Typing Equations in MS Word 2010

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1 CM3215 Fundamentals of Chemical Engineering Laboratory Typing Equations in MS Word Professor Faith Morrison Department of Chemical Engineering Michigan Technological University 1 Where are we in our discussion of error analysis? Let s revisit: 2 1

2 From Lecture 4 Error Propagation: Summary: Error Analysis with Real Numbers To understand the accuracy of our numbers, we need to determine a confidence interval. 2 with 95.0% confidence For replicate data with 7, replace 2 with., The Standard error for a measured quantity is the largest of: determined by replicates / or by estimate of reading error / 3 or by estimate of calibration error max error/2 Standard error for derived quantities (arrived at from equations), is obtained at through error propagation, which is a combination of variances. 3 From Lecture 4 Error Propagation: Error Propagation Taylor series:,, We use an analysis based on the Taylor series expansion of a nonlinear function.... A calculation of the function,, from uncertain values of,, is a random variable of mean and variance : (higher order terms) Covariance terms, if are correlated 4 2

3 From Lecture 4 Error Propagation: Error Propagation Taylor series:,, We use an analysis based on the Taylor series expansion of a nonlinear function.... A calculation of the function,, from uncertain values of,, is a random variable of mean and variance : neglect (higher order terms) Covariance terms, if are correlated Note: covariance terms are not always zero or small; but they often are. For now, this is fine. 5 From Lecture 4 Error Propagation: Worksheet for error propagation 6 3

4 From Lecture 4 Error Propagation: Example 1: What is the uncertainty (95% confidence interval) in as determined in the lab? / / / / / / / / / / 7 From Lecture 4 Error Propagation: Example 1: What is the uncertainty (95% confidence interval) in as determined in the lab? Excel is an excellent tool for error propagation Error propagation Worksheet f(x 1,x 2,x 3 ) f BF g/ml 2e s g/ml x i value df/dx i (df/dx i ) 2 e xi 2 e xi (df/dx i ) 2 2 e xi x 1 M F g E E E 11 g 2 /ml 2 x 2 M E g E E E 11 g 2 /ml 2 x 3 V pyc ml E E 05 g 2 /ml 2 2 e s 1.21E 05 g 2 /ml 2 e s g/ml 8 4

5 From Lecture 4 Error Propagation: Summary: Error Analysis with Real Numbers To understand the accuracy of our numbers, we need to determine a confidence interval. with 95.0% confidence For replicate data with, replace 2 with The Standard error for a measured quantity is the sum, in quadrature, of: determined by replicates by estimate of reading error by estimate of calibration error Standard error for derived quantities (arrived at from equations), is obtained through error propagation, which is a combination of variances. Replication improves the estimation of the mean. The prediction interval of the next value of x should encompass 95% of all measured values. The answer from replicates is more reliable than single values (if no systematic errors). 95% PI: or The weighting values indicate the impact of individual errors on the final value. Estimates for (particularly those obtained through ) may need to be re evaluated, if unreasonably narrow confidence intervals are identified. if 68 9 Now, how do we determine uncertainty from numbers that we obtain as parameters in a curve fit? 10 5

6 CM3215 Fundamentals of Chemical Engineering Laboratory Uncertainty in Least Squares Curve Fitting: Excel s LINEST Professor Faith Morrison Department of Chemical Engineering Michigan Technological University Reference: ertaintyslopeinterceptofleastsquaresfit.pdf 1. Quick start Replicate error 2. Reading Error 3. Calibration Error 4. Error Propagation 5. Least Squares Curve Fitting 11 Question: For a dataset of data pairs, that is expected to show a linear relationship between and, what are the parameters and of the equation for the line? 1 2 slope intercept 12 6

7 Solution: Assume you know the with certainty ( ordinary least squares) Guess a line, Create a measure of the error between the guess and the data (error measure should always be positive, so square it) Add these individual error measures to calculate a sum of squared errors, Use calculus (derivatives) to find the values of and that result in the least sum of squared error. 1 2 data slope intercept line 13 slope intercept Result: 1 2 Least squares slope Least squares intercept In Excel: SLOPE(y range, x range) INTERCEPT(y range,x range) 14 7

8 Result: and are calculated from the, These are the formulas used in Excel trendlines. Least squares slope Least squares intercept slope intercept In Excel: SLOPE(y range, x range) INTERCEPT(y range,x range) Result: slope intercept But, what 1 are the 2 error limits on and? Least squares slope Least squares intercept 16 8

9 slope intercept slope? Intercept? But, what 1 are the 2 error limits on and? 17 slope intercept slope? Intercept? slope 2 Intercept 2 (Later we will correct the 2 for small ) But what is? But, what 1 are the 2 error limits on and? 18 9

10 19 Error limits on 2,,,,,,,, 10

11 Error limits on Only the are variables; we assumed we knew the with certainty,,,, 2,,,, Error limits on Assume that the variances of the are the same for all. 2,, (, is the standard deviation of at a given value of,,,,,, 11

12 , The variance of, given, (, is the standard deviation of at a given value of ; ordinary least squares assumes it is constant) 1 2 slope intercept (This formula comes from the definition of variance) The variance of the mean value of at a given In Excel:, STEYX(y range, x range), or use LINEST 23, (, is the standard deviation of at a given value of ; ordinary least squares assumes it is constant) slope intercept The variance of, given, 1 2 Best value of at a given (This formula comes from the definition of variance) The variance of the mean value of at a given, is calculated from the, In Excel:, STEYX(y range, x range), or use LINEST 24 12

13 What are the error limits on? slope intercept slope 2, (This is the final result of the algebra indicated on the error propagation slide) for 26: slope., In Excel: STEYX(y range, x range, or (DEVSQ(x range) use LINEST 25 What are the error limits on? slope intercept intercept 2? Solve the same way, error propagation on the formula for 26 13

14 Error limits on 2,,,,,,,, What are the error limits on? slope intercept intercept 2 1, (This is the final result of the algebra indicated on the error propagation slide) for 26: intercept., In Excel: Calculate from STEYX(y range, x range) and DEVSQ(x range) and the formula above, or use LINEST 28 14

15 slope intercept For instructions on how to use Microsoft Excel s LINEST function, see the handout on the web: SlopeInterceptOfLeastSquaresFit.pdf (the appendix has some derivations, if you re interested) 29 What are the error limits on a value of obtained from the equation? slope intercept At a chosen, 2? 30 15

16 Error limits on Error limits on But, and are not independent (both are calculated from the ). 16

17 Error limits on Cov, What are the error limits on a value of obtained from the equation? slope intercept at, 2 1, for 26, replace 2 with., Use this for error limits on values obtained from the fit. (This is the final result of the algebra indicated on previous slide; see Appendix B of the handout.) In Excel:, STEYX(y range,x range) DEVSQ(x range) AVERAGE(x range) 34 17

18 What are the error limits on a predicted next experimental value of? slope intercept at, we predict a new measurement of will fall in the prediction interval: 2? 35 What are the error limits on a predicted next experimental value of? slope intercept at, we predict a new measurement of will fall in the prediction interval: 2? Solve with same approach as we have been using: write the equation to calculate the quantity, then propagate the error. (See Appendix B of the handout.) 36 18

19 What are the error limits on a predicted next experimental value of? slope intercept at, we predict a new measurement of will fall in the prediction interval: (See Appendix B of the handout.) 2, 1 1 for 26, replace 2 with., 37 Confidence interval for values from the fit: 1.40 Aqueous Sugar Solutions, 20 o C, 2014, (for large, the values of at each are well predicted (CI is narrow)) Prediction interval of data:, 0.80 density, g/ml CM3215 Fall 2014 data +95%CI 95%CI trendline 95%PI 95%PI (Notice that 95% of the data points fall within the PI; that s what it means to be a PI. The next data point likely will fall here too.) wt % sugar 38 19

20 1.40 Aqueous Sugar Solutions, 20 o C, density, g/ml Note: if your data are replicates 0.90 (data taken repeatedly at chosen values), do not pre average the data and follow up with a least squares curve fit. Instead, use all the replicates as individual values, and let LINEST find the least squared error wt % sugar CM3215 Fall 2014 data +95%CI 95%CI trendline 95%PI 95%PI 39 Summary: Uncertainty The Ordinary Least Squares Linear method provides the equations needed to obtain model parameters slope and intercept. The equations for the parameters may be used with error propagation to obtain the variances associated with the parameters and. 95% confidence intervals on the parameters are constructed with 2 for large For 26, the 95% CI is constructed as., We can construct 95% CI on the best values of at a chosen. These CI are used for error range on the fit. We can construct 95% prediction intervals (PI) on a next value of at a chosen ; use to evaluate next experimental point acquired. 40 slope intercept 20

21 slope intercept Excel Summary: Uncertainty AVERAGE(range) VAR.S(range) STDEV.S(range) COUNT(range) DEVSQ(x range) SLOPE(y range, x range) INTERCEPT(y range,x range), STEYX(y range, x range) LINEST (see handout) LOGEST (look it up) Use for CI error bars on values obtained from a fit Use for PI of next measured value of,,,, 1 41 slope intercept Excel Handy List: Uncertainty TREND(known y s, known x s, ) for and related by GROWTH(known y s, known x s, ) for and related by 42 21

22 slope intercept One final piece of advice: Uncertainty Often, you can transform your data to make it linear, allowing you to use linear regression. For example, if you know the data vary as the square root of the data, then versus will be linear. If data plotted with log log scaling (using scatterplot) look quadratic, then log versus log will be quadratic, and we can use trendline to obtain a fit: log log log Transforming data can greatly broaden our ability to fit empirical models to data. 43 Professor Faith Morrison Department of Chemical Engineering Michigan Technological University Done! 44 22

23 Comment on Curve Fitting: Coefficient of Determination, Which data set has a larger? y data x data 45 Comment on Curve Fitting: Coefficient of Determination, Which data set has a larger? y = x R² = y data y = x R² = x data 46 23

24 Comment on Curve Fitting: Coefficient of Determination, is a measure of the comparison of the hypothesized linear relationship and the relationshiop constant(horizontal line). So, if it is a horizontal line, will be zero. From page 6: 47 Which is the correct fit? y = 2.005x R² = y data 15.0 y data y = x x x x R² = x data x data 48 24

25 Which is the correct fit? y = 2.005x R² = y data 15.0 y data y = x x x x R² = x data x data (it depends on the error bars) Likely that the linear fit is a truer relationship to be used for interpolation 49 25