7/29/2019 Lecture 14 - Curve Fitting (Intro&Regression)

1/17

PE 3723

Numerical MethodsLecture 14: Curve Fitting

(Introduction & Regression)Maysam Pournik

Assistant Professor

Spring 20121

7/29/2019 Lecture 14 - Curve Fitting (Intro&Regression)

2/17

Curve Fitting Applications

2

Data given for discrete values along acontinuum

Objective: Need to estimate values betweendiscrete values

1. Techniques to fit curves to the data toobtain intermediate values

2. Need simplified version of a complicatedfunction

Compute values at discrete points & derive asimpler function to fit these values

7/29/2019 Lecture 14 - Curve Fitting (Intro&Regression)

3/17

Approaches to Curve Fitting

3

Two general approaches for curve fitting:

1. Data with error: Least-Squares Regression

Single curve that represents

general trend designed to followpattern, not intersect every point

Applied to data exhibiting

significant error

2. Data with no error: Interpolation

Fit a curve that pass directly through

each of the points

Applied to very precise data

7/29/2019 Lecture 14 - Curve Fitting (Intro&Regression)

4/17

Simple Method - Noncomputer

4

Plot and fit a line that virtually conforms to data Valid for quick estimate

Subjective viewpoint

Need for systematic and objective methods

Two types of application in fitting experimental data:A. Trend analysis: Process of using the data pattern for predictions.

B. Hypothesis testing: An existing mathematical model is

compared with experimental data.

7/29/2019 Lecture 14 - Curve Fitting (Intro&Regression)

5/17

Mathematical Background

5

Use statistics to convey as much information aspossible about specific characteristics of data set

Location of the center of distribution Arithmetic mean: sum of individual points divided by

total number of data points

7/29/2019 Lecture 14 - Curve Fitting (Intro&Regression)

6/17

Mathematical Background

6

Degree of spread of data set

Standard deviation: square root of total sum of residuals

between data points and mean value divided by total

number of data points minus 1

Variance: square of standard deviation

Coefficient of variation: ratio of standard deviation tothe mean

Provides a normalized measure of spread

Similar to relative error as is ratio of a measure of error to an

estimate of true value

7/29/2019 Lecture 14 - Curve Fitting (Intro&Regression)

7/17

Mathematical Background

7

Shape of data distribution

Histogram:

visual representation of

distribution

Constructed by sorting

measurements into intervals

Plotted as units of measurement

versus frequency of occurrence Can be represented by a single

smooth curve

One type is Normal distribution

symmetric, bell-shaped curve

7/29/2019 Lecture 14 - Curve Fitting (Intro&Regression)

8/17

Mathematical Background

8

Quantify confidence on a measurement

For normal distribution:

to + : contains 68% of total data

to + : contains 95% of total data

Need to estimate properties of a population based on

limited sample of the populationStatistical

Inference

Define a confidence interval around our estimate

& : sample information (estimated)

& : population information (true)

7/29/2019 Lecture 14 - Curve Fitting (Intro&Regression)

9/17

Mathematical Background

9

Interval estimator: gives range of values within

which the parameter is expected to lie with a given

probability

One-sided intervalless than or greater than true

Two-sided intervalno consideration to sign of

discrepancy

7/29/2019 Lecture 14 - Curve Fitting (Intro&Regression)

10/17

Mathematical Background

10

Probability that true mean of y, , falls within the

bound from L to U is 1- ( is significance level)

Standard normal estimate

Normalized distance between and

Normally distributed with mean of 0 and variance of 1

Probability that it lies withinz/2 and z/2 is 1-

Tabulated in books and software

7/29/2019 Lecture 14 - Curve Fitting (Intro&Regression)

11/17

Mathematical Background

11

Estimation of L and U

As true variance is not known, define based on

estimated variance using new variable

Based on t-distribution (not normal) & tabulated

Represents interval around mean of width t/2,n-1

times the standard deviation encompassing 95% of

distribution

7/29/2019 Lecture 14 - Curve Fitting (Intro&Regression)

12/17

Linear Regression

12

Systematic method to derive a curve that

minimize discrepancy between data and curve

Fit a straight line to data points

y = a0 + a1x + e

a0 = intercept

a1 = slope

e = error or residual betweenmodel and data

7/29/2019 Lecture 14 - Curve Fitting (Intro&Regression)

13/17

Linear Regression Best Fit

13

Selection of best fit based on:

Minimax method: Minimize the maximum distance that

an individual point falls from the line.

7/29/2019 Lecture 14 - Curve Fitting (Intro&Regression)

14/17

Linear Regression Least Square

14

Sum of the squares of residuals

Yields unique solution

7/29/2019 Lecture 14 - Curve Fitting (Intro&Regression)

15/17

Linear Regression Least Square

15

Standard deviation of regression line

Standard error of estimate

If sy/x < sy, then linear regression is valid and has merit

7/29/2019 Lecture 14 - Curve Fitting (Intro&Regression)

16/17

Linear Regression Least Square

16

Goodness of the fit quantify improvement indescribing data by a straight line rather than an

average value

Coefficient of determination, r2 (r is correlation

coefficient)

r = 1 : perfect fit and explains data variability

r = 0 : fit represents no improvement

7/29/2019 Lecture 14 - Curve Fitting (Intro&Regression)

17/17

Example: Linear Regression

17

Perform linear regression for this data set