ME EN 363 Elementary Instrumentation

Transcription

1 ME E 6 Elementary Instrumentation Curve Fitting Least squares regression analysis Standard error of fit Correlation coefficient Error in static sensitivity

2 Regression According to various Internet resources, regression can mean [Broad]: Movement backward to a previous and especially worse or more primitive state or condition [Marine]: Coastal advance due to falling sea level [Medicine]: A characteristic of diseases to epress lighter symptoms without disappearing totally [Psychology]: A defensive reaction to some unaccepted impulses [Statistics]: A statistical technique for estimating the relationships among variables

3 Merriam-Webster Regression Analysis Regression: A functional relationship between two or more correlated variables that is often empirically determined from data and is used especially to predict values of one variable when given values of the others <the regression of y on is linear> Essentially curve fitting Curve approimated by polynomial or Fourier series Most software packages (Matlab, Ecel, etc.) have curve fitting software Used in calibration, reducing data, etc.

4 Bad Eample of Curve Fitting (i) Source: Scientific Visualization and Information Architecture, J. P. Boyd, 000 [adapted]

5 Bad Eample of Curve Fitting (ii) Source: Scientific Visualization and Information Architecture, J. P. Boyd, 000

6 Bad Eample of Curve Fitting (iii) Source: Scientific Visualization and Information Architecture, J. P. Boyd, 000

7 Least-Squares Regression Analysis () Consider an m th -order polynomial fit y c = a 0 + a + a + + a m m : Independent variable value a 0 a m : Linear coefficients y c : Dependent variable value for given value of Goal: Minimize differences between data points and polynomial fit by adjusting coefficients For m th -order polynomial, need at least m + data points. ote: there are methods for functions of multiple variables (see last paragraph of section 4.6).

8 Least-Squares Regression Analysis () Find coefficients by measuring i and y i If there are n data points in and y ( and y are vectors consisting of i and y i ), then we have n equations and (m + ) unknowns. Usually n > (m + ), resulting in needing to solve an overdetermined system of equations. Most common technique in solving this system is the method of least squares.

9 Least-Squares Regression Analysis () If we call y ci the value obtained from the polynomial fit for the input i, then we can define the error in the fit for the i th input as ε = y i y ci The goal is to minimize the error in the fit over all the data n Choose a 0, a,, a m to minimize ( y i y ci ) i=

10 Least-Squares: Matri Method () Suppose we want to fit an m th order polynomial to the n data points,,, n, and y, y,, y n. We know i, so we can calculate i, i,, im. For the data point (, y ) we have: y = a 0 + a + a + + a m m For the data point (, y ) we have: y = a 0 + a + a + + a m m And so on up to the n th data point: y n = a 0 + a n + a n + + a m n m We write this system of n eqns. in matri form:

11 Least-Squares: Matri Method () = m m n n n n m m m n a a a a a y y y y 0 Y X a Y = Xa We want to know a n (m + ) (m + )

12 Least-Squares: Matri Method () Y = Xa We want to know a If X were square, we could possibly invert it. But this typically won t be the case. Solve for a Premultiply both sides by X T. Xa = Y X T Xa = X T Y a = (X T X) X T Y (X T X is square) (X T X) X T is known as the Moore-Penrose pseudoinverse of X

13 Fit Confidence Interval y() = y c ± t ν,p S y / (P %) S y : Standard error of a fit S y = υ ( yi yci ) i= Measure of the precision with which a polynomial describes the behavior of the data set Standard deviation of each data point and the fit Has ν = (m + ) degrees of freedom Accounts for variability in y only. For variability in and y, see Eq. 4.6.

14 Correlation Coefficient For linear fits, a correlation coefficient, R, is used as an indicator of goodness of fit S y R = S = S ( ) y yi y y Rule of thumb: a linear regression can be considered to be a reliable relation between y and for 0.9 R. R (estimate of variance in y) also often used R : also called coefficient of determination i=

15 Error in Static Sensitivity For a linear fit, y = a + a 0, a measure of the error in static sensitivity is given by: The true sensitivity (with a 95% probability) would be given by sensitivity = a ± t ν,95 S a = = = i i i i y a S S

16 Error in Static Sensitivity () A precision estimate of the zero intercept is given by: The true zero intercept would be given by intercept = a 0 ± t ν,95 S a0 = = = = 0 i i i i i i y a S S

17 What order of polynomial to use? Satisfy the following conditions: Lowest order of fit that reduces S y to acceptable value Maintain any known physical relationships

18 Least-Squares Eample Find and analyze the linear fit for the following data: = {.0,.0,.0, 4.0, 5.0} [cm] y = {.,.9,., 4., 5.} [V] a) Obtain the linear fit of the y vs. data. b) Show the equation for the linear fit with its 95% confidence interval. c) Estimate the static sensitivity error d) Estimate the y-intercept error

19 a 0 = 0.0 a =.04 S y = 0.59 t ν,95 = t,95 =.8 Answers t ν,95 S y / sqrt() = / sqrt(5) One approimation of uncertainty (ok for all curves): y c = a 0 + a ± t ν,95 S y Second approimation of uncertainty (for linear curves): y c = (a 0 ± t ν,95 S a0 ) + (a ± t ν,95 S a )

20 Eample Curve Fit 6 5 Measurement data Linear curve fit Curve fit precision interval using t*sy y = y (Voltage) y = y = (cm)

21 Eample Curve Fit Measurement data Linear curve fit Curve fit precision interval using Sa0, Sa y = ( ) + ( ) y (Voltage) y = ( ) + ( ) y = (cm)