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Name:Sujit Kumar Saha
Lecturer at Varendra University Rajshahi
Name: Istiaque Ahmed ShuvoId: 1413110575th batch, 7th SemesterSec-B
Dept. Of CseVarendra University, Rajshahi
Submitted By: Submitted To
11-Apr-16
Curve Fitting
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TOPICS ARE
Linear Regression Multiple Linear Regression Polynomial Regression Example of Newton’s Interpolation
Polynomial And example
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Fitting a straight line to a set of paired observations: (x1, y1), (x2, y2),…,(xn, yn).
y = a0+ a1 x + ea1 - slopea0 - intercepte - error, or residual, between the model and the observations
Linear Regression
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210
10
11
1
0
0
0)(2
0)(2
iiii
ii
iioir
ioio
r
xaxaxy
xaay
xxaayaS
xaayaS
210
10
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iiii
ii
xaxaxy
yaxna
naa
2 equations with 2 unknowns, can be solved
simultaneously
Linear Regression: Determination of ao and a1
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221
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xxn
yxyxna
xaya 10
Linear Regression: Determination of ao and a1
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• Another useful extension of linear regression is the case where y is a linear function of two or more independent variables:
• Again, the best fit is obtained by minimizing the sum of the squares of the estimate residuals:
Multiple Linear Regression
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• The least-squares procedure from Chapter 13 can be readily extended to fit data to a higher-order polynomial. Again, the idea is to minimize the sum of the squares of the estimate residuals.
• The figure shows the same data fit with:
a) A first order polynomialb) A second order polynomial
Polynomial Regression
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Many times, data is given only at discrete points such as (x0, y0), (x1, y1), ......, (xn−1, yn−1),(xn, yn). So, how then does one find the value of y at any other value of x ? Well, acontinuous function f (x) may be used to represent the n +1 data values with f (x)
passing through the n +1 points (Figure 1). Then one can find the value of y at anyother value of x . This is called interpolation.
Of course, if x falls outside the range of x for which the data is given, it is nolonger interpolation but instead is called extrapolation.
So what kind of function f (x) should one choose? A polynomial is a commonchoice for an interpolating function because polynomials are easy to
(A) evaluate,(B) differentiate, and
(C) integrate,relative to other choices such as a trigonometric and exponential series.
Polynomial interpolation involves finding a polynomial of order n that passesthrough the n +1 points. One of the methods of interpolation is called Newton’s divided
difference polynomial method. Other methods include the direct method and theLagrangian interpolation method. We will discuss Newton’s divided difference
polynomial method in this
What is interpolation?
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Newton’s Divided-Difference Interpolating Polynomials?
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Example of Newton’s Interpolation Polynomial
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