Error Covariance Propagation
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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 repetitions of the measurement result. To contrast this with a propagation of error approach, consider the
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simple example where we estimate the area of a rectangle from replicate measurements of length trajectory propagation model and width. The area $$ area = length \cdot width $$ can be computed from each replicate. The standard deviation of the
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reported area is 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 standard error covariance that would emerge if measurements covered a range of operating conditions and a sufficiently long time period independence from propagation of error model 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 error propagation standard deviation 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 \(s^2\) for width and length, respectively, in the approximate formula) \( E_{ij} = {(\Delta x)^i, (\Delta y)^j}\) where \( \Delta x = x - X \) and \( \Delta y = y - Y \) \( Cov((\Delta x)^2, (\Delta y)^2) = E_{22} - V(x) V(y) \) To obtain the standard deviation, simply take the square root of the above formula. Also, an estimate of the statistic is obtained by substituting sample estimates for the corresponding population values on the right hand side of the equation. Approximate formula assumes indpendence The approximate formula assumes that length and width are indepe
or more quantities, each with their individual uncertainties, and then combine the information from these quantities in order to come up with a final result of our experiment. How
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can you state your answer for the combined result of these measurements error propagation addition and their uncertainties scientifically? The answer to this fairly common question depends on how the individual measurements are combined
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in the result. We will treat each case separately: Addition of measured quantities If you have measured values for the quantities X, Y, and Z, with uncertainties dX, dY, and http://www.itl.nist.gov/div898/handbook/mpc/section5/mpc55.htm dZ, and your final result, R, is the sum or difference of these quantities, then the uncertainty dR is: Here the upper equation is an approximation that can also serve as an upper bound for the error. Please note that the rule is the same for addition and subtraction of quantities. Example: Suppose we have measured the starting position http://lectureonline.cl.msu.edu/~mmp/labs/error/e2.htm as x1 = 9.3+-0.2 m and the finishing position as x2 = 14.4+-0.3 m. Then the displacement is: Dx = x2-x1 = 14.4 m - 9.3 m = 5.1 m and the error in the displacement is: (0.22 + 0.32)1/2 m = 0.36 m Multiplication of measured quantities In the same way as for sums and differences, we can also state the result for the case of multiplication and division: Again the upper line is an approximation and the lower line is the exact result for independent random uncertainties in the individual variables. And again please note that for the purpose of error calculation there is no difference between multiplication and division. Example: We have measured a displacement of x = 5.1+-0.4 m during a time of t = 0.4+-0.1 s. What is the average velocity and the error in the average velocity? v = x / t = 5.1 m / 0.4 s = 12.75 m/s and the uncertainty in the velocity is: dv = |v| [ (dx/x)2 + (dt/t)2 ]1/2 = 12.75 m/s [(0.4/5.1)2 + (0.1/0.4)2]
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