# Logarithmic And Power Models Assignment Report

Added on 2022-09-09

11 Pages1321 Words25 Views

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

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

60.000

70.000

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

Step One: Creating Graphs

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

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

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

60.000

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

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

60.000

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

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