Propagation Error Approach
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The approach to uncertainty analysis that has been followed up to this point in the discussion has been what is called a top-down approach. Uncertainty components are estimated from direct propagation of error division repetitions of the measurement result. To contrast this with a propagation
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of error approach, consider the simple example where we estimate the area of a rectangle from replicate error propagation physics measurements of length and width. The area $$ area = length \cdot width $$ can be computed from each replicate. The standard deviation of the reported area is
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estimated directly from the replicates of area. Advantages of top-down approach This approach has the following advantages: proper treatment of covariances between measurements of length and width proper treatment of unsuspected sources of error that would emerge if measurements covered a range of operating conditions and a sufficiently long time period independence from propagation of error model error propagation excel Propagation of error approach combines estimates from individual auxiliary measurements The formal propagation of error approach is to compute: standard deviation from the length measurements standard deviation from the width measurements and combine the two into a standard deviation for area using the approximation for products of two variables (ignoring a possible covariance between length and width), $$ s_{area} = \sqrt{width^2 \cdot s_{length}^2 + length^2 \cdot s_{width}^2} $$ Exact formula Goodman (1960) derived an exact formula for the variance between two products. Given two random variables, \(x\) and \(y\) (correspond to width and length in the above approximate formula), the exact formula for the variance is: $$ V(\bar{x} \bar{y}) = \frac{1}{n} \left[ X^2 V(y) + Y^2 V(x) + 2XYE_{11} + 2X\frac{E_{12}}{n} + 2Y\frac{E_{21}}{n} + \frac{V(x) V(y)}{n} + \frac{Cov((\Delta x)^2, (\Delta y)^2) -E_{11}^2 }{n^2} \right] $$ with \(X = E(x)\) and \(Y = E(y)\) (corresponds to width and length, respectively, in the approximate formula) \(V(x)\) is the variance of \(x\) and \(V(y)\) is the variance \(y\) (corresponds to
| tags: | View Comments Updated April 23, 2013 at 09:28 PM In the previous section we examined an analytical approach to error propagation, and a simulation based approach. There is
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another approach to error propagation, using the uncertainties module (https://pypi.python.org/pypi/uncertainties/). You have to
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install this package, e.g. pip install uncertainties. After that, the module provides new classes of numbers and functions that incorporate error propagation definition uncertainty and propagate the uncertainty through the functions. In the examples that follow, we repeat the calculations from the previous section using the uncertainties module. Addition and subtraction import uncertainties as u A http://www.itl.nist.gov/div898/handbook/mpc/section5/mpc55.htm = u.ufloat((2.5, 0.4)) B = u.ufloat((4.1, 0.3)) print A + B print A - B >>> >>> >>> 6.6+/-0.5 -1.6+/-0.5 Multiplication and division F = u.ufloat((25, 1)) x = u.ufloat((6.4, 0.4)) t = F * x print t d = F / x print d >>> >>> >>> 160.0+/-11.8726576637 >>> >>> 3.90625+/-0.289859806243 Exponentiation t = u.ufloat((2.03, 0.0203)) print t**5 from uncertainties.umath import sqrt A = http://kitchingroup.cheme.cmu.edu/blog/2013/03/07/Another-approach-to-error-propagation/ u.ufloat((16.07, 0.06)) print sqrt(A) # print np.sqrt(A) # this does not work from uncertainties import unumpy as unp print unp.sqrt(A) 34.4730881243+/-1.72365440621 >>> >>> >>> >>> 4.00874045057+/-0.00748364738749 ... >>> >>> 4.00874045057+/-0.00748364738749 Note in the last example, we had to either import a function from uncertainties.umath or import a special version of numpy that handles uncertainty. This may be a limitation of teh uncertainties package as not all functions in arbitrary modules can be covered. Note, however, that you can wrap a function to make it handle uncertainty like this. import numpy as np wrapped_sqrt = u.wrap(np.sqrt) print wrapped_sqrt(A) >>> >>> 4.00874045057+/-0.00748364738774 Propagation of errors in an integral import numpy as np import uncertainties as u x = np.array([u.ufloat((1, 0.01)), u.ufloat((2, 0.1)), u.ufloat((3, 0.1))]) y = 2 * x print np.trapz(x, y) >>> >>> ... ... >>> >>> >>> >>> 8.0+/-0.600333240792 Chain rule in error propagation v0 = u.ufloat((1.2, 0.02)) a = u.ufloat((3.0, 0.3)) t = u.ufloat((12.0, 0.12)) v = v0 + a * t print v >>> >>> >>> >>> 37.2+/-3.61801050303 A real example? This is what I would setup for a real working example. We try to compute the exit concentration from a CSTR.
Επιλέξτε τη γλώσσα σας. Κλείσιμο Μάθετε περισσότερα View this message in English Το YouTube εμφανίζεται στα https://www.youtube.com/watch?v=N0OYaG6a51w Ελληνικά. Μπορείτε να αλλάξετε αυτή την προτίμηση παρακάτω. Learn more You're viewing YouTube in Greek. http://ieeexplore.ieee.org/iel5/20/19737/00914383.pdf You can change this preference below. Κλείσιμο Ναι, θέλω να τη κρατήσω Αναίρεση Κλείσιμο Αυτό error propagation το βίντεο δεν είναι διαθέσιμο. Ουρά παρακολούθησηςΟυράΟυρά παρακολούθησηςΟυρά Κατάργηση όλωνΑποσύνδεση Φόρτωση... Ουρά παρακολούθησης Ουρά __count__/__total__ Calculating the Propagation of Uncertainty Scott Lawson ΕγγραφήΕγγραφήκατεΚατάργηση εγγραφής3.7133 χιλ. Φόρτωση... Φόρτωση... Σε λειτουργία... Προσθήκη σε... propagation error approach Θέλετε να το δείτε ξανά αργότερα; Συνδεθείτε για να προσθέσετε το βίντεο σε playlist. Σύνδεση Κοινή χρήση Περισσότερα Αναφορά Θέλετε να αναφέρετε το βίντεο; Συνδεθείτε για να αναφέρετε ακατάλληλο περιεχόμενο. Σύνδεση Μεταγραφή Στατιστικά στοιχεία 48.418 προβολές 182 Σας αρέσει αυτό το βίντεο; Συνδεθείτε για να μετρήσει η άποψή σας. Σύνδεση 183 11 Δεν σας αρέσει αυτό το βίντεο; Συνδεθείτε για να μετρήσει η άποψή σας. Σύνδεση 12 Φόρτωση... Φόρτωση... Μεταγραφή Δεν ήταν δυνατή η φόρτωση της διαδραστικής μεταγραφής. Φόρτωση... Φόρτωση... Η δυνατότητα αξιολόγησης είναι διαθέσιμη όταν το βίντεο
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