The document discusses the Kalman filter, an algorithm used to estimate unknown variables using measurements observed over time that contain noise. It provides three key points:
1) The Kalman filter is an optimal estimator that recursively infers parameters from indirect, noisy measurements by fusing predictions with new measurements.
2) It is conceptualized using an example of estimating a boat's position over time based on noisy sextant and GPS measurements.
3) The filter works by predicting the next state, taking a new measurement, and updating the estimate by weighing the prediction and measurement based on their uncertainties.
Cornell University’s Hod Lipson is seeking to understand if machines can learn analytical laws automatically. For centuries, scientists have attempted to identify and document analytical laws underlying physical phenomena in nature. Despite the prevalence of computing power, the process of finding natural laws and their corresponding equations has resisted automation. Lipson has developed machines that take in information about their environment and discover natural laws all on their own, even learning to walk.
Cornell University’s Hod Lipson is seeking to understand if machines can learn analytical laws automatically. For centuries, scientists have attempted to identify and document analytical laws underlying physical phenomena in nature. Despite the prevalence of computing power, the process of finding natural laws and their corresponding equations has resisted automation. Lipson has developed machines that take in information about their environment and discover natural laws all on their own, even learning to walk.
Lesson 12: Linear Approximation and Differentials (Section 21 handout)Matthew Leingang
The line tangent to a curve is also the line which best "fits" the curve near that point. So derivatives can be used for approximating complicated functions with simple linear ones. Differentials are another set of notation for the same problem.
Apresentação do professor Pedro Grande, da seção UFRGS do Instituto Nacional de Engenharia de Superfície. Palestra convidada do Simpósio Engenharia de Superfície do X Encontro da SBPMAT. Realizada no dia 26 de setembro de 2011 em Gramado (RS).
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALLKees Nieuwstad
The motion induced by gravity of solid spheres in a vessel filled with fluid has been investigated experimentally at Reynolds numbers in the range from 1-74 and Stokes numbers ranging from 0.2-17. Trajectories of the spheres have been measured with a focus on start-up behavior, and on impact with a horizontal wall. Two models have been investigated. The first describes the accelerating motion of the sphere. The second model predicts the distance from the wall at which the sphere starts decelerating.
The flow in the vicinity of the sphere was measured by means of PIV. The time scales and flow structures strongly depend on the Reynolds number. Measurements performed are in good agreement with simulations performed at the Kramers Laboratorium.
Lesson 12: Linear Approximation (Section 41 handout)Matthew Leingang
The line tangent to a curve, which is also the line which best "fits" the curve near that point. So derivatives can be used for approximating complicated functions with simple linear ones. Differentials are another set of notation for the same problem.
AACIMP 2010 Summer School lecture by Dmitry Bibichkov. "Physics, Chemistry and Living Systems" stream. "Models of Synaptic Transmission" course. Part 2.
More info at http://summerschool.ssa.org.ua
Lesson 12: Linear Approximation and Differentials (Section 21 handout)Matthew Leingang
The line tangent to a curve is also the line which best "fits" the curve near that point. So derivatives can be used for approximating complicated functions with simple linear ones. Differentials are another set of notation for the same problem.
Apresentação do professor Pedro Grande, da seção UFRGS do Instituto Nacional de Engenharia de Superfície. Palestra convidada do Simpósio Engenharia de Superfície do X Encontro da SBPMAT. Realizada no dia 26 de setembro de 2011 em Gramado (RS).
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALLKees Nieuwstad
The motion induced by gravity of solid spheres in a vessel filled with fluid has been investigated experimentally at Reynolds numbers in the range from 1-74 and Stokes numbers ranging from 0.2-17. Trajectories of the spheres have been measured with a focus on start-up behavior, and on impact with a horizontal wall. Two models have been investigated. The first describes the accelerating motion of the sphere. The second model predicts the distance from the wall at which the sphere starts decelerating.
The flow in the vicinity of the sphere was measured by means of PIV. The time scales and flow structures strongly depend on the Reynolds number. Measurements performed are in good agreement with simulations performed at the Kramers Laboratorium.
Lesson 12: Linear Approximation (Section 41 handout)Matthew Leingang
The line tangent to a curve, which is also the line which best "fits" the curve near that point. So derivatives can be used for approximating complicated functions with simple linear ones. Differentials are another set of notation for the same problem.
AACIMP 2010 Summer School lecture by Dmitry Bibichkov. "Physics, Chemistry and Living Systems" stream. "Models of Synaptic Transmission" course. Part 2.
More info at http://summerschool.ssa.org.ua
Introduction to Statistical Clustering
Specifically K-Means and Gaussian Mixture Models (GMM). We also look at how Expectation Maximization (EM) can be used to fit GMMs.
A short introduction to time frequency analysis / wavelets with only pictures! This was a lunch and learn talk, so there was a great deal of spoken word that accompanied this. But you get the picture.
1. An ISO 9001:2008 Registered Company
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Kalman Filter
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2. Kalman Filter Facts
Dr. Rudolf Kalman is alive and well today (82 years old)
Important and used everywhere: GPS (predict update
location), surface to air missiles (hit target), machine
vision (track targets), brain computer interface
Not really a filter, it is an optimal estimator (infers
parameters of interest from indirect, noise
measurements)
It is recursive – so when a new measurement arrives it is
processed and you get a new estimate
Performs Data Fusion usually between measured and
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estimated states
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document.
3. Conceptual Overview – Example Definition
y
Lost on the 1-dimensional line, boat is not moving
Imagine that you are guessing your position by looking at
the stars using sextant
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Position function of time: y(t)
Assume Gaussian distributed measurements (errors)
3
4. Conceptual Overview - Prediction
0.16
Sextant Measurement 0.14
at t1: Mean = z1 and 0.12
Variance = z1 0.1
probability
ŷ(t1) = z1
Optimal estimate of 0.08 Predicted Position
position is: ŷ(t1) = z1 0.06
Variance of error 0.04
[y(t1) - ŷ(t1)] estimate: 0.02
2 (t ) = 2 0
0 10 20 30 40 50 60 70 80 90 100
e 1 z1 z
Boat in same position
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at time t2 - Predicted What if we also had a
position is z1 GPS unit?
4
5. Conceptual Overview - Measurement
0.16
0.14 prediction ŷ-(t2)
State (by looking
0.12 at the stars at t2)
0.1
0.08 Measurement
using GPS z(t2)
0.06
0.04
0.02
0
0 10 20 30 40 50 60 70 80 90 100
• So we have the prediction ŷ-(t2)
• GPS Measurement at t2: Mean = z2 and Variance = z2
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• Need to correct the prediction by Sextant due to
measurement to get ŷ(t2)
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6. Conceptual Overview – Data Fusion
0.16 Kalman filter: fuse
corrected optimal
0.14
estimate ŷ(t2) measurement and
0.12 prediction ŷ-(t2) prediction based on
0.1
confidence
0.08 measurement
z(t2) Corrected mean is
0.06
the new optimal
0.04
0.02
estimate of position
0
0 10 20 30 40 50 60 70 80 90 100
New variance is
smaller than either
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What if the boat is of the previous two
moving? variances
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7. Conceptual Overview – Prediction Model
0.16
ŷ(t2)
At time t3, boat
0.14
moves with velocity
0.12
Naïve Prediction
dy/dt=u
0.1 (sextant) ŷ-(t3)
Naïve approach:
0.08
Shift probability to
0.06
0.04
the right to predict
0.02
This would work if
0
0 10 20 30 40 50 60 70 80 90 100
we knew the velocity
exactly (perfect
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Try and predict where model)
it winds up.
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8. Conceptual Overview – Prediction Model
0.16
ŷ(t2)
But you may not be
0.14
so sure about the
0.12
Naïve Prediction
exact velocity
0.1 (sextant) ŷ-(t3)
Better to assume
0.08
Prediction ŷ-(t3)
imperfect model by
0.06
0.04
adding Gaussian
0.02
noise
0
0 10 20 30 40 50 60 70 80 90 100
dy/dt = u + w
Distribution for
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Assumptions: prediction is prediction moves
linear, noise is Gaussian and spreads out
8
9. Conceptual Overview – Update
0.16
Corrected optimal estimate ŷ(t3)
• Now we take a
0.14 Updated Sextant position using
GPS
measurement (GPS)
0.12
at t3
0.1 Measurement z(t3) GPS
• Need to once again
0.08
correct the
0.06
Prediction ŷ-(t3) Sextant
0.04
prediction (fusion)
0.02
• Recursive – rinse
0
0 10 20 30 40 50 60 70 80 90 100
and repeat as time
goes on
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Update, recursively
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10. Conceptual Overview
Optimal estimator only if:
Prediction model is linear (function of measurements)
All error (noise) is Gaussian: model error, measurement
error
Why is Kalman Filter so popular
Good results in practice due to optimality and structure.
Convenient form for online real time processing.
Easy to formulate and implement given a basic
understanding.
Measurement equations need not be inverted.
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11. State Space Equations
Estimated Estimated Control
State State Input
(now) (before)
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Observed
Measurement How do you find A,B,H?
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AWGN?
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12. Input
Output
Update Equations Place holder
Description Equation
State Prediction
Where do we end up
Covariance Prediction
When we get there, how much error
Innovation
Compare Reality to Prediction
Innovation Covariance
Compare real error to predicted error
Kalman Gain
What do you trust more?
State Update
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New estimate of where we are
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Covariance Update
New estimate of error
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13. Algorithm
Correction (Measurement Update)
Prediction (Time Update)
(1) Compute the Kalman Gain
(1) Project the state ahead
(2) Update estimate with measurement zk
(2) Project the error covariance ahead
(3) Update Error Covariance
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14. Measuring Constant Voltage (Classic Example 1)
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15. Predicting Trajectory of Projectile (Angry Bird)
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16. Equations
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document.
17. Simulation Results
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18. Modified TWS example
State : y
{ x , y , x , y}
Cov : Q E [ ww *]
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19. Derivation
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20. What if Assumptions don’t hold
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