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University of Colorado Boulder ASEN 5070 Statistical Orbit determination I Fall 2012 Professor George H. Born Professor Jeffrey S. Parker Lecture 5: Stat.

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Presentation on theme: "University of Colorado Boulder ASEN 5070 Statistical Orbit determination I Fall 2012 Professor George H. Born Professor Jeffrey S. Parker Lecture 5: Stat."— Presentation transcript:

1 University of Colorado Boulder ASEN 5070 Statistical Orbit determination I Fall 2012 Professor George H. Born Professor Jeffrey S. Parker Lecture 5: Stat OD 1

2 University of Colorado Boulder  Homework 2 due Thursday  Homework 3 out today ◦ Basic dynamical systems relationships ◦ Studies of the state transition matrix ◦ Linear algebra  I’m unavailable this Wednesday. Use those TAs and email is great of course. 2

3 University of Colorado Boulder 3

4 University of Colorado Boulder 4

5 University of Colorado Boulder 5

6 University of Colorado Boulder 6 ~N( 0.0, 1.41 )

7 University of Colorado Boulder 7

8 University of Colorado Boulder 8 ~N( 1.0, 0.41 )

9 University of Colorado Boulder 9

10 University of Colorado Boulder 10

11 University of Colorado Boulder  Some popular questions and answers  Energy with Drag 11

12 University of Colorado Boulder  Some popular questions and answers  Computation of Time of Perigee 12

13 University of Colorado Boulder 13

14 University of Colorado Boulder 14

15 University of Colorado Boulder 15

16 University of Colorado Boulder  Chapter 4, Problems 1-6 ◦ Solving ODEs ◦ Linear Algebra ◦ Studying the state transition matrix 16

17 University of Colorado Boulder  Review of Differential Equations ◦ Laplace Transforms  Review of Statistics 17

18 University of Colorado Boulder  Stat OD dynamics:  Solve for given A and 18

19 University of Colorado Boulder  Stat OD dynamics:  Solve for given A and 19

20 University of Colorado Boulder  Solve for  w/ 20

21 University of Colorado Boulder  Solve for  w/ 21

22 University of Colorado Boulder  Solve the ODE  We can solve this using “traditional” calculus: 22 Check your answer by plugging it back in

23 University of Colorado Boulder  Laplace Transforms are useful for analysis of linear time-invariant systems: ◦ electrical circuits, ◦ harmonic oscillators, ◦ optical devices, ◦ mechanical systems, ◦ even orbit problems.  Transformation from the time domain into the frequency domain.  Inverse Laplace Transform converts the system back. 23

24 University of Colorado Boulder 24

25 University of Colorado Boulder  Solve the ODE  We can solve this using “traditional” calculus: 25

26 University of Colorado Boulder  Solve the ODE  Or, we can solve this using Laplace Transforms: 26

27 University of Colorado Boulder  Solve the ODE 27

28 University of Colorado Boulder  Solve the ODE 28

29 University of Colorado Boulder  Solve the ODE 29

30 University of Colorado Boulder  Solve the ODE 30

31 University of Colorado Boulder  Solve the ODE 31

32 University of Colorado Boulder  Solve the ODE 32

33 University of Colorado Boulder  Solve the ODE 33

34 University of Colorado Boulder  Questions on Diff EQ?  Quick Break  Review of Statistics to follow 34

35 University of Colorado Boulder  X is a random variable with a prescribed domain.  x is a realization of that variable.  Example: ◦ 0 < X < 1 ◦ x 1 = 0.232 ◦ x 2 = 0.854 ◦ x 3 = 0.055 ◦ etc 35

36 University of Colorado Boulder Axioms of Probability 2. p(S)=1, S is the certain event

37 University of Colorado Boulder

38 University of Colorado Boulder Axioms of Probability

39 University of Colorado Boulder For the continuous random variable, axioms 1 and 2 become Probability Density & Distribution Functions

40 University of Colorado Boulder For the continuous random variable, axioms 1 and 2 become The third axiom becomes Which for a < b < c Probability Density & Distribution Functions

41 University of Colorado Boulder Probability Density & Distribution Functions Using axiom 2 as a guide, solve the following for k:

42 University of Colorado Boulder Probability Density & Distribution Functions Using axiom 2 as a guide, solve the following for k:

43 University of Colorado Boulder Probability Density & Distribution Functions

44 University of Colorado Boulder Example: From the definition of the density and distribution functions we have: From axioms 1 and 2, we find: Probability Density & Distribution Functions

45 University of Colorado Boulder Expected Values Note that:

46 University of Colorado Boulder Expected Values

47 University of Colorado Boulder Expected Values

48 University of Colorado Boulder Expected Values

49 University of Colorado Boulder Expected Values

50 University of Colorado Boulder The Gaussian or Normal Density Function

51 University of Colorado Boulder The Gaussian or Normal Density Function

52 University of Colorado Boulder

53 University of Colorado Boulder

54 University of Colorado Boulder

55 University of Colorado Boulder Two Random Variables

56 University of Colorado Boulder Marginal Distributions

57 University of Colorado Boulder Marginal Distributions

58 University of Colorado Boulder Independence of Random Variables

59 University of Colorado Boulder Conditional Probability

60 University of Colorado Boulder Expected Values of Bivariate Functions

61 University of Colorado Boulder Expected Values of Bivariate Functions

62 University of Colorado Boulder Expected Values of Bivariate Functions

63 University of Colorado Boulder

64 University of Colorado Boulder

65 University of Colorado Boulder The Variance-Covariance Matrix

66 University of Colorado Boulder Properties of the Correlation Coefficient

67 University of Colorado Boulder Properties of Covariance and Correlation

68 University of Colorado Boulder Properties of Covariance and Correlation

69 University of Colorado Boulder Central Limit Theorem

70 University of Colorado Boulder  Addition of multiple variables taken from any single distribution  Gaussian  Example: Uniform [0,1] 70 ~N( 1.0, 0.41 ) ~N( 0.5, 0.29 ) ~N( 1.5, 0.50 ) ~N( 2.0, 0.58 ) Sum of 1 var Sum of 2 varsSum of 3 vars Sum of 4 vars

71 University of Colorado Boulder  Addition of multiple variables taken from any single distribution  Gaussian  Example: Uniform {0,1,2} (Quiz Question #4) 71 ~N( 2.0, 1.16 ) ~N( 1.0, 0.82 ) ~N( 3.0, 1.42 ) ~N( 10, 2.58 ) Sum of 1 var Sum of 2 varsSum of 3 vars Sum of 10 vars

72 University of Colorado Boulder  Addition of multiple variables taken from any single distribution  Gaussian  Example: Skewed distribution 72 ~N( 0.5, 0.40 ) ~N( 0.25, 0.28 ) ~N( 0.75, 0.49 ) ~N( 1.0, 0.57 ) Sum of 1 var Sum of 2 varsSum of 3 vars Sum of 4 vars

73 University of Colorado Boulder

74 University of Colorado Boulder  Questions on Statistics?  I’ll go through example problems at the beginning of Thursday’s lecture  Homework 2 due Thursday  Homework 3 out today  Next quiz active tomorrow at 1pm. 74

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