advertisement

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 Rm 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