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MatLab – Palm Chapter 5 Curve Fitting. Class 14.1 Palm Chapter: 5.5-5.7. RAT 14.1. As in INDIVIDUAL you have 1 minute to answer the following question and another 30 seconds to turn it in. Ready? When (day and time) and where is Exam #3? The answer is: Thursday at 6:30 pm,

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Matlab palm chapter 5 curve fitting

MatLab – Palm Chapter 5Curve Fitting

Class 14.1 Palm Chapter: 5.5-5.7

ENGR 111A - Fall 2004


Rat 14 1
RAT 14.1

  • As in INDIVIDUAL you have 1 minute to answer the following question and another 30 seconds to turn it in. Ready?

  • When (day and time) and where is Exam #3?

  • The answer is: Thursday at 6:30 pm,

    Bright 124

  • Do we have any schedule problems?

ENGR 111A - Fall 2004


Learning objectives
Learning Objectives

  • Students should be able to:

    • Use the Function Discovery (i.e., curve fitting) Techniques

    • Use Regression Analysis

ENGR 111A - Fall 2004


5 5 function discovery
5.5 Function Discovery

  • Engineers use a few standard functions to represent physical conditions for design purposes. They are:

    • Linear: y(x) = mx + b

    • Power: y(x) = bxm

    • Exponential: y(x) = bemx (Naperian)

      y(x) = b(10)mx (Briggsian)

  • The corresponding plot types are explained at the top of p. 299.

ENGR 111A - Fall 2004


Steps for function discovery
Steps for Function Discovery

  • Examine data and theory near the origin; look for zeros and ones for a hint as to type.

  • Plot using rectilinear scales; if it is a straight line, it’s linear. Otherwise:

    • y(0) = 0 try power function

    • Otherwise, try exponential function

  • If power function, log-log is a straight line.

  • If exponential, semi-log is a straight line.

ENGR 111A - Fall 2004


Example function calls
Example Function Calls

  • polyfit( ) will provide the slope and y-intercept of the BEST fit line if a line function is specified.

  • Linear: polyfit(x, y, 1)

  • Power: polyfit(log10(x),log10(y),1)

  • Exponential: polyfit(x,log10(y),1); Briggsian

    polyfit(x,log(y),1); Naperian

    Note: the use of log10( ) or log( ) to transform the data to a linear dataset.

ENGR 111A - Fall 2004


Example 5 5 1 cantilever beam deflection
Example 5.5-1:Cantilever Beam Deflection

  • First, input the data table on page 304.

  • Next, plot deflection versus force (use data symbols or a line?)

  • Then, add axes and labels.

  • Use polyfit() to fit a line.

  • Hold the plot and add the fitted line to your graph.

ENGR 111A - Fall 2004


Solution
Solution

ENGR 111A - Fall 2004


Straight line plots
Straight Line Plots

ENGR 111A - Fall 2004


Why do these plot as lines
Why do these plot as lines?

Exponential function: y = bemx

Take the Naperian logarithm of both sides:

ln(y) = ln(bemx)

ln(y) = ln(b) + mx(ln(e))

ln(y) = ln(b) + mx

Thus, if the x value is plotted on a linear scale and the y value on a log scale, it is a straight line with a slope of m and y-intercept of ln(b).

ENGR 111A - Fall 2004


Why do these plot as lines1
Why do these plot as lines?

Exponential function: y = b10mx

Take the Briggsian logarithm of both sides:

log(y) = log(b10mx)

log(y) = log(b) + mx(log(10))

log(y) = log(b) + mx

Thus, if the x value is plotted on a linear scale and the y value on a log scale, it is a straight line. (Same as Naperian.)

ENGR 111A - Fall 2004


Why do these plot as lines2
Why do these plot as lines?

Power function: y = bxm

Take the Briggsian logarithm of both sides:

log(y) = log(bxm)

log(y) = log(b) + log(xm)

log(y) = log(b) + mlog(x)

Thus, if the x and y values are plotted on a on a log scale, it is a straight line. (Same can be done with Naperian log.)

ENGR 111A - Fall 2004


In class assignment 14 1 1
In-class Assignment 14.1.1

Given:

x=[1 2 3 4 5 6 7 8 9 10];

y1=[3 5 7 8 10 14 15 17 20 21];

y2=[3 8 16 24 34 44 56 68 81 95];

y3=[8 11 15 20 27 36 49 66 89 121];

  • Use MATLAB to plot x vs each of the y data sets.

  • Chose the best coordinate system for the data.

  • Be ready to explain why the system you chose is the best one.

ENGR 111A - Fall 2004


Solution1
Solution

ENGR 111A - Fall 2004


Be careful
Be Careful

  • What value does the first tick mark after 100 represent? What about the tick mark after 101 or 102?

  • Where is zero on a log scale? Or -25?

  • See pages 282 and 284 of Palm for more special characteristics of logarithmic plots.

ENGR 111A - Fall 2004


How to use polyfit command
How to use polyfit command.

  • Linear: pl = polyfit(x, y, 1)

    • m = pl(1); b = pl(2) of BEST FIT line.

  • Power: pp = polyfit(log10(x),log10(y),1)

    • m = pp(1); b = 10^pp(2) of BEST FIT line.

  • Exponential: pe = polyfit(x,log10(y),1)

    • m = pe(1); b = 10^pe(2), best fit line using Briggsian base.

      OR pe = polyfit(x,log(y),1)

    • m = pe(1); b = exp(pe(2)), best fit line using Naperian base.

ENGR 111A - Fall 2004


In class assignment 14 1 2
In-class Assignment 14.1.2

  • Determine the equation of the best-fit line for each of the data sets in In-class Assignment 14.1.1

  • Hint: use the result from ICA 14.1.1 and the polyfit( ) function in MatLab.

  • Plot the fitted lines in the figure.

ENGR 111A - Fall 2004


Solution2
Solution

ENGR 111A - Fall 2004


5 6 regression analysis
5.6 Regression Analysis

  • Involves a dependent variable (y) as a function of an independent variable (x), generally: y = mx + b

  • We use a “best fit” line through the data as an approximation to establish the values of: m = slope and b = y-axis intercept.

  • We either “eye ball” a line with a straight-edge or use the method of least squares to find these values.

ENGR 111A - Fall 2004


Curve fits by least squares
Curve Fits by Least Squares

  • Use Linear Regression unless you know that the data follows a different pattern: like n-degree polynomials, multiple linear, log-log, etc.

  • We will explore 1st (linear), … 4th order fits.

  • Cubic splines (piecewise, cubic) are a recently developed mathematical technique that closely follows the “ship’s” curves and analogue spline curves used in design offices for centuries for airplane and ship building.

  • Curve fitting is a common practice used my engineers.

ENGR 111A - Fall 2004


T5 6 1
T5.6-1

  • Solve problem T5.6-1 on page 318.

  • Notice that the fit looks better the higher the order – you can make it go through the points.

  • Use your fitted curves to estimate y at x = 10. Which order polynomial do you trust more out at x = 10? Why?

ENGR 111A - Fall 2004


Solution3
Solution

ENGR 111A - Fall 2004


Solution4
Solution

ENGR 111A - Fall 2004


Assignment 14 1
Assignment 14.1

  • Prepare for Exam #3.

  • Group Projects are due at Exam #3(parts 1 through 3 required; parts 4 and 5 as extra credit)

ENGR 111A - Fall 2004


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