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 • 291186790 • 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.
Consider this 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:
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
A simple, Power Dissipating Electrical Circuit. The
Value of RL is UNKnown
E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
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 • 291186790 • Page 2
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 = 29-May-16/04:00
© Bruce Mayer, PE • Chabot College • 291186790 • Page 3