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Exploiting HMM Sparsity to Perform Online
Real-Time Nonintrusive Load Monitoring (NILM)
Stephen Makonin, Senior Member, IEEE, Fred Popowich, Ivan V. Baji´
c, Senior Member, IEEE,
Bob Gill, Senior Member, IEEE and Lyn Bartram, Member, IEEE
Abstract—Understanding how appliances in a house consume
power is important when making intelligent and informed deci-
sions about conserving energy. Appliances can turn ON and OFF
either by the actions of occupants or by automatic sensing and
actuation (e.g., thermostat). It is, also, difficult to understand
how much a load consumes at any given operational state.
Occupants could buy sensors that would help, but this comes
at a high financial cost. Power utility companies around the
world are now replacing old electro-mechanical meters with
digital meters (smart meters) that have enhanced communication
capabilities. These smart meters are essentially free sensors that
offer an opportunity to use computation to infer what loads
are running and how much each load is consuming (i.e., load
disaggregation). We present a new load disaggregation algorithm
that uses a super-state hidden Markov model and a new Viterbi
algorithm variant which preserves dependencies between loads
and can disaggregate multi-state loads, all while performing
computationally efficient exact inference. Our sparse Viterbi
algorithm can efficiently compute sparse matrices with a large
number of super-states. Additionally, our disaggregator can run
in real-time on an inexpensive embedded processor using low
sampling rates.
Index Terms—load disaggregation, nonintrusive load moni-
toring, NILM, energy modeling, hidden Markov model, HMM,
sparsity, Viterbi algorithm, sustainability.
I. INTRODUCTION
LOAD DISAGGREGATION is a topic studied by re-
searchers who investigate algorithms that try to discern
what electrical loads (i.e., appliances) are running within a
physical area where power is supplied from the main power
meter. Such physical areas can include communities, industrial
sites, office towers/buildings, homes, and even within an
appliance. When focusing on homes, this area of research is
often referred to as nonintrusive (appliance) load monitoring
(NILM or NIALM). NILM research was first published by
Hart [1] in 1992. By knowing what loads are running, when
they are running, and how much power they are consuming,
we can begin to make informed choices that can lead to a
reduction in power consumption [2]. For load disaggregation to
be practical, it will need to run in real-time, use low-frequency
S. Makonin and F. Popowich are with the School of Computing Science,
Simon Fraser University, Burnaby, BC, Canada e-mail: smakonin@sfu.ca and
popowich@sfu.ca.
I. Baji´
c is with the School of Engineering Science, Simon Fraser University,
Burnaby, BC, Canada e-mail: ibajic@sfu.ca.
B. Gill with the School of Energy, British Columbia Institute of Technology,
Burnaby, BC, Canada e-mail: Bob Gill@bcit.ca.
L. Bartram with the School of Interactive Arts and Technology, Simon
Fraser University, Surrey, BC, Canada e-mail: lyn@sfu.ca.
Manuscript accepted October 7, 2015. Copyright c
2015 IEEE.
The original publication is available for download at ieeexplore.ieee.org.
sampling data, and have a high degree of accuracy. A good
disaggregator helps provide rich information about loads from
a given aggregate meter reading.
A. Our Contributions
Our main contribution is a disaggregation algorithm that:
(1) is agnostic of low-frequency sampling rates (1
3Hz, per
minute, and per hour) and measurement types (A, W, and
Wh); (2) is highly accurate at load state classification and load
consumption estimation; (3) can disaggregate appliances with
complex multi-state power signatures; (4) is the first hidden
Markov model (HMM) solution that preserves dependencies
between loads; and (5) can perform computationally efficient
exact inference, while other methods only use approximate
methods that are computationally more complicated.
The extent of matrix sparsity found in HMMs for house
loads is an integral issue that motivated our particular disag-
gregator. Others have examined matrix sparsity in the past [3].
However, their disaggregators implement complex solutions
to take advantage of sparsity that is not always efficient. We
present an efficient way to take advantage of sparsity in matrix
storage and processing and show how to do this by using
a super-state hidden Markov model, which was previously
dismissed because of state exponentiality. Our disaggregator
presents a new Viterbi algorithm variant, called sparse Viterbi
algorithm that can efficiently process very large sparse matri-
ces (over eight billion states).
Our disaggregator was first presented at the 2nd NILM
Workshop [4] and discussed in Makonin’s thesis [5]. Our
previous workshop presentation showed only preliminary test
results. This paper presents a more optimized algorithm than
what was discussed in Makonin’s thesis. We can now disaggre-
gate HMMs that have billions of states (>18 loads) as opposed
to smaller HMMs with of 1–2 million states (11 loads).
B. Super-State Definition
Our super-state HMM is in all respects a basic HMM.
However, we have chosen the prefix term super-state because
each HMM state is the Cartesian product of the different
possible states of each appliance/load we want to disaggregate.
In other words, the super-state can be viewed as the house’s
state. At each time interval, the super-state describes each
appliance – whether it is ON or OFF – and if ON, its
current operational state. There is a unique super-state for
each combination of appliance/load states. For example, if we
have two appliances each having two appliance-states there
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would be a total of four super-states. Super-state 0 would mean
both appliances are OFF. Super-state 1 would mean the first
appliance is ON and the second appliance is OFF. Super-state
2 is the opposite of super-state 1. Super-state 3 would mean
that both appliances are ON.
C. Paper Organization
The remainder of this article is organized as follows. We first
review the recent research in the area of load disaggregation
(Section II). We continue with an in-depth discussion of
our algorithm presenting the methodology (Section III) and
an analysis of complexity and efficiency (Section IV). This
is followed by details of our experimental setup and many
tests and their accuracy results (Section V) and discussion of
the results (Section VI). We conclude with a discussion on
significance and limitations (Section VII).
II. BACKGROUND
Many researchers have published vastly different ap-
proaches to load disaggregation. Artificial Neural Networks
(e.g., [6]), Support Vector Machines (e.g., [7]), and Nearest
Neighbour algorithms (e.g., [8]) have been popular methods
for disaggregation in the past. Artificial Neural Networks
(ANN) are easy to use. However, both the construction and
training of ANNs is arbitrary, and tuning can often result
in the convergence on local maxima and overfitting. Support
Vector Machines (SVM) use optimal line separation between
classifications and do not suffer from the same problems
of ANNs provided the dataset used is not large. Nearest
Neighbour (k-NN) is used to classify unlabeled data that is
nearest to each other (based on a distance function). However,
this method can be memory intensive having large storage
requirements.
A. Factorial HMM
Since 2011, methods that use HMMs have become a focal
point for most researchers. HMMs are a natural fit for disag-
gregation because they have the ability to model time series
data and represent the unobservable state of each load. To
curtail the state exponentiality problem of HMMs, researchers
have used factorial HMMs for disaggregation to lower com-
plexity. For instance, 8 two-state loads would have 256 states
(28) where a factorial HMM (FHMM) would have 8 chains.
Each load is a separate multi-state Markov chain that can
evolve in parallel to the others. Since each load is a separate
chain, load dependency information is lost. Additionally, there
is the added complexity of training these chains [9] with
approximation methods because exact inference is not possible
[10].
Kim et al. [11] used a combination of four different factorial
HMM variants to provide an unsupervised learning technique.
They achieved classification accuracies of between 69%–98%
(for 10 homes) using their M-fscore accuracy measure. Their
results seem to suggest the accuracy of the disaggregator
quickly decreases (from 98% to 69%) as more appliances
were added for disaggregation and required a high degree
of computational power to disaggregate. This work has many
issues, including their use of 4 FHMMs to disaggregate, a
configuration which would not be able to run online in real-
time without a large computational cost.
Kolter et al. [10] used a combination of two FHMM variants
(additive FHMM and difference FHMM) using high-frequency
sampling. Four of seven loads scored with moderate to high
accuracy, but loads such as electronics scored very low. This
approach would produce even lower accuracy results on low-
frequency data due to signal feature loss.
Kolter’s factorial models reduce the state exponentiality
problem of HMMs but at the expense of exact inference and
the loss of load dependencies. In other words, while trying
to avoid the complexity problems, these researchers have
introduced new inaccuracies. It is difficult to see these factorial
algorithms can run in real-time – which our disaggregator is
capable of doing.
Our solution preserves load dependence information and a
way to perform exact inference in a computationally efficient
manner, which is not possible when using factorial HMM or
VAST (discussed below). For example, consider the condition
when a heat pump turns ON. The HVAC fan (on a separate
breaker) increases its rotation speed and does the reverse when
the heat pump turns OFF (as we have observed in our dataset
is lost [12]).
B. Combining HMMs
Zia et al. [13] built their models using the sub-metering
data for each load. They determined load states and created
an HMM for each load by hand, then combined two loads to
make a larger HMM. This paper is arguably one of the first
papers to make use of HMMs for disaggregation. A two load
combinatorial search was performed until the total observed
load was processed. Combining and testing for only two loads
is not a realistic scenario. Accuracy results were not reported,
so it is hard to judge how successful their disaggregator was.
Unlike Zia et al., our disaggregator can determine load states
automatically during model building, can combine all loads
into one super-state HMM, and does not require the added
off-line pattern matching step.
C. Viterbi Algorithm with Sparse Transitions
Zeifman [3], [14] proposed Viterbi Algorithm with Sparse
Transitions (VAST), a modified version of the Viterbi algo-
rithm, on multiple transition matrices (each a triple of loads).
However, he limited the number of internal load states to only
two (ON or OFF). Zeifman reported the accuracy of VAST
to an average of 90.2% on 9 (simple ON/OFF) loads. Such
an approach would require the approximation of a many-
state load (e.g., the dishwasher) to be a simple ON/OFF
load; however, the majority of modern appliances are many-
state. Zeifman used triples of loads to avoid one large sparse
transitions matrix, but his transitions matrices could still have
zero-probability elements and sparsity in the emission matrix
was not dealt with.
Like Zeifman, we take advantage of matrix sparsity, but in
a different way. Unlike Zeifman, we can disaggregate loads
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with complex multi-state power signatures – not just ON/OFF
loads. Additionally, we have provided a more efficient and
simple method to do so.
D. Semi-Supervised & Unsupervised Learning
Parson et al. [15] used a variant of the difference FHMM
from Kolter et al. [10]. They proposed a disaggregator that
would train generic appliance models to specific appliance
models. This method shows promise for disaggregating a
small number of cyclical type appliances such as fridges and
freezers. It requires a lengthly training window making it more
suited for running off-line.
To have their solution run online, they have focused on
using a cloud computing based solution [15] to perform algo-
rithm execution, which could result in data privacy concerns
amongst consumers. While they have demonstrated that there
is promise in being able to take general appliance models
and actively tune them, this method is limited to cyclical
appliances – which has practical limits. Additionally, this
algorithm still needs a form of priors which is referred to
as a general appliance model. Loads that are more complex,
such as HVAC systems, operate quite differently based on
make/model and efficiency. Such complexity means that it
would not be possible to have a general appliance model for all
HVAC systems, unlike fridges and freezers that operate very
similarly regardless of the make/model and efficiency level.
Our solution, which needs priors, can disaggregate multi-state
appliances. We believe this is a more practical solution that
can be implemented in houses now to assist homeowners in
realizing energy savings.
Recently, Johnson et al. [16] considered using the factorial
variant of a hidden semi-Markov model (HSMM) because they
provided a means of representing state durations in a load
model. Although the authors claimed this was unsupervised
learning, they were incorrect [15] because they used labeled
data to build the appliance models from the same dataset used
to test. Rather than having their algorithm run on the entire
dataset, they hand-picked a number of specific segments for
testing and evaluation – this does not constitute a real-world
scenario. Our evaluations test on the entire dataset.
One of our main goals is to achieve high accuracy scores.
We believe disaggregators with active tuning (unsupervised
learning) may not work best for occupants. It is our opinion the
disaggregator would take too long to learn what loads are in a
house. Long learning times cause occupants to lose confidence
in the disaggregator’s ability to work properly because of
incorrect results. However, these problems provide direction
and motivation for future work.
III. METHODOLOGY
Our disaggregator (Figure 1) first analyzes the sub-metered
data from load priors and creates a probability mass function
for each. Load states are then determined by quantizing the
probability mass function (PMF) further. Each of the load’s
states is then combined to create a super-state HMM. The
super-state HMM is very sparse, and matrices are compressed
to take advantage of this. Once the super-state HMM is built,
there is no need for further sub-metering. The act of model
building creates a super-state HMM. We then take the last
times observation and the current observation and use our
sparse Viterbi algorithm to disaggregate the state of each load
and estimate load consumption. The compression technique we
use provides a computationally efficient way to perform exact
inference in a way that is both space and time optimized to
alleviate the state exponentiality problem HMMs have.
A. Nomenclature
A- the transition matrix (K×K)
B- the emission matrix (K×N)
K- the number of whole-house states (or super-states)
K(m)- the number of states for the m-th appliance
kt- the super-state at time t
k(m)
t- the m-th appliance state at time t
M- the number of loads/appliances
N- the number of possible observations
P0- initial prior probabilities vector t
Pt- the posterior probability vector at time t
PMF - probability mass function
Pr[·]- the probability of
pYm(·)- the PMF of the m-th appliance
S- the super-state
T- the length of time
X- the hidden load/appliance state
x(m)
t- the m-th appliance state at time t
ˆx(m)
t- the m-th appliance estimated state at time t
Y- discrete random variable for current draw
yt- the aggregate smart meter current draw at time t
ˆyt- the estimated aggregate current draw at time t
y(·)- current draw of the m-th appliance
y(m)
peak - the PMF current peaks for each state
B. Load Consumption Modelling
The Model Builder (second block in Figure 1) uses the
following steps to build a super-state HMM:
1) LOAD DATA: Prior to calling Model Builder, a dataset
is loaded into memory and split into priors data and
testing data. Only the priors are given to Model Builder
to build our model. The priors contain the aggregate
reading and all sub-meter readings of the loads targeted
for disaggregation.
2) CREATE PMF: Using the priors build a PFM for each
load. Discussed in Section III-C.
3) QUANTIZE: For each load, the Model Builder quan-
tizes each PMF to find the load’s states and the peak
value within that state. Discussed in Section III-D.
4) CREATE HMM: Using priors and load, state data
provided by Model Builder to build the super-state
HMM (Section III-D). Each prior observation’s sub-
meter data is used to find each load’s states that are used
to calculated the super-state. The previously observed
super-state and the currently observed super-state are
used to update the HMM data structures. Once all priors
are processed the HMM vectors and matrix rows are then
normalized to sum to 1.0.
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Model Builder
Smart Meter
Super-State
HMM
Sparse Viterbi
Algorithm
Sub-Meter
Data
Indicates process only required at initial startup/setup of disaggregator
Consumption
Estimator
Fig. 1. Block diagram of our disaggregator.
C. Probability Mass Functions
We represent an appliance as a discrete distribution by using
aprobability mass function (PMF). Let there be Mindepen-
dent discrete random variables Y1, Y2, . . . , YM, corresponding
to current draws from Mloads. Each Ymis the current or
power measurement of a metered electric load with a PMF
of pYm(n), where mis the load index i∈ {1,2, ..., M },y
is a number from a discrete set of possible measurements
y∈ {0,1, ..., Nm}, and Nmis the upper bound imposed by the
breaker that the m-th load is connected to. For example, with
current measurements (in dA) on a 15A breaker, we would
have Nm= 150. The PMF pYm(n)is defined as follows:
pYm(n) = (Pr[Ym=n],if n∈ {0,1, . . . , Nm},
0,otherwise,(1)
where Pr[Ym=n]is the probability that the current/power
draw of the m-th load is n. For example, using Table I, for
n= 3, Pr[Ym= 3] = 0.31, so the probability of the m-th load
drawing 3 dA (i.e., 0.3 A) is 0.31.
D. Super-State HMM
We model a house with Mloads as an HMM λ=
{P0,A,B}having a row-vector of initial prior probabilities
P0of length K, a K×Ktransition matrix A, and a K×N
emission matrix B, where Kis the number of whole-house
states (or super-states), and Nis the number of possible
observations. If t−1and trepresent the previous and the
present time instants, the entries of Aand Bare defined as
A[i, j] = p(St=j|St−1=i),B[j, n] = p(yt=n|St=j);
where Stis the super-state at time t, and ytthe observation
at time t.
The super-state is composed of the states of all Mappli-
ances: St= (X(1)
t, X(2)
t, ..., X(M)
t), where random variable
X(m)
tis the internal state of the m-th load at time t. For
example, a dishwasher may have 4 internal states that consist
of {OFF, WASH, RINSE, DRY}. The total number of super-
states is K=QM
m=1 K(m)where K(m)is the number of
internal states of the m-th appliance.
As the state of a load changes so too does its power or
current draw. In this work, we consider current draw as the
observation. Let y(·)be the current draw of the corresponding
internal appliance state, so that yx(m)
tis the current draw of
the m-th appliance in state x(m)
t. For notational convenience,
we assume that the current values are non-negative integers,
i.e., yx(m)
t∈ {0,1, ..., N }. In practice, these would not
necessarily be integers, but would still be constrained to a
discrete set of possible readings of the current meter. The
observed measurement at time tfrom the smart meter is the
sum of the current draws of individual appliances:
yt=
M
X
m=1
yx(m)
t. (2)
Model parameters, such as load state probabilities
p(X(m)
t=x(m)
t)as well as conditional probabilities in Aand
B, can be obtained from existing load disaggregation datasets
(e.g., AMPds [12], see Section III.A). In general, even though
the assumed full set of possible current draws is {0,1, ..., N },
most appliances have fewer than Nstates. To model this for
the m-th appliance, we quantize the set {0,1, ..., N }into K(m)
bins such that the first bin contains 0(and corresponds to the
OFF state). Other bins are centered around the peaks of the
empirical probability mass function of the current draw
pYm(n) = Pr hyX(m)
t=ni,n∈ {0,1, ..., N }, (3)
obtained from the dataset. A peak in the PMF is identified
when the slope on the left, pYm(n)−pYm(n−1), is positive,
the slope on the right, pYm(n+ 1) −pYm(n), is negative, and
pYm(n)> , where = 0.00021 used to ensure that small
peaks (noise) are not quantized as states. We chose the value
of by observing the PMFs and finding that any spike that
had <=110 out of (524,544) occurrences was not a state.
A simple example of such quantization is given in Table I,
where N= 5, but only two states are identified after
quantization, hence K(m)= 2. These states are indexed by
k(m)∈ {0,1}and are centred around the values n= 0 and
n= 3. The quantized states are denoted b
X(m)
t. The current
draws of the quantized states are stored as a K(m)-dimensional
vector, denoted y(m)
peak, whose elements are the locations of
the peaks in the original PMF. For the example in Table I,
y(m)
peak[0] = 0 and y(m)
peak[1] = 3. The probability of a given
quantized state is simply the sum of probability masses in the
corresponding bin, hence for the above example,
Pr hyb
X(m)
t= 0i=Pr hk(m)= 0i= 0.45
and
Pr hyb
X(m)
t= 3i=Pr hk(m)= 1i= 0.55 .
Since K(m)≤N, it can be seen that state quantization will
increase the sparsity of Aand B.
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TABLE I
AN EXAMP LE OF PMF AND STATE QUAN TIZATIO N
IEEE TRANSACTIONS ON SMART GRID, VOL. ?, NO. ?, SEP 2015 5
TABLE I
ANEXAMPLEOFPMF AND STATE QUANTIZATION
n012345
count(n)900 80 100 620 200 100
pYm(n)0.45 0.04 0.05 0.31 0.10 0.05
k(m),y(m)
peak 0, 0 1, 3
Pr[k(m)]0.45 0.55
The super-state corresponding to quantized internal states is
b
St=(b
X(1)
t,b
X(2)
t, ..., b
X(M)
t). The quantized super-states can
be indexed linearly in terms of the indices of the quantized
internal states k(1),k
(2),...,k
(M)as follows:
k=k(M)+
M1
X
m=1 k(m)·
M
Y
i=m+1
K(i)!. (4)
Based on (4), one can also extract individual load state indices
k(m)from kby iteratively extracting the remainder of division
of kby partial products Qm
i=1 K(i), starting with k(M).
The sparsity of Aand Bcan be used to simplify com-
putations involved in Viterbi-based state decoding, as will be
described in the next section. In addition, storage requirements
can be significantly reduced. Previously, we employed a mod-
ified version of the Harwell-Boeing sparse matrix format [17]
(or CCS, compressed column storage) to store Aand Bin
compressed form. Our modified CCS format uses hash mapped
data structures (e.g., Python dict datatype) for P0,A,B, and
Viterbi algorithm path vectors Ptwhich resulted in the ability
to disaggregation greater than 11 loads. We elaborate more on
this in the Space Complexity Subsection (IV.A).
E. Sparse Viterbi Algorithm
The standard Viterbi algorithm is well suited for well-
populated matrices. However, when used with sparse matrices,
there is an extensive amount of naive probability calculations
involving zero-probability terms. Matrix compression not only
reduces the amount of storage, but it also allows us to avoid
calculating zero-probability terms. Taking advantage of matrix
sparsity to avoid unnecessary calculations forms the basis of
our algorithm called SPA RSE-VITERBI(·)(see Algorithm 1),
a greedy version of the Viterbi algorithm.
Let yt1and ytbe the total current measurements at times
t1and t. The goal is to infer the quantized super-state bst(or,
equivalently, its index kt), from which we will determine the
quantized internal states. This will be achieved by decoding
the internal states’ indices from the super-state index using
(4). These posterior probabilities are stored in vector Pt1as
part of initialization (Algorithm 1, lines 1–7).
Pt1[j]=P0[j]·B[j, yt1],j =1,2,...,K . (5)
The computation is reduced by only considering non-zero
elements of B[j, yt1]in (5), to be clarified below. We now
calculate the posterior probabilities for the current time period
(the recursion step, Algorithm 1, lines 9–14)
Pt[j]= K
max
i=1 (Pt1[i]·A[i, j]·B[j, yt]),j =1,2,...,K . (6)
We terminate (Algorithm 1, line 17) to find the most likely
current super-state index kt= arg max(Pt). This algorithm
is called each time we need to disaggregate a reading, using a
sliding window of observations. For example, we disaggregate
t={1,2}, then t={2,3},t={3,4}, and so on.
Disaggregation only begins when the first 2 observations are
received from the meter. For our purposes we are not interested
in the prediction of the super-state from t1(the backtracking
step). Once the ktis determined feedback is sent to the
occupant – this makes it final, no turning back time. Again,
zero-probability terms are avoided in the calculation. The
zero-probability terms are effectively ignored due to matrix
compression method, which only stores non-zero elements
of the corresponding matrices. Using hash mapped structures
(Algorithm 1, lines 5, 10, and 11) only returns non-zero
elements, which removes zero probability terms from being
calculated.
Algorithm 1 SPAR SE-VITERBI(K, P0,A,B,y
t1,y
t)
1: # Set empty posterior path hash mapped structures
2: Pt1 {},Pt {}
3:
4: # Viterbi Step 1: initialization
5: for (j, pb)2B[yt1]do
6: Pt1[j] P0[j]·pb
7: end for
8:
9: # Viterbi Step 2: recursion
10: for (j, pb)2B[yt]do
11: for (i, pa)2A[j]do
12: Pt[j] max(Pt1[i]·pa·pb)
13: end for
14: end for
15:
16: # Step 3: terminate
17: return arg max(Pt)
F. Load Consumption Estimation
If we know the current/power draws of each appliance y(m)
t,
we could sum these levels yt⌘PM
m=1 y(m)
t, and the total
should equal the aggregate reading from the smart meter yt.
However, the consumption amount of each load is hidden
because the state of each load is hidden. We can estimate
the amount of consumption draw for each load by y(ˆx(m)
t).
The current draw of the decoded quantized state ˆx(m)
tis
assumed to be the location of the corresponding PMF peak,
i.e., y(ˆx(m)
t)=y(m)
peak[k(m)
t], where k(m)
tis the index of the
decoded quantized internal state of appliance mat time t.To
find the estimate of the whole-house we simply sum the load
estimates
ˆyt=
M
X
m=1
y(ˆx(m)
t)=
M
X
m=1
y(m)
peak[k(m)
t]. (7)
The super-state corresponding to quantized internal states is
b
St= ( b
X(1)
t,b
X(2)
t, ..., b
X(M)
t). The quantized super-states can
be indexed linearly in terms of the indices of the quantized
internal states k(1), k(2), ..., k(M)as follows:
k=k(M)+
M−1
X
m=1 k(m)·
M
Y
i=m+1
K(i)!. (4)
Based on (4), one can also extract individual load state indices
k(m)from kby iteratively extracting the remainder of division
of kby partial products Qm
i=1 K(i), starting with k(M).
The sparsity of Aand Bcan be used to simplify com-
putations involved in Viterbi-based state decoding, as will be
described in the next section. In addition, storage requirements
can be significantly reduced. Previously, we employed a mod-
ified version of the Harwell-Boeing sparse matrix format [17]
(or CCS, compressed column storage) to store Aand Bin
compressed form. Our modified CCS format uses hash mapped
data structures (e.g., Python dict datatype) for P0,A,B, and
Viterbi algorithm path vectors Ptwhich resulted in the ability
to disaggregation greater than 11 loads. We elaborate more on
this in the Space Complexity Subsection (IV.A).
E. Sparse Viterbi Algorithm
The standard Viterbi algorithm is well suited for well-
populated matrices. However, when used with sparse matrices,
there is an extensive amount of naive probability calculations
involving zero-probability terms. Matrix compression not only
reduces the amount of storage, but it also allows us to avoid
calculating zero-probability terms. Taking advantage of matrix
sparsity to avoid unnecessary calculations forms the basis of
our algorithm called SPARSE-VITERBI(·)(see Algorithm 1),
a greedy version of the Viterbi algorithm.
Let yt−1and ytbe the total current measurements at times
t−1and t. The goal is to infer the quantized super-state bst(or,
equivalently, its index kt), from which we will determine the
quantized internal states. This will be achieved by decoding
the internal states’ indices from the super-state index using
(4). These posterior probabilities are stored in vector Pt−1as
part of initialization (Algorithm 1, lines 1–7).
Pt−1[j] = P0[j]·B[j, yt−1], j = 1,2, ..., K . (5)
The computation is reduced by only considering non-zero
elements of B[j, yt−1]in (5), to be clarified below. We now
calculate the posterior probabilities for the current time period
(the recursion step, Algorithm 1, lines 9–14)
Pt[j] = K
max
i=1 (Pt−1[i]·A[i, j]·B[j, yt]), j = 1,2, ..., K . (6)
We terminate (Algorithm 1, line 17) to find the most likely
current super-state index kt= arg max(Pt). This algorithm
is called each time we need to disaggregate a reading, using a
sliding window of observations. For example, we disaggregate
t={1,2}, then t={2,3},t={3,4}, and so on.
Disaggregation only begins when the first 2 observations are
received from the meter. For our purposes we are not interested
in the prediction of the super-state from t−1(the backtracking
step). Once the ktis determined feedback is sent to the
occupant – this makes it final, no turning back time. Again,
zero-probability terms are avoided in the calculation. The
zero-probability terms are effectively ignored due to matrix
compression method, which only stores non-zero elements
of the corresponding matrices. Using hash mapped structures
(Algorithm 1, lines 5, 10, and 11) only returns non-zero
elements, which removes zero probability terms from being
calculated.
Algorithm 1 SPARS E-VI TE RBI(K, P0,A,B, yt−1, yt)
1: # Set empty posterior path hash mapped structures
2: Pt−1← {},Pt← {}
3:
4: # Viterbi Step 1: initialization
5: for (j, pb)∈B[yt−1]do
6: Pt−1[j]←P0[j]·pb
7: end for
8:
9: # Viterbi Step 2: recursion
10: for (j, pb)∈B[yt]do
11: for (i, pa)∈A[j]do
12: Pt[j]←max(Pt−1[i]·pa·pb)
13: end for
14: end for
15:
16: # Step 3: terminate
17: return arg max(Pt)
F. Load Consumption Estimation
If we know the current/power draws of each appliance y(m)
t,
we could sum these levels yt≡PM
m=1 y(m)
t, and the total
should equal the aggregate reading from the smart meter yt.
However, the consumption amount of each load is hidden
because the state of each load is hidden. We can estimate
the amount of consumption draw for each load by y(ˆx(m)
t).
The current draw of the decoded quantized state ˆx(m)
tis
assumed to be the location of the corresponding PMF peak,
i.e., y(ˆx(m)
t) = y(m)
peak[k(m)
t], where k(m)
tis the index of the
decoded quantized internal state of appliance mat time t. To
find the estimate of the whole-house we simply sum the load
estimates
ˆyt=
M
X
m=1
y(ˆx(m)
t) =
M
X
m=1
y(m)
peak[k(m)
t]. (7)
PUBLISHED IN: IEEE TRANSACTIONS ON SMART GRID, COPYRIGHT c
2015 IEEE, DOWNLOAD @ IEEEXPLORE.IEEE.ORG 6
IV. ALGORITHM COMPLEXITY & EFFICIENCY
Table II and Figure 2 summarizes the discussion below. We
did not test more then 19 loads as AMPds [12] only has 19
sub-meters.
A. Space Complexity
As the number of loads to disaggregate increases, the
number of super-states Kgrows exponentially, and so too
do the dimensions of P0,A,B, and Viterbi algorithm path
vectors Pt. However, in the case of load disaggregation these
vectors and matrices are very sparse. In fact, so sparse that
using an HMM with full super-state space is a practical option.
The theoretical sparsity of Bis clear from its definition. Since
for each specific super-state st= (x(1)
t, x(2)
t, ..., x(M)
t)there
is exactly one output
yt=yx(1)
t+yx(2)
t+· ·· +yx(M)
t, (8)
each row of Bshould contain exactly one non-zero element
(and that element is equal to 1). However, with quantization,
this might not hold exactly, as the current draw may fluctuate
slightly within a quantized state even if the super-state does
not change. So we may observe several non-zero values in a
given row of B, which of course should still sum up to 1.
Thus the best-case sparsity of B, defined as the fraction of
zero entries, can be calculated as
1−K
K·N= 1 −1
N. (9)
The sparsity in Areflects the fact there are relatively few
possible transitions from any given super-state. While this is
not as obvious as in the case of B, it can be appreciated by
realizing that multi-state appliances usually operate in cycles
that determine the sequence of their states. For example, a
possible state sequence for a dishwasher could be OFF →
WASH →RINSE →DRY →OFF. Meanwhile, DRY →
WASH →OFF would not make much sense.
Originally we used CCS [17] for matrix compression but
found that this was not sufficient. The column pointer vector
requires space of O(K)and grows exponentially with the
number of super-states. Not compressing the column pointer
vector prevented us from disaggregating more than 11 loads.
We can dramatically reduce the space requirements by either
moving to a hash mapped data structure or a run length
encoding (RLE)1of the column pointer vector. We chose to
use a hash mapped (dict in Python) mainly due to our tests
being run in Python. Figure 2 (left) demonstrates that when
using the AMPds dataset [12] the approximate space cost for
Acan be given as O(k√klog2K2), where kis the sum of
the states of all loads PmK(m). Similar decreases are noticed
when using the much smaller REDD dataset [18].
1see https://en.wikipedia.org/wiki/Run-length encoding.
B. Time Complexity
The Viterbi algorithm runs in O(K2T). Our algorithm
runs online where the length of Tis always 1 (or T= 1)
reducing the time complexity to O(K2)which is susceptible
to exponentiality of K. Using compressed matrices eliminated
any zero-probability states from being calculated which further
reduces the computation time cost. Although the worst case
time cost would be O(K2), in practice this is not the case. Fig-
ure 2 (right) demonstrates that when using the AMPds dataset
the approximate time cost can be given as O(klog2K2),
where kis the sum of the states of all loads PmK(m). Similar
decreases are noticed when using the much smaller REDD
dataset.
C. Algorithm Efficiency
To show the benefits of taking advantage of sparsity we
ran our disaggregator using the AMPds dataset. Our tests
showed that when disaggregating 2 loads (16 super-states),
for every 1 section of code execution in the Recursion Step
(Algorithm 1, line 12) of our sparse Viterbi algorithm, the
basic Viterbi algorithm would, on average, execute the same
section 20 times.
Figure 3 shows that amount of time (in milliseconds) it
takes to disaggregate one reading for both a basic Viterbi
algorithm and our sparse Viterbi algorithm. These tests ran
using Python 3.4 on a 64-bit Debian 8 PC with an Intel Core
i7-4790 3.6GHz processor and 32GB of memory. All tests ran
in a single thread. As expected the time taken to disaggregate
using the basic Viterbi algorithm is exponential following K.
However, the sparse Viterbi algorithm avoids this having far
better execution times. Results seem to suggest that the more
loads that are modeled, the more sparse the data structures
in the HMM. We can efficiently process an 8.2 billion state
sparse HMM in an average of 4.5 milliseconds.
V. ACCURACY EXPERIMENTATION
To support our contribution claims in Section I-A, we chose
two low-frequency datasets: AMPds [12] and REDD [18].
These datasets differ in terms of low-frequency sampling
rates (per minute vs per 3-seconds) and measurement types
(current vs apparent power). A prototype of our disaggregator
was first coded in Python 3.4. We chose Python because of
list manipulation capabilities and its ease in creating rapid
prototypes that are easily translated into C (at a later point).
Our prototype ran as a single threaded process and would
disaggregate the specified loads in all the readings of a given
dataset. All tests ran on a Mac Pro (Late 2013 model) with a
3.5GHz 6-core Intel Xeon E5 processor and 64GB of memory.
Each test ran used 10-fold cross-validation to mitigate testing
accuracy bias.
A. Nomenclature
There are a number of standard terms we use when reporting
experimental results in our findings tables. We now explain.
PUBLISHED IN: IEEE TRANSACTIONS ON SMART GRID, COPYRIGHT c
2015 IEEE, DOWNLOAD @ IEEEXPLORE.IEEE.ORG 7
Fig. 2. Space complexities (left) and time (right) complexities for comparing standard HMM space and Viterbi algorithm time with their sparse implementations
as the number of super-states increases. Test used AMPds disaggregating from 1 to 19 loads. A graphical depiction of Table II.
TABLE II
TIME AND SPACE COMPLEXITY RES ULTS US ING DATA FROM AMPDS: UN/COMPRESSED HMMS AND BA SIC/SPARS E VITERB I ALGORITHMS
IEEE TRANSACTIONS ON SMART GRID, VOL. ?, NO. ?, SEP 2015 7
Fig. 2. Space complexities (left) and time (right) complexities for comparing standard HMM space and Viterbi algorithm time with their sparse implementations
as the number of super-states increases. Test used AMPds disaggregating from 1 to 19 loads. A graphical depiction of Table II.
TABLE II
TIME AND SPACE COMPLEXITY RESULTS USING DATA FROM AMPDS:UN/COMPRESSED HMMSANDBASIC/SPARS E VITERBI ALGORITHMS
Loads
Super-States,
P0(K),
Init Step
A(K⇥K),
Recurs Step
B(K⇥N)
Basic
Disagg
Time (ms)
P0(!0) A(!0) B(!0) Sparse
Init Step
Sparse
Recurs Step
Sparse
Disagg
Time (ms)
1 4 16 8k 0.1 4 16 2.1k 3 9 0.02
2 16 256 32k 1.4 13 95 2.7k 4 13 0.03
3 48 2.3k 96k 13 36 283 4.2k 8 41 0.07
4 192 36.9k 384.2k 198 90 659 6.4k 12 73 0.12
5 768 589.8k 1.5M 3,1501 258 2.1k 11.9k 24 214 0.28
6 3,072 9.4M 6.1M 50,409 386 2.7k 13.7k 28 242 0.34
7 9,216 84.9M 18.4M 451,869 411 2.8k 13.9k 29 244 0.37
8 36,864 1.4G 73.8M 674 4.1k 17.2k 39 304 0.51
9 147,456 21.7G 295.1M 1.2k 6.9k 23.6k 55 467 0.74
10 589,824 347.9G 1.2G 2k 10.2k 31.1k 80 575 1.05
11 2,359,296 5.6T 4.7G 2.1k 10.6k 31.9k 83 578 1.10
12 7,077,888 50.1T 14.2G 2.7k 12.8k 36.1k 97 637 1.28
13 28,311,552 801.5T 56.7G 4.7k 21.4k 50k 151 1,001 2.10
14 56,623,104 3.2P 113.3G 5.5k 24.1k 54.1k 175 1,089 2.38
15 169,869,312 28.9P 339.9G 6.2k 25.8k 57.1k 187 1,046 2.51
16 339,738,624 115.4P 679.8G 6.2k 25.9k 57.1k 187 1,047 2.65
17 679,477,248 461.7P 1.4T 8.9k 36.4k 67.1k 233 1,438 3.33
18 2,038,431,744 4.2E 4.1T 9.8k 38.8k 70.1k 253 1,357 3.78
19 8,153,726,976 66.5E 16.3T 18k 67.1k 79.7k 306 1,133 4.55
1) Test Type: Each test ran on the same dataset and with
the same loads, but was configured slightly differently. These
different test configurations were: Denoised, Noisy, Modelled.
ADenoised configuration removes any noise (as defined in
[19]) from the aggregate reading so that the aggregate observed
value yis equal to the sum of the loads values y(m).A
Noisy configuration does not remove the noise in the aggregate
observed value y, nor does it try to model the noise as a load.
To us, this represents a more realistic configuration to test
against. A Modeled configuration treats the noise in ground
truth as a load we label unmetered which is then modeled as
an additional load to disaggregate.
2) Noise: Here we report the percent-noisy measure (%-
NM) [19] defined as
%-NM =PT
t=1|ytPM
m=1 y(m)
t|
PT
t=1 yt⇥100 , (10)
where ytis the aggregate observed current/power amount at
time tand y(m)
tis the ground truth current/power amount for
each appliance mto be disaggregated. For example, a denoised
test would result in 0%; whereas, a %-NM of 40% would
mean that 40% of the aggregate observed current/power for
the whole test was noise.
1) Test Type: Each test ran on the same dataset and with
the same loads, but was configured slightly differently. These
different test configurations were: Denoised, Noisy, Modelled.
ADenoised configuration removes any noise (as defined in
[19]) from the aggregate reading so that the aggregate observed
value yis equal to the sum of the loads values y(m). A
Noisy configuration does not remove the noise in the aggregate
observed value y, nor does it try to model the noise as a load.
To us, this represents a more realistic configuration to test
against. A Modeled configuration treats the noise in ground
truth as a load we label unmetered which is then modeled as
an additional load to disaggregate.
2) Noise: Here we report the percent-noisy measure (%-
NM) [19] defined as
%-NM =PT
t=1|yt−PM
m=1 y(m)
t|
PT
t=1 yt×100 , (10)
where ytis the aggregate observed current/power amount at
time tand y(m)
tis the ground truth current/power amount for
each appliance mto be disaggregated. For example, a denoised
test would result in 0%; whereas, a %-NM of 40% would
mean that 40% of the aggregate observed current/power for
the whole test was noise.
PUBLISHED IN: IEEE TRANSACTIONS ON SMART GRID, COPYRIGHT c
2015 IEEE, DOWNLOAD @ IEEEXPLORE.IEEE.ORG 8
3) Acc: Here we report the basic accuracy measure as
defined Acc =correct
correct+incorr ect . Although this measure is
not the best way to report accuracy, we include it because a
majority of disaggregation researchers use this measure.
4) FS-fscore: Here we report our Finite State version of
f-score [19] rather than the usual binary measures f-score.
5) Est Acc: Here we report the consumption estimation
accuracy measure developed by Kolter and Johnson [18].
B. Testing Deferrable Loads
The use of large appliances, such as a clothes dryer can
be deferred until the cost of electricity is cheaper, called off-
peak hours. Large load deferral should be the focus of NILM
because there is a direct benefit to the occupants (saving
money) and may have a direct benefit (brownout avoidance) to
the power grid as a whole. Deferring large loads from running
during peak times of usage can ensure power grid stability
and avoid grid brownouts. This idea is often discussed as
peak shaving [20]. There are other benefits to having NILM
only disaggregate deferrable loads. Firstly, large consumption
can be more easily identified resulting in better accuracy.
Secondly, disaggregators can run faster as there are fewer loads
to disaggregate. However, the disaggregator must be robust
enough to handle a large percentage of noise – many other
loads that would be running in the house.
To test the accuracies for disaggregating deferrable loads we
used AMPds [12] choosing the clothes dryer, the dishwasher,
the HVAC system, the heat pump, and the kitchen wall
oven. We ran all three test types on one year worth of data
(524,544 readings). These loads are multi-state loads. The
HVAC system has a continuous variable fan motor and a
constantly on thermostat. The readings within APMds do not
contain errors, so there was no need to clean or convert the
data. We ran tests using per minute sampling (minute tests) and
per hour sampling (hour tests). Denoised tests do not represent
a realistic scenario; we only report them for comparison as to
what the ideal result would be.
All three test types scored very high with the lowest score
being 94.1% for the estimation accuracy of the Noisy test
(see Table III). Looking at the specific load results for the
noisy and model tests, we noticed the accuracy results for the
dishwasher scored very low (see Figure 4), while other loads
scored quite high. The low results were due to having two load
Fig. 3. Runtimes (in milliseconds) of a single disaggregation for both the
basic and sparse Viterbi algorithms as the number of super-states increases.
Runtime numbers listed in Table II
states (at peaks 0.4A and 1.2A) similar to other loads (2 loads
and 3 loads respectively). In the model tests, the unmetered
load scored low because we restricted a number of states to
a maximum of 4. During this experiment, 7,522 of 524,544
observations have multiple loads switch states simultaneously.
We also wanted to test how well our disaggregator would
disaggregate very low-frequency sampling data – hourly data.
In these tests, we used Watt-hour measurements (at daWh
or Wh÷10 precision). We accumulated the Watt-hours for
each hour over the entire year for a total of 17,520 readings.
Table IV shows the overall results of the hour tests. The Noisy
test performed with high classification accuracy (91%) having
the estimation accuracy score roughly 4% lower. Examining
the load specific scores, we noticed that the dishwasher,
again, scored low. However, we also found the oven scored
even lower often having an FS-fscore of 0%. There were
misclassifications that occurred because of the high level
of noise and the infrequent, nonuniform use of the oven.
We were surprised our disaggregator scored as well as it
did after reading Kolter’s [21] published results for hourly
disaggregation. His discriminative disaggregator achieved an
estimation accuracy of 55%, which is 31% lower than our
Noisy test. Kolter used a different dataset (Plugwise) which
contained a large number of houses, so it is hard to do a
direct comparison of results.
TABLE III
DEFERRABLE LOADS ACCU RAC Y RESULTS (/MINUTE, A)
IEEE TRANSACTIONS ON SMART GRID, VOL. ?, NO. ?, SEP 2015 8
3) Acc: Here we report the basic accuracy measure as
defined Acc =correct
correct+incor rect . Although this measure is
not the best way to report accuracy, we include it because a
majority of disaggregation researchers use this measure.
4) FS-fscore: Here we report our Finite State version of
f-score [19] rather than the usual binary measures f-score.
5) Est Acc: Here we report the consumption estimation
accuracy measure developed by Kolter and Johnson [18].
B. Testing Deferrable Loads
The use of large appliances, such as a clothes dryer can
be deferred until the cost of electricity is cheaper, called off-
peak hours. Large load deferral should be the focus of NILM
because there is a direct benefit to the occupants (saving
money) and may have a direct benefit (brownout avoidance) to
the power grid as a whole. Deferring large loads from running
during peak times of usage can ensure power grid stability
and avoid grid brownouts. This idea is often discussed as
peak shaving [20]. There are other benefits to having NILM
only disaggregate deferrable loads. Firstly, large consumption
can be more easily identified resulting in better accuracy.
Secondly, disaggregators can run faster as there are fewer loads
to disaggregate. However, the disaggregator must be robust
enough to handle a large percentage of noise – many other
loads that would be running in the house.
To test the accuracies for disaggregating deferrable loads we
used AMPds [12] choosing the clothes dryer, the dishwasher,
the HVAC system, the heat pump, and the kitchen wall
oven. We ran all three test types on one year worth of data
(524,544 readings). These loads are multi-state loads. The
HVAC system has a continuous variable fan motor and a
constantly on thermostat. The readings within APMds do not
contain errors, so there was no need to clean or convert the
data. We ran tests using per minute sampling (minute tests) and
per hour sampling (hour tests). Denoised tests do not represent
a realistic scenario; we only report them for comparison as to
what the ideal result would be.
All three test types scored very high with the lowest score
being 94.1% for the estimation accuracy of the Noisy test
(see Table III). Looking at the specific load results for the
noisy and model tests, we noticed the accuracy results for the
dishwasher scored very low (see Figure 4), while other loads
scored quite high. The low results were due to having two load
Fig. 3. Runtimes (in milliseconds) of a single disaggregation for both the
basic and sparse Viterbi algorithms as the number of super-states increases.
Runtime numbers listed in Table II
states (at peaks 0.4A and 1.2A) similar to other loads (2 loads
and 3 loads respectively). In the model tests, the unmetered
load scored low because we restricted a number of states to
a maximum of 4. During this experiment, 7,522 of 524,544
observations have multiple loads switch states simultaneously.
We also wanted to test how well our disaggregator would
disaggregate very low-frequency sampling data – hourly data.
In these tests, we used Watt-hour measurements (at daWh
or Wh÷10 precision). We accumulated the Watt-hours for
each hour over the entire year for a total of 17,520 readings.
Table IV shows the overall results of the hour tests. The Noisy
test performed with high classification accuracy (91%) having
the estimation accuracy score roughly 4% lower. Examining
the load specific scores, we noticed that the dishwasher,
again, scored low. However, we also found the oven scored
even lower often having an FS-fscore of 0%. There were
misclassifications that occurred because of the high level
of noise and the infrequent, nonuniform use of the oven.
We were surprised our disaggregator scored as well as it
did after reading Kolter’s [21] published results for hourly
disaggregation. His discriminative disaggregator achieved an
estimation accuracy of 55%, which is 31% lower than our
Noisy test. Kolter used a different dataset (Plugwise) which
contained a large number of houses, so it is hard to do a
direct comparison of results.
TABLE III
DEFERRABLE LOADS ACCURACY RESULTS (/MINUTE,A)
Test Type Noise Acc FS-fscore Est Acc
Denoised 0.0% 99.2% 99.2% 99.0%
Noisy 60.6% 98.0% 98.1% 94.1%
Modelled 53.4% 93.6% 94.5% 98.6%
TABLE IV
DEFERRABLE LOADS ACCURACY RESULTS (/HOUR,WH)
Test Type Noise Acc FS-fscore Est Acc
Denoised 0.0% 96.7% 95.2% 93.7%
Noisy 68.2% 94.6% 90.6% 86.2%
Modelled 51.4% 91.7% 87.0% 88.0%
C. Comparing Disaggregators
To compare the accuracy of our disaggregator to that of oth-
ers [16], [18] we used the REDD dataset [18] (a smaller dataset
compared to AMPds, but data sampled every 3 seconds) and
used the same estimation accuracy measure. We wanted to
compare our results against the Kolter 2011 [18] results and
the Johnson 2013 [16] results. There has been a long-standing
issue in the field of NILM where the use of different datasets
and accuracy measure makes comparing different algorithms
near impossible. However, Kolter 2011 and Johnson 2013 use
the same dataset and accuracy measures and publish results
that are easy to compare to. The majority of NILM papers do
not do this, and we are hoping to continue a trend started by
the Johnson 2013 paper. Both papers use more complex types
of Markov model, and we wanted to show how much more
TABLE IV
DEFERRABLE LOADS ACCURACY RESULTS (/HOU R, WH)
IEEE TRANSACTIONS ON SMART GRID, VOL. ?, NO. ?, SEP 2015 8
3) Acc: Here we report the basic accuracy measure as
defined Acc =correct
correct+incor rect . Although this measure is
not the best way to report accuracy, we include it because a
majority of disaggregation researchers use this measure.
4) FS-fscore: Here we report our Finite State version of
f-score [19] rather than the usual binary measures f-score.
5) Est Acc: Here we report the consumption estimation
accuracy measure developed by Kolter and Johnson [18].
B. Testing Deferrable Loads
The use of large appliances, such as a clothes dryer can
be deferred until the cost of electricity is cheaper, called off-
peak hours. Large load deferral should be the focus of NILM
because there is a direct benefit to the occupants (saving
money) and may have a direct benefit (brownout avoidance) to
the power grid as a whole. Deferring large loads from running
during peak times of usage can ensure power grid stability
and avoid grid brownouts. This idea is often discussed as
peak shaving [20]. There are other benefits to having NILM
only disaggregate deferrable loads. Firstly, large consumption
can be more easily identified resulting in better accuracy.
Secondly, disaggregators can run faster as there are fewer loads
to disaggregate. However, the disaggregator must be robust
enough to handle a large percentage of noise – many other
loads that would be running in the house.
To test the accuracies for disaggregating deferrable loads we
used AMPds [12] choosing the clothes dryer, the dishwasher,
the HVAC system, the heat pump, and the kitchen wall
oven. We ran all three test types on one year worth of data
(524,544 readings). These loads are multi-state loads. The
HVAC system has a continuous variable fan motor and a
constantly on thermostat. The readings within APMds do not
contain errors, so there was no need to clean or convert the
data. We ran tests using per minute sampling (minute tests) and
per hour sampling (hour tests). Denoised tests do not represent
a realistic scenario; we only report them for comparison as to
what the ideal result would be.
All three test types scored very high with the lowest score
being 94.1% for the estimation accuracy of the Noisy test
(see Table III). Looking at the specific load results for the
noisy and model tests, we noticed the accuracy results for the
dishwasher scored very low (see Figure 4), while other loads
scored quite high. The low results were due to having two load
Fig. 3. Runtimes (in milliseconds) of a single disaggregation for both the
basic and sparse Viterbi algorithms as the number of super-states increases.
Runtime numbers listed in Table II
states (at peaks 0.4A and 1.2A) similar to other loads (2 loads
and 3 loads respectively). In the model tests, the unmetered
load scored low because we restricted a number of states to
a maximum of 4. During this experiment, 7,522 of 524,544
observations have multiple loads switch states simultaneously.
We also wanted to test how well our disaggregator would
disaggregate very low-frequency sampling data – hourly data.
In these tests, we used Watt-hour measurements (at daWh
or Wh÷10 precision). We accumulated the Watt-hours for
each hour over the entire year for a total of 17,520 readings.
Table IV shows the overall results of the hour tests. The Noisy
test performed with high classification accuracy (91%) having
the estimation accuracy score roughly 4% lower. Examining
the load specific scores, we noticed that the dishwasher,
again, scored low. However, we also found the oven scored
even lower often having an FS-fscore of 0%. There were
misclassifications that occurred because of the high level
of noise and the infrequent, nonuniform use of the oven.
We were surprised our disaggregator scored as well as it
did after reading Kolter’s [21] published results for hourly
disaggregation. His discriminative disaggregator achieved an
estimation accuracy of 55%, which is 31% lower than our
Noisy test. Kolter used a different dataset (Plugwise) which
contained a large number of houses, so it is hard to do a
direct comparison of results.
TABLE III
DEFERRABLE LOADS ACCURACY RESULTS (/MINUTE,A)
Test Type Noise Acc FS-fscore Est Acc
Denoised 0.0% 99.2% 99.2% 99.0%
Noisy 60.6% 98.0% 98.1% 94.1%
Modelled 53.4% 93.6% 94.5% 98.6%
TABLE IV
DEFERRABLE LOADS ACCURACY RESULTS (/HOUR,WH)
Test Type Noise Acc FS-fscore Est Acc
Denoised 0.0% 96.7% 95.2% 93.7%
Noisy 68.2% 94.6% 90.6% 86.2%
Modelled 51.4% 91.7% 87.0% 88.0%
C. Comparing Disaggregators
To compare the accuracy of our disaggregator to that of oth-
ers [16], [18] we used the REDD dataset [18] (a smaller dataset
compared to AMPds, but data sampled every 3 seconds) and
used the same estimation accuracy measure. We wanted to
compare our results against the Kolter 2011 [18] results and
the Johnson 2013 [16] results. There has been a long-standing
issue in the field of NILM where the use of different datasets
and accuracy measure makes comparing different algorithms
near impossible. However, Kolter 2011 and Johnson 2013 use
the same dataset and accuracy measures and publish results
that are easy to compare to. The majority of NILM papers do
not do this, and we are hoping to continue a trend started by
the Johnson 2013 paper. Both papers use more complex types
of Markov model, and we wanted to show how much more
C. Comparing Disaggregators
To compare the accuracy of our disaggregator to that of oth-
ers [16], [18] we used the REDD dataset [18] (a smaller dataset
compared to AMPds, but data sampled every 3 seconds) and
used the same estimation accuracy measure. We wanted to
compare our results against the Kolter 2011 [18] results and
the Johnson 2013 [16] results. There has been a long-standing
issue in the field of NILM where the use of different datasets
and accuracy measure makes comparing different algorithms
near impossible. However, Kolter 2011 and Johnson 2013 use
the same dataset and accuracy measures and publish results
that are easy to compare to. The majority of NILM papers do
not do this, and we are hoping to continue a trend started by
PUBLISHED IN: IEEE TRANSACTIONS ON SMART GRID, COPYRIGHT c
2015 IEEE, DOWNLOAD @ IEEEXPLORE.IEEE.ORG 9
Table 1
AMP01b
Dryer
0.1825
Dishwasher
0.0396
Heat Pump
0.4682
HVAC
0.2875
Oven
0.0222
Noisy Test Results
Oven
2.2%
HVAC
28.8%
Heat Pump
46.8%
Dishwasher
4.0%
Dryer
18.3%
Table 2
Untitled 1
Dryer
0.204
Dishwasher
0.0288
Heat Pump
0.4138
HVAC
0.3175
Oven
0.0359
Noisy - Ground Truth
Oven
3.6%
HVAC
31.8%
Heat Pump
41.4%
Dishwasher
2.9%
Dryer
20.4%
Table 3
AMP01b
Dryer
0.1651
Dishwasher
0.0124
Heat Pump
0.4211
HVAC
0.265
Oven
0.0202
Unmetered
0.1162
Modelled Test Results
Unmetered
12%
Oven
2.0%
HVAC
26.5%
Heat Pump
42.1%
Dishwasher
1.2%
Dryer
16.5%
Table 4
Untitled 1
Dryer
0.1726
Dishwasher
0.0244
Heat Pump
0.35
HVAC
0.2686
Oven
0.0304
Unmetered
0.1541
Model - Ground Truth
Unmetered
15%
Oven
3.0%
HVAC
26.9%
Heat Pump
35.0%
Dishwasher
2.4%
Dryer
17.3%
1
Fig. 4. The resulting estimation accuracy results for Noisy (left) and Modelled (right) tests as compared to their respective ground truth.
the Johnson 2013 paper. Both papers use more complex types
of Markov model, and we wanted to show how much more
accurate a simple HMM can be. We also wanted to show that
our observations about sparsity hold for other datasets.
Kolter and Johnson [18] initially reported accuracies on
Houses 1, 2, 3, 4, and 6 using a disaggregator that used
supervised learning. Johnson et al. [16] reported accuracies
on Houses 1, 2, 3, and 6 using a disaggregator that they
claimed used unsupervised learning. However, their claim of
unsupervised learning was incorrect. They used a supervised
learning solution because labeled data was used to build
appliance models (also the opinion of Parson [15]). We chose
to use the low-frequency sampling data from REDD that was
in apparent power measurements. The whole-house aggregate
power meter sampled at 1Hz (per second), opposed to the
load sub-meters that sampled at 1
3Hz (3-second intervals).
As a result, we needed to clean and convert the data for
our tests. For our tests, we downsampled the aggregate data
from 1Hz to 1
3Hz by discarding the between readings after
matching the timestamps. There was the additional problem
where the timestamps in some of the readings were out of
sync because two different metering systems were used (one
for aggregate and one for loads). To correct mismatches we
had our conversion program scan the aggregate data records
forward and backward until the aggregate reading matched the
sum of all the load readings. There were a few instances where
we added the difference to the aggregate reading so that the
noise cancelation would read zero.
TABLE V
COMPARING ACCURACY RESULTS WITH OTHER PUBLISHED WORK
IEEE TRANSACTIONS ON SMART GRID, VOL. ?, NO. ?, SEP 2015 9
Table 1
AMP01b
Dryer 0.1825
Dishwasher 0.0396
Heat Pump 0.4682
HVAC 0.2875
Oven 0.0222
Noisy Test Results
Oven
2.2%
HVAC
28.8%
Heat Pump
46.8%
Dishwasher
4.0%
Dryer
18.3%
Table 2
Untitled 1
Dryer 0.204
Dishwasher 0.0288
Heat Pump 0.4138
HVAC 0.3175
Oven 0.0359
Noisy - Ground Truth
Oven
3.6%
HVAC
31.8%
Heat Pump
41.4%
Dishwasher
2.9%
Dryer
20.4%
Table 3
AMP01b
Dryer 0.1651
Dishwasher 0.0124
Heat Pump 0.4211
HVAC 0.265
Oven 0.0202
Unmetered 0.1162
Modelled Test Results
Unmetered
12%
Oven
2.0%
HVAC
26.5%
Heat Pump
42.1%
Dishwasher
1.2%
Dryer
16.5%
Table 4
Untitled 1
Dryer 0.1726
Dishwasher 0.0244
Heat Pump 0.35
HVAC 0.2686
Oven 0.0304
Unmetered 0.1541
Model - Ground Truth
Unmetered
15%
Oven
3.0%
HVAC
26.9%
Heat Pump
35.0%
Dishwasher
2.4%
Dryer
17.3%
1
Fig. 4. The resulting estimation accuracy results for Noisy (left) and Modelled (right) tests as compared to their respective ground truth.
accurate a simple HMM can be. We also wanted to show that
our observations about sparsity hold for other datasets.
Kolter and Johnson [18] initially reported accuracies on
Houses 1, 2, 3, 4, and 6 using a disaggregator that used
supervised learning. Johnson et al. [16] reported accuracies
on Houses 1, 2, 3, and 6 using a disaggregator that they
claimed used unsupervised learning. However, their claim of
unsupervised learning was incorrect. They used a supervised
learning solution because labeled data was used to build
appliance models (also the opinion of Parson [15]). We chose
to use the low-frequency sampling data from REDD that was
in apparent power measurements. The whole-house aggregate
power meter sampled at 1Hz (per second), opposed to the
load sub-meters that sampled at 1
3Hz (3-second intervals).
As a result, we needed to clean and convert the data for
our tests. For our tests, we downsampled the aggregate data
from 1Hz to 1
3Hz by discarding the between readings after
matching the timestamps. There was the additional problem
where the timestamps in some of the readings were out of
sync because two different metering systems were used (one
for aggregate and one for loads). To correct mismatches we
had our conversion program scan the aggregate data records
forward and backward until the aggregate reading matched the
sum of all the load readings. There were a few instances where
we added the difference to the aggregate reading so that the
noise cancelation would read zero.
TABLE V
COMPARING ACCURACY RESULTS WITH OTHER PUBLISHED WORK
REDD
Kolter
2011 [18]
Johnson
2013 [16]
Noisy
Test
Modelled
Test
House 1 46.6% 82.1% 95.6% 99.3%
House 2 50.8% 84.8% 94.8% 99.0%
House 3 33.3% 81.5% 90.6% 97.5%
House 6 55.7% 77.7% 98.4% 99.7%
Average 46.6% 81.5% 94.9% 98.9%
Gain wrt. [18] — +34.9% +48.3% +52.3%
Gain wrt. [16] -34.9% — +13.3% +17.3%
We used the same previous three tests, and we selected the
same five loads to disaggregate as Johnson did: refrigerator,
lighting, dishwasher, microwave, and furnace. We compared
our results with the other two in Table V. Note the other
two papers did not report load specific accuracies, so we
cannot compare those. While the Johnson 2013 results were
significantly better than the Kolter 2011 results, our two testing
results were significantly better than the Johnson 2013 results.
Our Noisy test results were better than Johnson 2013 by
13.3%, and better than Kolter 2011 by 48.3%.
VI. ANALYSIS OF ACCURACY RESULTS
We have shown how our disaggregator achieves a high
degree of both load classification and consumption estimation
using different low-frequency sampling rates and different
measurement. Zeifman [14] previously identified six key re-
quirements he believed needed to be met in order for load
disaggregation to be solved. The six requirements are listed
below and we review how well we have met these require-
ments. We have confidently met four of Zeifman’s [14] six
key requirements. The two we have not confidently met are
no training and scalability.
A. Feature Selection
Feature selection constrains the measurements to those of
a typical smart meter: current, apparent power and energy
with low-frequency sampling. Our disaggregator has met this
requirement. We can disaggregate accurately at even lower
sampling frequencies: per minute and per hour.
B. Accuracy
Accuracy requires disaggregators to have a minimal accu-
racy measure score of 80%. Figure 5 shows our disaggregator
has met this requirement.
C. No Training
No training requires disaggregators to not involve significant
occupant efforts [14] to train. Our disaggregator minimizes the
involvement of occupants to just initial model building without
pre-tuning.
The idea of no training may be better rephrased as minimal
setup which could be handled in two ways: the collection
of priors or the active tuning of general load models. We
have chosen the collection of priors which provides more
We used the same previous three tests, and we selected the
same five loads to disaggregate as Johnson did: refrigerator,
lighting, dishwasher, microwave, and furnace. We compared
our results with the other two in Table V. Note the other
two papers did not report load specific accuracies, so we
cannot compare those. While the Johnson 2013 results were
significantly better than the Kolter 2011 results, our two testing
results were significantly better than the Johnson 2013 results.
Our Noisy test results were better than Johnson 2013 by
13.3%, and better than Kolter 2011 by 48.3%.
VI. ANA LYSIS OF ACCURACY RESULTS
We have shown how our disaggregator achieves a high
degree of both load classification and consumption estimation
using different low-frequency sampling rates and different
measurement. Zeifman [14] previously identified six key re-
quirements he believed needed to be met in order for load
disaggregation to be solved. The six requirements are listed
below and we review how well we have met these require-
ments. We have confidently met four of Zeifman’s [14] six
key requirements. The two we have not confidently met are
no training and scalability.
A. Feature Selection
Feature selection constrains the measurements to those of
a typical smart meter: current, apparent power and energy
with low-frequency sampling. Our disaggregator has met this
requirement. We can disaggregate accurately at even lower
sampling frequencies: per minute and per hour.
B. Accuracy
Accuracy requires disaggregators to have a minimal accu-
racy measure score of 80%. Figure 5 shows our disaggregator
has met this requirement.
C. No Training
No training requires disaggregators to not involve significant
occupant efforts [14] to train. Our disaggregator minimizes the
involvement of occupants to just initial model building without
pre-tuning.
The idea of no training may be better rephrased as minimal
setup which could be handled in two ways: the collection
of priors or the active tuning of general load models. We
PUBLISHED IN: IEEE TRANSACTIONS ON SMART GRID, COPYRIGHT c
2015 IEEE, DOWNLOAD @ IEEEXPLORE.IEEE.ORG 10
Axis
Test Type
A1
A60
Est Acc
Denoised
99.2%
95.2%
FS-fscore
99.0%
93.7%
Est Acc
Noisy
98.1%
90.6%
FS-fscore
94.1%
86.2%
Est Acc
Modelled
94.5%
87.0%
FS-fscore
98.6%
88.0%
FS-fscore Accuracy
50%
60%
70%
80%
90%
100%
Estimation Accuracy
50%
60%
70%
80%
90%
100%
Denoised
Noisy
Modelled
low accuracy area
1
Fig. 5. Compare the deferrable loads test scores (Table III and Table IV) for
FS-fscore and estimation accuracies using Zeifman’s minimum 80% accuracy
requirement.
have chosen the collection of priors which provides more
accurate results from the start. Active tuning can only provide
accurate results once the general models are tuned properly
to the specific loads in the house that could take an arbitrary
amount of time. How will the occupants know when this active
tuning period has completed and that the disaggregation results
can now be trusted as accurate? Low accuracy can cause
occupants to lose confidence in the disaggregator’s ability to
work properly.
D. Near Real-Time
Near real-time capabilities mean disaggregators must run
online and respond to events as they happen; i.e., the algorithm
must be robust and efficient. Our disaggregator has met this
requirement and can disaggregate in under one millisecond.
E. Scalability
Scalability requires disaggregators not to require additional
processing time and/or hardware to account for the identifica-
tion of new appliances. Our disaggregator currently cannot do
this; however, it can disaggregate accurately with high levels
of noise. One observation is the house used to create AMPds
did not have any major deferrable load changes in 10 years.
The idea of scalability is very much still an open research
question. It is difficult to identify new loads without devoting
some computational time to the act of finding them. This
also assumes we want to disaggregate every load a house
has. Disaggregation is very much a problem that is compu-
tationally exponential in both time and space. This means
we can mitigate some effects of exponentiality with different
strategies, such as our sparsity optimization or the use of
factorial models. We need to disaggregate loads that will allow
for the end goal of energy conservation and design robust
algorithms that handle large amounts of noise (unmetered
loads) like our disaggregator can. If, for instance, an occupant
wants to disaggregate lights, then it may be better to use home
automation systems to do this. Home automation systems can
be used to inform the occupant whether a light is ON or OFF
along with the ability to remotely control it.
F. Various Appliance Types
Various appliance types must be handled/detected by the
disaggregators. To simplify, our disaggregator only uses one
appliance type, multi-state (or finite state). Simple ON/OFF
and constantly on are simply special cases of multi-state. For
larger appliances such as a front-load washer, with variable
speed drum (continuously variable), we create a state for this
operation that seems to be sufficient for estimation purposes.
VII. CONCLUSIONS
Our disaggregator was designed to run in real-time on
an embedded processor using low-frequency sampling data
in an effort to show load disaggregation is indeed a viable
method for enabling occupants to understand how their home
consumes energy. This understanding would allow occupants
to make intelligent, informed decisions on how they conserve
the use of energy, which by all accounts is a very personal and
dynamic decision-making process. It is personal and dynamic
because the goals of the occupants involve many factors
such as home characteristics, occupant comfort levels, and
budgetary constraints (to name a few).
Our disaggregator can disaggregate a model with a large
number of super-states. However, there is still the exponential
limitation in time and space. We have only pushed back the
exponential problem through optimization, but have we pushed
it back far enough? No, if we want to try to disaggregate every
load in a home. Yes, if we want to disaggregate deferrable
loads that affect the amount owing on a power utility bill.
We believe the main goal for load disaggregation is to help
occupants conserve energy through the smarter use of high en-
ergy consumption loads. For example, occupants running large
consuming deferrable loads when the per kWh is low, instead
of when it is high, would see savings on their power utility
bill. The ability to disaggregate 18 loads, as we have shown,
goes far beyond the number of loads previously reported in
other publications (usually ranging from 6–9 loads).
Despite the proactive discounting of a model/algorithm
because of its theoretical limitations, promising mod-
els/algorithms often get overlooked or simply argued away.
While the theoretical limitation of exponentiality remains,
through the deep analysis of data, modifications can be made
to these models and algorithms that allow for use in practice.
We have demonstrated this through the design of our super-
state HMM and sparse Viterbi algorithm. We plan to extend
our disaggregator by providing an active tuning module that
would allow for a more general super-state HMM to be tuned
specifically to a given house.
SOURCE CODE
With the publication of the article, we have released our
disaggregator as open source for academic use (free of charge).
Disaggregator source code is available for download from
GitHub at https://github.com/smakonin/SparseNILM. In the
near future our algorithm will be added into NILMTK [22].
PUBLISHED IN: IEEE TRANSACTIONS ON SMART GRID, COPYRIGHT c
2015 IEEE, DOWNLOAD @ IEEEXPLORE.IEEE.ORG 11
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Stephen Makonin is currently a Postdoctoral Fellow
in Engineering Science at Simon Fraser University
(SFU) and previously a Postdoctoral Fellow in Com-
puting Science. He received his PhD in Computing
Science from SFU in 2014. His research interests
are in big data, machine learning, computational
sustainability, disaggregation, NILM, and visualiza-
tion. Previously, he worked in industry as a software
developer for over 12 years. He is an active member
of the IEEE, ACM, and AAAI and volunteers as
the IEEE Vancouver Joint Computing Chapter Chair
(includes both the Computer and Computational Intelligence Societies).
Fred Popowich received his PhD in Cognitive Sci-
ence from the University of Edinburgh in 1989, and
since then has been a faculty member in the School
of Computing Science at Simon Fraser University.
He is currently the President of the Canadian Net-
work for Visual Analytics and Founding Director of
the Vancouver Institute for Visual Analytics. Much
of his research is concerned with how computers can
be used to process human language, either to make it
easier for human beings to interact with computers,
or to make it easier for human beings to interact
with each other. As such, he has been concerned with how knowledge about
language and the world can be represented, maintained, and even learned
by computers. Typical real world applications of this research include smart
homes, the automatic translation of language, tools to assist people in learning
language, and technology to help people search and manage the information
contained on computer systems and networks.
Ivan V. Baji´
cis Associate Professor of Engineering
Science at Simon Fraser University, Burnaby, BC,
Canada. His research interests include signal, image,
and video processing and compression, multimedia
ergonomics, and communications. He has authored
about a dozen and co-authored another eight dozen
publications in these fields. He has served on the
organizing and/or program committees of various
conferences in the field, including GLOBECOM,
ICC, ICME, and ICIP, was the Chair of the Media
Streaming Interest Group of the IEEE Multimedia
Communications Technical Committee from 2010 to 2012, and is currently an
elected member of the IEEE Multimedia Systems and Applications Technical
Committee, Associate Editor of the IEEE Signal Processing Magazine, and
Chair of the Vancouver Chapter of the IEEE Signal Processing Society.
Bob Gill is a registered professional engineer in
the province of BC. He started his career with
IBM in Toronto designing computer systems, moved
on to British Columbia Hydro and power author-
ity and as part of a multi-skilled team worked in
telecommunication and powers systems technologies
for approximately 7 years. He completed his grad
studies at the Simon Fraser University in 1996.
He then moved on the TELUS and worked as a
Technology Manager. Since 1999, Bob has been
a faculty member at British Columbia Institute of
Technology and is currently the Program Head for the Telecommunication
and Networks Option. He is currently the IEEE Vancouver Section Chair and
is a Fellow of Engineers Canada.
Lyn Bartram is Associate Professor in the School of
Interactive Art and Technology at Simon Fraser Uni-
versity (SFU), where she is Director of the Human-
Centred Systems for Sustainable Living research
group. She holds a PhD in Computer Science from
SFU. Her work explores the intersecting potential
of information technologies, ubiquitous computing,
computational media and sustainable building design
in encouraging conservation and reducing our eco-
logical footprint in our homes and personal activi-
ties. She is a member of the ACM.
