Sebastian Wolf, Magnus Bitsch, Henrik...
Transcript of Sebastian Wolf, Magnus Bitsch, Henrik...
Hidden Markov Occupancy Modelling
Sebastian Wolf, Magnus Bitsch, Henrik Madsen
Dynamical Systems, DTU Compute
BackgroundModelling people’s presence in buildings
• improve HVAC control usinginformation about occupants’presence
• input for building simulations
• basis for modelling of other behaviour(windows, heating,...)
• presence often not directlymeasurable ⇒ indirect methods
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Occupancy ModellingHidden Markov Model - homogeneous
Yt Yt+1 Yt+2 . . .
Xt Xt+1 Xt+2 . . .
The model can be expressed by
p (Xt = i | Xt−1 = j) ∼ (Γ)i,j
Yt = ci + εi,t
where Γ ∈ Rm×m is a transition probability matrix, ci are the state meansand εi,t ∼ N(0, σ2
i ).
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Occupancy ModellingHidden Markov Model - inhomogeneous
Yt Yt+1 Yt+2 . . .
Xt Xt+1 Xt+2 . . .
The model can be expressed by
p (Xt = i | Xt−1 = j) ∼ (Γt)i,j
Yt = ci + εi,t
where Γt ∈ Rm×m is a inhomogeneous transition probability matrix, ci arethe state means and εi,t ∼ N(0, σ2
i ).
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Occupancy ModellingTransition Probabilities
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Occupancy ModellingMarkov-Switching AR(1) Model
Yt Yt+1 Yt+2 . . .
Xt Xt+1 Xt+2 . . .
The model can be expressed by
p (Xt = i | Xt−1 = j) ∼ (Γt)i,j
Yt = φiYt−1 + ci + εi,t
where Γt ∈ Rm×m is a transition probability matrix, ci are the state means,φi the auto-regressive parameters and εi,t ∼ N(0, σ2
i ).
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Occupancy ModellingComparison of Markov-switching Models
m=2 states parameters AIC BIC
homogeneous HMM 6 = (m2 +m) 7792 7832inhomogeneous HMM 12 = (m2 + 4m) 7627 7721Markov-Switching 14 = (m2 + 5m) -19187 -19080
states parameters AIC BIC2 16 -19187 -190803 27 -20326 -201464 40 -20771 -205035 55 -21392 -21023
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Occupancy ModellingGlobal decoding
(a) School data (b) Summer house data
Figure: global decoding of the 5 state Markov switching model. States representedby colours and by step function.
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Occupancy ModellingResidual analysis
(a) School data (b) Summer house data
Figure: Residual analysis of the 5 state Markov switching model
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Occupancy ModellingSimulation
(a) School data (b) Summer house data
Figure: 100 simulations with measured values (red).
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Occupancy ModellingOutlook
• ground truth validation
• further input variables- temperature- noise- humidity
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Thank [email protected]
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