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2145-391 Aerospace Engineering Laboratory I

2145-391 Aerospace Engineering Laboratory I. Measurement Measurement Errors / Models Measurement Problem and The Corresponding Measurement Model Measure with Single Instrument: Single Sample / Multiple Samples Measure with Multiple Instruments: Single Sample / Multiple Samples

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2145-391 Aerospace Engineering Laboratory I

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  1. 2145-391 Aerospace Engineering Laboratory I • Measurement • Measurement Errors / Models • Measurement Problem and The Corresponding Measurement Model • Measure with Single Instrument: Single Sample / Multiple Samples • Measure with Multiple Instruments: Single Sample / Multiple Samples • Uncertainty of A Measured Quantity (VS Uncertainty of A Derived Quantity – Next Week) • Measurement Statement • Single Sample Measurement • Multiple Samples Measurement

  2. Some Details of Contents • Where are we? • Measured Quantity VS Derived Quantity • Objectives and Motivation • Deterministic Phenomena VS Random Phenomena • Measurement Problems, Measurement Errors, Measurement Models • Population and Probability • Probability Distribution Function (PDF) • Probability Density Function (pdf) • Expected Value • Moments • Sample and Statistics • Sample Mean and Sample Variance • Interval Estimation • Terminologies for Measurement: Bias, Precise, Accurate • Error VS Uncertainty • Measurement Statement • Measured Variable as A Random Variable • Uncertainty of A Measured Quantity [VS Uncertainty of A Derived Quantity – Next Week] • Measurement Statement • Experimental Program: Test VS Sample • Some Uses of Uncertainty

  3. Week 1: Knowledge and Logic • Week 2: • Structure and Definition of an Experiment • DRD/DRE bottommost level bottommost level Week 3: Instruments Week 4: Measurement and Measurement Statement: Where are we on DRD?

  4. Measured Quantity VS Derived Quantity Recall the difference between Measured Quantityy (numerical value of yis determined from measurement with an instrument) VS Derived Quantityy (numerical value of y is determined from a functional relation) In this period we focus first on measured quantity.

  5. Objectives Measurement Statement: If you • know what this measurement statement says (regarding measurement result), • know its use [what good is if for, and when to use it] • know and understand its underlying ideas [why should we report a measurement result with this statement], • (know, understand, and) know how to report a measurement result of a measured quantity with this measurement statement for the cases of • A single-sample measurement • A multiple-sample measurement we can all go home. Activity: Class Discussion on the above

  6. Right. I can just simply close my eyes and bet against Class Activity and Discussion • 10 students come up and measure the resistance of a given resistor. • Then, report the measurement result. Discussions • Do they get the same result? • If not, what is, and who gets, the ‘correct’ value then? Wanna bet500 bahts? [on whom, or which value] • If the next 10 of your friends come up and measure the resistance, and less than 9 out of 10 of themdo not get the value you bet on, you lose and give me 500 bahts. Otherwise, I win. Or • I’ll let you set the term of our bet.[Of course, I have to agree on the term first.] What term should our bet be then? [Make it reasonable and “bettable.”]

  7. Motivation Given that there are some (random) variations in repeated measurements, how should we report a measurement result so that it makes some usable sense? The Why The Use The How of Measurement Statement

  8. Deterministic Phenomena VS Random Phenomena Deterministic Variable VS Random Variable

  9. In reality, however, chances are that we will not have that exact position Deterministic Phenomena Determinsitic Variable Deterministic Phenomena (1) The state of the system at time t, and (2) its governing relation, deterministically determine the state of the system – or the value of the deterministic variable y - at any later time. Example: Free fall (and Newton’s Second law)

  10. Random Phenomena [Random or Statistical Experiment]Random Variable A random or statistical experiment is an experiment in which [1] • all outcomes of the experiment are known in advance, • any realization (or trial) of the experiment results in an outcome that is not known in advance, and • the experiment can be repeated under nominally identical condition. [1]Rohatgi, V. K., 1976, An Introduction to Probability Theory and Mathematical Statistics, Wiley, New York, p. 20.

  11. Class Activity: Discussion Is tossing a coin a random experiment? • All possible outcomes are H and T and nothing more. • For any toss, we cannot know/predict the outcome in advance. H or T? 3. Care can be taken to repeat the toss under nominally identical condition.

  12. Measurement: • All possible outcomes are known in advance, e.g., • For any one realization w, we cannot know/predict the exact outcome with certainty in advance. • Care can be taken to repeat the measurement under nominally identical condition. Class Activity: Discussion Is measurement a random experiment?

  13. Measurement Problems Measurement Errors Measurement Models

  14. Measurement Problems,Measurement Errors,and The Corresponding Measurement Models • Measure with a single instrument • Single Sample • Multiple Samples • Measure with multiple instruments • Single Sample • Multiple Samples • Other models are possible, depending upon the nature of errors considered.

  15. X Measured value of X at reading/sample i : True value ofX : Total measurement error for reading i : Measurement Problem: Measure with A Single Instrument: Error at Measurement i Error at measurement i Measurement i We never know the true value.

  16. X Systematic/Bias error: Constant. Does not change with realization. Random/Precision error for reading i, Randomly varying from one realization to another. Measurement model Decomposition of ErrorSystematic/Bias Error VS Random/Precision Error Statistical Experiment: Repeated measurements under nominally identical condition Measurement i

  17. Measurement Model 1: The ith measured value The ith observed value of a random variableX True value [Can never be known for certainty, not a random variable] Systematic / Bias error [Constant. Does not change with realization,not a random variable] Random / Precision error [Randomly varying from one realization to another, a random variable] Measurement Problem: Measure with A Single Instrument: Measurement Model 1

  18. Frequency of occurrences the population distribution of measured valueX X Measured Value [Random Variable] How to Describe/Quantify A Random Variable: Distribution of Measured Value [Random Variable] Statistical Experiment: Repeated measurements under nominally identical condition Measurement i Repeated measurements under nominally identical condition Description of Deterministic Variable: State the numerical value of the variable under that condition. Description of Random Variable Since it randomly varies from one realization to the next [even under the same nominal condition and we cannot predict its value exactly in advance], a meaningful way to describe it is by describing its probability (distribution). VS

  19. Population Sample Probability and Statistics Probability – Deductive reasoning Given properties of population, extract information regarding a sample. Statistics – Inductive reasoning Given properties of a sample, extract information regarding the population.

  20. Population Sample Population and Probability Probability – Deductive Reasoning Given properties of population, extract information regarding a sample.

  21. Some Properties x The probability of an event is the value of Probability Distribution Function (PDF)

  22. Some Properties Area or = Area under the pdf curve from to x. x Probability Density Function (pdf) Thin / high VS Wide / low

  23. x Probability of An Event PDF FX(x) pdf fX(x) x • The probability of an event is • 1. the area under the fX(x) curve from to x, • 2. the value of FX(x).

  24. Area x2 x1 Probability of An Event FX(x2) PDF FX(x) pdf fX(x) FX(x2) - FX(x1) FX(x1) x • The probability of an event is • 1. the area under the fX(x) curve from x1 to x2, • 2. the value of [FX(x2) - FX(x1)].

  25. PDF: U(x;-1,1) PDF: U(x;-2,2) U(x;-1,1) U(x;-2,2) x Example of Some pdf: Uniform Density Function Function U(x) with parameters a and b: Thin / high VS Wide / low

  26. x Example of Some pdf: Normal Density Function Function N(x) with parameters m and s 2: PDF: N(x;0,1) PDF: N(x;0,2) N(x;0,1) N(x;0,2)

  27. x Example of Some pdf: Student’s t Density Function Function t(x) with parameter n : PDF: t(x; 20) PDF: t(x; 5) t(x; 5) t(x; 20)

  28. x Example of Some pdf: Chi-Squared Density Function Function c2(x) with parameter n : c 2(x; 5) c 2(x; 10) c 2(x; 15) c 2(x; 20)

  29. Expected Values of A Random Variable Definition: Expected Value of A Random Variable X The expected value (or the mathematical expectation or the statistical average) of a continuous random variable X with a pdf fX(x) is defined as Definition: Expected Value of A Function of A Random Variable X Let Y = g(X) be a function of a random variable X, then Yis also a random variable, and we have However, we can also calculate E(Y) from the knowledge of fX(x) without having to refer to fY(y)as

  30. Moments of A pdf Definition: Moment About The Origin The rth-ordermoment about the origin (of a df) of X, if it exists, is defined as where r = 0,1,2,…. Note that this is the rth-order moment of area under fX(x)about the origin. Definition: Central Moment The rth-order central momentof a df of X, if it exists, is defined as where r = 0,1,2,….and E[X] = mX. Note that this is the rth-order moment of area under fX(x)about mX.

  31. fX (x) x dx dA dA Interpretations of Moments moment arm for Mr = x(r) moment arm for mr = (x-mX)(r) Area dA = fX(x)dx NOTE: Due to the rth-power of the arm length, the values of fX (x) at further distance from the center (origin or mX) relatively contribute more to the moment than those at closer to the center. x x - mx mX

  32. Some Properties of Mr and mr Properties of Origin Moment Mr Moment order 0: Moment order 1 (Mean of rv X): Moment order 2 Properties of Central Moment mr Moment order 0: Moment order 1: Moment order 2: (Variance sX2) mXis the location of the centroid of the pdf. sX2is a measure of the width of the pdf.

  33. Population Sample Population Mean Population Variance Sample Mean Sample Variance How close is to in some sense? Sample and Statistics Since we do not know the properties of the population , we want to estimate them with the statistics drawn from a sample. Statistics – Inductive reasoning

  34. Sample Mean and Sample Variance Definition Let X1, X2, …, Xnbe a random sample from a distribution function fX(x). Then, the following statistics are defined. Sample Mean: Sample Variance: Sample Standard Deviation: Sample mean, sample variance, and sample standard deviation are statistics, hence, random variables, not simply numbers. Unbiased estimator of mX. Unbiased estimator of .

  35. Interval Estimation Assume X is a random variable whose pdf is normal and Let (X1, X2, …, Xn)be an iid random sample from • Interval Estimation: Probability Distributions of Random Variables

  36. pdf pdf Normal and Students t Chi Squared Area = a/2 Area = a/2 Convention on a

  37. Interval Estimation Theorem 1: Standard Normal Random Variable If , then . Zis called a standard normal random variable. In addition, we have or whereza/2 denotes the value on the z axis for which a/2of the area under the z curve lies to the right of za/2. magnitude of the deviation/distance from X to mX, or from mX to X.

  38. The probability that deviates from no more than ( times ) is .

  39. Theorem 2: Distribution for A Random Variable Let (X1, X2, …, Xn)be a sample from . Then, the random variable has Hence, or

  40. The probability that deviates from no more than ( times ) is .

  41. Theorem 3: Distribution for A Random Variable (Student’s t Distribution) Let (X1, X2, …, Xn)be a sample from . Then, the random variable has that is, T has a Student’s t distribution with degree of freedom n = n -1. Hence, or

  42. The probability that deviates from no more than ( times ) is .

  43. One More Sample from Previously Drawn n Samples (Large Sample Size Approximate, n large) Let (X1, X2, …, Xn)be a sample from and be the sample variance of this sample. Let be an additional single sample drawn from . Then, the random variable has that is, T has a Student-t distribution withdegree of freedom n = n-1. Hence, or

  44. The probability that deviates from no more than ( times ) is .

  45. Summary of Interval Estimation Scheme Diagram

  46. Students t pdf Area = a/ 2 Interval Estimation Assume X is a random variable whose pdf is normal and Let (X1, X2, …, Xn)be an iid random sample from • Interval Estimation: Probability Distributions of Random Variables

  47. Terminologies for Measurement Bias Precise Accurate

  48. X X XTrue , m X XTrue, m X XTrue, m X X X XTrue m X Terminologies for Measurement: Bias, Precise, and Accurate Frequency of occurrences Biased + Imprecise  Inaccurate Biased + Precise  Inaccurate Unbiased + Imprecise  Inaccurate Unbiased + Precise  Accurate

  49. Error VS Uncertainty

  50. Terminologies: Error VS Uncertainty • Error If the error is known for certainty, (it is the duty of the experimenter to) correct it and it is no longer an error. • Uncertainty For error that is not known for certainty, no correction scheme is possible to correct out these errors. In this respect, the termuncertainty is more suitable. • The two terms sometimes – if not often – are used without strictly adhere to this. Nonetheless, the above should be recognized.

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