Env E 421
Laboratory 1: Population Estimation and Water Supply
Matthew Scott
1055898
September 28, 2007
Part 1
Using the population data collected from 1945 to 1965 four models were calibrated.
These models were Arithmetic, Geometric, Declining Growth, and Logistic models.
The resultant population estimations are shown in Figure 1. It can be seen that certain
models follow the curve more closely than others. A large plot of Figure 1 can be found
at the end of the report.
Population Growth Estimates
400000
350000
Population
300000
250000
200000
Census Population
150000
Arithmetic growth Model
100000
Geometric Growth Model
Declining Growth Model
50000
Logistic Estimate Model
0
1945
1950
1955
1960
1965
Year
Figure 1: Population Growth Estimates
Arithmetic Model
To calculate the parameters of the arithmetic model, I used the method of least squares.
First assuming a value for Ka, I estimated the population for the years following 1945.
Next I summed the squares of the residuals (SSR) to get an estimation of the error. I used
excels solver tool to obtain the lowest value for SSR by changing Ka. This lead to a Ka
value of 11620.27, and the resultant population estimates are plotted in Figure 1. The
residuals for this Ka value are shown in Figure 2. The residuals appear to be random in
distribution leading to believe that the model does not have a pattern to the errors. The R2
value for this estimation is 0.9694, which suggests the model fits the data well. Taken
these factors into account, in addition to examining the actual population versus the
predicted, leads me to believe that this model is reasonably accurate for a short time
period but may not translate well when predicting a long term population from a small
data. This can be found in Appendix A: Arithmetic.
Arithm etic Residual Plot
30000
Residual Value
20000
10000
0
1940
-10000
1945
1950
1955
1960
1965
1970
-20000
-30000
Year
Figure 2: Arithmetic Residual Plot
Geometric
To calculate the parameters of the geometric model, I used the method of least squares.
I found the natural logarithm of the population data, and then assuming a value for Ka, I
estimated the population for the years following 1945. Next I summed the squares of the
residuals (SSR) to get an estimation of the error. I used excels solver tool to obtain the
lowest value for SSR by changing Kp. This lead to a Kp value of 0.05433, and the
resultant population estimates are plotted in Figure 1. The residuals for this KP value are
shown in Figure 3. The residuals plot appears to have the shape of a sin wave and this
leads me to believe that there may be a pattern to the errors in the model. The R2 value
for this estimation is 0.9985, which suggests the model fits the data well. If the source of
the residual pattern could be found, I believe that the Geometric model would be very
useful for predicting short term population growth, based on the R2 value, and the fit of
the corresponding population curves as shown in Figure 1. This can be found in
Appendix B: Geometric.
Residuals
Geometric Residuals Plot
0.05
0
1940
-0.05
1945
1950
1955
1960
Year
Figure 3: Geometric Residuals Plot
1965
1970
Declining Growth
For the declining growth model I first calculated the saturation population (Z), based on
the land use, and max area giving in the lab handout. I found Z to be 8139600. By using
this Z and assuming a KD, I was able to gather an initial prediction for the population.
Once again I used the excel solver to minimize SSR by changing KD. This resulted in a
KD value of 0.001232. The residuals for this model are random as shown in Figure 4,
however they are quite large. Looking at Figure 1, it is easy to see that this model has the
worst fit. This is reflected in the R2 value of 0.9479. I believe the fit is so poor due to
both the small sample data and the fact that the growth rate has not yet begun to decrease.
This can be found in Appendix C: Declining Growth.
Residuals for Declining Growth Rate
60000
Difference
40000
20000
0
-200001940
1945
1950
1955
1960
1965
1970
-40000
-60000
Year
Figure 4: Residuals for Declining Growth Rate
Logistic
For calibrating the logistic model for the time period from 1945 to 1965, I picked yo, y1,
and y2 as 1945, 1955 and 1965 respectively. Using these I calculated an initial K, a, and
b values for use in the model. I was able to gather an initial prediction for the population,
and I used the excel solver to minimize the SSR by changing K, a and b. This lead to a K
value of 795752.6931, an a value of 11.1716 and a b value of –0.05699. Although the
logistic model has an R2 value of 0.9844, which is lower than that of the geometric
model, it appears that the logistic model fits the actual population more accurately than
the geometric. This is likely due to the manner, in which the logistic model was
calibrated, using the 3 of 5 available data points to find K. The residuals plot shows no
discernable pattern, as shown in Figure 5. This can be found in Appendix D: Logistic.
Residuals for Logistic Growth Rate
6000
Difference
4000
2000
0
-20001940
1945
1950
1955
1960
1965
1970
-4000
-6000
-8000
Year
Figure 5: Residuals for Logistic Growth Rate
1975 Population Prediction
Table 1 shows the population estimated by each of the models, calibrated from 1945 to
1965, as well as an additional logistic model that was calibrated over 1925 until 2005.
The result show how the logistic model is much more accurate when giving more data to
work with. The arithmetic benefits from the fact that the population was consistently
growing during the post war boom. The geometric and 20 yr logistic models both suffer
from the small data set used for calibration. The declining growth rate successfully
captures the lower population growth rate that occurred as the baby boom ended, but
would likely not predict accurately much further into the future as the rate continues to
decrease in the model, but rises again in the census data.
Table 1:1975 Population Estimates
1975 Population Estimates
Model
Population Percent Difference
Actual
451635
N/A
Arithmetic
460353
1.9%
Geometric
614161
36.0%
Declining
431475
4.5%
Logistic (20)
606698
34.3%
Logistic (80)
483243
7.0%
When predicting population it is highly useful to have a large data set to rely on if using
any of these models. The models give a decent rough estimate, but more in depth analysis
would lead to a higher degree of accuracy. Looking at land use agreements and economic
stability and factoring both into the models would improve the models significantly.
Part 2
For predicting the 2017 population of Edmonton I calibrated the arithmetic, declining
growth and geometric models using the population from 1989 until 2006. The current 10year trend shows much more growth compared to the late to mid nineties but I felt that
four data points would again lead to inaccurate estimations for most of the models. I used
the logistic model twice, once with a 20-year (1986 to 2006) calibration and once with an
80-year (1925 to 2005) year calibration. (This can be found in Appendix E: 80-year
Logistic.) The 20-year interval was chosen to mimic the other models, while still
providing easy data to calibrate the model with. The 80-year was chosen to compare how
the model would work given an abundance of data. The results are shown in Table 2,
while the trend is shown in Figure 6. A larger plot of Figure 6 can be found at the end of
the report.
Table 2: 2017 Population Estimates
2017 Population Estimates
Model
Population
Arithmetic
803668
Geometric
821710
Declining
840148
Logistic (20)
1133639
Logistic (80)
751379
2017 Population Estim ation
1200000
Declining Grow th Model
Geometric Grow th Model
Population
1100000
Logistic Grow th Model 20 yr
Arithmetic Grow th Model
1000000
Logistic Grow th Model 80 yr
900000
800000
700000
600000
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
Year
Figure 6: 2017 Population Estimation
Figure 6 shows that most of the models follow the same basic trend of a gradual increase
in population at a constant rate. The exception is the 20-year logistic curve, the rate of
which increases sharply as the years progress. This is likely following the same trend that
was found in the calibration data, a period of slow growth (1986 to 1996) followed by a
period of high growth (1996 to 2006). Figure 7 shows the historic population trend since
1921. The population of Edmonton is experiencing another period of rapid growth.
Because of this I would recommend using the geometric model to predict the population
in 2017. I believe this is a valid choice as Edmonton currently has the economic strength
to maintain this growth for that period of time. Over the short term the geometric model
should prove accurate, however if economic conditions change, it is likely that this would
result in a gross overestimation of population. This is because the geometric model does
not handle decreases in growth rate well.
City of Edmonton Historic Populations
800000
700000
Population
600000
500000
400000
300000
200000
100000
06
05
20
01
20
96
20
91
19
89
19
86
19
80
19
75
19
70
19
65
19
60
19
55
19
50
19
45
19
40
19
35
19
30
19
25
19
19
19
21
0
Year
Figure 7: City of Edmonton Historic Populations
Part 3
To predict Edmonton’s per-capita water demand I used the arithmetic model calibrated
using the data from 1990 until 2005. I feel this would give the most accurate picture of
the current declining water usage, due to conservation practices that have been
implemented. Using this method I found the per-capita water use to be 363 L/person/day
in 2017. (This can be found in Appendix F: Water Use.) The trend in predicted water
usage from 2006 until 2017 is shown in Figure 8.
Estimate Per-Capita Water Demand
395
390
Litres/person/day
385
380
375
370
365
360
355
350
345
2006
2007
2008
2009
2010
2011 2012
2013
2014
2015
2016
2017
Year
Figure 8: Estimate Per-Capita Water Demand
More information would be necessary to more accurately predict future water demands.
Both potential industrial and residential growth should be considered. If industry
introduces users with an abnormally high water demand it may result in an increase in per
capita demand, even though personal use may have dropped. Residential growth should
not be forgotten because as more new homes are built, most will take advantage of newer
water saving appliances decreasing the residential per-capita demand. It is important to
note that the trend shown in Figure 8 cannot sustain itself as it would eventually reach a
limit of water usage governed by the efficiency of appliances, industrial processes, and
basic personal needs.