MAT523 MINI Project- Recycling Rates 1 PDF

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UNIVERSITI TEKNOLOGI MARAKAMPUS SEREMBANCAWANGAN NEGERI SEMBILANFACULTY OF COMPUTER AND MATHEMATICAL SCIENCES (FSKM)CSBACHELOR OF SCIENCE (HONS) STATISTICSMATLINEAR ALGEBRA IITITLEMINI PROJECT ASSIGNMENT-PAPER AND GLASS PACKAGING RECYCLINGRATESNAME STUDENT ID GROUPNOOR AIDA IZATI BINTI BORHANUDDIN 2...

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UNIVERSITI TEKNOLOGI MARA KAMPUS SEREMBAN CAWANGAN NEGERI SEMBILAN

FACULTY OF COMPUTER AND MATHEMATICAL SCIENCES (FSKM) CS241 BACHELOR OF SCIENCE (HONS) STATISTICS

MAT523 LINEAR ALGEBRA II TITLE MINI PROJECT ASSIGNMENT-PAPER AND GLASS PACKAGING RECYCLING RATES NAME NOOR AIDA IZATI BINTI BORHANUDDIN ABDULLAH MUZAKKIR BIN AHMAD SHAIPUDDIN AHMAD RAMADHAN BIN SAAD

STUDENT ID 2019637442 2019467892 2019636956

PREPARE FOR: PUAN NORAIMI AZLIN BINTI MOHD NORDIN

SUBMISSION DATE: 7TH DECEMBER 2020

GROUP N4CS2415T2 N4CS2415T3 N4CS2415T3

TABLE OF CONTENTS

1.0

INTRODUCTION .................................................................................................................. 1

2.0

LITERATURE REVIEW ....................................................................................................... 2

2.1

LINEAR MODEL ................................................................................................................... 5

2.2

QUADRATIC MODEL .......................................................................................................... 8

2.3

CUBIC MODEL ................................................................................................................... 11

3.0

ANALYSIS AND CONCLUSION ...................................................................................... 14

4.0

REFERENCES ................................................................................................................... 15

5.0

APPENDIX .......................................................................................................................... 16

Figure 2.1.1: Linear Graph.................................................................................................................. 7 Figure 2.2.1: Quadratic Graph ......................................................................................................... 10 Figure 2.3.1: Cubic Graph................................................................................................................. 13 Figure 3.0.1: Graph that produces least error................................................................................ 14

1.0

INTRODUCTION

The recycling of paper is the process by which waste paper is turned into new paper products. It has a number of important benefits: It saves waste paper from occupying homes of people and producing methane as it breaks down. Because paper fibre contains carbon (originally absorbed by the tree from which it was produced), recycling keeps the carbon locked up for longer and out of the atmosphere. Next, glass recycling is the processing of waste glass into usable products. Glass that is crushed and ready to be remelted is called cullet. There are two types of cullet: internal and external. Internal cullet is composed of defective products detected and rejected by a quality control process during the industrial process of glass manufacturing, transition phases of product changes (such as thickness and colour changes) and production offcuts. External cullet is waste glass that has been collected or reprocessed with the purpose of recycling. External cullet (which can be pre- or post-consumer) is classified as waste. The word "cullet", when used in the context of end-ofwaste, will always refer to external cullet. In this study, we would like to see if there is a predictable relationship between the percentages of each material that a country recycles. For this study, the least squares method will be used. Least square method is a form of mathematical regression analysis used to determine the line of best fit for a set of data, providing a visual demonstration of the relationship between the data points. We will place the paper recycling rates on the horizontal axis and those for glass on the vertical axis. Following is a data table that includes both percentages for 20 countries: Country % of Paper Packaging Recycled % of Glass Packaging Recycled Estonia 34 64 New Zealand 40 72 Poland 40 27 Cyprus 42 4 Mexico 45 20 Portugal 56 39 United States 59 21 Italy 62 56 Spain 63 41 Australia 66 44 Greece 70 34 Finland 70 56 Ireland 70 55 Netherlands 70 76 Sweden 70 100 France 76 59 Germany 83 81 Austria 83 44 Belgium 83 98 Japan 98 96 Table 1.0: Paper and Glass Packaging Recycling Rates for 20 Countries 1

2.0

LITERATURE REVIEW

In linear algebra, the best approximation is least square solutions to an inconsistent system. Least square solution (lsq) of linear system Consider a linear Ax=b that has consistent solutions. However, in some physical problems, even though theoretically, the solution is consistent ‘measurement errors’ may cause the system to be inconsistent. Thus we need to find x close enough so that it minimize ||Ax-b||. Least square problems states that for a linear system Ax=b, we need to find x that minimize ||Ax-b||and x = least squares solution to Ax=b. Least square fitting The result about orthogonal projection in inner product spaces shall be use to obtain a technique for fitting a line or curve to a set of experimentally determined points in the plane. Least square fit of a polynomial Let y be a polynomial of degree m, y = 𝑎0 + 𝑎1 𝑥 + 𝑎2 𝑥 2 +……. +𝑎𝑚 𝑥 𝑚 and let (𝑥1 , 𝑦1 ) (𝑥2 , 𝑦2 ) (𝑥3 , 𝑦3 ), ….. (𝑥𝑛 , 𝑦𝑛 ) be 𝑛 points that we are trying to fit on the curve y. Substituting, rearranging and simplified into matrix form: 1 𝑥1 𝑥12 𝑥13 ⋯ 𝑥1𝑛

𝑦1

M= 1 𝑥2 𝑥22 𝑥23 ⋯ 𝑥2𝑛

y = 𝑦2

⋮

⋮

1 𝑥𝑛 𝑥𝑛2 𝑥𝑛3 ⋯ 𝑥𝑛𝑛

𝑎1 and

v= 𝑎0

⋮

⋮ 𝑦3

𝑎𝑛

The solution of this normal system M T Mv = M T y is to determine coefficients 𝑎1, 𝑎2, …. 𝑎𝑚 and minimize || y − Mv ||. Provided (M T M)−1 exist, then there is unique solution v = v ∗ given by −1

𝑣 ∗ = (M T M) (MT y)

Magnitude of error vector for least square solution (inconsistent) 𝑒 2 = ||𝑦 − 𝑀𝑣|| = 𝑒12 + 𝑒22 + ⋯ + 𝑒𝑛2

2

According to the article written by Richard Routledge, Least Squares Method, also called least squares approximation is a method for estimating the true value of some quantity based on a consideration of errors in observations or measurements. In particular, the line (the function yi = a + bxi, where xi are the values at which yi is measured and i denotes an individual observation) that minimizes the sum of the squared distances (deviations) from the line to each observation is used to approximate a relationship that is assumed to be linear. That is, the sum over all i of ( yi − a − bxi)2 is minimized by setting the partial derivatives of the sum with respect to a and b equal to 0. The method can also be generalized for use with nonlinear relationships. The article also states that one of the first applications of the method of least squares was to settle a controversy involving Earth’s shape. The English mathematician, Isaac Newton asserted in the Principia (1687) that Earth has an oblate (grapefruit) shape due to its spin-causing the equatorial diameter to exceed the polar diameter by about 1 part in 230. In 1718 the director of the Paris Observatory, Jacques Cassini, asserted on the basis of his own measurements that Earth has a prolate (lemon) shape. Next, article written by Kumar Molugaram and G. Shanker Rao (2017) states that the method of least squares assumes that the best fit curve of a given type is the curve that has the minimal sum of deviations such as least square error from a given set of data. According to the method of least squares, the best fitting curve has the property that ∑ 1𝑛 𝑒𝑖2 =∑𝑛1 [𝑦𝑖 − 𝑓(𝑥𝑖)]2 is minimum / minimum error in terms of its magnitude. It is also states that the method of least squares is a widely used method of fitting curve for a given data. It is the most popular method used to determine the position of the trend line of a given time series. The trend line is technically called the best fit. In addition, based on the article written by Steven J. Miller from Mathematics Department at Brown University, The Method of Least Squares is a procedure to determine the best fit line to data where the proof uses simple calculus and linear algebra. The basic problem is to find the best fit straight line y = ax + b for n equals to any real number. However, it is extremely unlikely that we will observe a perfect linear relationship as there are two reasons for this. The first is experimental error and the second is that the underlying relationship may not be exactly linear but rather only approximately linear. It is clearly states from the article that The Method of Least Squares is a procedure, requiring just some calculus and linear algebra to determine what the “best fit” line is to the data. Of course, we need to quantify what we mean by “best fit”, which will require a brief review of some probability and statistics. A careful analysis of the proof will show that the method is capable of great generalizations. In another article written by Will Kenton (2019), the least squares method is a form of mathematical regression analysis used to determine the line of best fit for a set of data, providing a visual demonstration of the relationship between the data points. He also stated that the least squares method provides the overall rationale for the placement of the line of best fit among the data points being studied. Next, it is also states that the most common application of this method, which is sometimes referred to as "linear" or "ordinary", aims to create a straight line that minimizes the sum of the squares of the errors that are generated by differences in the observed value, and the value anticipated, based on that model. An analyst using the least squares method will generate a line of best fit that explains the potential relationship between independent and dependent variables. In contrast to a linear 3

problem, a non-linear least squares problem has no closed solution and is generally solved by iteration. The discovery of the least squares method is attributed to Carl Friedrich Gauss, who discovered the method in 1795.

4

2.1

LINEAR MODEL

1. Setup Mv=Y, Where M=% of paper packaging recycled and Y=% of glass packaging recycled

1 34

64

1 40

72

1 40

27

1 42

4

1 45

20

1 56

39

1 59

21

1 62

56

1 63

41

1 66

44

1 70

34

1 70

56

1 70

55

1 70

76

1 70

100

1 76

59

1 83

81

1 83

44

1 83

a

1 98

b

5

98 =

96

2. Its normal equation is 20 1280

1280 87478

a b

=

1087 74457

3. The least square solution is 𝑟=

a b

=

0.7870 -0.0115 1087 -0.0115 0.0002 74457

=

-1.9465 0.8796

∴ The best fit linear curve is y = −1.9465 + 0.8796𝑥 4. Error of vector e = Y – Mv’

𝑒=

36.039 38.7612 -6.2388 -30.9981 -17.637 -8.3129 -28.9518 3.4093 -12.4704 -12.1093 -25.6278 -3.6278 -4.6278 16.3722 40.3722 -5.9056 9.937 -27.063 26.937 11.7425

∴ ||𝑒|| = 98.2447

6

Figure 2.1.1: Linear Graph

7

2.2

QUADRATIC MODEL

1. Setup Mv=Y, Where M=% of paper packaging recycled and Y=% of glass packaging recycled

1

34

1156

64

1

40

1600

72

1

40

1600

27

1

42

1764

4

1

45

2025

20

1

56

3136

39

1

59

3481

21

1

62

3844

56

1

63

3969

41

1

66

4356

44

1

70

4900

34

1

70

4900

56

1

70

4900

55

1

70

4900

76

1

70

4900

100

1

76

5776

59

1

83

6889

81

1

83

6889 a

44

1

83

6889 b

1

98

9604 c

8

=

98 96

2. Its normal equation is 20 1280 87478

1280 87478 a 87478 6299912 b 6299912 473148502 c

1087

= 74457

5416917

3. The least square solution is 𝑟=

a b c

=

7.5224 -0.2411 0.0018 1087 80.9938 = -0.2411 0.0080 -0.0001 74457 -1.9476 0.0018 -0.0001 0.0000 5416917 0.0224

∴ The best fit linear curve is y = 80.9938 − 1.9476𝑥 + 0.0224𝑥 2 4. Error of vector e = Y – Mv’ 23.3224 33.0596 -11.9404 -34.7198 -18.725 -3.1945 -23.0818 9.6276 -6.2255 -6.0538 -20.4523 𝑒= 1.5477 0.5477 21.5477 45.5477 -3.3943 7.3012 -29.6988 24.3012 -9.3168 ∴ ||𝑒|| = 92.901

9

Figure 2.2.1: Quadratic Graph

10

2.3

CUBIC MODEL 1. Setup Mv=Y, Where M=% of paper packaging recycled and Y=% of glass packaging recycled 1

34

1156

39304

64

1

40

1600

64000

72

1

40

1600

64000

27

1

42

1764

74088

4

1

45

2025

91125

20

1

56

3136

175616

39

1

59

3481

205379

21

1

62

3844

238328

56

1

63

3969

250047

41

1

66

4356

287496

44

1

70

4900

343000

34

1

70

4900

343000

56

1

70

4900

343000

55

1

70

4900

343000

76

1

70

4900

343000

100

1

76

5776

438976

59

1

83

6889

571787

a

81

1

83

6889

571787

b

44

1

83

6889

571787

c

1

98

9604

941192

d

11

=

98 96

2. Its normal equation is

0.0000

0.0000

0.0000

0.0000

a

0.0000

0.0000

0.0000

0.0005

b

74457

0.0000

0.0000

0.0005

0.0368

c

5416917

0.0000

0.0005

0.0368

2.9461

d

412245927

=

1087

3. The least square solution is

𝑟=

a b c d

124.0547 -6.1655 0.0959 -0.0005 = -6.1655 0.3092 -0.0048 0.0000 0.0959 -0.0048 0.0001 -0.0000 -0.0005 0.0000 -0.0000 0.0000

1087 74457 5416917 412245927

443.4383 -20.3738 = 0.3149 -0.0015

∴ The best fit linear curve is y = 443.4383 − 20.3738𝑥 + 0.3149𝑥2 − 0.0015𝑥 3 4. Error of vector e = Y – Mv’

𝑒=

6.9434 33.6293 -11.3707 -30.4551 -10.5078 6.8089 -15.0079 15.1856 -1.5954 -4.3512 -22.6605 -0.6605 -1.6605 19.3395 43.3395 -10.3392 -1.1847 -38.1847 15.8153 6.917 ∴ ||𝑒|| = 86.6222

12

Figure 2.3.1: Cubic Graph

13

3.0

ANALYSIS AND CONCLUSION

Based on the model above, the following error magnitude in the approximation are obtained Types of best fit curve

Error magnitude

Linear

98.2447

Quadratic

92.9015

Cubic

86.6222

∴ The best fit curves in this case which will give the best approximation is cubic best fit curve because it has minimum error in terms of its magnitude.

Figure 3.0.1: Graph that produces least error

14

4.0

REFERENCES

1. Almukkahal, R., Ottman, L., DeLancey, D., Evans, A., Lawsky, E., & Meery, B. (2019, November 20). Displaying Bivariate Data. Retrieved March 25, 2020, from https://www.ck12.org/statistics/displaying-bivariate-data/lesson/Displaying-BivariateData-ADV-PST/ 2. Routledge, R. (2018, February 19). Least squares method. Retrieved March, 31, 2020 from https://www.britannica.com/topic/least-squares-approximation 3. Method of Least Square. (n.d.). Retrieved March 31, 2020, from https://www.sciencedirect.com/topics/engineering/method-of-least-square 4. Kenton, W. (2019, September 2). Least Squares Method Definition. Retrieved March 31, 2020, from https://www.investopedia.com/terms/l/least-squares-method.asp 5. Miller, S. J. (n.d.). The Method of Least Squares. Retrieved March 31, 2020, from https://web.williams.edu/Mathematics/sjmiller/public_html/BrownClasses/54/handouts /MethodLeastSquares.pdf

15

5.0

APPENDIX

1. LINEAR MODEL CODING & OUTPUT

16

17

18

2. QUADRATIC MODEL CODING & OUTPUT

19

20

21

22

3. CUBIC MODEL CODING & OUTPUT

23

24

25

26...