Time Series Analysis L. Torgo ltorgo@fc.up.pt Faculty of Sciences, University of Porto May 2014 3, Time Series Data A Definition Introduction Definition • A time series is a set of observations of a variable that are ordered by time. • E.g., X\ iX2 j ' * * j Xt-\ ? Xti Xt+-\ j ' * * ; *A7 where x? is the observation of variable X at time t. 9 A multivariate time series is a set of observations of a set of variables over a certain period of time. L.Torgo (FCUP/UP) TimeSeries May 2014 2/57 Introduction The Main Goals of Time Series Analysis Explanation Obtaining a Time Series Model help us to have a Deeper Understanding of the Mechanism that Generated the Observed Time Series Data. The Past! Forecasting • Given: • Obtain: a time series model • Which is able to make predictions concerning: xt+-\, • • • , xn The Future! Other Goals Time Series Data Mining © L.Torgo (FCUP / UP) TimeSeries May 2014 3/57 Introduction Main Time Series Data Mining Tasks • Indexing (Query by Content) Given a query time series Q and a similarity measure D(Q, X) find the most similar time series in a database D • Clustering Find the natural goupings of a set of time series in a database D using some similarity measure D(Q, X) • Classification Given an unlabelled time series Q, assign it a label C from a set of pre-defined labels (classes) L.Torgo (FCUP/UP) TimeSeries May 2014 4/57 Introduction fime Series Data in R R has several data structures capable of handling time series data In our illustration we will use the infra-structure provided by package xts > library(xts) > data{ice.river,package='tseries' ) > ice.river [ 1:4,] flow.vat flow.jok prec temp [1, ] 16.1 30 . 2 8 . . 1 0 . . 9 [2, ] 19.2 29.0 4 . . 4 1 . . 6 [3, ] 14 . 5 28 . 4 7 . . 0 0 . . 1 [4, ] 11 . 0 27 . 8 0 . . 0 0 . . 6 > ir <- xts (ice. river[, 1] , seq.Date(as.Date{' 1972-01-01') by='day', len=nrow{ice.river))) © L.Torgo (FCUP / UP) TimeSeries May 2014 5/57 Introduction Time Series Data in R - indexing examples > ir[l:3] [,1] 1972-01-01 16.1 1972-01-02 19.2 1972-01-03 14 . 5 > ir[' 1973- -05-02' ] [,1] 1973-05-02 11.6 > ir[' 1972- -01' ] [, 1] 1972-01-01 16.10 1972-01-02 19.20 > ir[' 1972- -01-23/1972-02-02' [,1] 1972-01-23 6 . 90 1972-01-24 6 . 90 > ir ['/1972-01-10'] [, 1] 1972-01-01 16.10 1972-01-02 19.20 > ir [' 1974-12-21/'] [, 1] 1974-12-30 5.34 1974-12-31 5.34 > ir [' 1972-01-23/1972-02' [, 1] 6 . 90 6 . 90 1972-01-23 1972-01-24 1974-12-30 5.34 1974-12-31 5.34 liaadO *h L.Torgo (FCUP/UP) TimeSeries May 2014 6/57 Exploratory Analysis Summaries Summaries of Time Series Data • Standard descriptive statistics (mean, standard deviation, etc.) do not allways work with time series (TS) data. • TS may contain trends, seasonality and some other systematic components, making these stats misleading. o So, for proving summaries of TS data we will be interested in concepts like trend, seasonality and correlation between sucessive observations of the TS. © L.Torgo (FCUP / UP) TimeSeries May 2014 7/57 Exploratory Analysis Variation Types of Variation Seasonal Variation Some time series exhibit a variation that is annual in period, e.g. demand for ice cream. Other Cyclic Variation Some time series have periodic variations that are not related to seasons but to other factors, e.g. some economic time series. Trends A trend is a long-term change in the mean level of the time series. J © L.Torgo (FCUP / UP) TimeSeries May 2014 Exploratory Analysis Stationarity An Informal Definition A time series is said to be stationary if there is no systematic change in mean (no trend), if there is no systematic change in variance and if strictly periodic variations have been removed. Note that in these cases statistics like mean, standard deviation, variance, etc., bring relevant information! © L.Torgo (FCUP / UP) TimeSeries May 2014 9/57 Exploratory Analysis Time Plots Time Plots -j 1974.5 1975.0 Time Ploting the time series values against time is one of the most important tools for analysing its behaviour. Time plots show important features like trends, seasonality, outliers and discontinuities. L.Torgo (FCUP/UP) TimeSeries May2014 10/57 Exploratory Analysis rime Plots in R ir > plot(ir) o LO o o CO o CM 1 1 1 ........ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 if 1 1 1 1 1 1 1 1 1 1 1 1 1 1 , ____L____ I: J H4\i J Jan 01 Jul 01 Jan 01 Jul 01 1972 1972 1973 1973 Jan 01 Jul 01 Dec 31 1974 1974 1974 öi liaadO wb © L.Torgo (FCUP / UP) TimeSeries May 2014 11/57 Exploratory Analysis Transformations - I Plotting the data may suggest transformations To stabilize the variance Symptoms: trend with the variance increasing with the mean. Solution: logarithmic transformation. To make the seasonal effects additive Symptoms: there is a trend and the size of the seasonal effect increases with the mean(multiplicative seasonality). Solution: logarithmic transformation. To remove trend Symptoms: there is systematic change on the mean. Solution 1: first order differentiation (VXt = Xt - Xt_^). Solution 2: model the trend and subtractit from the original series (Yt = Xt-rt). L.Torgo (FCUP/UP) TimeSeries May 2014 12/57 Exploratory Analysis Transformations - an example (1) Transformations - a simple example (1) An example time series with trend and a multiplicative seasonality effect. © L.Torgo (FCUP / UP) TimeSeries May 2014 13/57 Exploratory Analysis Transformations - an example (2) Transformations - a simple example (2) yt = xt- (7.708 + 42.521 xt) i I I I I r 0 20 40 60 80 100 time zt = iogi0(yt) \-1-1-1-1-r 0 20 40 60 80 100 time >. o — 7 T 1-1-1-1-r 0 20 40 60 80 100 time zt-z w, \-1-1-1-1-r 0 20 40 60 80 100 time L.Torgo (FCUP/UP) TimeSeries May 2014 14/57 Exploratory Analysis Transformations - an example (2) Some useful functions in R > (s <- ir[l: 10] ) > diff(s,diff=2) [, 1] [, 1] 1972-01-01 16 . 10 1972-01-01 NA 1972-01-02 19 .20 1972-01-02 NA 1972-01-03 14 .50 1972-01-03 -7 . 80 1972-01-04 11 . 00 1972-01-04 1.20 1972-01-05 13 . 60 1972-01-05 6.10 1972-01-06 12 .50 1972-01-06 -3.70 1972-01-07 10 .50 1972-01-07 -0.90 1972-01-08 10 . 10 1972-01-08 1.60 1972-01-09 9 . 68 1972-01-09 -0.02 1972-01-10 9 . 02 1972-01-10 -0.24 > diff(s) > loglO(s) [, 1] [, 1] 1972-01-01 NA 1972-01-01 1.2068259 1972-01-02 3 . 10 1972-01-02 1.2833012 1972-01-03 -4 .70 1972-01-03 1.1613680 1972-01-04 -3 .50 1972-01-04 1.0413927 1972-01-05 2 . 60 1972-01-05 1.1335389 1972-01-06 -1 . 10 1972-01-06 1.0969100 1972-01-07 -2 . 00 1972-01-07 1.0211893 1972-01-08 -0 .40 1972-01-08 1.0043214 1972-01-09 -0 .42 1972-01-09 0.9858754 1972-01-10 -0 . 66 1972-01-10 0 . 9552065 © L.Torgo (FCUP / UP) TimeSeries May 2014 15/57 Exploratory Analysis Autocorrelation Autocorrelation Sample Autocorrelation Coefficients They measure the correlation between observations different distances apart. Or liaadO *h L.Torgo (FCUP/UP) TimeSeries May2014 16/57 Correlogram Exploratory Analysis Time i i i r 5 "1—I—r ------ 10 Lag J_i_I_L 15 Plot the sample autocorrelation coefficients against the lags, /c = 0,1,- - ,M. Correlograms in R © L.Torgo (FCUP / UP) TimeSeries May 2014 17/57 Exploratory Analysis Autocorrelation > par(mfrow=c(2 , 1) ) > plot(ir) > acf(ir) L.Torgo (FCUP/UP) Jan 01 Jul 01 Jan 01 Jul 01 Jan 01 Jul 01 Dec 31 1972 1972 1973 1973 1974 1974 1974 0 5 10 15 20 25 TimeSeries May 2014 18/57 Exploratory Analysis Interpreting the Correlogram Random Series Most rk's near 0. Still, it is possible that 1 on 20 is significant... Short-Term Correlation Fairly large value of r\ with sucessive values rapidly tending to non-significant. \-1-1-1-1-r 0 100 200 300 400 500 i1.!1!1"!!11''1'1!1 i-1-1-r 10 15 20 25 I 10 © L.Torgo (FCUP / UP) TimeSeries May 2014 19/57 Exploratory Analysis Autocorrelation Interpreting the Correlogram (cont.) Alternating Series Similar pattern on the values of rk. Non-Stationary Series For series with a trend the values of rk will not go down till very large values of the lag. Seasonal Series The correlagram tends to exhibit the same periodicity as the original series. TT n-1—r n—i— 8 10 12 1955 1 L.Torgo (FCUP/UP) TimeSeries May 2014 20/57 Exploratory Analysis Handling Real World Data A Check List of Common Sense Things to Do (taken from Chatfield, 2004) • Do you understand the context? Have the right variables been measured? • Have all the time series been plotted? • Are there missing values? If so, what should be done about them? • Are there any outliers? If so, what should be done about them? • Are there any discontinuities? If so, what do they mean? • Does it make sense to transform the variables? • Is trend present? If so, what should be done about it? • Is seasonality present? If so, what should be done about it? Chatfield, C. (2004): The Analysis of Time Series - an introduction. CRC. © L.Torgo (FCUP / UP) TimeSeries May 2014 21 /57 Evaluation Goals of an Evaluation Method • The golden rule: The data used for evaluating (or comparing) any models cannot be seen during model development. • The goal of any evaluation procedure: ► Obtain a reliable estimate of some evaluation measure. High probability of achieving the same score on other samples of the same population. • Evaluation Measures ► Predictive accuracy. ► Model size. ► Computational complexity. L.Torgo (FCUP/UP) TimeSeries May 2014 22/57 Evaluation Reliable Estimates Performance Estimation for Time Series Models • The usual techniques for model evaluation revolve around resampling. ► Simulating the reality. * Obtain an evaluation estimate for unseen data. • Examples of Resampling-based Methods ► Holdout. ► Cross-validation. ► Bootstrap. Time Series Data Are Special! Any form of resampling changes the © L.Torgo (FCUP / UP) TimeSeries May 2014 23/57 Evaluation Evaluation Methodology Correct Evalution of Time Series Models • General Guidelines ► Do not "forget" the time tags of the observations. ► Do not evaluate a model on past data. • A possible method ► Divide the existing data in two time windows * Past data (observations till a time t). * "Future" data (observations after t). ► Use one of these three learn-test alternatives * Fixed learning window. * Growing window. * Sliding window. a liaadO *h © L.Torgo (FCUP / UP) TimeSeries May 2014 24/57 Evaluation Evaluation Methodology Learn-Test Strategies Fixed IZ -r— Fixed Window A single model is obtained with the available "training" data, and applied to all test period. Growing Window Every wv test cases a new model is obtained using all data available till then. Sliding Window Every wv test cases a new model is obtained using the previous ws observations of the time series. © L.Torgo (FCUP / UP) TimeSeries May 2014 25/57 Evaluation Evaluation Methodology Dealing with model selection • Most modelling techniques involve some form of parameters that usually need to be tunned. • The following describes an evaluation methodology considering this issue: yi • • • ys • • • > • • • yn Stage 1 Data used for obtaining the model alternatives Model tunning and selection period Stage 2 Data used for obtaining the selected model alternative / variant Final Evaluation Period L.Torgo (FCUP/UP) TimeSeries May 2014 26/57 Evaluation Evaluation Measures Some Metrics for Evaluating Predictive Performance Absolute Measures • Mean Squared Error (MSE) MSE = 1 f> - Xi)2 /=1 • Mean Absolute Deviation (MAD) MAD 1 n -> \Xj-Xj\ n ^ /=1 Relative Measures • Theil Coefficient U yjY.U&i-Xi)2 ^ElÁXi-x^r • Mean Absolute Percentage Error (MAPE) MAPE 1 /'=1 © L.Torgo (FCUP / UP) TimeSeries May 2014 27/57 Evaluation Experimental Comparisons The Goal of an Experimental Comparison • Given a set of observations of a time series X. • Given a set of alternative modelling approaches M. • Obtain estimates of the predictive performance of each at?/ for this time series. More specifically, given a forecasting period size, wtest, and a predictive performance statistic, Err, we want to obtain a reliable estimate of the value of Err for each m,. L.Torgo (FCUP/UP) TimeSeries May 2014 28/57 Experimental Comparisons A possible approach would be to use our proposed method of Model Selection. This would give us estimate of Err. More reliability is achievable if more repetitions of the process are carried out. TimeSeries May 2014 29/57 Experimental Comparisons Monte Carlo Estimates for Time Series Forecasting Given: a time series, a training window size, wtrain, a testing window size, wtest, and a number of repetitions, r, - randomly generate r points in the interval ]wtraim (n - wtest)l - for each point learn a model with data in interval [r - wtrairh r] and test it with the data in the interval [r + 1, r + wtest] Time Series Models Linear Approaches Assumptions of "Classical" Linear Approaches • Linearity The model of the time series behaviour is linear on its inputs. • Stationarity The underlying equations governing the behaviour of the system do not change with time. Most "classical" approaches assume stationary time series, thus one usually needs to transform non-stationary time series into stationary ones before using these tools. © L.Torgo (FCUP / UP) TimeSeries May 2014 31 /57 Time Series Models Linear Approaches Integrated ARMA (or ARIMA) Models Definition An integrated ARMA (or ARIMA) model of order p, c/, q is a series given by where Wt = VdXt is a d order difference. L.Torgo (FCUP/UP) TimeSeries May 2014 32/57 Time Series Models Linear Approaches ARIMA models in R > train <- ir ['/1973-12-31'] > test <- ir [' 1974-01-01/'] > > mad <- function (t,p) mean(abs (t-p)) > > > prevsARIMA <- function (tr, ordem) { + modelo <- arimaO(as.vector(tr), ordem) + as.vector(predict(modelo, n.ahead = l)$pred) + } > > ARIMA <- function (train, test, ord) { + pre <- rollapply (c(train, test), length(train), prevsARIMA, align = "right", ordem = ord) as.xts(lag(pre, -1)) > prevs <- ARIMA (train,test,c (3,1,2)) > mad(test,prevs) [1] 1.040762 © L.Torgo (FCUP / UP) TimeSeries May 2014 33/57 Time Series Models Linear Approaches ARIMA models in R - 2 test > plot(test) > lines (prevs, col= red' ) o LO o O CO o CM "i-r T "i-1-1-r "i-r Jan 01 Mar 01 May 01 1974 1974 1974 Jul 01 Sep 01 Nov 01 Dec 3;Ucf^l 1974 1974 1974 1974s" L.Torgo (FCUP/UP) TimeSeries May 2014 34/57 Time Series Models Non-Linear Approaches Delay-Coordinate Embedding Theorem (Takens, 1981) Informally, it states we can uncover the dynamics of any time series given the information on e past values of the series. For that to be possible we need to know the correct embed size (how far back in time to look) © L.Torgo (FCUP / UP) TimeSeries May 2014 35/57 Time Series Models Non-Linear Approaches An Example of Delay-Coordinate Embedding Example Given the time series, y-\, 72,73, the resulting embed vectors are, yioo, an embed dimension of 5, '5 re ri r8 < 75,74,73,72,71 > < 76,75,74,73,72 > < 77,76,75,74,73 > < 78,77,76,75,74 > L.Torgo (FCUP/UP) TimeSeries May 2014 36/57 Time Series Models Non-Linear Approaches Consequences of Delay-Coordinate Embedding If the system dynamics can be captured by a certain embed, then we may try to model the relationship between the state of the system and the future values of the series. That is, we can try to obtain a model of the form, Yt+h = m This modelling task can be handled by any multiple regression tool we have studied before! © L.Torgo (FCUP / UP) TimeSeries May 2014 37/57 Time Series Models Non-Linear Approaches An Example Using Artificial Neural Networks (ANNs) o ANNs are highly non-linear modelling techniques. • Given the embedding representation we may use an ANN for our modelling task. • Such network can be used to obtain predictions for Yt+h, where h is the forecasting horizon, given the current embed. L.Torgo (FCUP/UP) TimeSeries May 2014 38/57 Time Series Models Non-Linear Approaches An Example with SVMs > create.data <- function(ts,embed) { + t <- index(ts) [-(1: (embed-1))] + e <- embed(ts,embed) + colnames (e) <- paste ('V , embed: 1, sep='' ) + xt s (e, t) + } > ds <- create.data(ir,5) > train <- ds['/1973-12-31'] > test <- ds['1974-01-01/'] > > library(el071) > > m <- svm(V5 ~ .,train,cost=10) > p.s <- predict(m,test) > mad(test,p.s) [1] 1.409197 © L.Torgo (FCUP / UP) TimeSeries May 2014 39/57 Time Series Models Non-Linear Approaches SVMs- 2 test[, "V5"] o LO o plot (test [,'V5'] ) n lines (xts(p.s,index(test)),col o CM red' "i-r T "i-1-1-r "i-1-r Jan 01 Mar 01 May 01 Jul 01 Sep 01 Nov 01 Dec3;uAI 1974 1974 1974 1974 1974 1974 1974s © L.Torgo (FCUP / UP) TimeSeries May 2014 40/57 2D-lnterval Predictions for Time Series 2D-lnterval Predictions for Time Series What? Goal Forecast the range of plausible values of a time series for a future time interval Point Predictions Interval Predictions 2D-Interval Predictions t+m L.Torgo (FCUP/UP) TimeSeries May 2014 41 /57 2D-lnterval Predictions for Time Series 2D-lnterval Predictions for Time Series Why? Motivation • Several applications require planning ahead based on forecasts (e.g. production planning based on sales forecast) • Other applications require decisions to be made based on predictions of expected scenarios for some future time interval (e.g. financial investments) • Our work was driven by a particular application : water quality control on a large distribution network L.Torgo (FCUP/UP) TimeSeries May 2014 42/57 2D-lnterval Predictions for Time Series The AdDP application Monitoring Water Quality Parameters AdDP company □ EsUivio ľl" Cbragern » ETA • EsCa^Ěo Elevatůrla ft Reserva tóría q Reseruatorío s Elevatória • Fronto de Entrega Hundreds of parameters have to be monitored Legal limits with heavy fines In-company tighter limits for generating alarms and inspection decisions Very dynamic system -notion of "normality" varies along the year L.Torgo (FCUP/UP) TimeSeries May 2014 43/57 2D-lnterval Predictions for Time Series The AdDP application Monitoring Water Quality Parameters Application Goals At the beginning of each month provide an interval of expected values for a set of parameters (interval of "normality") Values outside these intervals should generate alarms and lead to inspection actions These alarms and inspections may lead to preventive actions to avoid surpassing the legal limits L.Torgo (FCUP/UP) TimeSeries May 2014 44/57 2D-lnterval Predictions for Time Series The AdDP application Monitoring Water Quality Parameters Defining the Data Mining Task • What we want: provide an interval of expected values for a set of time series (interval of "normality") • Can be seen as a form of summary statistic of the unknown future distribution of values of the series • We will use the interquartile range as target summary statistic ► Based on the 1st and 3rd quartiles ► Roughly 50% of the cases are supposed to be inside that range • Summarizing: our task will be to obtain estimates of the 1st and 3rd quartiles for a future time interval © L.Torgo (FCUP / UP) TimeSeries May 2014 45/57 2D-lnterval Predictions for Time Series State of the Art ► ► ► Obtaining Predictions for a Future Time Interval Possible Approaches with Existing Work • Iterated Predictions At time t obtain a prediction for time t + 1 Use this prediction as if it was past and obtain a prediction for t + 2 Iterate this process until we have predictions for the target interval ► With the k predictions calculate the 1st and 3rd quartiles to obtain the interval of values ► Potential Drawback: Accumulate errors ^-models ► Obtain k different models, each "designed" to predict the value t + /', where / e [1, k] ► With the k predictions of the k models calculate the 1st and 3rd quartiles to obtain the interval of values ► Potential Drawback: Computational complexity for large values of k or online scenarios o L.Torgo (FCUP/UP) TimeSeries May 2014 46/57 2D-lnterval Predictions for Time Series Our Proposal in a Nutshell Our Proposal The Key Idea Directly predict the summary statistics instead of the future values of the series Motivation Quantiles are robust statistics with a distribution that is smoother than the original series. Our hypothesis is that predicting them should be easier. © L.Torgo (FCUP / UP) TimeSeries May 2014 47/57 2D-lnterval Predictions for Time Series Our Proposal More Formally... Formalization Let and be the a and (3 unknown quantiles of the time series values for a future time window of size k. Define the following prediction problems: Qa = f(vu--- ,va) and Qp = f(v^--- , va), where v-\, • • • , va are a set of descriptor variables. L.Torgo (FCUP/UP) TimeSeries May 2014 48/57 2D-lnterval Predictions for Time Series Experimental Setup Goal Compare our approach (quantiies) with the two other approaches (iterated and k-models) Used Predictive Models Random Walk (rw), Regression Trees (rt), SVMs (svm), Random Forests (rf) and Quantile Random Forests (qrf) liaadO w b © L.Torgo (FCUP / UP) TimeSeries May 2014 49/57 2D-lnterval Predictions for Time Series Experimental Evaluation Experimental Setup (2) Estimation Method Monte Carlo simulation with 10 repetitions at randomly selected points in time. Estimates for different values of the future time window (k) Data All alternatives using the same predictor variables and model settings. Only difference is on the way the predictions for the 1st and 3rd quartiles are obtained. L.Torgo (FCUP/UP) TimeSeries May 2014 50/57 2D-lnterval Predictions for Time Series Experimental Setup (3) Evaluation Metrics • Total Quantile Error (TQE) • Mean Absolute Quantile error (MAQ) • Total Utility of predictions {Utility) a-{y- y)^ ify > y (1 — a) • (y — y) otherwise n TQE = J2 /'=1 i+k i+k E Qd.25,/) + E ^-75(yy» Pd.75,/) j=i j=i MAQ = ^~ 2/7 n IQo.25,/' _ Öq.25,/ + IQo.75,/' _ a liaadü wb © L.Torgo (FCUP / UP) TimeSeries May 2014 51 /57 low normal high 2D-lnterval Predictions for Time Series Experimental Evaluation Experiments with Artificial Time Series TGT = f(Ytl--- , ^-9,Q0.25,P °0.75,P ^ L.Torgo (FCUP/UP) TimeSeries May 2014 52/57 2D-lnterval Predictions for Time Series Experimental Evaluation Results with Artificial Time Series E I P3 TQE for Serles=1, k=10 RW sco re =127 MAQ for Series= 4, k=20 RW score=36 Utility for Serless?, k=30 RW score =-6 ORF ■ RF ■ SVM □ RT 10 15 20 © L.Torgo (FCUP / UP) TimeSeries May 2014 53/57 2D-lnterval Predictions for Time Series Experimental Evaluation Results with Water Quality Parameters E p3 TQE for Iron, k=30 RWscore=203.13 MAO for pH, k=30 RW score=0.09 Utility for Turbidity, k=30 RW score=16.93 ORF ■ RF ■ SVM □ RT liaadö *h © L.Torgo (FCUP / UP) TimeSeries May 2014 54/57 ^528542 2D-lnterval Predictions for Time Series Results with Water Demand Forecasting Goal Forecast the interval of plausible values for the water demand in the network for the next 12 and 24 hours. TQE for Water Consumption, k=12 MAQ for Water Consumption, k=12 Utility for Water Consumption, k=24 RW score=S1 RW score=4 RW score=16 _I _I _ G 20 40 SO SO 0 1 2 3 4 5 0 5 10 15 20 25 Ll ■ QRF ■ RF □ SVM □ RT © L.Torgo (FCUP / UP) TimeSeries May 2014 55/57 2D-lnterval Predictions for Time Series Experimental Evaluation Further Information Full details: L Torgo and O. Ohashi (2011) : 2D-lnterval Predictions for Time Series, in Proceedings of 17th ACM SIGKDD Conference on Knowledge Discovery and Data Mining (KDD'2011) • All code and data of the KDD paper and also the full report of all experiments that were carried out are available at http://www.dec.fc.up.pt/-Itorgo/KDD 2D-lnterval Predictions for Time Series Summary/Conclusions o New type of forecasting tasks for time series with high applicability • Proposed a method to address these tasks • Encouraging results in rather different setups when taking both accuracy and computation time into account L.Torgo (FCUP/UP) TimeSeries May 2014 57/57