Laboratory 1: Population Estimation and Water Supply

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.

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.

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.

Figure 3: Geometric Residuals Plot

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.

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.

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

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 10-year 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

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.

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.

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.