Logarithmic And Power Models Assignment Report

Table of Content
Step One: Creating Graphs.............................................................................................2
Step Two: Fitting the Data.............................................................................................4
Step Three: Making predictions.....................................................................................6
Step Four: Using additional data to update the model...................................................7
References....................................................................................................................11
List of Figures
Figure 1 : Linear Model Plot.............................................................................................2
Figure 2 : 2nd Order Polynomial Plot..............................................................................2
Figure 3 : 3rd Order Polynomial Plot...............................................................................3
Figure 4 : Exponential Model Plot....................................................................................3
Figure 5 : Quadratic Model Fitting...................................................................................4
Figure 6 : Cubic Model Fitting.........................................................................................5
Figure 7 : Exponential Model Fitting................................................................................5
Figure 8 : Quadratic Improved Model Fitting.................................................................8
Figure 9 : Cubic Improved Model Fitting........................................................................8
Figure 10 : Exponential Improved Model Fitting.............................................................9
List of Tables
Table 1 : Initial Models and their Performance................................................................6
Table 2 : Improved Models and their Performance..........................................................9

1
Step One: Creating Graphs
0 10 20 30 40 50 60 70
0.000
10.000
20.000
30.000
40.000
50.000
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80.000
Linear Model
Years after 1860
Population (in 1000s)
Figure 1: Linear Model Plot
This is not a possible type to fit the data. This is because the data under investigation
is the population data. It has been proven by many researchers that at no point can a
population growth take a linear model form (Stech, Peckham, & Pastor, 2012).
Further more, most of the data points are way out of the trend-line, especially around
1920.
0 10 20 30 40 50 60 70
0.000
10.000
20.000
30.000
40.000
50.000
60.000
70.000
80.000
2nd Order Polynomial
Years after 1860
Population (in 1000s)
Figure 2: 2nd Order Polynomial Plot

2
This could be a possible fit base on the trend-line. However, the tendency of the
population returning back to zero around the 10th year is abnormal and discredits this
curve as a possible fit (Patel, Khurana, Sharma, Kumar, & Ragumani, 2018).
0 10 20 30 40 50 60 70
0.000
10.000
20.000
30.000
40.000
50.000
60.000
70.000
80.000
3rd Order Polynomial
Years after 1860
Population (in 1000s)
Figure 3: 3rd Order Polynomial Plot
This is a possible fit since the trend-line seem to fit the data points well and does not
violate any expected trends in a normal population growth. Hence, this one is chosen
as the second choice fit for the data.
0 10 20 30 40 50 60 70
0.000
10.000
20.000
30.000
40.000
50.000
60.000
70.000
80.000
Exponential Model
Years after 1860
Population (in 1000s)
Figure 4: Exponential Model Plot

3
This is a possible fit based on the resulting curve. Again, researchers have found out
that population growth takes an exponential form, especially in the early stages
(Stech, Peckham, & Pastor, 2012). Hence this one is chosen as the first choice fit for
the data.
Note on the other functions:
The logarithmic and power models did not show any tend-lines. This could be
because the data range could not fit these curves by any means. As for the sine and
cosine functions, the are not in the option list among the curves to fit in the excel
model. This is expected, since these family of functions (commonly known as
sinusoidal functions) are periodic. Owing to their nature, they are well-defined and do
not fit random data.
Step Two: Fitting the Data
0 10 20 30 40 50 60 70
0.000
10.000
20.000
30.000
40.000
50.000
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70.000
80.000
f(x) = 0.03 x² − 0.68 x + 3.52
R² = 0.97
San Diego Population from 1860-1920
Years after 1860
Population ()in 1000s
Figure 5: Quadratic Model Fitting