sebastian wolf, magnus bitsch, henrik...
TRANSCRIPT
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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