case study of curve fitting this ppt includes the real life application of curve fitting in routine world ..

2. 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.
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3. Figure 1 – Data for polynomial regression in Example 1
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4. 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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5. Figure 2 – Quadratic regression output
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6. Figure 2 also shows that the regression quadratic that best fits
the data is
Hours 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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7. Figure 3 – Linear regression output
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8. The ratio of
the two light intensities, transmitted light (I) over the incident
light (I0) is known as the
transmittance 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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9. The figure below shows the UV-Visible absorption spectrum of
P3HT:PCBM blend.
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10. The energy band gap can be calculated
using absorption spectrum.
Absorption coefficient, α, can be extracted from absorbance:
α = A/t;
where t = thickness
A graph between (αhv)1/2 and Energy E can be
used to find the energy band gap by extrapolating the linear
region.
Here,
E (eV) = 1240/λ
The energy band gap of P3HT:PCBM lies between 1.85-2 eV.
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11. The results obtained using Lagrange’s
extrapolation formula are:
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12. As a rocket is launched from the ground, its upward
velocity, v(t) (m/s), is measured at certain
time 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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13. 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
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14. From the Lagrange interpolating polynomial P3(x) of
degree 3:
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.
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15. 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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