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# Error analysis statistics

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Error analysis statistics

Error analysis statistics

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• 1. Error Analysis - Statistics &#x2022; Accuracy and Precision &#x2022; Individual Measurement Uncertainty &#x2013; Distribution of Data &#x2013; Means, Variance and Standard Deviation &#x2013; Confidence Interval &#x2022; Uncertainty of Quantity calculated from several Measurements &#x2013; Error Propagation &#x2022; Least Squares Fitting of Data Slide 1
• 2. Accuracy and Precision &#x2022; Accuracy Closeness of the data (sample) to the &#x201C;true value.&#x201D; &#x2022; Precision Closeness of the grouping of the data (sample) around some central value. Slide 2
• 3. Accuracy and Precision &#x2022; Precise but Inaccurate Relative Frequency Relative Frequency &#x2022; Inaccurate &amp; Imprecise True Value X Value True Value X Value Slide 3
• 4. Accuracy and Precision &#x2022; Precise and Accurate Relative Frequency Relative Frequency &#x2022; Accurate but Imprecise True Value X Value True Value X Value Slide 4
• 5. Accuracy and Precision Q: How do we quantify the concept of accuracy and precision? -- How do we characterize the error that occurred in our measurement? Slide 5
• 6. Individual Measurement Statistics &#x2022; Take N measurements: X1, . . . , XN &#x2022; Calculate mean and standard deviation: 1 x&#xF03D; N Sx 2 N &#xF0E5;X i i &#xF03D;1 1 &#xF0E9;N 2&#xF0F9; &#xF03D; &#xF0E5; &#xF028;X i &#xF02D; &#xF06D; x &#xF029; &#xF0FA; N &#xF0EA; i &#xF03D;1 &#xF0EB; &#xF0FB; &#x2022; What to use as the &#x201C;best value&#x201D; and uncertainty so we can say we are Q% confident that the true value lies in the interval xbest &#xF0B1; &#xF020;&#xF020;&#xF020;&#xF044;x. &#x2022; Need to know how data is distributed. Slide 6
• 7. Population and Sample &#x2022; Parent Population The set of all possible measurements. &#x2022; Sample Samples Handful of marbles from the bag A subset of the population measurements actually made. Population Bag of Marbles Slide 7
• 8. Histogram (Sample Based) &#x2022; Histogram &#x2013; A plot of the number of times a given value occurred. &#x2022; Relative Frequency &#x2013; A plot of the relative number of times a given value occurred. Histogram 20 Relative Frequency Plot 0.25 Relative Frequency 0.3 Number of Measurements 25 15 10 5 0 0.2 0.15 0.1 0.05 0 30 35 40 45 50 55 60 65 70 75 80 30 35 40 45 50 55 60 65 70 75 80 X Value (Bin) X Value (Bin) Slide 8
• 9. Probability Distribution (Population Based) &#x2022; Probability Density Function (pdf) (p(x)) &#x2013; Describes the probability distribution of all possible measures of x. &#x2013; Limiting case of the relative frequency. Probability Density Function Probability per unit change in x 0.3 &#x2022; Probability Distribution Function (P(x)) P&#xF028; x &#xF029; &#xF03D; P[ X &#xF0A3; x] X &#xF0A3;x Probability that &#x2013; Probability Distribution Function is the integral of the pdf, i.e. x P &#xF028; x &#xF029; &#xF03D; &#xF0F2; p &#xF028; x &#xF029; dx &#xF02D;&#xF0A5; 0.25 Q: Plot the probability distribution function vs x. Q: What is the maximum value of P(x)? 0.2 0.15 0.1 0.05 0 30 35 40 45 50 55 60 65 70 75 80 x Value (Bin) Slide 9
• 10. Probability Density Function &#x2013; The probability that a measurement X takes value between (-&#xF0A5;&#xF02C;&#xF020;&#xF0A5;) is 1. &#xF0DE; &#xF0A5; &#xF0F2;&#xF02D;&#xF0A5; p&#xF028; x &#xF029; dx &#xF03D; 1 &#x2013; Every pdf satisfies the above property. Ex: 1 &#xF02D; p&#xF028; x &#xF029; &#xF03D; e A x2 B is a probability density function. Find the relationship between A and B. &#xF0E6; &#xF0E7; Hint: &#xF0E8; &#xF0A5; &#xF0F2;0 2 e - a x dx &#xF03D; 1 2 &#xF070;&#xF0F6; &#xF0F7; a&#xF0F8; Q: Given a pdf, how would one find the probability that a measurement is between A and B? Slide 10
• 11. Common Statistical Distributions &#x2022; Gaussian (Normal) Distribution p &#xF028; x&#xF029; &#xF03D; where: x &#xF06D;x &#xF073;x &#xF073;x2 1 &#xF073; x 2&#xF070; &#xF02D; e &#xF028; x &#xF02D; &#xF06D; x &#xF029;2 p&#xF028; x&#xF029; 2 &#xF073; x2 = measured value = true (mean) value = standard deviation = variance Q: What are the two parameters that define a Gaussian distribution? x Value Q: How would one calculate the probability of a Gaussian distribution between x1 and x2? ( See Chapter 4, Appendix A ) Slide 11
• 12. Common Statistical Distributions &#x2022; Uniform Distribution p &#xF028;x &#xF029; &#xF03D; 1 x2 &#xF02D; x1 &#xF03D;0 x1 &#xF0A3; x &#xF0A3; x2 p&#xF028; x &#xF029; otherwise where: x = measured value x1 = lower limit x2 = upper limit x Value Q: Why do x1 and x2 also define the magnitude of the uniform distribution PDF? Slide 12
• 13. Common Statistical Distributions Ex: A voltage measurement has a Gaussian distribution with mean 3.4 [V] and a standard deviation of 0.4 [V]. Using Chapter 4, Appendix A, calculate the probability that a measurement is between: (a) [2.98, 3.82] [V] Ex: The quantization error of an ADC has a uniform distribution in the quantization interval Q. What is the probability that the actual input voltage is within &#xF0B1;&#xF020;Q/8 of the estimated input voltage? (b) [2.4, 4.02] [V] Slide 13
• 14. Statistical Analysis &#x2022; Standard Deviation (&#xF073;x and Sx ) &#x2013; Characterize the typical deviation of measurements from the mean and the width of the Gaussian distribution (bell curve). &#x2013; Smaller &#xF073;x , implies better ______________. &#x2013; Population Based 1 &#xF0F9;2 &#xF0A5; 2 &#xF073; x &#xF03D; &#xF0E9; &#xF0F2; &#xF028; x &#xF02D; &#xF06D; x &#xF029; p &#xF028; x &#xF029; dx &#xF0EA; &#xF02D;&#xF0A5; &#xF0FA; &#xF0EB; &#xF0FB; &#x2013; Sample Based (N samples) Sx &#xF03D; 1 N N &#xF0E5; &#xF028;X 2 i &#xF02D; &#xF06D;x &#xF029; i &#xF03D;1 Q: Often we do not know &#xF06D;x , how should we calculate Sx ? Slide 14
• 15. Statistical Analysis &#x2022; Standard Deviation (&#xF073;x and Sx ) (cont.) Common Name for "Error" Level Error Level in Terms of &#xF073; % That the Deviation from the Mean is Smaller Odds That the Deviation is Greater Standard Deviation &#xF0B1;&#xF020;&#xF073; 68.3 about 1 in 3 "Two-Sigma Error" &#xF0B1;&#xF020;&#xF031;&#xF02E;&#xF039;&#xF036;&#xF073; 95 1 in 20 "Three-Sigma Error" &#xF0B1;&#xF020;&#xF033;&#xF073; 99.7 1 in 370 "Four-Sigma Error" &#xF0B1;&#xF020;&#xF034;&#xF073; 99.994 1 in 16,000 &#xF06D; x &#xF02D; Z&#xF073; x &#xF0A3; x &#xF0A3; &#xF06D; x &#xF02B; Z&#xF073; x Slide 15
• 16. Statistical Analysis &#x2022; Sampled Mean x is the best estimate of &#xF06D;x . &#xF0A5; 1 N &#xF0BE;Best&#xF0BE; &#xF0BE;&#xF0BE;&#xF0AE; &#xF06D; x &#xF03D; E &#xF05B; X &#xF05D; &#xF03D; &#xF0F2; &#xF0BE; Estimate x p &#xF028; x &#xF029; dx x &#xF03D; &#xF0E5; Xi &#xF02D;&#xF0A5; N i &#xF03D;1 Degree of Freedom &#x2022; Sampled Standard Deviation ( Sx ) &#x2013; Use x when &#xF06D;x is not available. &#xF0AE; reduce by one degree of freedom. Sx &#xF03D; 1 N N &#xF0E5; &#xF028;X i &#xF03D;1 2 i &#xF02D; &#xF06D;x &#xF029; N 1 2 &#xF0BE;&#xF0BE; &#xF0BE; &#xF0BE; &#xF0BE; &#xF0AE; S x &#xF03D; &#xF0BE; &#xF0E5; &#xF028;X i &#xF02D; x &#xF029; &#xF028;N &#xF02D; 1&#xF029; i &#xF03D;1 When &#xF06D; x not known Q: If the sampled mean is only an estimate of the &#x201C;true mean&#x201D; &#xF06D;x , how do we characterize its error? Q: If we take another set of samples, will we get a different sampled mean? Q: If we take many more sample sets, what will be the statistics of the set of sampled means? Slide 16
• 17. Statistical Analysis Ex: The inlet pressure of a steam generator was measured 100 times during a 12 hour period. The specified inlet pressure is 4.00 MPa, with 0.7% allowable fluctuation. The measured data is summarized in the following table: Pressure (P)(MPa) Number of Results (m) 3.970 1 3.980 3 3.990 12 4.000 25 4.010 33 4.020 17 4.030 6 4.040 2 4.050 1 (1) Calculate the mean, variance and standard deviation. (2) Given the data, what pressure range will contain 95% of the data? Slide 17
• 18. Confidence Interval &#x2022; Sampled Mean Statistics &#x2013; If N is large, x will also have a Gaussian distribution. (Central Limit Theorem) &#x2013; Mean of x : &#xF06D;x &#xF03D; E&#xF05B; x &#xF05D; &#xF03D; &#xF06D;x x is an unbiased estimate. p( x ) p( x ) &#x2013; Standard Deviation of x : &#xF073;x &#xF03D; &#xF073;x N &#xF073; x is the best estimate of the error in estimating &#xF06D;x . p( x ) &#xF06D;x &#xF063; &#xF06D;x Q: Since we don&#x2019;t know &#xF073;x , how would we calculate &#xF073; x ? Slide 18
• 19. Confidence Interval &#x2022; For Large Samples ( N &gt; 60 ), Q% of all the sampled means x will lie in the interval p&#xF028; x &#xF029; &#xF073;x &#xF06D;x &#xF0B1; z Q&#xF073; x &#xF0BA; &#xF06D;x &#xF0B1; z Q N Equivalently, &#xF073; &#xF073; x &#xF02D; zQ x &#xF03C; &#xF06D;x &#xF03C; x &#xF02B; zQ x N N &#xF07B; &#xF07B; &#xF073;x &#xF073;x is the Q% Confidence Interval &#xF06D;x x zQ&#xF073; x zQ&#xF073; x When &#xF073;x is unknown, Sx will be a reasonable approximation. Slide 19
• 20. Confidence Interval Ex: 64 acceleration measurements were taken during an experiment. The estimated mean and standard deviation of the measurements were 3.15 m/s2 and 0.4 m/s2. (1) Find the 98% confidence interval for the true mean. (2) How confident are you that the true mean will be in the range from 2.85 to 3.45 m/s2 ? Slide 20
• 21. Confidence Interval &#x2022; For Small Samples ( N &lt; 60 ), the Q% Confidence Interval can be calculated using the Student-T distribution, which is similar to the normal distribution but depends on N. &#x2013; with Q% confidence, the true mean &#xF06D;x will lie in the following interval about any sampled mean: Sx Sx x &#xF02D; t &#xF06E; ,Q &#xF03C; &#xF06D;x &#xF03C; x &#xF02B; t &#xF06E; ,Q &#xF0AC; Q% confidence interval N N &#xF07B; &#xF07B; Sx Sx where &#xF06E; &#xF03D; N &#xF02D; 1 t&#xF06E;,Q is defined in class notes Chapter 4, Appendix B. Slide 21
• 22. Confidence Interval Ex: A simple postal scale is supplied with &#xBD; , 1, 2, and 4 oz brass weights. For quality check, 14 of the 1 oz weights were measured on a precision scale. The results, in oz, are as follows: 1.08 1.03 0.96 0.95 1.04 1.01 0.98 0.99 1.05 1.08 0.97 1.00 0.98 1.01 Based on this sample and that the parent population of the weight is normally distributed, what is the 95% confidence interval for the &#x201C;true&#x201D; weight of the 1 oz brass weights? Slide 22
• 23. Propagation of Error Q: If you measured the diameter (D) and height (h) of a cylindrical container, how would the measurement error affect your estimation of the volume ( V = &#xF070;D2h/4 )? Q: What is the uncertainty in calculating the kinetic energy ( mv2/ 2 ) given the uncertainties in the measurements of mass (m) and velocity (v)? &#xF0DE;&#xF020;How do errors propagate through calculations? Slide 23
• 24. Propagation of Error &#x2022; A Simple Example Suppose that y is related to two independent quantities X1 and X2 through y &#xF03D; C1 X 1 &#xF02B; C 2 X 2 &#xF03D; f &#xF028; X 1 , X 2 &#xF029; To relate the changes in y to the uncertainties in X1 and X2, we need to find dy = g(dX1, dX2): dy &#xF03D; The magnitude of dy is the expected change in y due to the uncertainties in x1 and x2: 2 2 &#xF0E6; &#xF0B6;f &#xF0F6; &#xF0E6; &#xF0B6;f &#xF0F6; &#xF073; y &#xF03D; &#xF044;y &#xF03D; &#xF0E7; &#xF044;x1 &#xF0F7; &#xF02B; &#xF0E7; &#xF044;x 2 &#xF0F7; &#xF03D; &#xF0E7; &#xF0B6;X &#xF0F7; &#xF0E7; &#xF0B6;X &#xF0F7; &#xF0E8; 1 &#xF0F8; &#xF0E8; 2 &#xF0F8; &#xF028;C &#xF073; &#xF029; &#xF02B; &#xF028;C &#xF073; &#xF029; 2 1 x1 2 2 x2 Slide 24
• 25. Propagation of Error &#x2022; General Formula Suppose that y is related to n independent measured variables {X1, X2, &#x2026;, Xn} by a functional representation: y &#xF03D; f &#xF028;X 1, X 2 ,&#xF04C; , X n &#xF029; Given the uncertainties of X&#x2019;s around some operating points: &#xF07B;x1 &#xF0B1; &#xF044;x 1 , x 2 &#xF0B1; &#xF044;x 2 ,&#xF04C; , x n &#xF0B1; &#xF044;x n &#xF07D; The expected value of y and its uncertainty &#xF044;y are: y &#xF03D; f &#xF028; x1 , x1 , &#xF04C; , xn &#xF029; 2 2 &#xF0E6; &#xF0B6;f &#xF0F6; &#xF0E6; &#xF0B6;f &#xF0F6; &#xF0E6; &#xF0B6;f &#xF0F6; &#xF044;y &#xF03D; &#xF0E7; &#xF044;x1 &#xF0F7; &#xF02B; &#xF0E7; &#xF044;x2 &#xF0F7; &#xF02B; &#xF04C; &#xF02B; &#xF0E7; &#xF044;x n &#xF0F7; &#xF0E7; &#xF0B6;X &#xF0F7; &#xF0E7; &#xF0B6;X &#xF0F7; &#xF0E7; &#xF0B6;X &#xF0F7; &#xF0E8; 1 &#xF0F8; &#xF0E8; 2 &#xF0F8; &#xF0E8; n &#xF0F8; 2 &#xF028; x1 , x1 ,&#xF04C;, x n &#xF029; Slide 25
• 26. Propagation of Error &#x2022;Proof: Assume that the variability in measurement y is caused by k independent zero-mean error sources: e1, e2, . . . , ek. Then, (y - ytrue)2 = (e1 + e2 + . . . + ek)2 = e12 + e22 + . . . + ek2 + 2e1e2 + 2e1e3 + . . . E[(y - ytrue)2] = E[e12 + e22 + . . . + ek2 + 2e1e2 + 2e1e3 + . . .] = E[e12 + e22 + . . . + ek2] &#xF073;y &#xF03D; &#xF05B; &#xF05D; &#xF05B; &#xF05D; &#xF05B; &#xF05D; E e1 2 &#xF02B; E e2 2 &#xF02B; &#xF04C; &#xF02B; E e k 2 &#xF03D; &#xF073; 1 2 &#xF02B; &#xF073; 2 2 &#xF02B; &#xF04C; &#xF02B; &#xF073; k 2 Slide 26
• 27. Propagation of Error &#x2022; Example (Standard Deviation of Sampled Mean) Given x &#xF03D; 1 &#xF028;X 1 &#xF02B; X 2 &#xF02B; X 3 &#xF02B; &#xF04C; &#xF02B; X N N &#xF029; Use the general formula for error propagation: 2 &#xF073;x &#xF03D; &#xF0E6; &#xF0B6;x &#xF0F6; &#xF0E6; &#xF0B6;x &#xF0E7; &#xF073; x1 &#xF0F7; &#xF02B; &#xF0E7; &#xF0E7; &#xF0B6;X &#xF0F7; &#xF0E7; &#xF0B6;X &#xF073; x2 1 2 &#xF0E8; &#xF0F8; &#xF0E8; &#xF0DE;&#xF073;x &#xF03D; 2 2 &#xF0F6; &#xF0E6; &#xF0B6;x &#xF0F6; &#xF0E6; &#xF0B6;x &#xF0F7; &#xF02B;&#xF0E7; &#xF073; x3 &#xF0F7; &#xF02B; &#xF04C; &#xF02B; &#xF0E7; &#xF0F7; &#xF0E7; &#xF0B6;X &#xF0F7; &#xF0E7; &#xF0B6;X &#xF073; x N 3 N &#xF0F8; &#xF0E8; &#xF0F8; &#xF0E8; &#xF0F6; &#xF0F7; &#xF0F7; &#xF0F8; 2 &#xF073;x N Slide 27
• 28. Propagation of Error Ex: What is the uncertainty in calculating the kinetic energy ( mv2/ 2 ) given the uncertainties in the measurements of mass (m) and velocity (v)? 2 &#xF0B6;KE &#xF0B6;KE &#xF0F6; &#xF044;KE &#xF03D; &#xF0E6; &#xF044;m&#xF0F6; &#xF02B; &#xF0E6; &#xF044;v&#xF0F7; &#xF0E7; &#xF0F7; &#xF0E7; &#xF0E8; &#xF0B6;m &#xF0F8; &#xF0E8; &#xF0B6;v &#xF0F8; 2 1 &#xF0E6; &#xF044;m &#xF0F6; 2 &#xF0E6; &#xF044;v &#xF0F6; 2 &#xF0E7; mv 2 &#xF0F7; &#xF02B; &#xF0E7; 2 mv 2 &#xF0F7; &#xF03D; &#xF0E8; 2 &#xF0E8; m&#xF0F8; v&#xF0F8; 1 &#xF044;m &#xF0F6; 2 &#xF0E6; &#xF044;v &#xF0F6; 2 &#xF0E7; &#xF0F7; &#xF02B; &#xF0E7;2 &#xF0F7; &#xF03D; mv 2 &#xF0E6; &#xF0E8; m&#xF0F8; &#xF0E8; v&#xF0F8; 2 Slide 28
• 29. &#x2022; Best Linear Fit &#x2013;How do we characterize &#x201C;BEST&#x201D;? Fit a linear model (relation) Output Y Least Squares Fitting of Data best linear fit yest &#xF024; yi &#xF03D; ao &#xF02B; a1 xi to N pairs of [xi, yi] measurements. Given xi, the error between the &#xF024; estimated output y i and the measured output yi is: &#xF024; ni &#xF03D; yi &#xF02D; yi measured output yi Input X The &#x201C;BEST&#x201D; fit is the model that &#xF0E9; N 2&#xF0F9; &#xF0E9;N 2&#xF0F9; &#xF024; &#xF0DE; min &#xF0EA; &#xF0E5; ni &#xF0FA; &#xF03D; min &#xF0EA;&#xF0E5; &#xF028; yi &#xF02D; yi &#xF029; &#xF0FA; minimizes the sum of the ___________ &#xF0EB; i=1 &#xF0FB; &#xF031;&#xF034;&#xF034; &#xF032;&#xF034;&#xF034;&#xF034;&#xF0FB; &#xF0EB;i=1 &#xF034; &#xF033; of the error Least Square Error Slide 29
• 30. Least Squares Fitting of Data N &#xF0E9;N 2&#xF0F9; 2 Let &#xF024; J &#xF03D; &#xF0EA; &#xF0E5; &#xF028; yi &#xF02D; yi &#xF029; &#xF0FA; &#xF03D; &#xF0E5; &#xF028; yi &#xF02D; ao &#xF02D; a1 x i &#xF029; &#xF0EB; i=1 &#xF0FB; i=1 The two independent variables are? M inim ize J &#xF0DE; Find a o and a1 such that dJ &#xF03D; 0 &#xF0B6;J &#xF0B6;J &#xF03D; 0 &#xF0DE; &#xF03D; 0 &#xF0DE; &#xF0B6; a&#xF031; &#xF0B6;a o N &#xF0DE; &#xF02D; &#xF0E5; i &#xF03D; 1 2 &#xF028; y i &#xF02D; a o &#xF02D; a1 x i &#xF029; &#xF03D; 0 N &#xF0DE; &#xF02D; &#xF0E5; i &#xF03D; 1 2 x i &#xF028; y i &#xF02D; a o &#xF02D; a1 x i &#xF029; &#xF03D; 0 Q: What are we trying to solve? Slide 30
• 31. Least Squares Fitting of Data Rewrite the last two equations as two simultaneous equations for ao and a1: x &#xF03D; y &#xF0EC;a N &#xF02B; a &#xF0E5; i 1&#xF0E5; i &#xF0EF; o &#xF0ED; 2 &#xF0EF; a o &#xF0E5; x i &#xF02B; a1 &#xF0E5; x i &#xF03D; &#xF0EE; &#xF028; &#xF029; &#xF0E5; &#xF028; x i yi &#xF029; &#xF0DE; &#xF0E9; &#xF0EA; &#xF0EB; &#xF0F9; &#xF0E9; ao &#xF0F9; &#xF0E9; &#xF0E5; yi &#xF0F9; &#xF0FA; &#xF0EA; a &#xF0FA; &#xF03D; &#xF0EA; &#xF028; x y &#xF029;&#xF0FA; &#xF0FB; &#xF0EB; 1 &#xF0FB; &#xF0EB;&#xF0E5; i i &#xF0FB; &#xF05B; &#xF028; &#xF029;&#xF05D; &#xF0EC; &#xF0E5; xi 2 &#xF028;&#xF0E5; yi &#xF029; &#xF02D; &#xF028;&#xF0E5; xi &#xF029;&#xF05B;&#xF0E5; &#xF028; xi yi &#xF029;&#xF05D; &#xF0EF;ao &#xF03D; &#xF0EF; &#xF044; &#xF0DE;&#xF0ED; N&#xF05B; &#xF0E5; &#xF028; xi yi &#xF029;&#xF05D; &#xF02D; &#xF028;&#xF0E5; xi &#xF029;&#xF028; &#xF0E5; yi &#xF029; &#xF0EF; &#xF0EF;a1 &#xF03D; &#xF0EE; &#xF044; &#xF05B;&#xF0E5; &#xF028; xi 2 &#xF029;&#xF05D; &#xF02D; &#xF028;&#xF0E5; xi &#xF029; where &#xF044; &#xF03D; N Slide 31 2
• 32. Least Squares Fitting of Data &#x2022; Summary: Given N pairs of input/output measurements [xi, yi], the best linear Least Squares model from input xi to output yi is: &#xF024; yi &#xF03D; ao &#xF02B; a1 xi &#xF05B;&#xF0E5; &#xF028; x &#xF029;&#xF05D;&#xF028;&#xF0E5; y &#xF029; &#xF02D; &#xF028;&#xF0E5; x &#xF029;&#xF05B;&#xF0E5; &#xF028; x y &#xF029;&#xF05D; &#xF03D; 2 where ao a1 &#xF03D; i i i i i &#xF044; N&#xF05B; &#xF0E5; &#xF028; x i yi &#xF029;&#xF05D; &#xF02D; &#xF028;&#xF0E5; x i &#xF029;&#xF028;&#xF0E5; yi &#xF029; and &#xF044; &#xF03D; N &#xF05B;&#xF0E5; &#xF028; &#xF029;&#xF05D; x i 2 &#xF02D; &#xF028;&#xF0E5; xi &#xF029; &#xF044; &#x2022; The process of minimizing squared error can be used for fitting nonlinear models and many engineering applications. &#x2022; Same result can also be derived from a probability distribution point of view (see Course Notes, Ch. 4 - Maximum Likelihood Estimation ). Q: Given a theoretical model y = ao + a2 x2 , what are the Least Squares estimates for ao &amp; a2? Slide 32 2
• 33. Least Squares Fitting of Data &#x2022; Variance of the fit: &#xF024; &#xF073;n2 &#xF03D; 1 N &#xF02D;2 &#xF028; yi &#xF02D; ao &#xF02D; a1xi &#xF029;2 &#xF0E5; N i &#xF03D;1 &#x2022; Variance of the measurements in y: &#xF073;y2 &#x2022; Assume measurements in x are precise. &#x2022; Correlation coefficient: &#xF024; &#xF073; n2 &#xF073;n2 R &#xF03D;1&#xF02D; 2 &#xF0BB;1&#xF02D; 2 , &#xF073;y Sy 2 is a measure of how well the model explains the data. R2 = 1 implies that the linear model fits the data perfectly. Slide 33