ENGR-45: Chp5 Workshop on Fitting of Linear Data
SJSU↔Chabot Notation Correspondence
EAllen↔BMayer Linear Regression Derivation
Notation
EA
BM
1
Meaning
Number of DATA POINTS as ORDERED PAIRS
n
n
 xi , y i 
 xk , y k 
a0
b
General INTERCEPT Value for a LINEAR Equation
a1
m
General SLOPE Value for a LINEAR Equation
yp
yL
Value of y as PREDICTED by the Linear Equation using as it’s input an ACTUAL x value
ei
k
Error, or Residual, between the ACTUAL y-value and the y-valued PREDICTED by the linear equation
S
J
Sum of the SQUARED ERRORS
â 0
b0
LEAST SQUARED-ERROR (Best) INTERCEPT Value for a LINEAR Equation
â1
m0
LEAST SQUARED-ERROR (Best) SLOPE Value for a LINEAR Equation
−−−
S
Sum of Squared Differences ABOUT THE MEAN1
An arbitrary ACTUAL Data Point/Pair
Used in Goodness of Fit Analysis which is not addressed in Professor Allen’s Linear Regression Discussion.
© Bruce Mayer, PE • Chabot College • Document1 • Page 1
Linearization
Consider this set of statements by Professor Allen regarding SCATTERED (x,y) Data:
 Whether there appears to be any functional relationship at all among the data
 If there is a relationship, determine which is the independent variable and which is the
dependent variable.
 Determine whether there is a linear relationship between the data, and why you might
expect one.
 If the relationship is not linear, determine whether there is a way to transform the data
so that it is linear.
o See Chapter 6 in William J Palm III, Introduction to MATLAB for Engineers, 3rd
Edition, McGraw-Hill, ©2011, ISBN-13 9780073534879
Consider the Electrical Circuit Shown at Right. Say that we need to determine the
RESISTANCE value for RL. Furthermore, we have at our disposal TWO instruments:
 a POWER METER that is part of the Voltage-Based Power supply, Vs
o Measures the Power
dissipated by RL in
WATTS
 an AMMETER, A, that can be
inserted into to the circuit as
shown
o Measures the Current
flowing thru RL in
AMPERES, or “Amps’ for
short
From PHYS4B and/or ENGR43 we
expect this physical relationship
between the resistor Power, P, and the
resistor current, RL:
A simple, Power Dissipating Electrical Circuit. The
Value of RL is UNKnown
P  RI L2
Recognizing P as the dependent variable and I as the independent quantity the power
equation can be transformed to linear form by:
P y
Rm
I L2  x
Thus the NON-Linear (it’s quadratic) Power equation can be viewed, using the above
transformation as LINEAR in the current-Squared
P  RI L2
Linear Transform 
y  mx
© Bruce Mayer, PE • Chabot College • Document1 • Page 2
So a plot of P vs. (IL)2 should produce a STRAIGHT LINE, thru the ORIGIN, with a SLOPE
equal to the resistance value of RL
See the next page for an example of RAW and TRANSFORMED Resistor Power Plots
© Bruce Mayer, PE • Chabot College • Document1 • Page 3
Prand vs I
1000
900
800
700
Power (W)
600
500
400
300
200
100
0
-100
0
0.5
1
1.5
2
2.5
3
Current (Amps)
3.5
4
4.5
5
Prand vs I2
1000
900
800
700
Power (W)
600
500
400
300
200
100
0
-100
0
5
10
15
Current-Squared (Square-Amps)
20
25
Print Date/Time = 10-Feb-16/07:41
© Bruce Mayer, PE • Chabot College • Document1 • Page 4