Calibration  Curve Fitting

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Calibration Curve Fitting

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1. Calibration & Curve Fitting P M V Subbarao Professor Mechanical Engineering Department

3. 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).

5. 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.

6. Interpolation Vs Curve Fitting

7. 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.

8. Sensitivity of A Sensor

10. 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

11. 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?

13. 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

14. Hunting for A Shape & Geometry of A Data Set

22. The Least-Squares mth Degree Polynomials

26. Selection of Order of Fit

27. Under Fit or Over Fit: Picking An appropriate Order

28. 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

29. 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.

35. 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 ? = ± 1. The correlation is poor if ? ˜ 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.

38. 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

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