{"title": "Adaptive Learning of Smoothing Functions: Application to Electricity Load Forecasting", "book": "Advances in Neural Information Processing Systems", "page_first": 2510, "page_last": 2518, "abstract": "This paper proposes an efficient online learning algorithm to track the smoothing functions of Additive Models. The key idea is to combine the linear representation of Additive Models with a Recursive Least Squares (RLS) filter. In order to quickly track changes in the model and put more weight on recent data, the RLS filter uses a forgetting factor which exponentially weights down observations by the order of their arrival. The tracking behaviour is further enhanced by using an adaptive forgetting factor which is updated based on the gradient of the a priori errors. Using results from Lyapunov stability theory, upper bounds for the learning rate are analyzed. The proposed algorithm is applied to 5 years of electricity load data provided by the French utility company Electricite de France (EDF). Compared to state-of-the-art methods, it achieves a superior performance in terms of model tracking and prediction accuracy.", "full_text": "Adaptive Learning of Smoothing Functions:\nApplication to Electricity Load Forecasting\n\nAmadou Ba\n\nIBM Research - Ireland\nMulhuddart, Dublin 15\n\namadouba@ie.ibm.com\n\nmathsinn@ie.ibm.com\n\nMathieu Sinn\n\nIBM Research - Ireland\nMulhuddart, Dublin 15\n\nYannig Goude\n\nEDF R&D\n\nClamart, France\n\nyannig.goude@edf.fr\n\nPascal Pompey\n\nIBM Research - Ireland\nMulhuddart, Dublin 15\n\npapompey@ie.ibm.com\n\nAbstract\n\nThis paper proposes an ef\ufb01cient online learning algorithm to track the smoothing\nfunctions of Additive Models. The key idea is to combine the linear representa-\ntion of Additive Models with a Recursive Least Squares (RLS) \ufb01lter. In order to\nquickly track changes in the model and put more weight on recent data, the RLS\n\ufb01lter uses a forgetting factor which exponentially weights down observations by\nthe order of their arrival. The tracking behaviour is further enhanced by using an\nadaptive forgetting factor which is updated based on the gradient of the a priori\nerrors. Using results from Lyapunov stability theory, upper bounds for the learn-\ning rate are analyzed. The proposed algorithm is applied to 5 years of electricity\nload data provided by the French utility company Electricit\u00b4e de France (EDF).\nCompared to state-of-the-art methods, it achieves a superior performance in terms\nof model tracking and prediction accuracy.\n\n1\n\nIntroduction\n\nAdditive Models are a class of nonparametric regression methods which have been the subject of\nintensive theoretical research and found widespread applications in practice (see [1]). This con-\nsiderable attention comes from the ability of Additive Models to represent non-linear associations\nbetween covariates and response variables in an intuitive way, and the availability of ef\ufb01cient train-\ning methods. The fundamental assumption of Additive Models is that the effect of covariates on\nthe dependent variable follows an additive form. The separate effects are modeled by smoothing\nsplines, which can be learned using penalized least squares.\nA particularly fruitful \ufb01eld for the application of Additive Models is the modeling and forecasting\nof short term electricity load. There exists a vast body of literature on this subject, covering methods\nfrom statistics (Seasonal ARIMA models [2, 3], Exponential Smoothing [4], regression models\n[5, 6, 7]) and, more recently, also from machine learning [8, 9, 10]. Additive Models were applied,\nwith good results, to the nation-wide load in France [11] and to regional loads in Australia [12].\nBesides electricity load, Additive Models have also been applied to natural gas demand [13].\nSeveral methods have been proposed to track time-varying behaviour of the smoothing splines in\nAdditive Models. Hoover et al. [14] examine estimators based on locally weighted polynomials and\nderive some of their asymptotic properties. In a similar vein, Eubank et al. [15] introduce a Bayesian\napproach which can handle multiple responses. A componentwise smoothing spline is suggested by\nChiang et al. [16]. Fan and Zhang [17] propose a two-stage algorithm which \ufb01rst computes raw\n\n\festimates of the smoothing functions at different time points and then smoothes the estimates. A\ncomprehensive review can be found in [18]. A common feature of all these methods is that they\nidentify and estimate the time-varying behaviour a posteriori.\nAdaptive learning of Additive Models in an online fashion is a relatively new topic. In [19], an\nalgorithm based on iterative QR decompositions is proposed, which yields promising results for the\nFrench electricity load but also highlights the need for a forgetting factor to be more reactive, e.g., to\nmacroeconomic and meteorological changes, or varying consumer portfolios. Harvey and Koopman\n[20] propose an adaptive learning method which is restricted to changing periodic patterns. Adaptive\nmethods of a similar type have been studied in the \ufb01eld of neural networks [21, 22].\nThe contributions of our paper are threefold: First, we introduce a new algorithm which combines\nAdditive Models with a Recursive Least Squares (RLS) \ufb01lter to track time-varying behaviour of the\nsmoothing splines. Second, in order to enhance the tracking ability, we consider \ufb01lters that include a\nforgetting factor which can be either \ufb01xed, or updapted using a gradient descent approach [23]. The\nbasic idea is to decrease the forgetting factor (and hence increase the reactivity) in transient phases,\nand increasing the forgetting factor (thus decreasing the variability) during stationary regimes. Using\nresults from Lyapunov stability theory [24], we provide a theoretical analysis of the learning rate in\nthe gradient descent approach. Third, we evaluate the proposed methodology on 5 years of electricity\nload data provided by the French utility company Electricit\u00b4e de France (EDF). The results show that\nthe adaptive learning algorithm outperforms state-of-the-art methods in terms of model tracking\nand prediction accuracy. Moreover, the experiments demonstrate that using an adaptive forgetting\nfactor stabilizes the algorithm and yields results comparable to those obtained by using the (a priori\nunknown) optimal value for a \ufb01xed forgetting factor. Note that, in this paper, we do not compare our\nproposed algorithm with existing online learning methods from the machine learning literature, such\nas tracking of best experts (see [25] for an overview). The reason is that we are speci\ufb01cally interested\nin adaptive versions of Additive Models, which have been shown to be particularly well-suited for\nmodeling and forecasting electricity load.\nThe remainder of the paper is organized as follows. Section 2 reviews the de\ufb01nition of Additive\nModels and provides some background on the spline representation of smoothing functions. In Sec-\ntion 3 we present our adaptive learning algorithms which combine Additive Models with a Recursive\nLeast Squares (RLS) \ufb01lter. We discuss different approaches for including forgetting factors and an-\nalyze the learning rate for the gradient descent method in the adaptive forgetting factor approach.\nA case study with real electricity load data from EDF is presented in Section 4. An outlook on\nproblems for future research concludes the paper.\n\n2 Additive Models\n\nIn this section we review the Additive Models and provide background information on the spline\nrepresentation of smoothing functions. Additive Models have the following form:\n\nI(cid:88)\n\nyk =\n\nfi(xk) + \u0001k.\n\ni=1\n\nIn this formulation, xk is a vector of covariates which can be either categorical or continuous, and\nyk is the dependent variable, which is assumed to be continuous. The noise term \u0001k is assumed\nto be Gaussian, independent and identically distributed with mean zero and \ufb01nite variance. The\nfunctions fi are the transfer functions of the model, which can be of the following types: constant\n(exactly one transfer function, representing the intercept of the model), categorical (evaluating to 0\nor 1 depending on whether the covariates satisfy certain conditions), or continuous. The continuous\ntransfer functions can be either linear functions of covariates (representing simple linear trends), or\nsmoothing splines. Typically, smoothing splines depend on only 1-2 of the continuous covariates.\nAn interesting possibility is to combine smoothing splines with categorical conditions; in the context\nof electricity load modeling this allows, e.g., for having different effects of the time of the day\ndepending on the day of the week.\n\n\fJi(cid:88)\n\nIn our experiments, we use 1- and 2-dimensional cubic B-splines, which allows us to write the\nsmoothing splines in the following form:\n\nfi(xk) = \u03b2T\n\ni bi(xk) =\n\n\u03b2ijbij(xk),\n\n(1)\n\nj=1\n\nwhere \u03b2ij are the spline coef\ufb01cients and bij are the spline basis functions which depend on 1 or 2\ncomponents of xk. Note that the basis functions are de\ufb01ned by a (\ufb01xed) sequence of knot points,\nwhile the coef\ufb01cients are used to \ufb01t the spline to the data (see [1] for details). The quantity Ji in\nequation (1) is the number of spline coef\ufb01cients associated with the transfer function fi. Now, let \u03b2\ndenote the stacked vector containing the spline coef\ufb01cients, and b(xk) the stacked vector containing\nthe spline basis functions of all the transfer functions. This allows us to write the Additive Models\nin the following linear form:\n\nyk = \u03b2T b(xk) + \u0001k.\n\n(2)\n\n2.1 Learning Additive Models\n\n(cid:98)\u03b2K = min\n\n\u03b2\n\nThe linear representation of Additive Models in (2) is the starting point for ef\ufb01cient learning algo-\nrithms. Consider K samples (xk, yk), k = 1, . . . , K of covariates and dependent variables. Then an\nestimate of the model coef\ufb01cients \u03b2 can be obtained by solving the following weighted penalized\nleast squares problem:\n\n(cid:110)\n(yK \u2212 BK\u03b2)T \u2126K (yK \u2212 BK\u03b2) + \u03b2T SK\u03b2\n\n(3)\nHere yK = (y1, y2, . . . , yK)T is the K \u00d7 1 vector containing all the dependent variables, BK is the\nmatrix with the rows b(x1)T , b(x2)T , . . . , b(xK)T containing the evaluated spline basis functions.\nThe matrix \u2126K puts different weights on the samples. In this paper, we consider two scenarios: \u2126K\nis the identity matrix (putting equal weight on the K regressors), or a diagonal matrix which puts\nexponentially decreasing weights on the samples, according to the order of their arrival (thus giving\nrise to the notion of forgetting factors). The different weighting schemes are discussed in more detail\nin Section 3. The matrix SK in (3) introduces a penalizing term in order to avoid over\ufb01tting of the\nsmoothing splines. In this paper, we use diagonal penalizers not depending on the sample size K:\n\n(cid:111)\n\n.\n\n(4)\nwhere \u03b3 > 0. Note that this penalizer shrinks the smoothing splines towards zero functions, and\nthe strength of this effect is tuned by \u03b3. As a well-known fact (see [1]), provided that the matrix\n(BT\n\nK\u2126KBK + S) is non-singular, the above least squares problem has the closed-form solution\n\nS = diag(\u03b3, \u03b3, . . . , \u03b3),\n\n(cid:98)\u03b2K = (BT\n\nK\u2126KBK + S)\u22121BT\n\nK\u2126KyK.\n\n(5)\n\n3 Adaptive learning of smoothing functions\n\nEquation (5) gives rise to an ef\ufb01cient batch learning algorithm for Additive Models. Next, we\npropose an adaptive method which allows us to track changes in the smoothing functions in an\nonline fashion. The basic idea is to combine the linear representation of Additive Models in (2) with\nclassical Recursive Least Squares (RLS) \ufb01lters. To improve the tracking behaviour, we introduce a\nforgetting factor which puts more weight on recent samples. See Algorithm 1 for details. As starting\n\nvalues, we choose(cid:98)\u03b20 equal to an initial estimate of \u03b2 (e.g., obtained in previous experiments), or\n(cid:98)\u03b2k can be used to compute predictions for new given covariates.\n\nequal to a zero vector if no prior information is available. The initial precision matrix P 0 is set equal\nto the inverse of the penalizer S in (4). Anytime while the algorithm is running, the current estimate\n\nLet us discuss the role of the forgetting factor \u03c9 in the adaptive learning algorithm. First, note\nthat Algorithm 1 is equivalent to the solution of the weighted least squares problem in (5) with the\nweighting matrix \u2126K = diag(\u03c9K\u22121, \u03c9K\u22122, . . . , \u03c92, \u03c9, 1) and the penalizer S as de\ufb01ned in (4). If\n\u03c9 = 1, all samples are weighted equally. For \u03c9 < 1, samples are discounted exponentially according\nto the order of their arrival. In general, a smaller forgetting factor improves the tracking of temporal\nchanges in the model coef\ufb01cients \u03b2. This reduction of the bias typically comes at the cost of an\nincrease of the variance. Therefore, \ufb01nding the right balance between the forgetting factor \u03c9 and the\nstrength \u03b3 of the penalizer in (4) is crucial for a good performance of the forecasting algorithm.\n\n\fAlgorithm 1 Adaptive learning (\ufb01xed forgetting factor)\n\n1: Input: Initial estimate(cid:98)\u03b20, forgetting factor \u03c9 \u2208 (0, 1], penalizer strength \u03b3 > 0.\n\n2: Compute the initial precision matrix P 0 = diag(\u03b3\u22121, \u03b3\u22121, . . . , \u03b3\u22121).\n3: for k = 1, 2, . . . do\n4:\n5:\n6:\n\nObtain new covariates xk and dependent variable yk.\nCompute the spline basis functions bk = b(xk).\nCompute the a priori error and the Kalman gain:\n\n7:\n\nUpdate the estimate and the precision matrix:\n\n(cid:98)\u0001k = yk \u2212 bT\n\nk(cid:98)\u03b2k\u22121,\n\ngk =\n\nP k\u22121bk\n\n\u03c9 + bT\n\nk P k\u22121bk\n\n.\n\n(cid:98)\u03b2k = (cid:98)\u03b2k\u22121 + gk(cid:98)\u0001k,\nP k = \u03c9\u22121(cid:2)P k\u22121 \u2212 gkbT\n\n(cid:3).\n\nk P k\u22121\n\n8: end for\n\nAlgorithm 2 Adaptive learning (adaptive forgetting factor)\n\n1: Input: Initial estimate(cid:98)\u03b20, initial forgetting factor \u03c90 \u2208 (0, 1], lower bound for the forgetting\n\nfactor \u03c9min \u2208 (0, 1], learning rate \u03b7 > 0, penalizer strength \u03b3 > 0.\n\n2: Same as Step 2 in Algorithm 1.\n3: Set \u03c80 equal to a zero vector and \u03a80 to the identity matrix.\n4: for k = 1, 2, . . . do\n5:\n6:\n\nSame as Steps 4-6 in Algorithm 1, with \u03c9k\u22121 instead of \u03c9.\nUpdate the forgetting factor:\n\n\u03c9k = \u03c9k\u22121 + \u03b7 bT\n\nk \u03c8k\u22121(cid:98)\u0001k.\n(cid:1) \u2212 \u03c9\u22121\n\nIf \u03c9k > 1, then set \u03c9k equal to 1. If \u03c9k < \u03c9min, then set \u03c9k equal to \u03c9min.\nSame as Step 7 in Algorithm 1, with \u03c9k instead of \u03c9.\nCompute the updates (where I denotes the identity matrix):\n\n7:\n8:\n\n(cid:0)I \u2212 gkbT\n\u03c8k = (cid:0)I \u2212 gkbT\n\n\u03a8k = \u03c9\u22121\n\n(cid:1)\u03a8k\u22121\n(cid:0)I \u2212 bkgT\n(cid:1)\u03c8k\u22121 + \u03a8kbk(cid:98)\u0001k.\n\nk\n\nk\n\nk\n\nk\n\nk P k + \u03c9\u22121\n\nk gkgT\nk ,\n\n9: end for\n\n3.1 Adaptive forgetting factors\n\nIn this section we present a modi\ufb01cation of Algorithm 1 which uses adaptive forgetting factors\nin order to improve the stability and the tracking behaviour. The basic idea is to choose a large\nforgetting factor during stationary regimes (when the a priori errors are small), and small forgetting\nfactors during transient phases (when the a priori error is large). In this paper we adopt the gradient\ndescent approach in [23] and update the forgetting factor according to the following formula:\n\n\u03c9k = \u03c9k\u22121 \u2212 \u03b7\n\n\u2202 E[(cid:98)\u0001 2\n\nk ]\n\u2202 \u03c9k\u22121\n\n.\n\nSearching in the direction of the partial derivative of E[(cid:98)\u0001 2\nNote that \u03c9k is updated in an iterative fashion based on \u03c8k (the gradient of the estimate (cid:98)\u03b2k with\n\nk ] with respect to \u03c9k\u22121 aims at minimizing\nthe expected value of the a priori errors. The learning rate \u03b7 > 0 determines the reactivity of the\nalgorithm: if it is high, then the errors lead to large decreases of the forgetting factor, and vice versa.\nThe details of the adaptive forgetting factor approach are given in Algorithm 2.\n\nrespect to \u03c9k\u22121), and on \u03a8k (the gradient of the precision matrix P k with respect to \u03c9k\u22121).\n\n\f3.2 Stability analysis\n\nIn the following, we apply results from Lyapunov stability theory to analyze the effect of the learning\nrate \u03b7. We show how to derive analytical bounds for \u03b7 that guarantee stability of the algorithm.\n\nk /2.\nClearly, the following conditions are satis\ufb01ed: if x = 0 then V (x) = 0; if x (cid:54)= 0 then V (x) > 0;\n\nk(cid:98)\u03b2k\u22121. As equilibrium point of our algorithm,\nRecall the de\ufb01nition of the a priori error,(cid:98)\u0001k = yk\u2212 bT\nwe consider the ideal situation(cid:98)\u0001k = 0. We choose the candidate Lyapunov function V ((cid:98)\u0001k) =(cid:98)\u0001 2\nand V (x) \u2192 \u221e as x \u2192 \u221e. Consider the discrete time derivative \u2206V ((cid:98)\u0001k) = V ((cid:98)\u0001k+1) \u2212 V ((cid:98)\u0001k)\nof the candidate Lyapunov function. According to Lyapunov stability theory, if V ((cid:98)\u0001k) > 0 and\n\u2206V ((cid:98)\u0001k) < 0, then V ((cid:98)\u0001k) converges to zero as k tends to in\ufb01nity.\nLet us analyze \u2206V ((cid:98)\u0001k) more closely. Using the relation(cid:98)\u0001k = \u2206(cid:98)\u0001k+1 +(cid:98)\u0001k we arrive at\n(cid:17)\n\u2206(cid:98)\u0001k +(cid:98)\u0001k\nNext we approximate \u2206(cid:98)\u0001k by its \ufb01rst order Taylor series expansion:\n\n(6)\n\n2\n\n.\n\n\u2206\u03c9k.\n\n\u2202\u03c9k\n\nk \u03c8k\u22121\n\n(cid:16) 1\n\u2206V ((cid:98)\u0001k) = \u2206(cid:98)\u0001k\n\u2202(cid:98)\u0001k\n\u2206(cid:98)\u0001k =\nand \u2206\u03c9k = \u03b7(cid:98)\u0001kbT\n(cid:17)(cid:20) 1\n(cid:16)\u2212bT\n(cid:17)(cid:16)\n\u03b7(cid:98)\u0001kbT\n(cid:1)2(cid:1).\n(cid:1)2(cid:0) \u2212 2 + \u03b7(cid:0)bT\n(cid:1)2 .\n\n(cid:17)(cid:16)\n\u03b7(cid:98)\u0001kbT\n(cid:0)bT\n\u03b7(cid:98)\u0001 2\n\n0 < \u03b7 <\n\nk \u03c8k\u22121.\n\nk \u03c8k\u22121\n\nk \u03c8k\u22121\n\nk \u03c8k\u22121\n\nk \u03c8k\u22121\n\n1\n2\n\n2\n\n(cid:0)bT\n\n2\n\nk \u03c8k\u22121\n\nk\n\n= \u2212bT\n\n\u2202\u03c9k\n\n\u2202(cid:98)\u0001k\n(cid:16)\u2212bT\n\u2206V ((cid:98)\u0001k) =\n\nk \u03c8k\u22121\n\n(cid:21)\n\n.\n\n+(cid:98)\u0001k\n\n(7)\n\n(8)\n\n(9)\n\n(cid:17)\n\nk \u03c8k\u22121\n\nNow it is easy to see that an (approximate) equivalent condition for Lyapunov stability is given by\n\nFurthermore, note that\n\n\u2206V ((cid:98)\u0001k) =\n\nSubstituting the expressions in (7) and (8) back into (6), we obtain the approximation\n\nAfter some basic algebraic manipulations we arrive at the approximation\n\n4 Case study: Forecasting of electricity load\n\nIn this section, we apply our adaptive learning algorithms to real electricity load data provided by\nthe French utility company Electricit\u00b4e de France (EDF). Modeling and forecasting electricity load\nis a challenging task due to the non-linear effects, e.g., of the temperature and the time of the day.\nMoreover, the electricity load exhibits many non-stationary patterns, e.g., due to changing macroe-\nconomic conditions (leading to an increase/decrease in electricity demand), or varying customer\nportfolios resulting from the liberalization of European electricity markets. The performance on\nthese highly complex, non-linear and non-stationary learning tasks is a challenging benchmark for\nour adaptive algorithms.\n\n4.1 Experimental data\n\nThe dependent variables yk in the data provided by EDF represent half-hourly electricity load mea-\nsurements between February 2, 2006 and April 6, 2011. The covariates xk include the following\ninformation:\n\n, xTimeOfDay\n\nk\n\n, xTimeOfYear\n\nk\n\n, xTemperature\n\nk\n\n, xCloudCover\n\nk\n\n, xLoadDecrease\n\nk\n\nLet us explain these components in more detail:\n\n\u2022 xDayType\n\nis a categorical variable representing the day type: 1 for Sunday, 2 for Monday, 3\n\nk\nfor Tuesday-Wednesday-Thursday, 4 for Friday, 5 for Saturday, and 6 for bank holidays.\n\nxk = (cid:0)xDayType\n\nk\n\n(cid:1).\n\n\f\u2022 xTimeOfDay\n\n\u2022 xTimeOfYear\n\n\u2022 xTemperature\n\nis the index (in half-hourly time steps) of the current time within the day. Its\n\nk\nvalues range from 0 for 0.00 am to 47 for 11.30 pm.\n\nis the position of the current day within the year (taking values from 0 for January\n\nk\n1, to 1 for December 31).\n\nk\n\nand xCloudCover\n\nrepresent the temperature and the cloud cover (ranging from 0 for a\nk\nblue sky to 8 for overcast). These meteorological covariates have been provided by M\u00b4et\u00b4eo\nFrance; the raw data include temperature and cloud cover data recorded every 3 hours from\n26 weather stations all over France. We interpolate these measurements to obtain half-\nhourly data. A weighted average \u2013 the weights re\ufb02ecting the importance of a region in\nterms of the national electricity load \u2013 is computed to obtain the national temperature and\ncloud cover covariates.\n\ncontains information about the activation of contracts between EDF and some\n\nk\nbig customers to reduce the electricity load during peak days.\n\n\u2022 xLoadDecrease\n\n6(cid:88)\n\nWe partition the data into two sets: a training set from February 2, 2006 to August 31, 2010, and a\ntest set from September 1, 2010 to April 6, 2011.\n\n4.2 Modeling the electricity load\n\nWe use the following Additive Model for the electricity load:\n\nyk = \u03b2Intercept + f Trend(k) + f LagLoad(yk\u221248) +\n\n1(xDayType\n\nk\n\n= l)(\u03b2DayType\n\nl\n\n+ f TimeOfDay\n\nl\n\n(xk))\n\n+ f CloudCover(xk) + f Temperature/TimeOfDay(xk) + f LagTemperature(xk\u221248)\n+ f TimeOfYear(xk) + xLoadDecrease\nLet us explain the model in more detail:\n\nf LoadDecrease(xk) + \u0001k.\n\nk\n\nl=1\n\nl\n\nl\n\n\u2022 The intercept \u03b2Intercept models the base load, and f Trend(k) captures non-linear trends, e.g.,\n\nand f TimeOfDay\n\n(xk) capture the day-type speci\ufb01c effects of the time of the day.\n\ndue to the economic crisis and changes in the customer portfolios of EDF.\n\u2022 f LagLoad(yk\u221248) takes into account the electricity load of the previous day.\n\u2022 \u03b2DayType\n\u2022 f CloudCover(xk) and f Temperature/TimeOfDay(xk) represent respectively the effect of the cloud cover\n\u2022 The term f LagTemperature(xk\u221248) takes into account the temperature of the previous day, which\nis important to capture the thermal inertia of buildings.\n\u2022 f TimeOfYear(xk) represents yearly cycles, and xLoadDecrease\n\nand the bivariate effect of the temperature and the time of the day.\n\nf LoadDecrease(xk) models the effect of\n\nk\n\ncontracts to reduce peak loads depending on the time of the day.\n\nTo \ufb01t the model we use the R package mgcv (see [26, 27]). For more information about the de-\nsign of models for electricity data we refer to [19, 11]. Figure 1 shows the estimated joint effect\nof the temperature and the time of the day, and the estimated yearly cycle. As to be expected,\nlow (resp. high) temperatures lead to an increase of the electricity load due to electrical heating\n(resp. cooling), whereas temperatures between 10\u25e6 and 20\u25e6 Celsius have almost no effect on the\nelectricity load. Due to the widespread usage of electrical heating and relatively low usage of air\nconditioning in France, the effect of heating is approximately four times higher than the effect of\ncooling. The yearly cycle reveals a strong decrease of the electricity load during the summer and\nChristmas holidays (around 0.6 and 1 of the time of the year). Note that the scales of the effects\nhave been normalized because of data con\ufb01dentiality reasons.\nThe \ufb01tted model achieves a good performance on the training data set with an adjusted R-square of\n0.993, a Mean Absolute Percentage Error (MAPE) of 1.4%, and a Root Mean Square Error (RMSE)\nof 835 MW. All the incorporated effects yield signi\ufb01cant improvements in terms of the Generalized\nCross Validation (GCV) score, so the model size cannot be reduced. The \ufb01tted model consists of\n268 spline basis coef\ufb01cients, which indicates the complexity of modeling electricity load data.\n\n\fFigure 1: Effect of the temperature and the time of the day (left), and yearly cycle (right).\n\n4.3 Adaptive learning and forecasting\n\nWe compare the performance of \ufb01ve different algorithms:\n\ndata without updating the model parameters.\n\n\u2022 The of\ufb02ine method (denoted by o\ufb02) uses the model learned in R and applies it to the test\n\u2022 The \ufb01xed forgetting factor method (denoted by fff) updates the Additive Model using a\n\ufb01xed forgetting factor (see Algorithm 1). The value of the \ufb01xed forgetting factor and the\nstrength of the penalizer are determined in the following way: We divide the test set into\ntwo parts of equal length, a calibration set (September 1, 2010 - November 15, 2010)\nand a validation set (November 16, 2010 - April 6, 2011). We choose the combination of\nforgetting factor and penalizer strength which yields the best results on the calibration set\nin terms of MAPE and RMSE, and evalute the performance on the validation set.\n\u2022 The post-\ufb01xed forgetting factor method (denoted by post-fff) uses the \ufb01xed forgetting fac-\ntor and strength of the penalizer which yield the best performance on the validation set. This\n\u201cideal\u201d parameterization gives us an upper bound for the performance of the fff method and\na benchmark for the adaptive forgetting factor approaches.\n\n\u2022 The adaptive forgetting factor method (denoted by aff) uses Algorithm 2.\n\u2022 Finally, we evaluate an adaptive approach that optimizes the values of the forgetting factor\nand the penalizer strength on a grid (denoted by affg): For each combination on the grids\n(0.995, 0.996, ..., 0.999) and (1000, 2000, ..., 10000), we run \ufb01xed forgetting factor algo-\nrithms in parallel. At each time point, we choose the combination which so far has given\nthe best performance in terms of MAPE.\n\n4.4 Results\n\nThe performance of all \ufb01ve algorithms is evaluted on the validation set from November 16, 2010 to\nApril 6, 2011. Table 1 shows the results in terms of MAPE and RMSE. As can be seen, the adaptive\nforgetting factor method (aff) achieves the best performance. It even outperforms the post-fff method\nwhich uses the (a priori unknown) optimal combination of penalizer strength and \ufb01xed forgetting\nfactor. The improvements over the of\ufb02ine approach (which doesn\u2019t update the model parameters)\nare signi\ufb01cant both in terms of the MAPE (about 0.2%) and the RMSE (about 100 MW). This\ncorresponds to an improvement of approximatively 10% in terms of the day-ahead forecasting error.\nFigure 2 (left) shows the cumulative sum of the errors of the \ufb01ve forecasting algorithms. As can be\nseen, the of\ufb02ine approach suffers from a strong positive bias and tends to overestimate the electricity\nload over time. In fact, there was a decrease in the electricity demand over the considered time\nhorizon due to the economic crisis. The adaptive forgetting factor shows a much better tracking\nbehaviour and is able to adapt to the change in the demand patterns.\n\nInstant010203040Temperature01020300.00.51.0llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll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llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll0.00.20.40.60.81.0\u22121.0\u22120.50.00.51.0Time Of YearYearly Cycle Effect\fThe graph on the right hand side of Figure 2 illustrates the roles of the forgetting factor and of\nthe strength of the penalizer. Values of the forgetting factor close to 1 result in reduced tracking be-\nhaviour and less improvement over the of\ufb02ine approach. Choosing too small values for the forgetting\nfactor can lead to loss of information and instabilities of the algorithm. Increasing the penalizer re-\nduces the variability of the smoothing splines, however, it also introduces a bias as the splines are\nshrinked towards zero.\n\nMethod\n\nMAPE (%)\nRMSE (MW)\n\no\ufb02\n1.83\n1185\n\nfff\n2.28\n1869\n\naffg\n1.7\n1124\n\naff\n1.63\n1071\n\npost-fff\n1.64\n1073\n\nTable 1: Performance of the \ufb01ve different forecasting methods\n\nFigure 2: Cumulative sum of the errors (left) and results for different choices of the forgetting factor\nand the strength of the penalizer (right)\n\n5 Conclusions and future work\n\nWe have presented an adaptive learning algorithm that updates the smoothing functions of Addi-\ntive Models in an online fashion. We have introduced methods to improve the tracking behaviour\nbased on forgetting factors and analyzed theoretical properties using results from Lyapunov stability\ntheory. The signi\ufb01cance of the proposed algorithms was demonstrated in the context of forecasting\nelectricity load data. Modeling and forecasting electricity load data is particularly challenging due\nto the high complexity of the models (the Additive Models in our experiments included 268 spline\nbasis functions), the non-linear relation between the covariates and dependent variables, and the\nnon-stationary dynamics of the models. Experiments on 5 years of data from Electricit\u00b4e de France\nhave shown the superior performance of algorithms using an adaptive forgetting factor. As it turned\nout, a crucial point is to \ufb01nd the right combination of forgetting factors and the strength of the pe-\nnalizer. While forgetting factors tend to reduce the bias of models evolving over time, they typically\nincrease the variance, an effect which can be compensated by choosing stronger penalizer. Our fu-\nture research will follow two directions: \ufb01rst, we plan to consider dynamic penalizers which can\nautomatically adapt to changes in the model complexity. Second, we will develop methods for in-\ncorporating prior information on model components, e.g., by integrating beliefs for the initial values\nof the adaptive algorithms.\n\nReferences\n[1] Trevor Hastie, Robert Tibshirani, and Jerome Friedman. The Elements of Statistical Learning. Second\n\nEdition, Springer, 2009.\n\n[2] J Nowicka-Zagrajek and R Weron. Modeling electricity loads in California: ARMA models with hyper-\n\nbolic noise. Signal Processing, pages 1903\u20131915, 2002.\n\n01000200030004000\u22121.2\u22121.0\u22120.8\u22120.6\u22120.4\u22120.20.0TimeNormalized Cumulative Errorsoflfffaffgaffpost\u2212fff\f[3] Shyh-Jier Huang and Kuang-Rong Shih. Short-Term Load Forecasting Via ARMA Model Identi\ufb01cation\nIncluding Non-Gaussian Process Considerations. IEEE Transactions on Power Systems, 18(2):673\u2013679,\n2003.\n\n[4] James W. Taylor. Short-Term Load Forecasting with Exponentially Weighted Methods. IEEE Transac-\n\ntions on Power Systems, 27(1):673\u2013679, 2012.\n\n[5] Derek W. Bunn and E. D. Farmer. Comparative models for electrical load forecasting. Eds. Wiley, New\n\nYork, 1985.\n\n[6] R Campo and P. Ruiz. Adaptive Weather-Sensitive Short Term Load Forecast . IEEE Transactions on\n\nPower Systems, 2(3):592\u2013598, 1987.\n\n[7] Ramu Ramanathan, Robert Engle, Clive W. J. Granger, Farshid Vahid-Araghi, and Casey Brace. Short-run\n\nforecasts of electricity loads and peaks . International Journal of Forecasting, 13(3):161\u2013174, 1997.\n\n[8] Bo-Juen Chen, Ming-Wei Chang, and Chih-Jen Lin. Load Forecasting Using Support Vector Machines:\nA Study on EUNITE Competition 2001 . IEEE Transaction on Power Systems, 19(3):1821\u20131830, 2004.\nIEEE\n\n[9] Shu Fan and Luonan Chen. Short-term load forecasting based on an adaptive hybrid method .\n\nTransaction on Power Systems, 21(1):392\u2013401, 2006.\n\n[10] V. H Hinojosa and A Hoese. Short-Term Load Forecasting Using Fuzzy Inductive Reasoning and Evolu-\n\ntionary Algorithms . IEEE Transaction on Power Systems, 25(1):565\u2013574, 2010.\n\n[11] A Pierrot and Yannig Goude. Short-term electricity load forecasting with generalized additive models. In\n\nProceedings of ISAP Power, pages 593\u2013600, 2011.\n\n[12] Shu Fan and R Hyndman. Short-Term Load Forecasting Based on a Semi-Parematetric Additive Model .\n\nIEEE Transaction on Power Systems, 27(1):134\u2013141, 2012.\n\n[13] M Brabek, O Konr, M Mal, M Pelikn, and J Vondrcek. A statistical model for natural gas standardized\nload pro\ufb01les. Journal of the Royal Statistical Society: Series C (Applied Statistics), 58(1):123\u2013139, 2009.\n[14] Donald R. Hoover, John A. Rice, Colin O. Wu, and Li-Ping Yang. Nonparametric smoothing estimates\n\nof time-varying coef\ufb01cient models with longitudinal data. Biometrika, 85(4):809\u2013822, 1998.\n\n[15] R. L. Eubank, Chunfeng Huang, Y. Munoz. Maldonado, and R. J. Buchanan. Smoothing spline estimation\n\nin varying coef\ufb01cient models. Journal of the Royal Statistical Society, 66(3):653\u2013667, 2004.\n\n[16] Chin-Tsang Chiang, John A. Rice, and Colin O. Wu. Smoothing spline estimation for varying coef\ufb01cient\nmodels with repeatedly measured dependent variables. Journal of the American Statistical Asociation,\n96(454):605\u2013619, 2001.\n\n[17] Jianqing Fan and Jin-Ting Zhang. Two-Step Estimation of Functional Linear Models with Applications\n\nto Longitudinal Data. Journal of the Royal Statistical Society, 62:303\u2013322, 2000.\n\n[18] Jianqing Fan and Wenyang Zhang. Statistical methods with varying coef\ufb01cient models. Statistics and Its\n\nInterface, 1:179\u2013195, 2008.\n\n[19] S. Wood, Y. Goude, and S. Shaw. Generalized Additive Models. Preprint, 2011.\n[20] A Harvey and S. J Koopman. Forecasting Hourly Electricity Demand Using Time-Varying Splines. Jour-\n\nnal of the American Statistical Association, 88(424):1228\u20131236, 1993.\n\n[21] Herbert Jaeger. Adaptive non-linear system identi\ufb01cation with echo state networks. In Proc. Advances\n\nNeural Information Processing Systems, pages 593\u2013600, 2002.\n\n[22] Mauro Birattari, Gianluca Bontempi, and Hugues Bersini. Lazy learning meets the recursive least squares\n\nalgorithm. In Proc. Advances Neural Information Processing Systems, pages 375\u2013381, 1999.\n\n[23] S-H Leung and C. F So. Gradient-Based Variable Forgetting Factor RLS Algorithm in Time-Varying\n\nEnvironments. IEEE Transaction on Signal Processing, 53(8):3141\u20133150, 2005.\n\n[24] Z Man, H. R Wu, S Liu, and X Yu. A New Adaptive Backpropagation Algorithm Based on Lyapunov\n\nStability Theory for Neural Network. IEEE Transaction on Neural Networks, 17(6):1580\u20131591, 2006.\n\n[25] Nicol`o Cesa-Bianchi and G\u00b4abor Lugosi. Prediction, Learning, and Games. Cambridge University Press,\n\n2006.\n\n[26] Simon Wood. Generalized Additive Models an Introduction with R. Chapman and Hall Eds., 2006.\n[27] Simon Wood. mgcv :GAMs and Generalized Ridge Regression for R. R News, 1(2):20\u201325, 2001.\n\n\f", "award": [], "sourceid": 1205, "authors": [{"given_name": "Amadou", "family_name": "Ba", "institution": null}, {"given_name": "Mathieu", "family_name": "Sinn", "institution": null}, {"given_name": "Yannig", "family_name": "Goude", "institution": null}, {"given_name": "Pascal", "family_name": "Pompey", "institution": null}]}