer Watkins Silicon Valley & Tornado Alley USA 


A COMPARISON OF POPULATION PROJECTIONS FOR
USING VARIOUS EXTRAPOLATION METHODS 

The data for past populations
YEAR  POPULATION  

SAN JOSE  CALIFORNIA  U.S.  
1970  1,072,600  20,039,000  203,302,031 
1980  1,299,700  23,780,100  226,542,199 
1990  1,477,000  29,839,250  248,718,291 
B =(POP_{LAST}  POP_{FIRST})/(LAST.YEARFIRST.YEAR)
SAN JOSE
B = (1,477,001,072,600)/(19901970)=(404400)/(20) = 20,220 PER YEAR
PROJECTION FOR 2000 AD = 1,477,000 + (20,220)(10) = 1,679,200
CALIFORNIA
B =(29,839,25020,039,000)/20 = 490,012.5 PER YEAR
PROJECTION FOR 2000 AD = 29,839,250 + 4,900,125 = 34,739,375
(1+GROWTH.RATE) = (POP_{LAST}/POP_{FIRST})^{1/(LAST.YEARFIRST.YEAR)}
SAN JOSE
(1+GROWTH.RATE) = (1,477,000/1,072,600)^{ 1/20}
= (1.377)^{ .05} = 1.016125
POP_{2000} = (1,477,000)(1.016125)^{ 10}
= (1,477,000)(1.17468) = 1,733,213
CALIFORNIA
(1+GROWTH.RATE) = (1.48906)^{ .05} = 1.0201
POP_{2000} = (29,839,250)(1.0201)^{ 10 }
= (29,839,250)(1.2203) = 36,411,940
FOR THREE EQUALLY SPACED YEARS
Let TIME be coded as:
FIRST YEAR= 1, MIDDLE YEAR = 0, and LAST YEAR = +1.
Then:
Subtracting the second equation from the third yields:
(1)
(POP_{LAST}  POP_{MIDDLE}) = A(B1)
Subtracting the first equation from the second yields:
(2)
(POP_{MIDDLE}  POP_{FIRST}) = A(1  1/B) = A(B1)/B.
IF the previous equation (2) is divided into the equation before that (1) the result, after the cancellation of the factor A(B1) is just B.
Thus:
B = (POP_{LAST}  POP_{MIDDLE})/(POP_{MIDDLE}  POP_{FIRST})
Since B is now known the value of A can be determined from equation (1):
A = (POP_{LAST}  POP_{MIDDLE})/(B1)
With a knowledge of A the value of C can be determined from the equation for P_{MIDDLE}; i.e.,
C = P_{MIDDLE}  A
MODIFIED EXPONENTIAL PROJECTIONS FOR:
SAN JOSE
B = (1,477,000  1,299,700)/(1,299,700  1,072,600)
= 0.7807
A = 808,530
C = 2,108,200
Since time is coded: YEAR 1970 = 1, YEAR 1980 = 0, and YEAR 1990 = +1, this means that YEAR 2000 = +2.
Therefore:
POP_{2000} = 2,108,200  (808,530)(0.7807)^{2}
= 2,108,200  (808,530)(0.6095) = 1,615,407
The years 2010 and 2020 correspond to TIME=+3 and TIME=+4, respectively.
MODIFIED EXPONENTIAL PROJECTIONS FOR:
CALIFORNIA
B = (248718291  23,780,100)/(23,780,100  20,039,000)
= 1.6196
A = 9,779,132
C = 14,000,968
Therefore:
POP_{2000} = 14,000,968 + (9,779,132)(1.6196)^{2}
= 14,000,968 + (9,779,132)(2.6231)
= 39,652,650
MODIFIED EXPONENTIAL PROJECTIONS FOR:
UNITED STATES
B = (29,839,250  226,542,199)/(226,542,199  203,402,031)
= 0.95421
A = 484,000,000
C = 710,883,640
Therefore:
POP_{2000} = 710,883,640  (484,000,000)(0.95421)^{2}
= 710,883,640  (484,000,000)(0.91052)
= 269,879,027
The projections are compiled in the following table:
YEAR  PROJECTED POPULATION Modified Exponential 


SAN JOSE  CALIFORNIA  U.S.  
2000  1,615,407  39,652,650  269,879,027 
2010  1,723,477  55,469,524  290,372,745 
2020  1,807,846  81,288,001  309,627,939 
There are two other extrapolation curves that are like the modified exponential but more complicated mathematically:
The general shape of the Gompertz curve is the same as the Logistic curve.
If the logistics curve is expressed in terms of reciprocal population; i.e.,
the form is the same as that of the modified exponential and the same method used for the modified exponential can be used to get the projection of the reciprocal population.
If the logarithms are taken of both sides of the Gompertz equation the result is
This also is mathematically the same form as the modified exponential and the same method can be used to project the logarithm of population.
Projections and Actual Values for San Jose Population 2000  

Linear  Exponential  Modified Exponential  Logistics  Gompertz  Actual 
1,679,200  1,733,213  1,615,420  1,600,810  1,608,279  1,683,000 
HOME PAGE OF Thayer Watkins 