Least Square Regression

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09 ME [126,127,129,X-06 &09 ME [126,127,129,X-06 &

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Presentation On Least SquarePresentation On Least SquareLinear RegressionLinear Regression

Concerned Teacher: SirConcerned Teacher: SirZeeshan Ali MemonZeeshan Ali Memon


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LEAST SQUARE LINEARLEAST SQU

ARE LINEAR

REGRESSIONREGRESSION

LINEAR REGRESSIONLINEAR REGRESSION

GRAPHICAL REPRESENTATIONGRAPHICAL REPRESENTATION

LEAST SQUARE METHODLEAST SQ

UARE METHOD

LEAST SQUARE REGRESSION ANALYSISLEAST SQ

UARE REGRESSION ANALYSIS

EXAMPLEEXAMPLE

APPLICATIONSAPPLICATIONS


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LINEAR REGRESSIONINEAR REGRESSION In statistics,In statistics, l inear regressioninear regression is an approachis an approach

to modeling the relationshipto modeling the relationshipbetweenbetweena scalara scalarvariablevariableyy and one or more variablesand one or more variablesdenoteddenotedbyby

XX.. NoteNotethat one variable is dependent on thethat one variable is dependent on theother.other.

In linear regression, data is modeled using linearIn linear regression, data is modeled using linearfunctions, and unknown model parameters arefunctions, and unknown model parameters areestimated from the data.estimated from the data.Such models areSuch models arecalledcalledl inear modelsinear models ..


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GRAPHICAL REPRESENTATIONRAPHICAL REPRESENTATION


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The red line in the graphThe red line in the graph

is known asis known as regressionegression l ineine joining the pointsjoining the pointswhich are best fit forwhich are best fit for

the data given.the data given.

Taking a simple example,Taking a simple example, Y=aXY=aXis a simple linear equationis a simple linear equation

wherewhere YY is dependent onis dependent on X.X. By using given values ofBy using given values of XX,,putting them in the equation ,getting different values ofputting them in the equation ,getting different values of YY,,

we get different points in the graph. Of them the best fitwe get different points in the graph. Of them the best fit

points are those which make a straight line as shown in thepoints are those which make a straight line as shown in the

graph above.graph above.


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LEAST SQUARE METHODEAST SQUARE METHOD The least-squares method was first described byThe least-squares method was first described byCarl Friedrich Gaussarl Friedrich Gauss around 1794. Thearound 1794. The

method of least squares is a standard approach tomethod of least squares is a standard approach tothe approximate solution of over determinedthe approximate solution of over determinedsystems, i.e. sets of equations in which there aresystems, i.e. sets of equations in which there aremore equations than unknowns.more equations than unknowns.

The most important application is in data fitting.The most important application is in data fitting.It is often used for regression analysis.It is often used for regression analysis.


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LEAST SQUARE REGRESSIONEAST SQUARE REGRESSIONANALYSISNALYSIS

Consider a straight line equation y = a + bx. WhereConsider a straight line equation y = a + bx. Where aa andandbbare coefficients representing the intercept and the slopeare coefficients representing the intercept and the sloperespectively.respectively.

From above equation a = y bx .....(i)From above equation a = y bx .....(i)

b = (nb = (n xy - x yxy - x y)) [n x [n x22 ((x)x)22]. This is a least]. This is a leastsquare linear equation for a straight line.square linear equation for a straight line.

Then using the value ofThen using the value of bb in (i) we getin (i) we get a.a.

Finally usingFinally using aa andandbb in the main equation we get the leastin the main equation we get the leastsquares fit.squares fit.


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EXAMPLEXAMPLE The following data was obtained in a PhysicsThe following data was obtained in a Physics

experiment and plotted. Determine if a straightexperiment and plotted. Determine if a straight

line is a good model for this data and if so,line is a good model for this data and if so,

determine the line of best fit.determine the line of best fit.

Time (s)Time (s) Distance (cm)Distance (cm)

11 22

33 55

55 88

77 1111

99 1515


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When this data is plotted as a graph, it appears that aWhen this data is plotted as a graph, it appears that astraight line is the best model:straight line is the best model:

The next step is to calculate 'a' and 'b', which areThe next step is to calculate 'a' and 'b', which arecalculated to becalculated to bea= 1.6 s and b=0.2 cm bya= 1.6 s and b=0.2 cm by least square equations.east square equations.

Therefore the e uation of the line of best fit is d=1.6t+0.2.Therefore, the e uation of the line of best fit is d=1.6t+0.2.


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APPLICATIONSPPLICATIONS If the goal is prediction, or forecasting, linear regressionIf the goal is prediction, or forecasting, linear regression

can be used to fit a predictive model to an observed datacan be used to fit a predictive model to an observed dataset ofset ofyyandand XXvalues. After developing such a model, ifvalues. After developing such a model, if

an additional value ofan additional value ofXXis then given without itsis then given without itsaccompanying value ofaccompanying value ofyy, the fitted model can be used, the fitted model can be usedto make a prediction of the value ofto make a prediction of the value ofyy..

Given a variableGiven a variable yyand a number of variablesand a number of variables XX11, ... ,, ... ,XXpp that may be related tothat may be related to yy, linear regression analysis, linear regression analysiscan be applied to quantify the strength of thecan be applied to quantify the strength of therelationship betweenrelationship between yyand theand the XXjj, to assess which, to assess which XXjj

may have no relationship withmay have no relationship with yyat all.at all.


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THE ENDTHE END

Thanks for yourThanks for your

concentration.concentration.

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Least Square Regression