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7/29/2019 Lecture 14 - Curve Fitting (Intro&Regression)
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PE 3723
Numerical MethodsLecture 14: Curve Fitting
(Introduction & Regression)Maysam Pournik
Assistant Professor
Spring 20121
7/29/2019 Lecture 14 - Curve Fitting (Intro&Regression)
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Curve Fitting Applications
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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
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Approaches to Curve Fitting
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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
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Simple Method - Noncomputer
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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.
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Mathematical Background
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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
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Mathematical Background
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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
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Mathematical Background
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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
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Mathematical Background
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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)
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Mathematical Background
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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
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Mathematical Background
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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
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Mathematical Background
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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
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Linear Regression
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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
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Linear Regression Best Fit
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Selection of best fit based on:
Minimax method: Minimize the maximum distance that
an individual point falls from the line.
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Linear Regression Least Square
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Sum of the squares of residuals
Yields unique solution
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Linear Regression Least Square
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Standard deviation of regression line
Standard error of estimate
If sy/x < sy, then linear regression is valid and has merit
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Linear Regression Least Square
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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
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Example: Linear Regression
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Perform linear regression for this data set
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