REAL-TIME COMPRESSION OF TIME SERIES BUILDING … · ments are considerably shorter than t res, then the output segments will have an average width of approx-imately t res. The output

REAL-TIME COMPRESSION OF TIME SERIESBUILDING PERFORMANCE DATA

Rhys Goldstein, Michael Glueck, and Azam KhanAutodesk Research, 210 King St. East, Toronto, ON, Canada

ABSTRACTIf building performance simulations are to fully bene-fit from increasing quantities of sensor data, subsets oflarge datasets must be efficiently extracted at varyinglevels of detail. A key issue with time series data is thatrelevant time scales vary by orders of magnitude de-pending on the desired analysis. To ensure that a sub-set of a time series is available when needed at an ap-propriate resolution, lossy compression can be appliedin real time as data is acquired. We propose an algo-rithm that compresses a piecewise constant time seriesby merging segments within a sliding time window.This procedure tends to preserve prominent edges andspikes. While building control system dashboards andsimulation tools often average data over fixed timeperiods (e.g. hourly averages), the proposed methodachieves lower errors for the same compression ratioand provides better support for signal processing, datavisualization, and simulation.

INTRODUCTIONThe accuracy of a simulation-based building energyanalysis depends largely on whether the modeler hasconvenient access to an adequate amount of suffi-ciently detailed input data acquired under relevant con-ditions. The critical importance of simulation inputdata is leading to increasingly large datasets.Weather data provides a good example of why largedatasets are necessary. First, weather data includesseveral properties, including air temperature, humid-ity, direct and diffuse solar irradiance, direct and dif-fuse solar illuminance, wind velocity and direction,among others. Barnaby and Crawley (2011) indi-cate that this data must be collected over many years,for numerous cities, and possibly multiple locationswithin each city due to urban heat island effects. Atthe same time they call for finer resolutions than thehourly averages typically used for simulation.An increasing amount of data is being collected andrecorded within buildings as well. The recently con-structed Y2E2 Building at Stanford University fea-tures 2370 HVAC system measurement points, eachsampled every minute (Kunz et al., 2009). Detailedsensor networks like these can be used to detect prob-lems in building control systems (Eisenhower et al.,2010) and to improve the accuracy of building energymodels (Raftery et al., 2009).

For building performance simulations to fully bene-fit from increasing quantities of sensor data, advanceddata management techniques may be needed to effi-ciently extract subsets of large datasets. In the caseof time series data, relevant time scales vary by or-ders of magnitude depending on the desired analysis.Sub-minute data may be needed, for example, to in-vestigate brief power quality disturbances. To studyannual trends in energy use, however, a low-resolutionversion of the same time series would likely suffice.Our interest in data compression is not aimed at reduc-ing overall data storage requirements, as we believeraw data should be preserved whenever possible. In-stead, we regard compression as a means of supportingthe efficient extraction, processing, and exploration oftime series data. The idea is to apply a lossy com-pression algorithm repeatedly to precompute and storecopies of every time series at different resolutions. Ifan original time series is unnecessarily detailed for aparticular analysis, a copy at more appropriate resolu-tion will then be readily available.A simple method for reducing the resolution of a timeseries is to average the data within consecutive timeperiods of equal width. But while the resulting fixed-width segments are common in building control sys-tem dashboards and simulation tools, they are poor atpreserving prominent edges and spikes seen frequentlyin, for instance, light or electrical data.We present an algorithm that compresses piecewiseconstant time series data by maintaining a limitednumber of segments within a sliding time window offixed width. If the limit on the number of segments isexceeded, then the most similar pair of adjacent seg-ments is merged. We show how this selective mergingprocedure preserves prominent edges and spikes, andintroduces smaller compression errors for the samecompression ratios than the common fixed-width ap-proach. The algorithm can be applied in real time tosensor data, and may also be useful for the output ofdetailed, long-term simulations.The compression algorithm is the main focus and con-tribution of this paper. However, we also hope to in-spire other applications of time series represented us-ing constant-valued, variable-width segments with er-ror values. We describe how this way of representingbuilding performance data improves support for signalprocessing, data visualization, and simulation.

Proceedings of Building Simulation 2011: 12th Conference of International Building Performance Simulation Association, Sydney, 14-16 November.

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BACKGROUNDTime Series RepresentationFigure 1 shows two common techniques for represent-ing a time series, both of which involve sets of consec-utive linear segments. The piecewise linear represen-tation constrains the endpoints of each segment to co-incide with those of neighboring segments (Figure 1a).The piecewise constant representation constrains theslope of each segment to zero, but allows discontinu-ities between adjacent segments (Figure 1b).

(a) Piecewise linear (b) Piecewise constant

Figure 1: Time series representations

This paper focuses on the piecewise constant repre-sentation, which we regard as the more flexible of thetwo. Although the piecewise linear approach seemsappropriate for physical properties like temperaturethat tend to change gradually over time, its taperededges can be misleading if used to represent dramatictransitions and fluctuations commonly found in lightand electrical power data. With enough segments apiecewise constant time series can approximate a con-tinuous quantity, and discrete quantities like numbersof occupants are piecewise constant by nature.In some cases time series are represented such that allsegments must have the same width, while in othercases the segment widths may differ. For example,some sensors produce fixed-width segments by record-ing values at regular time intervals. Other sensorsrecord a new value only when it differs from the pre-vious by a certain minimum amount. A light sensor ofthe latter type in an office environment, for example,may record numerous segments during the day whenindoor luminance patterns change over time. At night,when lighting conditions tend to remain constant, veryfew segments would be recorded.Time series output by simulations can also have fixed-width or variable-width segments. Consider the simplemodel below, in which T (t) and T (t) represent tem-perature and its rate of change at time t.

T (t) = 0.5·(

5·sin(π·t12

)− T (t)

)T (0) = 0

The temperature is approximated by repeatedly evalu-ating the following, with Tapprox being some suitableapproximation for T .

T (t+ ∆t) = T (t) + Tapprox·∆t

The time step ∆t is arbitrary. It is common practiceto use the same time step throughout a simulation, as

done to produce the time series with fixed-width seg-ments shown in Figure 2a. Here Tapprox was obtainedwith the Modified Euler method, which is describedin a similar context in dos Santos and Mendes (2004).The approximation converges on the smooth red lineas the time step is reduced.

(a) Fixed-width segments

(b) Variable-width segments

Figure 2: Time series produced by simulation

If the state of a model changes at fixed time steps, oneis said to be using discrete time simulation. Anotheroption is to vary the time steps: to shorten them for im-proved efficiency when data is changing rapidly, whilelengthening them for computational efficiency whendata is changing slowly. Figure 2b shows another timeseries produced by a Modified Euler temperature sim-ulation. In this case the segments vary in width, asthe time step was repeatedly calculated according tothe following formula. Here a, b, and c are arbitraryconstants.

∆t = max

(a,min

(b,

∣∣∣∣∣ c

Tapprox

∣∣∣∣∣))

Variable time steps are a key feature of discrete eventsimulation, an approach that is often used to integratemodels with different time advancement schemes.

Time Series CompressionGiven a time series y, the task is to produce a new timeseries y while striving to minimize both the memoryrequired to store y and some measure δ of the averagecompression error. We measure this error over the timeperiod ta ≤ t < tb using (1).

δ =

√∫ tb

ta

(y(t)− y(t))2dt (1)

Relevant work on time series compression can befound in the field of data mining. Morbitzer (2003)


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and others have investigated applications of data min-ing to building performance simulation, though in thiscommunity there has been little focus on general timeseries compression techniques. Our impression is that,if a time series of building performance data con-tains too many segments for a particular analysis, itis typically averaged within consecutive time periodsof equal width. This data reduction technique, whichwe refer to as Fixed-Width Compression, is expressedformally in (2).

y(t) =1

∆t·∫ (i+1)·∆t

i·∆t

y(τ)dτ

where i·∆t ≤ t < (i+ 1)·∆t, i ∈ N

(2)

In the data mining community there are many com-pression methods developed for various time seriesrepresentations. Our interest is in relatively simple al-gorithms for which y and y are both piecewise constantwith variable-width segments. Lazaridis and Mehrotra(2003) present one such technique, called Poor Man’sCompression (PMC), which constrains the output timeseries to be within an arbitrary tolerance h of the origi-nal. The algorithm repeatedly merges the two most re-cently acquired segments so long as |y(t)− y(t)| ≤ hremains true for all t. Choosing an appropriate valuefor h may be difficult in the case of building perfor-mance data due to the wide range of possible sensortypes and makes.Keogh et al. (2001a) describe a method that involves“converting the problem into a wavelet compressionproblem, for which there are well known optimal so-lutions, then converting the solution back [to a piece-wise constant representation].” But because it operateson only recently acquired segments, PMC seems eas-ier to apply in real time as data is acquired.The Sliding Window And Bottom-up method (SWAB)from Keogh et al. (2001b) is similar to PMC in thatit operates on recently acquired segments. Instead ofmerging only the two most recent segments, the al-gorithm searches all pairs of adjacent segments ac-quired within a sliding time window. If any pairs canbe merged without violating an arbitrary maximum er-ror level, the pair associated with the least additionalerror is merged. The selective merging procedure iscompelling and can be applied in real time. However,SWAB operates on piecewise linear time series.

PROPOSED ALGORITHMOverviewOur method involves the application of a compressionbuffer that contains a limited number of segments andspans a limited length of time. The algorithm is simi-lar to SWAB in that recently acquired segments enter abuffer where they are selectively merged with adjacentsamples. Unlike SWAB, both y and y are piecewiseconstant. The input time series may have fixed-width

or variable-width segments, but in either case the out-put segments will generally vary in width.The algorithm requires two parameters: the buffer sizem and the target resolution ∆tres. If all input seg-ments are considerably shorter than ∆tres, then theoutput segments will have an average width of approx-imately ∆tres. The output resolution is constrainedsuch that there will never be more output segmentsthan input segments in any time window, and there willnever be more than m+1 output segments completelycontained in a time window of width m·∆tres.Unlike many preexisting compression algorithms, oursdoes not constrain the quality of the output time series.Recall that SWAB limits the compression error asso-ciated with each output segment, while PMC restrictsthe distance between all corresponding input and out-put segments. These quality guarantees conform withthe vision that, given an original time series, a singlecompressed time series is produced and used for allsubsequent operations. Our vision is different. We in-tend our compression algorithm to be used multipletimes with the same time series but different values of∆tres. The result would be several archived copiesof the original time series, each at a different reso-lution. When data is extracted for an analysis, oneselects the resolution that best balances the need forprecision with technological limitations related to pro-cessing power or bandwidth.For our data management strategy, it is more practi-cal for the compression process to restrict resolutionthan error. However, one must still be able to deter-mine whether an analysis was compromised by exces-sive data compression. We therefore formulate the al-gorithm such that a compression error is stored witheach segment of the output time series.

Segment Merging CalculationsWe store a time series as a sequence of datapoints[ti, yi, δi], where ti is the start time of the ith segment,yi is its value, and δi is its error value. For the ith seg-ment to be a complete segment, the (i+ 1)th datapointmust also be available to provide the end time ti+1.Note that a time series can be expressed as a function:y(t) = yi, with i chosen such that ti ≤ t < ti+1.As mentioned, our algorithm compresses a time se-ries by repeatedly merging adjacent segments. Merg-ing the ith and (i+ 1)th segments means replacing theith and (i+ 1)th datapoints with a new one, [ti, yi, δi].This merged datapoint is given by (3) below, where∆ti = ti+1 − ti and ρi2 = yi

2 + δi2.

ti = ti

yi =∆ti·yi + ∆ti+1·yi+1

∆ti + ∆ti+1

δi =

√∆ti·ρi2 + ∆ti+1·ρi+1

2

∆ti + ∆ti+1− y2

i

(3)


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Figure 3: Example of real time compression. Segments shown on a blue background are in the compression buffer.Transition A – B shows the arrival of a datapoints within the buffer’s time window, C – D illustrates the selectivemerging of segments in a full buffer, and E – F shows the arrival of a datapoint beyond the end of the time window.

The compression procedure involves applying (3) tomultiple pairs of segments before deciding which pairto merge. The chosen pair is the one that gives thelowest cost C, which depends on both the error valuesand the widths of the segments.

C = (∆ti + ∆ti+1)·δi − (∆ti·δi + ∆ti+1·δi+1)

One may reduce memory requirements by excludingδi from all datapoints in an original time series, sincethere we assume δi = 0. Whenever two segmentswith different values are merged, however, the result-ing error must be recorded as it influences subsequentmerging operations. A pair of adjacent segments withlarge error values is more likely to be merged thanan otherwise congruent pair with smaller error values.It might be beneficial to assign measurement errorsor simulation-estimated uncertainty values to δi in anoriginal time series, but we leave that idea for futureresearch.

Compression ProcedureStarting with an empty compression buffer, newly ac-quired datapoints are added in real time while ob-serving two constraints. The first constraint is thatthe number of complete segments in the compressionbuffer cannot exceed m. If a new datapoint completesthe (m+1)th segment within the buffer, then two seg-ments must be merged to bring the total back downto m. The second constraint is that the combinedduration of all segments in the buffer cannot exceedm·∆tres. If a new datapoint breaks this condition, oneor more of the oldest segments are released from thebuffer and become part of the output time series.

Figure 3 illustrates the procedure with a size-4 com-pression buffer. The first datapoint is added to thebuffer at stage A. At stage B the second datapoint ar-rives and completes the first segment. The processcontinues until stage C, when the buffer gains a fifthsegment completely contained in the m·∆tres timewindow. Because m = 4, a pair must now be merged.As shown, there are four pairs of adjacent segments toconsider. In this case the third pair yields the smallestvalue of C, and is therefore selected as the least costlyto merge. At stage D this pair has been merged, andonly four segments remain in the buffer.When the next datapoint arrives at stage E, it com-pletes a segment that extends beyond the m·∆trestime window. In this case, in order to keep the buffer’scombined segment width under m·∆tres, the two old-est segments must be released. As shown in stage F,the buffer slides forward to encompass the new data-point. The two released segments have become part ofthe output time series, and may no longer be altered.There are two basic ways to use the algorithm to pro-duce multiple copies of a time series at different reso-lutions. The first option is to apply multiple compres-sion buffers directly to the original time series withdifferent values of ∆tres. The second option is toapply the algorithm once to the original time series,then repeatedly to its own output with successivelycoarser target resolutions (i.e. larger values of ∆tres).Both strategies seem reasonable. The second optionrequires less computation but increases latency, as asegment can enter a compression buffer only after itexits the preceding one. We used the first option forall results shown in this paper.


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ResultsQualitative ObservationsRaw sensor data was acquired with a maximum sam-ple rate of 2 Hz, where each datapoint was recordedonly if its value differed by a certain minimum amountfrom that of the preceding datapoint. This producedpiecewise constant time series data with variable-width segments.In Figure 4, roughly two hours of light sensor datais shown in its original form and after applying com-pression buffers with two different target resolutions.Clearly information is lost as the resolution decreases.Note how the compressed segments contract in placesto better represent prominent features.

Figure 4: Light sensor data compressed with a size-12buffer at 1- and 5-minute target resolutions

A section of the same dataset is shown expanded inFigure 5. Four different compression schemes are ap-plied to the time series, but the target resolution is thesame in each case. The first compressed time series isproduced with Fixed-Width Compression (FWC), theusual means of reducing building performance data.The remaining time series are produced with size-mcompression buffers, abbreviated CB-m.

Figure 5: Light sensor data compressed with a 1-minute target resolution using different schemes

Looking at Figure 5, the prominent rising edge labeledB is apparently preserved by the simple FWC algo-rithm. This turns out to be incidental; the edge is pre-

served only because it happens to coincide with theboundary between one fixed-width output segment andthe next. By contrast, the edge labeled D coincideswith the middle of an output segment, and is conse-quently misrepresented in the compressed time series.Note that these two edges and others emerge from allthree compression buffers intact.With FWC, spikes in the original data that are nar-rower than ∆tres are flattened and widened to someextent. Depending on the situation, a compressionbuffer may preserve such spikes. Looking again atFigure 5, the spikes labeled C and F are considerablyflattened after passing through the size-4 compressionbuffer, yet well represented when the buffer size is in-creased to 12. This makes sense because, as the seg-ments that compose a spike pass through a compres-sion buffer, they will survive unaltered only if thereare other segments in the buffer that can be merged in-stead. When the buffer size is increased to 24, spike Ais preserved as well, though this may have come at theexpense of one or two segments near spike C. Notethat spike E is essentially neglected by all four com-pression schemes. The 1-minute target resolution maybe too coarse preserve this feature.

Analysis of Compression Ratios and ErrorsFrom visual observations we gain an intuitive sensethat the application of a compression buffer outper-forms FWC, and that a buffer with a maximum size ofa dozen or more segments produces better results thana buffer limited to a few segments. To test this percep-tion quantitatively, we applied FWC and compressionbuffers of different sizes to eight different time series.Each time series was acquired over a 4-day period bya different sensor, and compressed with the followingtarget resolutions: 1, 5, 15, and 30 seconds; 1, 5, 15,and 30 minutes; and 1 hour. We recorded the resultingcompression ratios and associated compression errors.The compression ratio is the number of output seg-ments divided by the number of input segments. Thecompression error δ was defined in (1), but becausewe store error values with each output segment, we areable to use the more convenient expression in (4). Herei, ti, and δi pertain to datapoints of the output time se-ries y. The input time series y is no longer needed tocompute the error.

δ =

√√√√ 1

tb − ta·b−1∑i=a

(ti+1 − ti)·δi2 (4)

Figure 6 shows a scatter plot of the compression ratiosand errors obtained using data from four temperaturesensors. As expected, relatively small compressionratios are associated with relatively high errors. Foreach compression scheme (i.e. FWC, CB-4, CB-12, orCB-24), the points in the scatter plot are quite spreadout. This indicates that the errors one encounters fora given compression ratio vary significantly from one


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Figure 6: Temperature data compression statistics.

time series to the next, even if both are collected usingthe same type of sensor. Despite large deviations, theerrors produced by FWC are noticeably higher thanthose produced by the compression buffers, particu-larly as compression ratios approach 1. There also ap-pears to be a slight tendency for compression errors todecrease with increasing buffer size.Because a compression buffer can transform a timeseries only by merging segments, the number of seg-ments cannot increase and the compression ratio can-not exceed 1. Furthermore, as reflected in Figure 6, thecompression error will converge on zero as the com-pression ratio approaches 1. With FWC, by contrast,the number of output segments can exceed the num-ber of input segments. Care must to taken with FWCto avoid the case of the rightmost points of Figure 6,where an excessively fine target resolution increasesthe size of a time series and still introduces error.For the light and electrical current data, the results foreach sensor are shown in a separate scatter plot. Forlight sensor 1, in Figure 7a, the errors produced byall three compression buffers are roughly four timessmaller than those produced by FWC over a widerange of compression ratios. For light sensor 2, in Fig-ure 7b, the compression buffers still appear to outper-form FWC, but the relative difference is only appre-ciable for compression ratios greater than 0.01.On the right-hand side of Figure 7b, and in a few othercases as well, the size-12 and size-24 compressionbuffers yield noticeably smaller errors than the size-4 buffer. This is consistent with our qualitative ob-servations, as larger compression buffers appear morelikely to preserve spikes. Somewhat surprisingly, thereare many cases in which all three compression buffersproduce essentially the same error. It is possible thatwhile certain spikes are prominent in appearance, theymay be too small in width to have a noticeable effecton our quantitative error measurements.

We note that for the two leftmost points of CB-24in Figure 7b, the compression error unexpectedly in-creases with target resolution. The reason this is pos-sible is that, given a finer target resolution, the com-pression buffer operates over a shorter time windowand may miss opportunities to better “distribute” theoutput segments.

(a) Light sensor 1

(b) Light sensor 2

Figure 7: Light sensor data compression statistics


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The electrical current sensors were set up to monitorpersonal computers in an office. As shown in Figure 8,the compression buffers again outperform FWC, par-ticularly for compression ratios greater than 0.1.

(a) Current sensor 1

(b) Current sensor 2

Figure 8: Electrical data compression statistics

FWC has one advantage not represented in the scatterplots: the individual segment start times ti need not bestored in the output time series. Assuming one storesonly the value yi of each segment, and assuming allscalar values require the same storage space, the FWCcompression ratios should all be reduced by a factor of2. With this correction the compression buffers wouldstill outperform FWC for light sensor 1, but in cer-tain other cases FWC would appear more competitive.The assumptions that lead to this 2-fold correction arequestionable, however. We argue that each output seg-ment should contain an error value δi, which impliesa 3:2 correction. If one were to store the minimumand maximum input values as well, there would be lessbenefit still in discarding the start times.

APPLICATIONSApplication to Signal Processing and VisualizationOur approach to time series compression supports fur-ther signal processing and visualization in a number ofways. First, by precomputing and storing copies of atime series at different resolutions, one supports inter-

active graphs like that of the well known web-basedapplication Google Finance. With this type of visual-ization tool, the resolution of the visible data changesautomatically as the user manipulates the time periodof interest. Second, time series with variable-widthsegments may be processed with fewer computations,as few segments are needed for relatively flat regions.Third, the error values stored in the compressed seg-ments provide useful information.Here we demonstrate how a common signal process-ing routine, a low-pass filter, can be applied to a com-pressed time series produced by the proposed algo-rithm. Filtering is typically formulated as a convolu-tion using a window function. We assume that a win-dow function f is defined only between −1 and 1, andnormalized such that

∫ 1

−1f(x)dx = 1. With variable-

width segments, we require the corresponding cumu-lative function F , where F (x) =

∫ x

−1f(u)du. Below

are f and F for the Hann function, a window functionfrequently used by low-pass filters.

f(x) =1 + cos(π·x)

2, −1 < x < 1

F (x) =1

2·(

1 + x+sin(π·x)

π

), −1 < x < 1

With errors values stored in the compressed segments,we can produce not only a filtered signal z, but alsoan approximation s of the signal’s variability. When zand s are evaluated at time t, the window function iscentered on t and scaled such that its half-width in thetime domain matches the parameter d. Both z and sare defined below, with ρi2 = yi

2 + δi2 and ∆ρi

2 =ρi+1

2− ρi2. The integers a and b are chosen such thatta ≤ (t− d) < ta+1 and tb ≤ (t+ d) < tb+1.

z(t) = yb −b−1∑i=a

(yi+1 − yi)·F(ti+1 − t

d

)

s(t) =

√√√√ρb2 −b−1∑i=a

∆ρi2·F(ti+1 − t

d

)− z(t)2

Figure 9 shows roughly five hours of electrical data,the same data compressed with a size-12 buffer at a2-minute target resolution, and the result of applyinga low-pass filter to both the original and compressedtime series. The filter uses a Hann window with a5-minute half-width. In the CB-12 plot, the regionbounded by yi ± δi is shaded. For the filtered data,the shaded regions are bounded by z(t)± s(t).The filtered data in Figure 9 can be sampled to approx-imate the average power consumption and noise levelof a personal computer near a certain time t. Directlysampling the original or compressed time series is lessinformative, since t may coincide with an anomalousspike in the data. Note that it is far more efficient to


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Figure 9: Electrical data with low-pass filtering ap-plied to the original and compressed time series

filter the compressed data than the original data, as inthis case the latter contains roughly a hundred timesmore datapoints. Yet because the filter parameter d issomewhat greater than ∆tres, applying the filter to thecompressed time series produces essentially the sameresult as applying it to the original.

Application to Building Performance SimulationThe proposed compression algorithm is appropriatefor building performance data because it can be ap-plied to a wide range of sensor types without the risk ofselecting inappropriate quality guarantees, and with-out the risk of increasing the number of segments inan unexpectedly sparse time series. A database wouldrecord a separate time series for each sensor and eachtarget resolution. Subsets of various time series wouldthen be extracted from the database at requested reso-lutions to serve as simulation input data.For a typical discrete time simulation, compressed in-put data would be requested at a resolution somewhatfiner than the simulation time step. Although the useof variable-width segments would reduce bandwidthrequirements between the database and the simulationplatform, the data would have to be converted into afixed-width representation before entering the simu-lation. For a discrete event simulation, as applied tobuilding performance in Zimmerman (2001), variable-width segments could be input directly. Simulations ofeither type should be able to input the compression er-rors, and use them to help estimate uncertainty.

Future building energy models may become so de-tailed that real time compression is required for sim-ulation output data as well as input data. A trendtowards longer simulation periods has been been ob-served by Morbitzer (2003), who also highlights theimportant information provided by short-term fluctua-tions found in long-term simulation results.

REFERENCESBarnaby, C. S. and Crawley, D. B. 2011. Weather data

for building performance simulation. In Hensen, J.L. M. and Lamberts, R., editors, Building Perfor-mance Simulation for Design and Operation, pages37–55. Spon Press.

dos Santos, G. H. and Mendes, N. 2004. Analysis ofnumerical methods and simulation time step effectson the prediction of building thermal performance.Applied Thermal Engineering, 24(8–9):1129–1142.

Eisenhower, B., Maile, T., Fischer, M., and Mezic,I. 2010. Decomposing Building System Data forModel Validation and Analysis Using the KoopmanOperator. In Proceedings of the National IBPSA-USA Conference, New York, USA.

Keogh, E., Chakrabarti, K., Pazzani, M., and Mehro-tra, S. 2001a. Locally adaptive dimensionality re-duction for indexing large time series databases.In Proceedings of the ACM SIGMOD InternationalConference on Management of Data, Santa Barbara,USA.

Keogh, E., Chu, S., Hart, D., and Pazzani, M. 2001b.An Online Algorithm for Segmenting Time Series.In Proceedings of the IEEE International Confer-ence on Data Mining, San Jose, USA.

Kunz, J., Maile, T., and Bazjanac, V. 2009. Summaryof the Energy Analysis of the First year of the Stan-ford Jerry Yang & Akiko Yamazaki Environment &Energy (Y2E2) Building. Technical report, CIFE,Stanford University.

Lazaridis, I. and Mehrotra, S. 2003. Capturing Sensor-Generated Time Series with Quality Guarantees.In Proceedings of the International Conference onData Engineering, Bangalore, India.

Morbitzer, C. 2003. Towards the Integration of Simu-lation into the Building Design Process. PhD thesis,University of Strathclyde, Glasgow, Scotland.

Raftery, P., Keane1and, M., and Costa, A. 2009. Cal-ibration of a Detailed Simulation Model to EnergyMonitoring System Data: A Methodology and CaseStudy. In Proceedings of the International IBPSAConference, Glasglow, Scotland.

Zimmerman, G. 2001. A New Approach to Build-ing Simulation Based on Communicating Objects.In Proceedings of the International IBPSA Confer-ence, Rio de Janeiro, Brazil.


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REAL-TIME COMPRESSION OF TIME SERIES BUILDING … · ments are considerably shorter than t res, then the output segments will have an average width of approx-imately t res. The output