Prediction and Indexing of Moving Objects with Unknown Motion Patterns

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Prediction and Indexing of Moving Objects with
Unknown Motion Patterns

• Y. Tao, C. Faloutsos, D. Papadias, B. Liu
• City University of Hong Kong
• Carnegie Mellon University
• HK University of Science and Technology

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Problem

• Predictive range query on moving objects
• Which flights are expected to enter the airspace
  of California in the next 10 minutes?

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Existing method

• Assume objects move linearly with the same speed
• o(t) o(0) v? t
• The system stores o(0), v
• d-dimensional vectors

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Why linear model?

• Common justifications
• Simple
• Arbitrarily complex movements can be approximated
  using piece-wise linear movements.
• NOT TRUE in prediction

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Failure of the linear model
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Goal

• How to model objects movements for prediction?
• Capture as many types of movements as possible.

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Movement types

• . . .

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Pitfall

• Quadratic?
• High-degree polynomial?
• Circular?
• Parabola?
• We should not assume any fixed type of movements.
• Otherwise, we miss the other types.

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Idea

• Let objects reveal their next steps
• Objects location at recent timestamps
• gt
• Its location at the next timestamp

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Simple example Linear movement

• o(t) o(0) v? t
• o(t) o(t?1) o(t?1) ? o(t?2)
• 2o(t?1) ? o(t?2)
• o(t?1), o(t?2) gt o(t)
• The new formula captures linear movements with
  any speed.

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More complex example Accelerative

• o(t) o(0) v? t ½ a? t2
• o(t)3o(t?1)?3o(t?2)o(t?3)
• o(t?1), o(t?2), o(t?3) gt o(t).
• The new formula captures any speed and
  acceleration.

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Even more complex movement Circular

• o(t?1), o(t?2) gt o(t).
• The new formula captures any radius.

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Recursive Linear Function

• C1, C2, , Cf are constant d?d matrices.
• Many non-linear movements actually have a
  recursive, linear form.
• f retrospect
• How many past timestamps should be considered
  (f5 captures all movements tested)

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Concise, and equivalent representation

• Where (d?f
  )1 vector
• so(t) the state of o at time t
• Ko motion matrix of o

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An example with f2, d2
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Property 1

• For each type of movements, Ko is unique.

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Property 2

• Predict the location at t timestamps later

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Deriving Ko from past location

• Given o(t), o(t-1), o(t-2), , o(t-h1)
• Decide all the coefficients kij

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Idea (assuming f2)

• o(t), o(t-1), o(t-2), , o(t-h1) define h-1
  states
• so(t)o(t), o(t-1)
• so(t-1)o(t-1), o(t-2)
• so(t-h2)o(t-h2), o(t-h1)

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Idea (cont.)

• The derived Ko should satisfy the following h-2
  equations as accurately as possible
• so(t) Ko so(t-1)
• so(t-2) Ko so(t-3)
• so(t-h2) Ko so(t-h1)
• gt Minimum squared error

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Indexing non-linear objects

• Can we still use the methods proposed before for
  linear movements?
• Yes!
• Assuming a farthest future timestamp that can
  be queried

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Conquer non-linearity with linearity

• The farthest query-able timestamp 4

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Power of RMF 1

• Each axis has range 0, 10000

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Power of RMF 2

• H10

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Power of RMF 3
f3 prediction
f2 prediction
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Power of RMF 4
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Conclusions

• A more powerful mathematical tool for modeling
  non-linear movements.
• Future work
• Can the server index the recursive motion
  function directly in order to answer queries
  faster?
• Other query types like NN search, joins?