Error Propagation 1. November 13, HMS, 2017, v1.0

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1 Error Propagation 1 November 13, HMS, 2017, v1.0

2 Chapter References Diez: None Navidi, Chapter 3 Chapter References 2

3 Motivation Measured Value = True Value + Bias + Random Error The bias refers to the systematic error due to bias in the instrument or bias in the human operator or even software. Bias can be corrected. Random errors however are not easily reduced and propagate into any additional calculations we may do with the data. Errors 3

4 Motivation Let us measure the area of a circle: A = πr 2 In measuring the area we note that there is uncertainty in our measurement of the radius, r. For example r = 2.5 ± 0.2 Given this uncertainty what is the uncertainty the computed area: A ±? Error Propagation 4

5 Motivation The reaction rate for an enzyme catalyzed reaction is given by: v = V ms K m + S where V m is the maximal velocity and K m the substrate concentration, S at half the maximal rate. The V m and K m are estimated to be 15.8 ± 0.9 mm/s and 0.5 ± 0.05 mm respectively. At a substrate concentration of 5 mm, what is the reaction velocity and associated uncertainty? Error Propagation 5

6 Linear Combination of Measurements If c is a constant, then: σ c1x 1 + σ c2x = σ cx = c σ X c 2 1 σ2 X 1 + c 2 1 σ2 X Important caveat: X 1, X 2,... are independent measurements. Error Propagation 6

7 Example The radius of a circle, R, is measured to be 3.0 ± 0.1 cm. Estimate the circumference and find the uncertainty, σ c, in the estimate. Using: R = 2πR σ cx = c σ X σ c = 2π σ R = 0.63 cm The circumference is therefore: ± 0.63 cm Error Propagation 7

8 Example A surveyor is measuring the perimeter of a rectangular plot. Two adjacent sides are measured to be ± 0.05m and ± 0.08 cm. Estimate the perimeter of the lot and the uncertainty. P = 2X + 2Y = cm Using: σ c1x 1 + σ c2x = c 2 1 σ2 X 1 + c 2 1 σ2 X σ P = σ 2X + σ 2Y = The perimeter is ± 0.19 m 4σ 2 X + 4σ2 2 = 0.19 m You cannot use P = X + X + Y + Y and form σ P = σ X + σ X + σ Y + σ Y because X + X is not the sum of independent quantities. Error Propagation 8

9 Dependent Measurements For dependent measurements, one can compute an upper bound on the uncertainty in a linear combination: σ c1x 1 + σ c2x c 1 σ X1 + c 2 σ X Error Propagation 9

10 Uncertainties for Functions of One Measurement What happens if your function is nonlinear rather than a simple linear combination? Given a random variable X, with known standard deviation σ X and given U = U(X) how do we compute σ U? Error Propagation 10

11 Uncertainties for Functions of One Measurement σ U du dx σ X Note the equation gives an approximate estimate. Error Propagation 11

12 Uncertainties for Functions of One Measurement Going back to the first example of finding the area of a circle: A = πr 2. If R = 5.00 ± 0.01 cm then find the area and the uncertainty in the Area. σ A da dr σ R da dr = 2πR σ R = 10π 0.01 = 0.31 cm 2 The area is therefore given by 78.5 ± 0.3 cm 2 Error Propagation 12

13 Relative Uncertainties of One Measurement If U is a measurement with true mean µ and uncertainty σ U, then the relative uncertainty is: σ U U The relative uncertainty is also called the coefficient of variation Error Propagation 13

14 Relative Uncertainties of One Measurement Recall the area of the circle in the previous example was 78.5 ± 0.3 cm 2. The absolute uncertainty was 0.3 cm 2. The relative uncertainty is: σ A A = = = 0.4% Error Propagation 14

15 Uncertainties for Functions of Several Measurements If A, B,..., are independent measurements whose uncertainties are σ A, σ B, and if U = U(A, B,...) then σ U = ( U A ) 2 ( ) 2 U σa 2 + σb 2 B +... Error Propagation 15

16 Uncertainties for Functions of Several Measurements Assume we have a function: U = U(A, B) The total deviation as a result of uncertainty in the measurements, A or B is given by: du = U U A + A B B This assumes the deviations are small. Error Propagation 16

17 How to Propagate U = U U A + A B B The variance in U is given by: σ 2 U = 1 N N ( U i ) 2 i=1 The variances in the individual measurements, A and B are given by σ 2 A = 1 N N ( A i ) 2, i=1 σ 2 B = 1 N N ( B i ) 2, i=1 Error Propagation 17

18 How to Propagate Inserting U into σ 2 U σ 2 U = 1 N ( U A A i + U ) 2 B B i Expanding the squared term yields: σu 2 = 1 [ ( ) 2 ( ) ] 2 U U N A A i + B B i + 2 U U A B A i B i Let us assume that the errors are random and independent, this means that the cross-term will on average equal zero (positive and well as negative deviations are equally possible) Error Propagation 18

19 How to Propagate We are therefore left with: σu 2 = 1 [ ( ) 2 ( ) ] 2 U U N A A i + B B i We can rearrange this equation by pulling out the derivatives: σ 2 U = ( ) 2 U 1 A N ( ) 2 U ( Ai ) ( Bi ) 2 B N Do you recognized the terms: 1 N ( Ai ) 2 and 1 N ( Bi ) 2? They are the variances for A and B, therefore... Error Propagation 19

20 How to Propagate σ 2 U = ( ) 2 U σa 2 + A ( ) 2 U σb 2 B A similar proof can also be made if the σ s are standard errors. ( U ) 2 ( ) 2 U σ U = σa 2 A + σb 2 B +... Error Propagation 20

21 General Expression for product/quotient For the expression, U = abc/(xyz) the uncertainty in U can be shown to be: σ U U = (σa a ) 2 + ( σb b ) 2 + ( σc c ) 2 + ( σx x ) ( ) 2 2 σy + + y ( σz ) 2 z Error Propagation 21

22 General Rules Addition/Subtraction x = a + b c σ x = Mutiplication/Division x = a b/c σ x x = Exponential x = a k σ ( x x = k σa a σ 2 a + σ 2 b + σ2 c ( σa ) 2 ( σb + a b ) ) 2 ( σc ) 2 + c Error Propagation 22

23 Standard Error Consider the mean x: x = x 1 + x N In measuring an individual x i there will be uncertainty in the x i by an amount σ xi. Let us propagate the uncertainties in x i into the mean standard deviation: ( ) 2 ( ) 2 x x σ x = σx x σx 1 x Error Propagation 23

24 Standard Error σ x = ( x x 1 ) 2 ( ) 2 x σx σx x Let us assume that the uncertainties in each x i are the same, that is σ x1 = σ x2 =... = σ x Since x = x i /N we can evaluate the partial derivatives: x = x =... = 1 x 1 x 2 N Inserting these into the propagation relationship yields: Error Propagation 24

25 Standard Error σ x = ( ) 2 ( ) N σ x + N σ x +... Rearranging yields: And finally: σ x = N σ2 x N 2 σ x = σ x N The standard error simply expresses how uncertainty from the sampling propagates into the random variable that describes the means. Error Propagation 25

26 Example Assume the mass of a rock is measured to be m = ± 1 gm and the volume is measured to be V = ± 0.1 ml. Estimate the density of the rock and the uncertainty of the estate. Apply Density = m V = = g σ D = ( ) 2 D σm m 2 + ( ) 2 D σv 2 V +... D m = 1 = ml 1 m D V = m = g/ml2 V 2 Error Propagation 26

27 Example Assume the mass of a rock is measured to be m = ± 1 g and the volume is measured to be V = ± 0.1 ml. ( ) 2 ( ) 2 D D σ D = σm m 2 + σv 2 V σ D = g/ml The density of the rock is: ± g/ml Error Propagation 27

28 Example Suppose a concentration of a given solution is 13.7 ± 0.3 moles L 1. A UV spec is used to measure the absorbance using a cuvette with a path length of 1.0 ± 0.1 cm. The absorbance is found to be ± Estimate the molar absorptivity, ε, using Beer s law ε = A/(lc). Since the expression is of the form product/quotient, we can use: σ (σa ) 2 ( ε ε = σl ) 2 ( σc ) A l c Error Propagation 28

29 Example Inserting the values into the equation yields: ( ) 2 ( ) 2 ( ) 2 σ ε ε = σ ε ε = Using Beer s law we can compute ε: Therefore: ε = /(1 13.7) = L mol 1 cm 1 ε = ± L mol 1 cm 1 Error Propagation 29

30 Effect on Error Bars The following data was measured for an enzymatic reaction with standard error of 0.5 for each measurement Substrate Reaction Rate Standard Error Error Propagation 30

31 Effect on Error Bars The enzymatic reaction is governed by: Transforming to: v = V m s K m + s 1 v = K m V m 1 s + 1 V m That is a plot of 1/v versus 1/s will yield a straight line. Error Propagation 31

32 Effect on Error Bars 1 v = K m V m 1 s + 1 V m Error Propagation 32

33 Effect on Error Bars Plot of reaction velocity versus substrate concentration. Error Propagation 33

34 Effect on Error Bars Left hand-side plot is reaction velocity versus substrate concentration. Left plot is plot of 1/v versus 1/s Error Propagation 34

35 Effect on Error Bars How is the error propagated into 1/v? σ (σa ) 2 x x = +... a Therefore σ 1/v 1/v = (σv v σ 1/v = σ v v 2 ) 2 = σ v v Error Propagation 35

36 Effect on Error Bars σ 1/v = σ v v 2 As at low v (high 1/v), the uncertainty in 1/v significantly increases. Error Propagation 36

37 Effect on Error Bars High v Low v Error Propagation 37

38 Error Propagation using Python Use the uncertainties package: te.installpackage( uncertainties ) Other Pythons use: pip install --upgrade uncertainties Error Propagation 38

39 Error Propagation using Python >>> from uncertainties import ufloat >>> x = ufloat(1, 0.1) # x = 1+/-0.1 >>> print 2*x 2.00+/-0.20 >>> v = ufloat (4.5, 0.25) >>> print 1/v / Error Propagation 39

40 Error Propagation using Python >>> from uncertainties import unumpy >>>y = unumpy.uarray ([0.3636, 0.533, 0.631, 0.695, , 0.774,0.8,0.821, 0.837, 0.851], [0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5]) >>>print 1/y [ / / / / / / / / / / ] >>>print 0.5/(0.3636*0.3636) Error Propagation 40

41 Linear Regression This leads eventually on to Linear Regression. But first we must introduce Analysis of Variance. Error Propagation 41

Biochemical Kinetics: the science that studies rates of chemical reactions An example is the reaction (A P), The velocity, v, or rate, of the

Biochemical Kinetics: the science that studies rates of chemical reactions An example is the reaction (A P), The velocity, v, or rate, of the reaction A P is the amount of P formed or the amount of A consumed