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Linear least square methods allow us to
study how variables are related. Let Yi =
α +
βXi
+ε i
for i = 1, 2,.. N are independent random variables with means
E(Yi) = α
+ βXi
& that the collection i is a random sample from a distribution with
mean 0 and standard deviation σ , and that all parameters ( α
, β &
ε
) are unknown. Independent variable are called
regressors /covariates; dependent variable is called response
variable/endogenous variable. ε is
the error. |
| Independent variable,dataset1 :x1 |
|
dependent variable-dataset1: y1 |
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| Independent variable,dataset2 :x2 |
|
dependent variable-dataset2: y2 |
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| Independent variable,dataset3 :x3 |
|
dependent variable-dataset3: y3 |
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| Independent variable,dataset4 :x4 |
|
dependent variable-dataset4: y4 |
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| δs/δβ = 0 |
β +
α =
|
δs/δα = 0 |
β +
α =
|
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β= |
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Line of Best Fit |
y=
x
+
|
|
α= |
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Discrepancy in y (expt.value-best fit value) |
ε1=ε2=ε3=ε4= |
| |
|
sum of square of deviation, S(α , β ) |
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| |
|
standard deviation( σ) |
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Fitting a curve of Quadratic Function -I |
y =bx2
; S(b)
=Σ [y-bx2 ]2 |
| δs/δb = 0 |
b= |
Best Fit Curve |
y= x2 |
| |
|
Discrepancy in y (expt.value-best fit value) |
ε1=ε2=ε3=ε4= |
| |
|
sum of square of deviation, S(α , β ) |
|
| reference:
http://en.wikipedia.org/wiki/List_of_statistical_packages |
standard deviation( σ) |
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Fitting a curve of Quadratic Function -II |
y =γx2
+βx +α
S(γ,β,α)=Σ [y-(γx2+βx+α)
]2 |
| δs/δγ = 0 |
γ
+ β
+ α =
|
|
| δs/δβ = 0 |
γ
+ β
+ α =
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|
| δs/δα = 0 |
γ
+ β
+ α =
|
|
|
γ = |
|
Best Fit Curve |
y =x2
+x + |
| β = |
|
Discrepancy in y (expt.value-best fit value) |
ε1=ε2=ε3=ε4= |
| α = |
|
sum of square of deviation, S(γ , β
,α ) |
|
| |
|
standard deviation( σ) |
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