project report probability

8/18/2019 Project Report Probability

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Project report

Course title:

ENGINEERING SATISTICS AND PROBABILITY

Submitted b:

GRO!P N" "#

ALI RA$A !%&'#&CE&Bsc&"''

NAEE( $A)AR !%&''&(E&Bsc&"'*

(ASOOD C+ANDIO !%&'#&CE&Bsc&",-

RA.A $!L/ERNAIN !%&'#&CE&Bsc&"00

Submitted to:

SIR TARI/

Dep1rtme2t o3 ci4il e25i2eeri25

%EC


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APPLICATION O) CORRELATION AND

REGRESSION

Ac62o7led5me2t:


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Countless gratitude to Almighty ALLAH, Who is then omnipotent, omnipresent & HE, who

blessed with the chance and choice, health and courage, and knowledge enabled us to complete

this project.

All respect or the H!L" #$!#HE% 'HAA( )*.A.W.W+, who is oreer a torch o

knowledge and guidance to humanity & enables us to shape our lie according to the teachings o

-*LA, & endowed us an eemplary guidance in eery sphere o lie.

- acknowledge the serices o r. %ari/ Hussain in helping and guiding me in compiling and

presenting the present report. -n act it would not hae been possible or me to accomplish this

task without his help.

- dedicate this work to my #arents, to whom - am ery thankul as they encouraged me and

proided me all the necessary resources that had made possible or me to be able to accomplish

this task.

$egards

Ali $A0A


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ContentsAbstract......................................................................................................................5

Introduction:............................................................................................................... 6

Brief History of Correlation......................................................................................... 7

Types of Correlation.................................................................................................... 8

Correlation coecient.............................................................................................. 1

Co!ariance............................................................................................................ 1

1or a population......................................................................................................1"

1or a sample...........................................................................................................15

#$y %se Correlation&...............................................................................................15

'e(ression................................................................................................................ 16

History...................................................................................................................... 16

%ses of Correlation and 'e(ression..........................................................................1

Assu)ptions............................................................................................................. 1

#$y %se 'e(ression.................................................................................................1

Application of correlation and re(ression.................................................................

Correlation and 'e(ression Conclusion....................................................................*

'eferences................................................................................................................"


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Abstract%he present reiew introduces methods o analy2ing the relationship between two /uantitatie

ariables. %he calculation and interpretation o the sample product moment correlation

coeicient and the linear regression e/uation are discussed and illustrated. Common misuses o

the techni/ues are considered. %ests and conidence interals or the population parameters are

described, and ailures o the underlying assumptions are highlighted.


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Introduction:

%he most commonly used techni/ues or inestigating the relationship between two /uantitatieariables are correlation and linear regression. Correlation /uantiies the strength o the linear

relationship between a pair o ariables, whereas regression epresses the relationship in the

orm o an e/uation. 1or eample, in patients attending an accident and emergency unit )A&E+,

we could use correlation and regression to determine whether there is a relationship between age

and urea leel, and whether the leel o urea can be predicted or a gien age


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Brief History of Correlation

*ir 1rancis 3alton pioneered correlation ),', #*, #8, #91, 0,, 0#+. 3alton, a cousin o Charles(arwin, did a lot4 he studied medicine, he eplored Arica, he published in psychology and

anthropology, he deeloped graphic techni/ues to map the weather )#91, 0,+. And, like others o

his era, 3alton stroe to understand heredity )'#, '0, '-, ,"+.

-n 5677, 3alton uneiled reersion, the earliest ancestor o correlation, and described it like this

)'#+4 $eersion is the tendency o that ideal mean type to depart rom the parent

type, reverting towards what may be roughly and perhaps airly described as the aerage

ancestral type.

%he empirical odder or this obseration8 %he weights o 9:; sweet peas. thus, the

stature o the ather is correlated to that o the adult son > the stature o the uncle to that o the

http://advan.physiology.org/content/34/4/186#ref-21http://advan.physiology.org/content/34/4/186#ref-34http://advan.physiology.org/content/34/4/186#ref-35http://advan.physiology.org/content/34/4/186#ref-39http://advan.physiology.org/content/34/4/186#ref-42http://advan.physiology.org/content/34/4/186#ref-43http://advan.physiology.org/content/34/4/186#ref-39http://advan.physiology.org/content/34/4/186#ref-39http://advan.physiology.org/content/34/4/186#ref-42http://advan.physiology.org/content/34/4/186#ref-13http://advan.physiology.org/content/34/4/186#ref-14http://advan.physiology.org/content/34/4/186#ref-17http://advan.physiology.org/content/34/4/186#ref-20http://advan.physiology.org/content/34/4/186#ref-13http://advan.physiology.org/content/34/4/186#ref-14http://advan.physiology.org/content/34/4/186#ref-14http://advan.physiology.org/content/34/4/186#ref-14http://advan.physiology.org/content/34/4/186#ref-17http://advan.physiology.org/content/34/4/186#ref-17http://advan.physiology.org/content/34/4/186#ref-20http://advan.physiology.org/content/34/4/186#ref-15http://advan.physiology.org/content/34/4/186#ref-21http://advan.physiology.org/content/34/4/186#ref-34http://advan.physiology.org/content/34/4/186#ref-35http://advan.physiology.org/content/34/4/186#ref-39http://advan.physiology.org/content/34/4/186#ref-42http://advan.physiology.org/content/34/4/186#ref-43http://advan.physiology.org/content/34/4/186#ref-39http://advan.physiology.org/content/34/4/186#ref-42http://advan.physiology.org/content/34/4/186#ref-13http://advan.physiology.org/content/34/4/186#ref-14http://advan.physiology.org/content/34/4/186#ref-17http://advan.physiology.org/content/34/4/186#ref-20http://advan.physiology.org/content/34/4/186#ref-13http://advan.physiology.org/content/34/4/186#ref-14http://advan.physiology.org/content/34/4/186#ref-14http://advan.physiology.org/content/34/4/186#ref-17http://advan.physiology.org/content/34/4/186#ref-20http://advan.physiology.org/content/34/4/186#ref-15


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adult nephew, and so on> but the inde o co@relation, which is what - there B$e. '0

calledregression, is dierent in the dierent cases.

Dy 566:, 3alton was writing co@relation as correlation )0,+, and he had become ascinated by

ingerprints )'8, '9+. 3alton?s 56:; account o his deelopment o correlation )'+ would be hislast substantie paper on the subject )0#+.

arl #earson, 3alton?s colleague and riend, and ather o Egon #earson, pursued the reinement

o correlation )##, #0, #-+ with such igor that the statistic r , a statistic 3alton called the inde o

co@relation )'*+ and #earson called the 3alton coeicient o reersion )#8+, is known today as

#earson?s r .

Correlation

Correlation and regression analysis are related in the sense that both deal with relationshipsamong ariables. %he correlation coeicient is a measure o linear association between two

ariables. Falues o the correlation coeicient are always between @5 and G5. A correlation

coeicient o G5 indicates that two ariables are perectly related in a positie linear sense, acorrelation coeicient o @5 indicates that two ariables are perectly related in a negatie linear

sense, and a correlation coeicient o ; indicates that there is no linear relationship between the

two ariables. 1or simple linear regression, the sample correlation coeicient is the s/uare rooto the coeicient o determination, with the sign o the correlation coeicient being the same as

the sign o b5, the coeicient o 5 in the estimated regression e/uation.


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Ne51ti4e Correl1tio2


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#erect correlation occurs when there is a uncional dependency between the ariables.

-n this case all the points are in a straight line.

Strong Correlation

A correlation is stronger the closer the points are located to one another on the line.

%e16 Correl1tio2

A correlation is weaker the arther apart the points are located to one another on the line.


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%hrough the coeicient o correlation, we can

measure the degree or etent o the correlation between two ariables.

!n the basis o the coeicient o correlation we

can also determine whether the correlation is

positie or negatie and also its degree or

etent.

Per3ect correl1tio2: - two ariables changes in

the same direction and in the same proportion,

the correlation between the two is per3ect

positi4e

Abse2ce o3 correl1tio2: - two series o two

ariables ehibit no relations between them or

change in variable does not lead to a change in

the other variable Limited de5rees o3 correl1tio2: - two

ariables are not perectly correlated or is there

a perect absence o correlation, then we term

the correlation as Limited correlation


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Correlation coecient#earson?s correlation coeicient is the coariance o the two ariables diided by the product o

their standard deiations. %he orm o the deinition inoles a product moment, that is, the

mean )the irst moment about the origin+ o the product o the mean@adjusted random ariables>

hence the modiier product-moment in the name.

Co!ariance

High degree, moderate degree or low degree are

the three categories o this kind o correlation.

%he ollowing table reeals the eect o

coeicient or correlation.

We shall consider the ollowing most

commonly used methods.)5+ *catter #lot

)I+ ar #earsonJs coeicient o correlation

https://en.wikipedia.org/wiki/Covariancehttps://en.wikipedia.org/wiki/Standard_deviationshttps://en.wikipedia.org/wiki/Covariancehttps://en.wikipedia.org/wiki/Standard_deviations


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Coariance indicates how two ariables are related. A positie coariance means the ariables

are positiely related, while a negatie coariance means the ariables are inersely related. %he

ormula or calculating coariance o sample data is shown below.

x K the independent ariable y K the dependent ariable

n K number o data points in the sample

K the mean o the independent ariable x

K the mean o the dependent ariable y

%o understand how coariance is used, consider the table below, which describes the rate o economic growth ) xi+ and the rate o return on the * ;; ) yi+.

'sing the coariance ormula, you can determine whether economic growth and * ;;returns hae a positie or inerse relationship. Deore you compute the coariance, calculate the

mean o x and y. )%he *ummary easures topic o the (iscrete #robability (istributions section

eplains the mean ormula in detail.+


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%he coariance between the returns o the * ;; and economic growth is 5.M. *ince the

coariance is positie, the ariables are positiely relatedOthey moe together in the same

direction.

)or 1 popul1tio2

#earson?s correlation coeicient when applied to a population is commonly represented by the

3reek letter ρ )rho+ and may be reerred to as the population correlation coefficient or

the population Pearson correlation coefficient . %he ormula or ρB7 is4

where4

• is the coariance

• is the standard deiation o

%he ormula or ρ can be epressed in terms o mean and epectation. *ince

•B7

%hen the ormula or ρ can also be written as

where4

• and are deined as aboe

• is the mean o

• is the epectation.

%he ormula or ρ can be epressed in terms o uncentered moments. *ince

•

•

https://en.wikipedia.org/wiki/Statistical_Populationhttps://en.wikipedia.org/wiki/Pearson_product-moment_correlation_coefficient#cite_note-RealCorBasic-7https://en.wikipedia.org/wiki/Covariancehttps://en.wikipedia.org/wiki/Standard_deviationhttps://en.wikipedia.org/wiki/Pearson_product-moment_correlation_coefficient#cite_note-RealCorBasic-7https://en.wikipedia.org/wiki/Meanhttps://en.wikipedia.org/wiki/Expected_Valuehttps://en.wikipedia.org/wiki/Expected_Valuehttps://en.wikipedia.org/wiki/Statistical_Populationhttps://en.wikipedia.org/wiki/Pearson_product-moment_correlation_coefficient#cite_note-RealCorBasic-7https://en.wikipedia.org/wiki/Covariancehttps://en.wikipedia.org/wiki/Standard_deviationhttps://en.wikipedia.org/wiki/Pearson_product-moment_correlation_coefficient#cite_note-RealCorBasic-7https://en.wikipedia.org/wiki/Meanhttps://en.wikipedia.org/wiki/Expected_Value


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•

•

•

%hen the ormula or ρ can also be written as

)or 1 s1mple

#earson?s correlation coeicient when applied to a sample is commonly represented by theletter r and may be reerred to as the sample correlation coefficient or the sample Pearson

correlation coefficient . We can obtain a ormula or r by substituting estimates o the coariances

and ariances based on a sample into the ormula aboe. *o i we hae one dataset P x5,..., xnQ

containing n alues and another dataset P y5,..., ynQ containing n alues then that ormula or r is4

Why Use Correlation?

We can use the correlation coeicient, such as the #earson #roduct oment Correlation

Coeicient, to test i there is a linear relationship between the ariables. %o /uantiy the strengtho the relationship, we can calculate the correlation coeicient )r+. -ts numerical alue ranges

rom G5.; to @5.;. rR ; indicates positie linear relationship, r S ; indicates negatie linear

relationship while r K ; indicates no linear relationship.

https://en.wikipedia.org/wiki/Sample_(statistics)https://en.wikipedia.org/wiki/Statistical_samplehttps://en.wikipedia.org/wiki/Sample_(statistics)https://en.wikipedia.org/wiki/Statistical_sample


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Regression

-n statistics, regression is a statistical process or estimating the relationships among ariables. -t

includes many techni/ues or modeling and analysing seeral ariables, when the ocus is on the

relationship between a dependent ariable and one or more independent ariables. orespeciically, regression analysis helps one understand how the typical alue o the dependent

ariable )or ?criterion ariable?+ changes when any one o the independent ariables is aried,

while the other independent ariables are held ied. ost commonly, regression analysis

estimates the conditional epectation o the dependent ariable gien the independent ariables =

that is, the aerage alue o the dependent ariable when the independent ariables are ied.

Less commonly, the ocus is on a /uantile, or other location parameter o the conditional

distribution o the dependent ariable gien the independent ariables. -n all cases, the

estimation target is a unction o the independent ariables called the re5ressio2 3u2ctio2. -n

regression analysis, it is also o interest to characteri2e the ariation o the dependent ariablearound the regression unction which can be described by a probability distribution.

$egression analysis is widely used or prediction and orecasting, where its use has substantial

oerlap with the ield o machine learning. $egression analysis is also used to understand which

among the independent ariables are related to the dependent ariable, and to eplore the orms

o these relationships. -n restricted circumstances, regression analysis can be used to iner causal

relationships between the independent and dependent ariables. Howeer this can lead to

illusions or alse relationships, so caution is adisable> or eample, correlation does not imply

causation.

History

%he earliest orm o regression was the method o least s/uares, which was published

by Legendre in 56;,and by 3auss in 56;:. Legendre and 3auss both applied the method to the

problem o determining, rom astronomical obserations, the orbits o bodies about the *un

)mostly comets, but also later the then newly discoered minor planets+. 3auss published a

urther deelopment o the theory o least s/uares in 56I5,including a ersion o the 3auss=

arko theorem.

%he term regression was coined by 1rancis 3alton in the nineteenth century to describe a

biological phenomenon. %he phenomenon was that the heights o descendants o tall ancestors

tend to regress down towards a normal aerage )a phenomenon also known as regression toward

the mean+. 1or 3alton, regression had only this biological meaning, but his work was later

etended by 'dny "ule and arl #earson to a more general statistical contet. -n the work o


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"ule and #earson, the joint distribution o the response and eplanatory ariables is assumed to

be 3aussian. %his assumption was weakened by $.A. 1isher in his works o 5:II and

5:I. 1isher assumed that the conditional distribution o the response ariable is 3aussian, but

the joint distribution need not be. -n this respect, 1isher?s assumption is closer to 3auss?s

ormulation o 56I5.

-n the 5:;s and 5:N;s, economists used electromechanical desk calculators to calculate

regressions. Deore 5:7;, it sometimes took up to I9 hours to receie the result rom one

regression.

$egression methods continue to be an area o actie research. -n recent decades, new methods

hae been deeloped or robust regression, regression inoling correlated responses such

as time series and growth cures, regression in which the predictor or response ariables are

cures, images, graphs, or other comple data objects, regression methods accommodating

arious types o missing data, nonparametric regression, Dayesian methods or regression,

regression in which the predictor ariables are measured with error, regression with more

predictor ariables than obserations, and causal inerence with regression.


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Regression analysis is a mathematical measureof the averages relationship between two or

more variable in terms of the original units ofdata.

Types of Regression(i) Simple Regression (Two

Variable at a time)(ii) Multiple Regression (More than

two variable at a time)

Linear Regression !f the regression curve is astraight line then there is a linear regressionbetween the variables .

"on#linear Regression$ %urvilinear Regression!f the regression curve is not a straight linethen there is a non#linear regression betweenthe variables.


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'e(ression analysis $elps int$ree i)portant +ays :,

• It pro!ides esti)ate of!alues of dependent!ariables fro) !alues ofindependent !ariables.

• It can be e-tended to or)ore !ariables +$ic$ is/no+n as )ultiplere(ression.

•

It s$o+s t$e nature ofrelations$ip bet+een t+oor )ore !ariable.

0r


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Algebraically ethod!:

"#$east %&uare 'ethod!:

T$e re(ression euation of 2 on 3 is :24 ab3

#$ere24ependent !ariable

34Independent !ariable

T$e re(ression euation of 3 on 2 is: 3 4 ab2

#$ere 34ependent !ariable 24Independent !ariable

And t$e !alues of a and b in t$e abo!e

euations are found by t$e )et$od of leastof uares,reference . T$e !alues of a and bare found +it$ t$e $elp of nor)al euations(i!en belo+: I 9 II 9


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olution,:


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TK;.9:G;.79"


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Uses of Correlation and Regression

%here are three main uses or correlation and regression.

• !ne is to test hypotheses about cause@and@eect relationships. -n this case, the

eperimenter determines the alues o the T@ariable and sees whether ariation in T causesariation in ". 1or eample, giing people dierent amounts o a drug and measuring their

blood pressure.

ubstitution t$e !alues fro) t$e table +e(et

45a"b;;;;;;;i91684"a1"b8"41a71b;;;;;;..ii9


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• %he second main use or correlation and regression is to see whether two ariables are

associated, without necessarily inerring a cause@and@eect relationship. -n this case, neither

ariable is determined by the eperimenter> both are naturally ariable. - an association isound, the inerence is that ariation in T may cause ariation in ", or ariation in " may

cause ariation in T, or ariation in some other actor may aect both T and ".

• %he third common use o linear regression is estimating the alue o one ariablecorresponding to a particular alue o the other ariable.

Assuptions

*ome underlying assumptions goerning the uses o correlation and regression are as ollows.

%he obserations are assumed to be independent. 1or correlation, both ariables should be

random ariables, but or regression only the dependent ariable " must be random. -n carryingout hypothesis tests, the response ariable should ollow


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I. Construction engineering

M. Enironmental engineering

9. 1ire protection engineering

. 3eotechnical engineering

N. Hydraulic engineering

7. aterials science

6. *tructural engineering

:. *ureying

5;. %imber Engineering

55. %ransportation engineering

5I. Water resources engineering

5M. Agricultural Engineering

59. Ciil Engineering

5. Chemical Engineering

5N. Electrical Engineering

57. Enironmental Engineering

56. -ndustrial Engineering

5:. arine Engineering

I;. aterial *cience


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I5. echanical & -ndustrial Engineering

II. echanical Engineering

Correlation and Regression Conclusion

Although they may not know it, most successul businessmen rely on regression analysis to

predict trends to ensure the success o their businesses. Consciously or unconsciously, they rely

on regression to ensure that they produce the right products at the right time. %hey use it to

measure the success o their marketing and adertising eorts. %hey rely on inerence to

predict uture market trends and react to them. %hat is also why statistical analysis is gaining in

popularity as a career. - you are interested in statistics and how you can help business predict

uture trends or measure current success, try this course in I2troductor st1tistics rom

'demy today.

https://www.udemy.com/introductory-statistics-part1-descriptive-statistics/?tc=blog.correlationandregression&couponCode=half-off-for-blog&utm_source=blog&utm_medium=udemyads&utm_content=post35516&utm_campaign=content-marketing-blog&xref=bloghttps://www.udemy.com/introductory-statistics-part1-descriptive-statistics/?tc=blog.correlationandregression&couponCode=half-off-for-blog&utm_source=blog&utm_medium=udemyads&utm_content=post35516&utm_campaign=content-marketing-blog&xref=blog


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project report probability

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