a hierarchical hmm framework for home appliance modelling 200301_a_hhmm... · menu 1 abstract 2...

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A hierarchical HMM framework for home appliance modelling

Weicong Kong

March 1, 2020

Weicong Kong A hierarchical HMM framework for home appliance modelling March 1, 2020 1 / 38

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1 Abstract

2 Introduction

3 Model Representations

4 MODEL FITTING

5 Modelling real world appliance with hhmm

6 Load disaggregation with HHMM

7 Conclusion


Abstract

Abstract

whyProper individual home appliance modeling is critical to the performance of NILM.

This model aims to provide better representation for those appliances that havemultiple built-in modes.

whatThe dynamic Bayesian network representation of such an appliance model is built.

A forward-backward algorithm, which is based on the framework of expectationmaximization, is formalized for the HHMM fitting process.

howTests on publically available data show that the HHMM and proposed algorithmcan effectively handle the modeling of appliances with multiple functional modes,as well as better representing a general type of appliances.

A disaggregation test also demonstrates that the fitted HHMM can be easilyapplied to a general inference solver(particle filter) to outperform conventionalhidden Markov model in the estimation of energy disaggregation.


Introduction

NILM problems during steps

Figure 1: NILM problems during steps

Problem statement yt =∑k

fk(xkt ) + et


Introduction

Introduction

HMM with low frequency data(first 3 paragraphs)The advent of smart meter infrastructure has provided plentiful information ofstudying load characteristics, but research in mining such big data to provideactionable insights is still at the early stage.

Load disaggregation is one of the key directions...

low frequency sampling data with HMM model is attracting attention.

HMM not ok with two/multiple mode 4th paragraphbut ... have difficulty to distinguish the standby state and the off state.

for multiple mode appliances,...not be captured, depending on user’s behaviour

re-usability of such model is compromised in the load disaggregation problem

HMM variant with hierarchical structurein the area of speech, Cocktail Party Problem

hierarchical HMM is adopted

two Markov chain, upper chain(the mode of appliance), lower chain(power profileof the specific mode)


Introduction

previous HHMM

Figure 2: prevois HHMM,model as DBN

Wong, Yung Fei, Thomas Drummond, and Y. A. Sekercioglu. ”Real-time loaddisaggregation algorithm using particle-based distribution truncation with stateoccupancy model.” Electronics Letters 50.9 (2014): 697-699.


Introduction

previous Sequential Monte Carlo/particle filter for localization simulation

Figure 3: previous Monte Carlo/particle filter for localization simulation


Introduction

Monte Carlo Localization flow

Figure 4: Monte Carlo Localization flow


Introduction

Why The particle filter solver for NILM with DBN model

Figure 5: NILM using particle filter

non-linear, non-Gaussian Ditribution


Model Representations

Hidden Markov Model

Figure 6: Graphical illustration of a HMM.

Hmm provides a good framework to describe the dynamic through discretised timeseries of individual appliances.

πi = P (X1 = i) (1)

Ai,j = P(Xt = j

∣∣Xt−1 = i)

(2)

P(yt∣∣Xt = i

)∼ N (µi, σi) (3)



Hierarchical Hidden Markov Model(HHMM)

an extension of HMM to hierarchical hidden markov model

dynamic bayesian network

Figure 7: The state transition diagram for a two-level HHMM, modelling a 3-state appliance.



HHMM

Figure 8: Graphical illustration of a two-level HHMM.

P(XLt = j

∣∣∣XLt−1 = i , Ft−1 = f,XU

t = κ)

=

{ALκ (i, j) if f = 0

πLκ ( j) if f = 1(4)

1 XUt , X

Lt−1: state variables of upper/lower level at time t

2 Ft−1: binary auxiliary variable, 1: lower level HMM finished,0: not finished3 ALκ (i, j) is rescaled version of ALκ (i, j)4 πLκ ( j) is the distribution for initial state for the low level



HHMM eq5-9

1 The relationship between the subHMM transition matrix and rescaled matrix:

ALκ (i, j)(1−ALκ (i, end)

)= ALκ (i, j) (5)

ALκ (i, end) is the probability of a state being terminiated from state iK∑j=1

ALκ (i, j) = 1

P(Ft = 1

∣∣XLt = i,XU

t = κ)= ALκ (i, end) (6)

P(Ft = 0

∣∣XLt = i,XU

t = κ)= 1−ALκ (i, end) (7)

P(XUt = κ

∣∣XUt−1 = η, Ft−1 = f

)=

{δ (η, κ) if f = 0

AU (η, κ) if f = 1(8)

P(XU

1 = κ)= πU (κ) (9)


MODEL FITTING

Model fitting

θ denote the of paramters, θ is the estimated value, y = y1:T is the wholeobserved sequence

θ = argmaxθP (y|θ) (10)

the famous algorithm is expectation maximization (EM) algorithm

EM for HMM

EM for HHMM

HMM fitting by adopting the dynamic bayesian network representation

for simplicity, P (y|θ) = P (y1:T )


MODEL FITTING

Expectation Maximization for HMM

the first version of forward variable

αt( j) = P (Xt = j, y1:t) (11)

The normalized version adopted for forward variable in this paper

αt( j) ,P (Xt = j, y1:t)

P (y1:t)= P

(Xt = j

∣∣y1:t)∝

(K∑i=1

αt−1(i)Ai,j

)P(yt∣∣Xt = j

)(12)

backward variable

βt(i) , P(yt+1:T

∣∣Xt = i)=

K∑j=1

Ai,jP(yt+1

∣∣Xt+1 = j)βt+1( j)

(13)

the base case for both forward and backward variables

α1( j) = P(y1∣∣X1 = j

)πj (14)

βT (i) = 1 (15)


MODEL FITTING

Expectation Maximization

The probability of each state at t given observation sequence

γt(i) , P(Xt = i

∣∣y1:T )=

1

P (y1:T )P(yt+1:T

∣∣Xt = i)P(Xt = i

∣∣y1:t)∝ αt(i)βt(i) (16)

ξt(i, j) , P(Xt−1 = i,Xt = j

∣∣y1:T )∝ P

(Xt = j

∣∣Xt−1 = i)× P

(Xt−1 = i

∣∣y1:t−1

)× P

(yt+1:T

∣∣Xt = j)× P

(yt∣∣Xt = j

)= Ai,jαt−1(i)βt( j)P

(yt∣∣Xt = j

)(17)

γt(i) and ξt(i, j) is the estimation step of EM


MODEL FITTING

Expectation Maximization

The maximization step of EM

πi = γ1(i) (18)

Ai,j =1

Z

T∑t=2

ξt (i, j) (19)

yt(i) = γt(i)× yt. (20)

parameters by fitting a Gaussian distribution using a maximum likelihood estimate

Z is a normalizing constant to ensure the transition matrix is a stochastic matrix


MODEL FITTING

Number Underflow

as the length of sequence grows, the joint probability keeps decreasing.

the value exceed the maximum length which the computer can represent, resultingthe value to be considered to 0

to deal with this issue, using log

normalization is also performed in log space to mitigate the effect,∑j

exp(αt(j)) = 1


MODEL FITTING

The log likelihood of model parameters

the forward variables be rewritten into

αt( j) = P(Xt = j

∣∣y1:t)=

1

P (yt| y1 : t−1)P(Xt = j, yt

∣∣y1 : t−1

)(21)

letting ct = P (yt|y1 : t−1),then

ct =∑j

P(Xt = j, yt

∣∣1 : t−1

)(22)

P(Xt = j, yt

∣∣y1:t−1

)=

(K∑i=1

αt−1(i)Ai,j

)P(yt∣∣Xt = j

)likelihood of observation can be calculated

P (y1:T ) = P (y1)P(y2∣∣y1) · · ·P (yT ∣∣y1:T−1

)=

T∏t=1

ct (23)


MODEL FITTING

Expectation Maximization for HHMM

the set of parameters for a HHMM

Θ ={AU , πU ,

{ALκ

},{πLκ

},{ALκ (:, end)

},O}

(24)

every level has its unique state labels

upper level ON,OFF

lower level 1,2,3

the vertical transition matrix is sparse because the lower level is 1 where the upperlevel is off

i.e.P (XLt = 1|XU

t = OFF ) = 1

P (XLt = 2|XU

t = OFF ) = 0, P (XLt = 3|XU

t = OFF ) = 0


MODEL FITTING

The forward variables of EM in HHMM

According to this stable labelling setting

αt ( j, κ, f) , P(XLt = j,XU

t = κ, Ft−1 = f∣∣y1:t)

∝ P(yt∣∣XL

t = j)∑

i

∑η

h (i, η)∑g

αt−1 (i, η, g)

(25)

κ is the state of the upper level

h is an auxiliary function

h (i, η) =

{(1−ALη (i, end)

)δ (η, κ) ALκ (i, j) if f = 0

ALη (i, end)AU (η, κ)πLκ ( j) if f = 1(26)

the initial forward variable

α1 ( j, κ, f) =

{0 if f = 0

πU (κ)πLκ ( j)P(y1∣∣XL

1 = j)

if f = 1(27)


MODEL FITTING

The backward variables

backward variable

βt (i, η, f) , P(yt+1:T

∣∣XLt = i,XU

t = η, Ft−1 = f)

=∑j

P(yt+1

∣∣XLt+1 = j

)∑κ

((1−ALη (i, end)

)× δ (η, κ)× ALκ (i, j)× βt+1 (j, κ, 0) +ALη (i, end)

×AU (η, κ)× πLκ ( j)× βt+1 (j, κ, 1))

(28)

the initial backward variableβT (i, η, f) = 1 (29)


MODEL FITTING

Sufficient Statistics 1

horizontal transition probability

ξht (i, j, κ) = P(XLt−1 = i,X

L

t = j,XUt = κ, Ft−1 = 0

∣∣y1:T)∝∑η

∑g

αt−1 (i, η, g)×(1−ALη (i, end)

)× δ (η, κ)× ALκ (i, j)× P

(yt∣∣XL

t = j)

× βt (j, κ, 0) (30)

vertical transition probability

ξvt (j, κ) = P(XLt = j,XU

t = κ, Ft−1 = 1∣∣y1:T)

∝ αt ( j, κ, 1)βt (j, κ, 1) (31)

smoothed state probability

γt (j, κ, f) = P(XLt = j,XU

t = κ, Ft−1 = f∣∣y1:T)

∝ αt (j, κ, f)βt (j, κ, f) (32)


MODEL FITTING


end transition probability

ξet (i, η, 1) = P(XLt = i,XU

t = η, Ft = 1∣∣y1:T)

∝∑j

∑κ

∑f

αt (i, η, f)×ALη (i, end)

×AU (η, κ)× πLκ ( j)

× P(yt∣∣XL

t = j)× βt+1 (j, κ, 1) (33)

ξet (i, η, 1) denotes the probability that subHMM endsξet (i, η, 0) denotes the probability that subHMM not ends

ξet (i, η, 0) = P(XLt = i,XU

t = η, Ft = 0∣∣y1:T)

∝∑j

∑κ

∑f

αt (i, η, f)

×(1−ALη (i, end)

)× δ (η, κ)

× ALκ (i, j)× P(yt+1

∣∣XLt+1 = j

)× βt+1 (j, κ, 0) (34)


MODEL FITTING


horizontal transition probability for the upper level

χt (η, κ) = P(XUt−1 = η,X

U

t = κ, Ft−1 = 1∣∣y1:T)

∝∑i

∑j

∑g

αt−1 (i, η, g)×ALη (i, end)

×AU (η, κ)× πLκ ( j)

× P(yt∣∣XL

t = j)× βt (j, κ, 1) (35)


MODEL FITTING

The maximization step

the contribution of an observation to a concrete state i

yt(i) =

(∑κ

γt (i, κ)

)× yt (36)

the lower level transition parameters

AL

κ (i, j) =1

Zh

T∑t=2

ξht (i, j, κ) (37)

Zh is the normalizing constant to ensure the is a stochastic matrixthe end transition parameters

ALκ (i, end) =

1

Ze

T−1∑t=1

ξet (i, κ, 1) (38)

Ze is the normalizing constant matrix for the end transitions

Ze (i, η) =

T−1∑t=1

∑g

ξet (i, κ, g) (39)


MODEL FITTING

The maximization step

the initial state distribution parameters at the lower level

πLκ ( j) =1

Zv

T∑t=1

ξvt (j, κ) (40)

the transition parameter at the upper level

AU (η, κ) =1

ZU

T∑t=2

χt (η, κ) (41)


MODEL FITTING

The proposed EM algorithm for HHMM

Figure 9: The proposed EM algorithm for HHMM


Modelling real world appliance with hhmm

fitting single mode appliances and its benefit

modelling real world appliances with our proposed model and fitting algorithm

in NILM area, most neither state the length of the power profiles nor used a ratherlengthy sequence for fitting model

lengthy sequence not represent good generality

so..only use one typical duty cycle which eliminating the possibility of including theextra information

test the proposed model using publically available data introduced by..

typical duty cycle power sequence

i.e. typical one of washing machine: normal wash, permanent press and delicatewash. The power consumption pattern differ from each other.

down-sampled to 1 reading per minute


Modelling real world appliance with hhmm

fitting multiple mode appliances and its benefit

the proposed algorithm is applied ...yield three HHMMs

three HHMMs assembled to one HHMMassembling rule

1 without further information2 upon finishing any of the modes, should go back to the OFF mode


Load disaggregation with HHMM

Synthesis test case design

the proposed HHMM can be learned via EM

then apply the learned models in the NILM

the dataset include typical cycles of 7 major appliance typesthe difficulty of NILM are how mixed at same period

I randomly mix a random set of 7 appliance into a short 3 hour peroidI generate sufficiently complicated test casesI in order to present the varying complexity and diversity of reallife situations, 30 test

cases



The particle filter solver

the particle filter solver adopted to NILM with HMM [12][24][25]

the pros of the PF solver: linear scalability with number of appliances

the con of the PF solver: slow

have not PF solver with HHMM, but in this paper

PF for HHMM should have an upper state sequence sampled in real time. Thelower state transition governed by the upper.

the upper state for appliance n at time t of particle m:

xU(n)t [m] =

xU(n)t−1 [m] if f = 0

∼ AU(n)

xU(n)t−1 [m],∗

if f = 1(42)

∼, the distribution of random variable, m is the particle index, A is the statetransition matrix

xL(n)t [m] ∼

AL(n)

xU(n)t−1 [m]

(xL(n)t [m], ∗

)if f = 0

πL(n)

κxU(n)t−1 [m]

if f = 1(43)



The particle filter solver

assuming a load disaggregation problem with N appliances, T time stepseach appliance with KU upper state and KL lower state

Figure 10: NILM using particle filter



The evaluation metric

there are numberous evaluation metrics for load disaggregation

not only the on/off events of each appliance, but also the amount of powerconsumption

The metric is calculated by

Acc = 1−

∑Tt=1

∑Nn=1

∣∣∣y(n)t − y(n)t

∣∣∣2∑Tt=1 yt

(44)

y(n)t denotes the estimate of power consumption for the ith appliance at time t

y(n)t denotes the actual power consumption for the ith appliance

yt denotes the observed power consumption sequence



results and comparison

30 cases 4 different scenarios using particle filter solver

the number of particles is fixed to 5000

from table v, HHMM outperformance the conventional HMM

from fig.11, HHMM more stabe than HMM

fig.12-16, more benefit of HHMM



real world test case REDD dataset

further justify proposed model, apply PF with HHMM to the REDD dataset1 not provide the detailed information about appliance modes2 real world dataset confirm the potential of HHMM3 although the performance decreases as the number of appliances increase4 one of the causes is that most appliances in real world pratice have muliple modes

but modelled as single-mode


Conclusion

Conclustion

this paperI inpired by the speech recognitionI more concise EM for HHMM fittingI comprehensive test verified better result in NILM

theoretical progressI the efficiency of sequential Monte Carlo (SMC) for performing inference in structured

probabilistic modelsI and the flexibility of deep neural networks to model complex conditional probability

distributionsI i.e.AUTO-ENCODING SEQUENTIAL MONTE CARLO (ICLR 2018)

Scenario, inference in embedded system, time consuming before the deadline, inorder real time control

I need the explainable inference modelI training or fitting model parameter in the cloud

future workI dataset format:I model:HHMM extend to combine with other sophisticated i.e. HSMMs, HMMsI fitting:more efficient solver with HHMMI solver:I evaluation:...plenty of spaceI loc-aware disaggregation


Conclusion

Thank You!


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