Case Study of Curve fitting

Subject :- NM(2140706)

CO (evening) -: Prepared Bye :-

PATEL ADARSH (36)KRISNA MISRA (30)BHUVA MAHESH (06)

Guided Faculty :-PROF. JIGISHA PANDYA

Sarvajanik College Of Engineering and Technology

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Polynomial regression

EXAMPLE :: 1 A group of senior citizens who have never used the Internet

before are given training. A sample of 5 people is chosen at random and the number of hours of Internet use is recorded for 6 months, as shown in the table on the upper left side of Figure 1. Determine whether a quadratic regression line is a good fit for the data.

3Figure 1 – Data for polynomial regression in Example 1

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We next create the table on the right in Figure 1 from this data, adding a second independent variable (MonSq) which is equal to the square of the month. We now run the Regression data analysis tool using the table on the right (quadratic model) in columns I, J and K as the input. The results are displayed in Figure 2.

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Figure 2 – Quadratic regression output

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Figure 2 also shows that the regression quadratic that best fits the data isHours of Use = 21.92 – 24.55 * Month + 8.06 * Month2

Thus to predict the number of hours that a particular senior will use the Internet after 3 months, we plug 3 into the model (or use the TREND function) to get 20.8 hours of use.

We can also run the Regression data analysis tool on the original data to compare the above results with the linear model studied in Regression Analysis The linear model is generated by using only columns I and K from Figure 1. The output is shown in Figure 3.

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Figure 3 – Linear regression output

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EXAMPLE :: 2 UV-visible absorption spectrophotometer

Extrapolation

The ratio ofthe two light intensities, transmitted light (I) over the incident light (I0) is known as thetransmittance of the sample. And the absorbance is calculated by

A = - log(I/I0)

where I and I0 are respectively the transmitted and incident light intensities

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Analysis of UV-Visible absorption spectrum:

The figure below shows the UV-Visible absorption spectrum of P3HT:PCBM blend.

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The energy band gap can be calculatedusing absorption spectrum.Absorption coefficient, α, can be extracted from absorbance:

α = A/t;where t = thicknessA graph between (αhv)1/2 and Energy E can beused to find the energy band gap by extrapolating the linearregion.Here,

E (eV) = 1240/λ

The energy band gap of P3HT:PCBM lies between 1.85-2 eV.

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The results obtained using Lagrange’s extrapolation formula are:

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A Case Study: Upward Velocity of a Rocket

As a rocket is launched from the ground, its upward velocity, v(t) (m/s), is measured at certaintime instants t(s). Suppose that one measures the upward velocity of a rocket for time 0 t 30 and the measurements are tabulated as follows:

T(S) V(T) (m/s)

0 0

10 250

15 350

22 655

25 890

30 910

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Using the above table, one would like to predict the velocity of the rocket at certain non tabulated times, say, t = 5s, t = 20s, t = 23s, t = 29s. Such a problem of predicting the values of the dependent variable at non tabulated values of the independent variable in a given interval is called interpolation.

For the above rocket example, if we can find a function v(t), that interpolates the above data, then it can be immediately used to predict its value for any value of t in that interval.

For this example, of course, once v(t) is determined, it can also be used to find the acceleration of the rocket at a certain given time just by differentiating the function. Similarly, the distance covered from time t = t1 to time t = t2 (t2 > t1) in the given interval, can be obtained by evaluating the integral

A = t1∫t2 v(t) dt

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From the Lagrange interpolating polynomial P3(x) of degree 3:

P3(x) = 7L0(x) + 13L1(x) + 21L2(x) + 43L3(x)

We will compute P3(3) and accept it as an interpolated value of f(3).

L0(3) = 1/4 , L1(3) = −1, L2(3) = 3/2 , L3(3) = 1/4 .

So, P3(3) = 7L0(3) + 13L1(3) + 21L2(3) + 43L3(3) = 31.

Interpolated value of f(3) = P3(3) = 31.

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Bibliography

Polynomial regression Example:1

http://www.real-statistics.com/multiple-regression/polynomial-regression/

Polynomial regression Example:2

Department of Physics & Astrophysics, University of Delhi

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