Calibration & Curve Fitting

P M V SubbaraoProfessor

Mechanical Engineering Department

How to obtain y=f(x) ???????

What are the units of xi & ui?

How to convert a Measured variable into Physical (Derived) Variable?

We have developed confirmed methods to measure xi & ui

Instrument & calibration

• An instrument is a device that transforms a physical variable of interest (the measurand) into a form that is suitable for recording (the measurement).

Simple Instrument Model

Advanced Instrument Model

Sensors

• Sensors convert physical variables to signal variables.

• Sensors are often transducers : They are devices that convert input energy of one form into output energy of another form.

• Sensors can be categorized into two broad classes depending on how they interact with the environment they are measuring.

• Passive sensors: do not add energy as part of the measurement process but may remove energy in their operation.

• Active sensors : add energy to the measurement environment as part of the measurement process.

Interpolation Vs Curve Fitting

Calibration

• The process of development of a relationship between the physical measurement variable input and the signal variable (output) for a specific sensor is known as the calibration of the sensor.

• Typically, a sensor (or an entire instrument system) is calibrated by providing a known physical input to the system and recording the output.

• The data are plotted on a calibration curve.

Linear range

Saturation Point

Sensitivity of A Sensor

Sensitivity of Thermistor

Curve Fitting Techniques

• Where does this given function

• Measured Variable = f (Physical Variable)

• come from in the first place?

• Analytical models of phenomena (e.g. equations from physics)

• Create an equation from observed data

• Curve fitting - capturing the trend in the data by assigning a single function across the entire range.

• A straight line is described generically by

baxxf The goal is to identify the coefficients ‘a’ and ‘b’ such that f(x) ‘fits’ the data well

Linear curve fitting (linear regression)

• Given the general form of a straight line• How can we pick the coefficients that best fits the

line to the data?• What makes a particular straight line a ‘good’ fit?

y

x

Quantifying error in a curve fit

• Assumptions:• positive or negative error have the same value (data point is

above or below the line)• Weight greater errors more heavily

• Denote data values as (x, y) •Name points on the fitted line as (x, f(x)).

The error is available at the four data points.

Hunting for A Shape & Geometry of A Data Set

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112 xfyxfyxfyxfydError i

Our fit is a straight line, so now substitute baxxf

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•The ‘best’ line has minimum error between line and data points•This is called the least squares approach, since square of the error is minimized.

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Take the derivative of the error with respect to a and b, set each to zero

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Solve for the a and b so that the previous two equations both = 0

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put these into matrix form

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Is a straight line suitable for each of these cases ?

The Least-Squares mth Degree Polynomials

When using an mth degree polynomial

mm xaxaxaay .........2

210

to approximate the given set of data, (x1,y1), (x2,y2)…… (xn,yn),

where n ≥ m, the best fitting curve has the least square error,

i.e., min1

2

n

i ii xfy

n

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22210 min......

To obtain the least square error, the unknown coefficients a0, a1, …. and am     must yield zero first derivatives.

Expanding the previous equations, we have

The unknown coefficients can hence be obtained by solving the above linear equations.

No matter what the order j, we always get equations LINEAR with respect to the coefficients.This means we can use the following solution method

Selection of Order of Fit

2nd and 6th order look similar, but 6th has a ‘squiggle to it. Is it Required or not?

Under Fit or Over Fit: Picking An appropriate Order

•Underfit - If the order is too low to capture obvious trends in the data•Overfit - over-doing the requirement for the fit to ‘match’ the data trend (order too high)• Polynomials become more ‘squiggly’ as their order increases. •A ‘squiggly’ appearance comes from inflections in function

General rule: pick a polynomial form at least several orders lower than the number of data points.

Start with linear and add order until trends are matched.

Linear Regression Analysis

• Linear curve fitting

• Polynomial curve fitting

• Power Law curve fitting: y=axb

• ln(y) = ln(a)+bln(x)

• Exponential curve fitting: y=aebx

• ln(y)=ln(a)+bx

Goodness of fit and the correlation coefficient

• A measure of how good the regression line as a representation of the data.

• It is possible to fit two lines to data by • (a) treating x as the independent variable : y=ax+b, y

as the dependent variable or by• (b) treating y as the independent variable and x as the

dependent variable. • This is described by a relation of the form x= a'y +b'. • The procedure followed earlier can be followed again.

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Recast the second fit line as:

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ais the slope of this second line, which not same as the first line

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•The ratio of the slopes of the two lines is a measure of how good the form of the fit is to the data.•In view of this the correlation coefficient ρ defined through the relation

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line Regression second of Slope

line Regressionfirst of slopeaa

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Correlation Coefficient

• The sign of the correlation coefficient is determined by the sign of the covariance.

• If the regression line has a negative slope the correlation coefficient is negative

• while it is positive if the regression line has a positive slope. • The correlation is said to be perfect if ρ = ± 1.• The correlation is poor if ρ ≈ 0.• Absolute value of the correlation coefficient should be greater

than 0.5 to indicate that y and x are related!• In the case of a non-linear fit a quantity known as the index of

correlation is defined to determine the goodness of the fit. • The fit is termed good if the variance of the deviates is much

less than the variance of the y’s. • It is required that the index of correlation defined below to be

close to ±1 for the fit to be considered good.

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2=1.000 2=0.991 2=0.904

2=0.821 2=0.493 2=0.0526

Multi-Variable Regression Analysis

• Cases considered so far, involved one independent variable and one dependent variable.

• Sometimes the dependent variable may be a function of more than one variable.

• For example, the relation of the form

• is a common type of relationship for flow through an Orifice or Venturi.

• mass flow rate is a dependent variable and others are independent variables.

pipe

orifice

d

dApTpfm ,,,,

Set up a mathematical model as:e

pipe

orificedcb

d

dAp

RT

pam

Taking logarithm both sides

pipe

orifice

d

deAdpc

RT

pbam lnlnlnlnlnln

Simply: eodncmblay ln

where y is the dependent variable, l, m, n, o and p are independent variables and a, b, c, d, e are the fit parameters.

The least square method may be used to determine the fit parameters.

Let the data be available for set of N values of y, l, m, n, o, p values.

The quantity to be minimized is given by

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What is the permissible value of N ?

The normal linear equations are obtained by the usual process of setting the first partial derivatives with respect to the fit parameters to zero.

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These equations are solved simultaneously to get the six fit parameters.

We may also calculate the index of correlation as an indicator of the quality of the fit. This calculation is left to you!