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ASEN 5070Statistical Orbit determination I
Fall 2012
Professor George H. BornProfessor Jeffrey S. Parker
Lecture 5: Stat OD
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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.
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~N( 0.0, 1.41 )
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~N( 1.0, 0.41 )
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Some popular questions and answers
Energy with Drag
Homework 2
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Some popular questions and answers
Computation of Time of Perigee
Homework 2
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Homework 2: Tp Calculation
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Homework 2: Tp Calculation
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Homework 2: Tp Calculation
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Chapter 4, Problems 1-6
◦ Solving ODEs◦ Linear Algebra◦ Studying the state transition matrix
Homework 3
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Review of Differential Equations◦ Laplace Transforms
Review of Statistics
Today’s Lecture
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Stat OD dynamics:
Solve for given A and
Review of Diff EQ
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Stat OD dynamics:
Solve for given A and
Review of Diff EQ
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Solve for w/
Review of Diff EQ
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Solve for w/
Review of Diff EQ
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Solve the ODE
We can solve this using “traditional” calculus:
Example
Check your answer by plugging it back in
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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.
Laplace Transforms
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Laplace Transform Tables
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Solve the ODE
We can solve this using “traditional” calculus:
Example
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Solve the ODE
Or, we can solve this using Laplace Transforms:
Example
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Solve the ODE
Applied to Stat OD
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Solve the ODE
Applied to Stat OD
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Solve the ODE
Applied to Stat OD
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Solve the ODE
Applied to Stat OD
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Solve the ODE
Applied to Stat OD
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Solve the ODE
Applied to Stat OD
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Solve the ODE
Applied to Stat OD
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Questions on Diff EQ?
Quick Break
Review of Statistics to follow
Questions?
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X is a random variable with a prescribed domain.
x is a realization of that variable.
Example:◦ 0 < X < 1
◦ x1 = 0.232◦ x2 = 0.854◦ x3 = 0.055◦ etc
Review of Statistics
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Axioms of Probability
2. p(S)=1, S is the certain event
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Venn Diagram
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Axioms of Probability
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• For the continuous random variable, axioms 1 and 2 become
Probability Density & Distribution Functions
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• For the continuous random variable, axioms 1 and 2 become
• The third axiom becomes
• Which for a < b < c
Probability Density & Distribution Functions
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Probability Density & Distribution Functions
Using axiom 2 as a guide, solve the following for k:
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Probability Density & Distribution Functions
Using axiom 2 as a guide, solve the following for k:
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Probability Density & Distribution Functions
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Example:
• From the definition of the density and distribution functions we have:
• From axioms 1 and 2, we find:
Probability Density & Distribution Functions
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Expected Values
Note that:
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Expected Values
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Expected Values
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Expected Values
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Expected Values
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The Gaussian or Normal Density Function
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The Gaussian or Normal Density Function
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Moment Generating Functions
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Moment Generating Functions
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Moment Generating Functions
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Two Random Variables
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Marginal Distributions
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Marginal Distributions
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Independence of Random Variables
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Conditional Probability
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Expected Values of Bivariate Functions
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Expected Values of Bivariate Functions
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Expected Values of Bivariate Functions
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Expected Values of Bivariate Functions
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Expected Values of Bivariate Functions
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The Variance-Covariance Matrix
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Properties of the Correlation Coefficient
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Properties of Covariance and Correlation
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Properties of Covariance and Correlation
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Central Limit Theorem
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Addition of multiple variables taken from any single distribution Gaussian
Example: Uniform [0,1]
Central Limit Theorem
~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 vars Sum of 3 vars Sum of 4 vars
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Addition of multiple variables taken from any single distribution Gaussian
Example: Uniform {0,1,2} (Quiz Question #4)
Central Limit Theorem
~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 vars Sum of 3 vars Sum of 10 vars
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Addition of multiple variables taken from any single distribution Gaussian
Example: Skewed distribution
Central Limit Theorem
~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 vars Sum of 3 vars Sum of 4 vars
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Central limit theorem
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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.
Questions