Error analysis   statistics
Upcoming SlideShare
Loading in...5
×
 

Error analysis statistics

on

  • 414 views

Error analysis statistics

Error analysis statistics

Statistics

Views

Total Views
414
Views on SlideShare
275
Embed Views
139

Actions

Likes
0
Downloads
5
Comments
0

7 Embeds 139

http://tarungehlots.blogspot.in 114
http://tarungehlots.blogspot.com 12
http://www.tarungehlots.blogspot.in 4
http://tarungehlots.blogspot.com.br 4
http://tarungehlots.blogspot.ru 3
http://tarungehlots.blogspot.ae 1
http://tarungehlots.blogspot.co.uk 1
More...

Accessibility

Categories

Upload Details

Uploaded via as Adobe PDF

Usage Rights

© All Rights Reserved

Report content

Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
  • Full Name Full Name Comment goes here.
    Are you sure you want to
    Your message goes here
    Processing…
Post Comment
Edit your comment

Error analysis   statistics Error analysis statistics Presentation Transcript

  • Error Analysis - Statistics • Accuracy and Precision • Individual Measurement Uncertainty – Distribution of Data – Means, Variance and Standard Deviation – Confidence Interval • Uncertainty of Quantity calculated from several Measurements – Error Propagation • Least Squares Fitting of Data Slide 1
  • Accuracy and Precision • Accuracy Closeness of the data (sample) to the “true value.” • Precision Closeness of the grouping of the data (sample) around some central value. Slide 2
  • Accuracy and Precision • Precise but Inaccurate Relative Frequency Relative Frequency • Inaccurate & Imprecise True Value X Value True Value X Value Slide 3 View slide
  • Accuracy and Precision • Precise and Accurate Relative Frequency Relative Frequency • Accurate but Imprecise True Value X Value True Value X Value Slide 4 View slide
  • 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
  • Individual Measurement Statistics • Take N measurements: X1, . . . , XN • Calculate mean and standard deviation: 1 x N Sx 2 N X i i 1 1 N 2   X i   x   N  i 1   • What to use as the “best value” and uncertainty so we can say we are Q% confident that the true value lies in the interval xbest  x. • Need to know how data is distributed. Slide 6
  • Population and Sample • Parent Population The set of all possible measurements. • Sample Samples Handful of marbles from the bag A subset of the population measurements actually made. Population Bag of Marbles Slide 7
  • Histogram (Sample Based) • Histogram – A plot of the number of times a given value occurred. • Relative Frequency – 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
  • Probability Distribution (Population Based) • Probability Density Function (pdf) (p(x)) – Describes the probability distribution of all possible measures of x. – Limiting case of the relative frequency. Probability Density Function Probability per unit change in x 0.3 • Probability Distribution Function (P(x)) P x   P[ X  x] X x Probability that – Probability Distribution Function is the integral of the pdf, i.e. x P  x    p  x  dx  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
  • Probability Density Function – The probability that a measurement X takes value between (-) is 1.    p x  dx  1 – Every pdf satisfies the above property. Ex: 1  p x   e A x2 B is a probability density function. Find the relationship between A and B.   Hint:   0 2 e - a x dx  1 2   a Q: Given a pdf, how would one find the probability that a measurement is between A and B? Slide 10
  • Common Statistical Distributions • Gaussian (Normal) Distribution p  x  where: x x x x2 1  x 2  e  x   x 2 p x 2  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
  • Common Statistical Distributions • Uniform Distribution p x   1 x2  x1 0 x1  x  x2 p x  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
  • 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 Q/8 of the estimated input voltage? (b) [2.4, 4.02] [V] Slide 13
  • Statistical Analysis • Standard Deviation (x and Sx ) – Characterize the typical deviation of measurements from the mean and the width of the Gaussian distribution (bell curve). – Smaller x , implies better ______________. – Population Based 1 2  2  x     x   x  p  x  dx      – Sample Based (N samples) Sx  1 N N  X 2 i  x  i 1 Q: Often we do not know x , how should we calculate Sx ? Slide 14
  • Statistical Analysis • Standard Deviation (x and Sx ) (cont.) Common Name for "Error" Level Error Level in Terms of  % That the Deviation from the Mean is Smaller Odds That the Deviation is Greater Standard Deviation  68.3 about 1 in 3 "Two-Sigma Error"  95 1 in 20 "Three-Sigma Error"  99.7 1 in 370 "Four-Sigma Error"  99.994 1 in 16,000  x  Z x  x   x  Z x Slide 15
  • Statistical Analysis • Sampled Mean x is the best estimate of x .  1 N Best   x  E  X     Estimate x p  x  dx x   Xi  N i 1 Degree of Freedom • Sampled Standard Deviation ( Sx ) – Use x when x is not available.  reduce by one degree of freedom. Sx  1 N N  X i 1 2 i  x  N 1 2      S x    X i  x  N  1 i 1 When  x not known Q: If the sampled mean is only an estimate of the “true mean” 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
  • 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
  • Confidence Interval • Sampled Mean Statistics – If N is large, x will also have a Gaussian distribution. (Central Limit Theorem) – Mean of x : x  E x   x x is an unbiased estimate. p( x ) p( x ) – Standard Deviation of x : x  x N  x is the best estimate of the error in estimating x . p( x ) x  x Q: Since we don’t know x , how would we calculate  x ? Slide 18
  • Confidence Interval • For Large Samples ( N > 60 ), Q% of all the sampled means x will lie in the interval p x  x x  z Q x  x  z Q N Equivalently,   x  zQ x  x  x  zQ x N N   x x is the Q% Confidence Interval x x zQ x zQ x When x is unknown, Sx will be a reasonable approximation. Slide 19
  • 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
  • Confidence Interval • For Small Samples ( N < 60 ), the Q% Confidence Interval can be calculated using the Student-T distribution, which is similar to the normal distribution but depends on N. – with Q% confidence, the true mean x will lie in the following interval about any sampled mean: Sx Sx x  t  ,Q  x  x  t  ,Q  Q% confidence interval N N   Sx Sx where   N  1 t,Q is defined in class notes Chapter 4, Appendix B. Slide 21
  • Confidence Interval Ex: A simple postal scale is supplied with ½ , 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 “true” weight of the 1 oz brass weights? Slide 22
  • 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 = 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)? How do errors propagate through calculations? Slide 23
  • Propagation of Error • A Simple Example Suppose that y is related to two independent quantities X1 and X2 through y  C1 X 1  C 2 X 2  f  X 1 , X 2  To relate the changes in y to the uncertainties in X1 and X2, we need to find dy = g(dX1, dX2): dy  The magnitude of dy is the expected change in y due to the uncertainties in x1 and x2: 2 2  f   f   y  y   x1    x 2    X   X   1   2  C    C   2 1 x1 2 2 x2 Slide 24
  • Propagation of Error • General Formula Suppose that y is related to n independent measured variables {X1, X2, …, Xn} by a functional representation: y  f X 1, X 2 , , X n  Given the uncertainties of X’s around some operating points: x1  x 1 , x 2  x 2 , , x n  x n  The expected value of y and its uncertainty y are: y  f  x1 , x1 ,  , xn  2 2  f   f   f  y   x1    x2      x n   X   X   X   1   2   n  2  x1 , x1 ,, x n  Slide 25
  • Propagation of Error •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] y        E e1 2  E e2 2    E e k 2   1 2   2 2     k 2 Slide 26
  • Propagation of Error • Example (Standard Deviation of Sampled Mean) Given x  1 X 1  X 2  X 3    X N N  Use the general formula for error propagation: 2 x   x   x   x1     X   X  x2 1 2    x  2 2   x   x    x3        X   X  x N 3 N         2 x N Slide 27
  • 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 KE KE  KE   m   v     m   v  2 1  m  2  v  2  mv 2    2 mv 2    2  m v 1 m  2  v  2    2   mv 2   m  v 2 Slide 28
  • • Best Linear Fit –How do we characterize “BEST”? Fit a linear model (relation) Output Y Least Squares Fitting of Data best linear fit yest  yi  ao  a1 xi to N pairs of [xi, yi] measurements. Given xi, the error between the  estimated output y i and the measured output yi is:  ni  yi  yi measured output yi Input X The “BEST” fit is the model that  N 2 N 2   min   ni   min   yi  yi   minimizes the sum of the ___________  i=1    i=1   of the error Least Square Error Slide 29
  • Least Squares Fitting of Data N N 2 2 Let  J     yi  yi      yi  ao  a1 x i   i=1  i=1 The two independent variables are? M inim ize J  Find a o and a1 such that dJ  0 J J  0   0   a a o N    i  1 2  y i  a o  a1 x i   0 N    i  1 2 x i  y i  a o  a1 x i   0 Q: What are we trying to solve? Slide 30
  • Least Squares Fitting of Data Rewrite the last two equations as two simultaneous equations for ao and a1: x  y a N  a  i 1 i  o  2  a o  x i  a1  x i       x i yi        ao    yi    a     x y    1   i i       xi 2  yi    xi   xi yi  ao     N   xi yi    xi   yi   a1      xi 2    xi  where   N Slide 31 2
  • Least Squares Fitting of Data • Summary: Given N pairs of input/output measurements [xi, yi], the best linear Least Squares model from input xi to output yi is:  yi  ao  a1 xi   x  y    x   x y   2 where ao a1  i i i i i  N   x i yi    x i  yi  and   N    x i 2   xi   • The process of minimizing squared error can be used for fitting nonlinear models and many engineering applications. • 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 & a2? Slide 32 2
  • Least Squares Fitting of Data • Variance of the fit:  n2  1 N 2  yi  ao  a1xi 2  N i 1 • Variance of the measurements in y: y2 • Assume measurements in x are precise. • Correlation coefficient:   n2 n2 R 1 2 1 2 , 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