02 Statistics review

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02 Statistics review

  1. 1. Bayesian Networks Unit 2 Statistics Review Wang, Yuan-Kai, 王元凱 ykwang@mails.fju.edu.tw http://www.ykwang.tw Department of Electrical Engineering, Fu Jen Univ. 輔仁大學電機工程系 2006~2011 Reference this document as: Wang, Yuan-Kai, “Statistics Review," Lecture Notes of Wang, Yuan-Kai, Fu Jen University, Taiwan, 2011.Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  2. 2. 王元凱 Unit - Statistics Review p. 2 Goal of this Unit  Review basic concepts of statistics in terms of  Image processing  Pattern recognitionFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  3. 3. 王元凱 Unit - Statistics Review p. 3 Related Units  Previous unit(s)  Probability Review  Next units  Uncertainty Inference (Discrete)  Uncertainty Inference (Continuous)Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  4. 4. 王元凱 Unit - Statistics Review p. 4 Self-Study  Artificial Intelligence: a modern approach  Russell & Norvig, 2nd, Prentice Hall, 2003. pp.462~474,  Chapter 13, Sec. 13.1~13.3  統計學的世界  墨爾著,鄭惟厚譯, 天下文化,2002  深入淺出統計學  D. Grifiths, 楊仁和譯,2009, O’ ReillyFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  5. 5. 王元凱 Unit - Statistics Review p. 5 Contents 1. Introduction …………………………… 6 2. Histogram ……………………………… 12 3. Central Tendency .............................. 18 4. Variance ............................................. 26 5. Frequency Distribution …………...... 34 6. Covariance ......................................... 52 7. Covariance Matrix …………………… 57 8. Correlation .......................................... 64 9. Chart and Graph …………………….. 68 10.References …………………………… 79Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  6. 6. 王元凱 Unit - Statistics Review p. 6 1. Introduction  Probability and statistics are about uncertainty  The world is full of uncertainty  Our hardware/software implementation needs to consider uncertaintyFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  7. 7. 王元凱 Unit - Statistics Review p. 7 Uncertainty by Probability  It summarizes the uncertainty that arises from laziness and ignorance  An example  P(your toothache is caused by a cavity) = 0.8  20% represents your laziness and ignorance – all other possible causesFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  8. 8. 王元凱 Unit - Statistics Review p. 8 Uncertainty by Statistics  It derives probabilistic facts from a set of data  Derive actual probability number  P(your toothache is caused by a cavity) = 0.8  Describe characteristics of data  Mean, variance, moment, ...  Build the statistic model of data  Gaussian, Gaussian Mixture  Reason new facts from the dataFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  9. 9. 王元凱 Unit - Statistics Review p. 9 What Is Statistics  Given a set of data from a random variable  A statistic is a number that provides information about the data  Descriptive statistics  Two way to describe data  Measures of central tendency  Measures of dispersionFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  10. 10. 王元凱 Unit - Statistics Review p. 10 Measures of Central Tendency  Mean  Average, expected value of the random variable  Median  Middle value of the R.V.  Mode  The variable value at the peak of the pmf/pdfFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  11. 11. 王元凱 Unit - Statistics Review p. 11 Measures of Dispersion  Dispersion  Variance  Covariance  Correlation  Moment  Others: range, percentiles, 95% percentile,Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  12. 12. 王元凱 Unit - Statistics Review p. 12 2. Histogram  This course has 15 students  Every student has a score with values: 0, 10, 20, ... 100  Random variable X = Students score  Scores of the 15 student  {20, 90, 90, 100, 50, 60, 70, 60 ,80, 70, 80, 90, 80, 70, 70}  20x1; 50x1; 60x2; 70x4; 80x3; 90x3; 100x1Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  13. 13. 王元凱 Unit - Statistics Review p. 13 The Histogram  20x1; 50x1; 60x2; 70x4; 80x3; 90x3; 100x1 No. of X X 10 20 30 40 50 60 70 80 90 100  20x1/15; 50x1/15; 60x2/15; 70x4/15; 80x3/15; 90x3/15; 100x1/15 P(X) X Histogram is P.D.F. 10 20 30 40 50 60 70 80 90 100Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  14. 14. 王元凱 Unit - Statistics Review p. 14 Definitions  For a random variable X  X has n possible values {x1, x2, ..., xn}  Now there are N random data of X  x1, x2, .., xN  Histogram & Distribution  The number of xi : N(xi)  The probabilities of xi : p(xi)= N(xi)/NFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  15. 15. 王元凱 Unit - Statistics Review p. 15 Histogram v.s. P.D.F.Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  16. 16. 王元凱 Unit - Statistics Review p. 16 2D Gaussian Histogram pdfFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  17. 17. 王元凱 Unit - Statistics Review p. 17 Histogram of an Image • Random variable X (Gray level) has n possible values {x1, x2, ..., xn}, n=256 • N random data x1, x2, .., xN of X, N=Width*Height • Histogram: N(xi) • Distribution: P(xi) = N(xi) / NFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  18. 18. 王元凱 Unit - Statistics Review p. 18 3. Central Tendency  Random variable X = Students score  Scores of the 15 student  {20, 90, 90, 100, 50, 60, 70, 60 ,80, 70, 80, 90, 80, 70, 70}  20x1; 50x1; 60x2; 70x4; 80x3; 90x3; 100x1 P(X)  Histogram X 10 20 30 40 50 60 70 80 90 100  Mean ?Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  19. 19. 王元凱 Unit - Statistics Review p. 19 Mean  Mean from the set of data 1 N E[ X ]  x  N x i 1 i  Mean from the p.d.f n E [ X ]  x   xi p( xi ) i 1Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  20. 20. 王元凱 Unit - Statistics Review p. 20 Mean of an Image 1 N n E[ X ]  x   xi E [ X ]  x   xi p( xi ) N i 1 i 1Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  21. 21. 王元凱 Unit - Statistics Review p. 21 Disadvantage of Mean  Mean is easily influenced by outlier (extreme values)  Mean may not be the real value P(X) 1 N x N  x i  72 i 1 X 10 20 30 40 50 60 70 80 90 100Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  22. 22. 王元凱 Unit - Statistics Review p. 22 Median & Mode  Median and mode are another measures of central tendency Median: (1) Sort the scores, (2) Select the middle {20, 50, 60, 60, 70, 70, 70, 70, 80 ,80, 80, 90, 90, 90, 100} Mode: select the score with the maximum N(X) or P(X) 20x1; 50x1; 60x2; 70x4; 80x3; 90x3; 100x1 P(X) X 10 20 30 40 50 60 70 80 90 100Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  23. 23. 王元凱 Unit - Statistics Review p. 23 Advantage of Median & Mode  Median and mode is not influenced by outlier  Median and mode will be the real valueFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  24. 24. 王元凱 Unit - Statistics Review p. 24 Expected Value  E[X] : mean n E[ X ]  x   xi p ( xi ) E[ X ]  x   i 1  xp( x)dx   E[Xr] – rth moment of X n  E[ X ]   xi p ( xi ) E[ X ]   x r p ( x)dx r r  i 1  E[(X-µ)r] – rthn central moment of X E[( X   ) ]   ( xi   ) p( xi ) r r i 1  E[( X   ) ]   ( x   ) p ( x)dx r r Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  25. 25. 王元凱 Unit - Statistics Review p. 25 Deviation about the Mean x xi  It indicates how far a value is from the center  It is a very important number to measure the dispersion of how a distribution spreads outFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  26. 26. 王元凱 Unit - Statistics Review p. 26 4. Variance  Variance and standard deviation come from the “deviation”  Average Deviation  Calculate all of the deviations and find their average  It is a measure of the typical amount any given data point might vary N  ( xi  x ) x x i AD  i 1 NFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  27. 27. 王元凱 Unit - Statistics Review p. 27 We need Absolute Deviation xi x x i N xi | xi  x | 1 1-3=-2  xi  x 1 |1-3|=2 2 2-3=-1 AAD  i 1 2 |2-3|=1 3 3-3=0 N 3 |3-3|=0 4 4-3=1 4 |4-3|=1 5 5-3=2 5 |5-3|=2 =15 =? N =15 =6 x  15/5  (x i  x) x  15/5 ABD=6/ AD  i 1 =3.0 N =3.0 5 =1.2Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  28. 28. 王元凱 Unit - Statistics Review p. 28 Or Square of the Deviation N Square the deviations  ( x i  x ) 2 Take the square root to remove Variance  i 1 to return to the minus signs N original scale N  x xi N  (x i  x) 2 AAD  i 1 N  i 1 N N  (x i  x) AD  i 1 NFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  29. 29. 王元凱 Unit - Statistics Review p. 29 Sample Mean and Sample Variance  We can approximate the expected value by the sample mean N x 1 N x i 1 i N  Sample variance s  i 2 1 N  (x  x) i 2 But, strangely enough, if you1want a good approximation of the true variance, you should use 2 N 2 1 N i   ˆ N 1 s   (x  x) N  1 i 1 2Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  30. 30. 王元凱 Unit - Statistics Review p. 30 Variance of an Image n 1 1  2   ( xi  x ) 2 p ( xi ), n  256 N   ˆ2  N  1 i 1 ( xi  x )2 ˆ i 0 1 n x N  x p( x ) Moments i 1 i iFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  31. 31. 王元凱 Unit - Statistics Review p. 31 An Example of Variance  Variance of the scores of 15 students in this course = ?  {20, 90, 90, 100, 50, 60, 70, 60 ,80, 70, 80, 90, 80, 70, 70} P(X) 1 N  x   x  72 i X N i 1 10 20 30 40 50 60 70 80 90 100 1 N i 2  ˆ  N  1 i 1 ( x  x )2 = 388.6Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  32. 32. 王元凱 Unit - Statistics Review p. 32 Ex. of Standard Deviation  Standard deviation (SD):   = (Var)1/2 1 N  ˆ  ( x  x ) = 19.7 N  1 i 1 i 2 P(X) 1 N | x X 10 20 30 40 50 60 70 80 90 100  i x| N i 1 52.3 72 91.7Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  33. 33. 王元凱 Unit - Statistics Review p. 33 Formal Definition of Variance Var ( X )   x  E[( X  E[ X ]) 2 ]   ( xi  x ) 2 p( xi ) 2 i Var ( X )    E[( X  E[ X ]) ]   ( xi  x ) p( xi )dx 2 x 2 2 xFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  34. 34. 王元凱 Unit - Statistics Review p. 34 5. Frequency Distributions • A graph or chart that shows the number of observations of a given value, or class intervalFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  35. 35. 王元凱 Unit - Statistics Review p. 35 Shape of Distribution  Modality  The number of peaks in the curve  Skewness  An asymmetry in a distribution where values are shifted to one extreme or the other.  Kurtosis  The degree of Peakedness/flatness in the curveFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  36. 36. 王元凱 Unit - Statistics Review p. 36 Modality  Unimodal  Bimodal  MultimodalFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  37. 37. 王元凱 Unit - Statistics Review p. 37 Skewness (1/3)  The third moment about the Mean  =0: symmetry distribution (Normal distribution )  >0 : Right Skew (Positive Skew)  <0 : Left Skew (Negative Skew) Right Left skew Symmetry skewFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  38. 38. 王元凱 Unit - Statistics Review p. 38 Skewness (2/3)  x   Skewness formula N 3 from data i x i 1 N  1  Skewness formula from p.d.f n  x  x  3 i p ( xi ) i 1Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  39. 39. 王元凱 Unit - Statistics Review p. 39 Skewness (3/3) Coefficient of Skewness (Normalized skewness)  x  x  N  Skewness formula i 3 from data i 1 3 N  1 n  x  x   Skewness formula 3 p ( xi ) from p.d.f i 1 i 3 Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  40. 40. 王元凱 Unit - Statistics Review p. 40 Kurtosis (1/3)  The normalized fourth moment about the Mean  K=3: normal peak (Gaussian)  K>3: sharp peak  K<3: flat peak K=3Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  41. 41. 王元凱 Unit - Statistics Review p. 41 Kurtosis (2/3)  x  N 4  From data i x i 1 4 N  1 n  x  x  4  From p.d.f i p ( xi ) i 1 4 Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  42. 42. 王元凱 Unit - Statistics Review p. 42 Kurtosis (3/3)  Another definition of kurtosis  K=0: normal peak (Gaussian)  K>0: sharp peak  K<0: flat peak  x  N n  x  x  4 i x i 4 p ( xi ) 4 3 i 1 N  1 i 1 4 3 Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  43. 43. 王元凱 Unit - Statistics Review p. 43 Demo  A demo of shape of frequency distributionFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  44. 44. 王元凱 Unit - Statistics Review p. 44 Shape of Distribution of an ImageFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  45. 45. 王元凱 Unit - Statistics Review p. 45 Frequency Distributions - Types  The Normal  The Uniform  The Log-normal  The Exponential  Statistical Distributions  t  -Square  FFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  46. 46. 王元凱 Unit - Statistics Review p. 46 Frequency Distributions – Types (cont.)  Hyper-geometric  Poisson  Binomial  Gamma  Weibull  Cauchy  BetaFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  47. 47. 王元凱 Unit - Statistics Review p. 47 Gaussian Dist. Of Data (1/5) Observe: Mean, Principal axes, implication of off-diagonal Common convention: show contour covariance term, max gradient corresponding to 2 standard zone of p(x) deviations from meanFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  48. 48. 王元凱 Unit - Statistics Review p. 48 Gaussian Dist. Of Data (2/5)Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  49. 49. 王元凱 Unit - Statistics Review p. 49 Gaussian Dist. Of Data (3/5) In this example, x and y are almost independentFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  50. 50. 王元凱 Unit - Statistics Review p. 50 Gaussian Dist. Of Data (4/5) In this example, x and “x+y” are clearly not independentFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  51. 51. 王元凱 Unit - Statistics Review p. 51 Gaussian Dist. Of Data (5/5) In this example, x and “20x+y” are clearly not independentFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  52. 52. 王元凱 Unit - Statistics Review p. 52 6. Covariance  In addition to know the mean & variance of random variables X & Y  We usually want to know  If X increase,  Will Y probably increase or decrease?  We want to know the probable relationship between X and Y  We will use covariance to identify the relationshipFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  53. 53. 王元凱 Unit - Statistics Review p. 53 Mean Vector  Random variable X: score of student of this course  Random variable Y: score of student of another course  Z=(X,Y) is called a random vector  Mean vector ( x , y )is the mean of Z Ny 1 Nx 1 x Nx x i y Ny y i 1 i i 1Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  54. 54. 王元凱 Unit - Statistics Review p. 54 Variance Vector ?  Variance of X and Y: x2, y2 1 Nx   VAR[ X ]  2 x N x  1 i 1  ( x i  x )2 Ny 1   VAR[Y ]  2 y  N y  1 i 1 ( y i  y )2 Variance vector (x2, y2)? No. Not Enough. •We want to know •If X increase, •Will Y probably increase or decrease?Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  55. 55. 王元凱 Unit - Statistics Review p. 55 Covariance  For two random variables X and Y  Covariance indicates the tendency of the two random variables X & Y to vary together  i.e., to co-vary Cov ( X , Y )   XY  E[( X  E[ X ])(Y  E[Y ])]   ( xi  x )( yi  y ) p( xi , yi ) i 1 N i   N  1 i 1 ( x  x )( y i  y )Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  56. 56. 王元凱 Unit - Statistics Review p. 56 Property of Covariance  If X and Y tend to increase together, then XY>0  If X tends to decrease when Y increases, then XY<0  If X and Y are uncorrelated, then  XY=0  |XY|≤ XY, where X is the standard deviation of X  XX = 2X = VAR(X)  XY = YXFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  57. 57. 王元凱 Unit - Statistics Review p. 57 7. Covariance Matrix  For two random variables X, Y  There are two variances XX, YY and two covariance XY ,YX  We usually write the four variances into a covariance matrix   E[(x  μ)(x  μ)T ]  E[( x μx ) ] 2 E[( x μx )( y μy )]    E[( x μx )( y μy )] E[( x μx ) ]  2     x  xy  2  2   xy  y   Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  58. 58. 王元凱 Unit - Statistics Review p. 58 QuizFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  59. 59. 王元凱 Unit - Statistics Review p. 59 Quiz  For 3 random variables: x,y,z  The random vector v=(x,y,z)  The covariance matrix of v?Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  60. 60. 王元凱 Unit - Statistics Review p. 60 Independence and CovarianceFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  61. 61. 王元凱 Unit - Statistics Review p. 61 Formal Definition (1/2)  For a random vector X = (X1, X2, ..., Xm) with n random variables  The probability of x =(x1, x2, ..., xn), P(x), is a joint probability P(x) = P(x1, x2, ..., xn)  Its expected values are a mean vector  = (1, 2, ..., m)  E[X] = (E[X1], E[X2], ..., E[Xm]) = (1, 2, ..., m) = Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  62. 62. 王元凱 Unit - Statistics Review p. 62 Formal Definition (2/2)  For the random vector X = (X1, X2, ..., Xm)  All of its covariance values are a covariance matrix Cov (X)  E[(X   )(X   )T ]  E[( X 1  1 )( X 1  1 )]  E[( X 1  1 )( X m   m )]         E[( X m   m )( X 1  1 )]  E[( X m   m )( X m   m )]     12  12   1m  •  is a square and    21  2   2 m  2  symmetric matrix       • Its diagonal elements  2   m1  m 2   m    are 2iFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  63. 63. 王元凱 Unit - Statistics Review p. 63 Property of Covariance Matrix  Symmetric matrix  Positive definition matrix  EigenvalueFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  64. 64. 王元凱 Unit - Statistics Review p. 64 8. Correlation  Correlation is another measure of the relationship of two random variables Cov( X , Y )  XY  XY   Var ( X )Var (Y )  X  Y • -1  XY  1 • XY = 0 means that the variables are uncorrelated • XY = 0 E[XY] = E[X] E[Y] • Independence means uncorrelated • But not vice versa!Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  65. 65. 王元凱 Unit - Statistics Review p. 65 Correlation & Covariance  |XY|≤ XY  -XY ≤ XY ≤ XY  XY = XYXY  -1  XY  1  XY is called the correlation coefficient  -1   xy  xy  +1  x  yFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  66. 66. 王元凱 Unit - Statistics Review p. 66 Covariance v.s. Correlation Y Y X X XY = XY XY = 1/2XY XY = +1 XY = 1/2 Y Y Y X X X XY = -XY XY = -1/2XY XY = 0 XY = -1 XY = -1/2 XY = 0Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  67. 67. 王元凱 Unit - Statistics Review p. 67 Demos  Flash demos of correlation with scatter plotFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  68. 68. 王元凱 Unit - Statistics Review p. 68 9. Charts and Graphs  Descriptive Graphs  Bar Chart  Pie Graph  Line Graph  Distributions  Histogram  Box Plot  Steam and LeafFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  69. 69. 王元凱 Unit - Statistics Review p. 69 Bar Charts  Best for displaying actual values.  Can handle moderate # of cases (bars)  Excel calls it a column chartFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  70. 70. 王元凱 Unit - Statistics Review p. 70 Bar Chart – An Example Vote for President in 2000 60 Millions 50 40 30 20 10 0 e r sh an e or ad Bu an G N ch BuFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  71. 71. 王元凱 Unit - Statistics Review p. 71 Pie Charts  Best used with small number of categories or cases to display  Good for showing relative distribution  Percentages, proportions  Use only one column of data  Plus one column of labelsFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  72. 72. 王元凱 Unit - Statistics Review p. 72 Pie Chart Examples Vote for President in 2004 Bush Nader Kerry Vote for President in 2000 Buchanan Gore Bush NaderFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  73. 73. 王元凱 Unit - Statistics Review p. 73 Line graphs  Best for showing data across time  Always give dates  Label X axis  Indicate units on Y axis  Use legend for multiple linesFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  74. 74. 王元凱 Unit - Statistics Review p. 74 Line Graphs - Example Defense Spending - 1940-2002 400000 300000 Millions $ Defense 200000 Spending 100000 0 1940 1948 1956 1964 1972 1980 1988 1996 YearFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  75. 75. 王元凱 Unit - Statistics Review p. 75 Box Plot  Quick picture of a distribution  Parts of box give distribution characteristics  Your Text is not quite accurate!Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  76. 76. 王元凱 Unit - Statistics Review p. 76 Stem and Leaf Plot  Good for showing distribution while preserving data  Figuring out stems can be trickyFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  77. 77. 王元凱 Unit - Statistics Review p. 77 Review for Next Unit  We learn  Histogram, mean, variance, covariance, correlation  Frequency distribution, Gaussian  We will learn in next unit  Moment, Gaussian Mixture, Linear Gaussian  Moment  mean, varianceFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  78. 78. 王元凱 Unit - Statistics Review p. 78 Mean & Variance of an Image 1 n 2  ˆ 1 N i  ( x  x )2 2  ˆ  N  1 i 1 ( xi  x ) 2 p ( xi ) N  1 i 1 1 n x N  x p( x ) Moments i 1 i iFu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  79. 79. 王元凱 Unit - Statistics Review p. 79 10. References  An excellent textbook 統計學的世界,鄭惟厚譯,天下文化2002  Statistics: concepts and controversies, 5th, D. S. Moore, 2001  Chart & graph: Chap. 10,11  Central tendency & dispersion: Chap. 12  Gaussian distribution: Chap. 13  Correlation: Chap. 14, 15Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
  80. 80. 王元凱 Unit - Statistics Review p. 80 透過Excel學統計  統計學,劉明德等著,全華,93  Chart & graph : Chap. 2  Central tendency & dispersion: Chap. 3  Frequency distribution: Chap. 6,7  統計學與Excel,王文中著,博碩,93  Central tendency (mean, mode, median) : Chap. 2  Dispersion (variance, standard deviation) : Chap. 3  Gaussian distribution : Chap. 4  Correlation: Chap. 11Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright

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