Explain the method of fitting a straight line of the form y=a+bx to a novitiate data
Answers
Step-by-step explanation:
The graphical method has the drawback in that the straight line drawn may not
be unique but principle of least squares provides a unique set of values to the
constants and hence suggests a curve of best fit to the given data. The method of
least square is probably the most systematic procedure to fit a unique curve through
the given data points.
We will consider some of the best fitting curves of the type:
1. A straight line.
2. A second degree curve.
3. The exponential curve y = aebx.
4. The curve y — a x 71.
1. Fitting a straight line by the method of least squares:
Let (x;,yj), t = 0,1,2,....,n be the n sets of observations and let the related
relation by y = ax + b. Now we have to select a and b so that the straight line is the
best fit to the data.
As explained earlier, the residual at x = x t is
d i= y i~ f O i) = yt~ Caxi + b),i = 1,2,, 71
e = ir= i df = Xf=i [yf - (axi + b)]2
By the principle of least squares, E is minimum.
AL-Mustansirriya University
College oE Engineering
Computer & Software Eng. Dep. AiL^I Course ( l) Lecture (4)
3rd Class
dE dE — = 0 and — = 0 da db
i-e-, 2 - (axt + b)] ( - ^ ) = 0 & 2 £ [y £ - (axt + b)] ( - 1 ) = 0
i.e., - ax? - bXi) = 0 & E?=i(yi - ax< - ft) = 0
i.e., aZ?=i3C? = Z " = i^ y i .... (eq.l)
And a £"=1 x t + nb = £ f=1 y; .....(eq.2)
Since, x it y t are known, equations (1) & (2) give two equations in a & b. Solve for a
& b from (1) & (2) & obtain the best fit y= ax + b.
Note:
• Equations (1) & (2) are called normal equations.
• Dropping suffix i from (1) & (2), the normal equations are
aY ,x + nb = £ y & a£x2 + b j ^ x = £xy
Which are get taking £ on both sides of y = ax + b & also taking £ on both sides
after multiplying by x both sides of y = ax + b.
• Transformation like X = ~ ~ , Y — ~ ~ reduce the linear equation y = ax + b to
the form Y = AX + B. Hence, a linear fit is another linear fit in both systems of
coordinates.
Example 1:
By the method of least squares find the straight line to the data given below:
X 5 10 15 20 25
y 16 19 23 26 30
Solution:
Let the straight line be y=ax+b
The nor
AL-Mustansirriya University
College of Engineering
Computer & Software Eng. Dep.
Course ( l) Lecture (4)
3rd Class
X y X 2 xy
5 16 25 80
10 19 100 190
15 23 225 345
20 26 400 520
25 30 625 750
Total 75 114 1375 1885
The normal equations are 75a+5b=l 14 .....(eq.l)
1375a+75b=1885 ..... (eq.2)
Eliminate b, multiply (1) by 15
1125a+75b=1710 .....(eq.3)
(eq.2) - (eq.3) gives, 250 a=T75 or a=0.7, hence b= 1 ? ^
Hence, the best fitting line is y=0.7x+12.3
Let X = x~xmid = *~15 y = y~ymid = y~23 h 5 ’ h 5
Let the line in the new variable by Y=AX+B
X y X X2 Y XY
5 16 -2 4 -1.4 2.8
10 19 -1 1 -0.8 0.8
15 23 0 0 0 0
20 26 1 1 0.6 0.6
25 30 2 4 1.4 2.8
Total 0 10 -0.2 7
The normal equations are A £ X + 55 = £ Y .....(eq.4)
A Y X 2 + B Y , X = Z X Y .....(eq.5)
Therefore, -SB = —0.2 —> B = —0.04
10A = 7 ->A = 0.7
The equations Y=0.7X - 0.04
i.e. = 0.7 ( - 0.04 y - 23 = 0.7x - 10.5 - 0.2
i.e. y=0.7x+33.3
Which is the same equation as seen before.
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AL-Mustansirriya University
College of Engineering
Computer & Software Eng. Dep. ■Miami
Course ( l) Lecture (4)
3rd Class
Example 2:
Fit a straight line to the data given below. Also estimate the value of y at x=2.5
x 0 l 2 3 4
y l 1.8 3.3 4.5 6.3
Solution:
Let the best fit be y= ax + b .....(eq.l)
The normal equations are a £ x + 5b = £ y .....(eq.2)
a£x2 + b'Zx = 2 > y .....(eq.3)
We prepare the table for easy use.
X y X 2 xy
0 l 0 0
1 1.8 1 1.8
2 3.3 4 6.6
3 4.5 9 13.5
4 6.3 16 25.2
Total 10 16.9 30 47.1
Substituting in (eq.2) and (eq.3), we get,
10a+5b=16.9
30a+10b=47.1
Solving eq.(2)-eq.(l), we get, a=l .33, b= 0.72
Hence, the equation is y =1.33x+0.72
y (at x=2.5) =1.33 (2.5) +0.72 = 4.045
Answer:
y=a+bx
Explanation:
Procedure:
(i) Straight line is represented by:
Y = a + bX
where Y =actual value
X =time
a, b =constants
(ii) ‘a’ and ‘b’ are estimated by solving the following two normal equations :
ΣY = n a + b ΣX
ΣXY = a ΣX + b ΣX²
Where n = number of years.
(iii) When ΣX = 0, the two normal equations reduces to
a=ΣY/n and b=ΣXY/ΣX²
(iv) By substituting, ‘a’ and ‘b’ in the equation which is given, we get the Line of Best Fit to novitiate the data.
Therefore, we get the line of best fit for y=a+bx.
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