Expanding Linear
Regression
Introduction
– Linear Regression
All
scientific calculators that have two-variable dimensions has curve
fitting. The most (and universal) curve
fitting mode is linear regression. Linear regression attempts to model bivariate
data ((x,y)) to a line by the equation:
y = a + bx
Where
a is the y-intercept (ITC) and b is the slope (SLP). The correlation coefficient, r, is calculated by:
r
= SXY / √(SX2 * SY2) where
SX2
= Σ(x – μx)^2
SY2
= Σ(y – μy)^2
SXY
= Σ( (x – μx)*(y – μy) )
μx
= mean of x values, μy = mean of y values
Source: HP 17B II Financial Calculator Owner’s
Manual, Corvallis, OR 1989.
The
best fits are were |r| = 1, or as close to 1 as we can get. If r is close to 0, the first is not good. Note that the formula for correlation does
not involve the slope or intercept. We
can use this to our advantage.
“Linearize”
the Equation
If
you can “linearize” equation, we can use the linear regression mode to fit a
variety of curves. The ultimate goal is
of course:
y
= a + bx
Now,
most calculators use the form for linear equations. However, some switch the a and b around and use
the form, like the TI-30X IIS,
y
= ax + b
Hewlett
Packard calculators has the b for the intercept and m for slope.
For
clarity, I will use the form Y = ITC + SLP*X.
Calculator Comparison
To
demonstrate how we can use the linear regression to fit a variety of curves, I
use two calculators:
Sharp
EL-W516T which offers 7 regressions.
TI-30X
IIS which offers just linear regression.
I use the TI-30X IIS to use the linearized form.
[screen
shot of the calcs]
|
|
Sharp
EL-W516T
|
TI-30X
IIS
|
|
SLP
|
B
|
a
|
|
ITC
|
A
|
b
|
|
Linear
Regression Equation
|
y
= a + bx
|
y
= ax + b
|
Screen
shots are generated from a Casio fx-CG50.
Logarithmic
Regression: y = a + b ln x
Translation
to Linear: This is pretty simple, we
already have the form that we need, except we note that we have ln x instead of
x. Set the following variables as such:
X
= ln x, Y = y, a = ITC, b = SLP
Example
data set:
|
x
|
y
|
X
= ln x (8 decimal places)
|
|
1.0
|
2.00
|
0
|
|
1.3
|
3.00
|
0.2623643
|
|
1.7
|
4.15
|
0.5306283
|
|
2.3
|
5.34
|
0.8329091
|
|
2.8
|
6.10
|
1.0296194
|
|
3.5
|
7.00
|
1.252763
|
To
use the linear regression mode, enter the data as such: (ln x, y).
Everything else remains the same.
Logarithmic
Regression Results:
ITC
= a = 1.990614146
SLP
= b = 4.003372076
r
= 0.99905634
Comparing
results between using the Logarithmic regression with the Sharp EL-W516T and
the Linear regression with the TI-30X IIS, I get the same results.
Exponential
Regression: y = a*e^(b*x)
Translation
to linear:
y
= a*e^(b*x)
ln
y = ln (a*e^(b*x))
ln
y = ln a + ln(e^(b*x))
ln
y = ln a + b*x
We
got our form of Y = ITC + SLP*X where:
X
= x, Y = ln y, a = e^ITC, b = SLP
When
using the linear regression mode, enter data as (x, ln y). When calculating a, note that
ITC
= ln a
e^ITC
= a
Example
data set:
|
X
|
y
|
Y
= ln y (8 decimal places)
|
|
1.0
|
109.2
|
4.693181
|
|
1.3
|
363.0
|
5.8944028
|
|
1.7
|
1795.6
|
7.4930945
|
|
2.3
|
19794.3
|
9.8931493
|
|
2.8
|
146260.8
|
11.893147
|
|
3.5
|
2405208.5
|
14.693147
|
Results:
ITC
= 0.693829923; a = 2.001365951
SLP
= b = 3.99773169
r
= 0.999999992
Inverse
Regression: y = a + b/x
Translation
to Linear: Like logarithmic regression,
we pretty much have the equation pretty much set, with the exception of 1/x
instead of x. Hence:
X
= 1/x, Y = y, a = ITC, b = SLP
When using the linear regression mode, enter
data as (1/x, y). Everything else remains.
Example
data set:
|
X
|
y
|
X
= 1/x (8 decimal places)
|
|
1.0
|
6.00
|
1
|
|
1.3
|
5.55
|
0.7692308
|
|
1.7
|
5.16
|
0.5882353
|
|
2.3
|
4.86
|
0.4347826
|
|
2.8
|
4.73
|
0.3571429
|
|
3.5
|
4.58
|
0.2857143
|
Results:
a
= ITC = 4.005529472
b
= SLP = 1.993191341
r
= 0.999733184
Power
Regression: y = a*x^b
Translation
to Linear: X = ln x, Y = ln y, A = e^a,
b
y
= a*x^b
ln
y = ln (a*x^b)
ln
y = ln a + ln(x^b)
ln
y = ln a + b * ln x
So
we have the following: X = ln x, Y = ln y, ITC = ln a, SLP = b.
In
the linear regression mode, enter data as (ln x, ln y). Also, a = e^ITC.
Example
data set:
|
x
|
y
|
X
= ln x
|
Y
= ln y
|
|
1.0
|
3.95
|
0
|
1.3737156
|
|
1.3
|
6.76
|
0.2623643
|
1.9110229
|
|
1.7
|
11.56
|
0.5306283
|
2.4475509
|
|
2.3
|
21.09
|
0.8329091
|
3.048799
|
|
2.8
|
31.40
|
1.0296194
|
3.4468079
|
|
3.5
|
49.00
|
1.252763
|
3.8918203
|
Results:
ITC
= 1.379195052; a = e^ITC = 3.971703326
SLP
= b = 2.007158681
CORR
= r = 0.999991481
Table of Linear
Regression Equivalents to Curve Fitting
Regression |
X |
Y |
ITC |
SLP |
Logarithmic: y = a + b * ln x |
ln x |
y |
a |
b |
Exponential: y = a*e^(b*x) |
x |
ln y |
e^a |
b |
Inverse: y = a + b/x |
1/x |
y |
a |
b |
Power: y = a*x^b |
ln x |
ln y |
e^a |
b |
General Exponential: y = a * b^x |
x |
ln y |
e^a |
e^b |
Simple Logistic: y = 1/(a + b*e^(-x)) |
e^(-x) |
1/y |
a |
b |
Square Root Linear: y = √(a + b*x) |
x |
y^2 |
a |
b |
Cosine: y = a + b*cos(ω(x – ϕ)) With ϕ = the point (x) nearest to zero where the trough or peak begins ω = (2*π)/period (radians) or ω = 360°/period (degrees) |
cos(ω(x – ϕ)) |
y |
a |
b |
Eddie
This
blog is property of Edward Shore, 2016