Quantile Regression Averaging (QRA) is a forecast combination approach to the computation of prediction intervals . It involves applying quantile regression to the point forecasts of a small number of individual forecasting models or experts. It has been introduced in 2014 by Jakub Nowotarski and Rafał Weron and originally used for probabilistic forecasting of electricity prices and loads. Despite its simplicity it has been found to perform extremely well in practice - the top two performing teams in the price track of the Global Energy Forecasting Competition (GEFCom2014) used variants of QRA.
43-600: QRA may refer to: Quantile regression averaging , in econometrics and forecasting Quantitative risk assessment , an estimation of risk Quaternary Research Association , associated with the Journal of Quaternary Science Queensland Regional Airlines , a defunct Australian airline Quick Reaction Alert , a NATO state of readiness in military aviation Rand Airport (IATA airport code: QRA), South Africa QRA locator , an obsolete geographic coordinate system Kra (letter) ,
86-466: A (large) panel of forecasts of the individual models, FQRA concentrates on a small number of common factors, which - by construction - are orthogonal to each other, and hence are contemporaneously uncorrelated. FQRA can be also interpreted as a forecast averaging approach. The factors estimated within PCA are linear combinations of individual vectors of the panel and FQRA can therefore be used to assign weights to
129-416: A distribution that minimizes expected squared error while the median minimizes expected absolute error. Least absolute deviations shares the ability to be relatively insensitive to large deviations in outlying observations, although even better methods of robust regression are available. The quantiles of a random variable are preserved under increasing transformations, in the sense that, for example, if m
172-497: A generalization that can cover as special cases the continuous distributions. For discrete distributions the sample median as defined through this concept has an asymptotically Normal distribution, see Ma, Y., Genton, M. G., & Parzen, E. (2011). Asymptotic properties of sample quantiles of discrete distributions. Annals of the Institute of Statistical Mathematics, 63(2), 227–243. Computing approximate quantiles from data arriving from
215-714: A priori , the relevant information contained in all forecasting models at hand is extracted using principal component analysis (PCA). The prediction intervals are then constructed on the basis of the common factors ( f t {\displaystyle f_{t}} ) obtained from the panel of point forecasts, as independent variables in a quantile regression. More precisely, in the FQRA method X t = [ 1 , f ^ 1 , t , . . . , f ^ k , t ] {\displaystyle X_{t}=[1,{\hat {f}}_{1,t},...,{\hat {f}}_{k,t}]}
258-404: A stream can be done efficiently using compressed data structures. The most popular methods are t-digest and KLL. These methods read a stream of values in a continuous fashion and can, at any time, be queried about the approximate value of a specified quantile. Both algorithms are based on a similar idea: compressing the stream of values by summarizing identical or similar values with a weight. If
301-492: Is a vector of k < m {\displaystyle k<m} factors extracted from a panel of point forecasts of m {\displaystyle m} individual models, not a vector of point forecasts of the individual models themselves. A similar principal component-type approach was proposed in the context of obtaining point forecasts from the Survey of Professional Forecasters data. Instead of considering
344-1418: Is a vector of parameters (for quantile q ). The parameters are estimated by minimizing the loss function for a particular q -th quantile: min β q [ ∑ { t : y t ≥ X t β q } q | y t − X t β q | + ∑ { t : y t < X t β q } ( 1 − q ) | y t − X t β q | ] = min β q [ ∑ t ( q − 1 y t < X t β q ) ( y t − X t β q ) ] {\displaystyle \min \limits _{\beta _{q}}\left[\sum \limits _{\{t:y_{t}\geq X_{t}\beta _{q}\}}q|y_{t}-X_{t}\beta _{q}|+\sum \limits _{\{t:y_{t}<X_{t}\beta _{q}\}}(1-q)|y_{t}-X_{t}\beta _{q}|\right]=\min \limits _{\beta _{q}}\left[\sum \limits _{t}(q-\mathbf {1} _{y_{t}<X_{t}\beta _{q}})(y_{t}-X_{t}\beta _{q})\right]} QRA assigns weights to individual forecasting methods and combines them to yield forecasts of chosen quantiles. Although
387-430: Is also used in peer-reviewed scientific research articles. The meaning used can be derived from its context. If a distribution is symmetric, then the median is the mean (so long as the latter exists). But, in general, the median and the mean can differ. For instance, with a random variable that has an exponential distribution , any particular sample of this random variable will have roughly a 63% chance of being less than
430-473: Is an extension beyond traditional statistics definitions. The following two examples use the Nearest Rank definition of quantile with rounding. For an explanation of this definition, see percentiles . Consider an ordered population of 10 data values [3, 6, 7, 8, 8, 10, 13, 15, 16, 20]. What are the 4-quantiles (the "quartiles") of this dataset? So the first, second and third 4-quantiles (the "quartiles") of
473-528: Is the conditional q -th quantile of the dependent variable ( y t {\displaystyle y_{t}} ), X t = [ 1 , y ^ 1 , t , . . . , y ^ m , t ] {\displaystyle X_{t}=[1,{\hat {y}}_{1,t},...,{\hat {y}}_{m,t}]} is a vector of point forecasts of m {\displaystyle m} individual models (i.e. independent variables) and β q
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#1732851620852516-444: Is the value of the p -quantile for 0 < p < 1 (or equivalently is the k -th q -quantile for p = k / q ), where μ is the distribution's arithmetic mean , and where σ is the distribution's standard deviation . In particular, the median ( p = k / q = 1/2) is never more than one standard deviation from the mean. The above formula can be used to bound the value μ + zσ in terms of quantiles. When z ≥ 0 ,
559-411: The cumulative distribution function ) to the values {1/ q , 2/ q , …, ( q − 1)/ q }. As in the computation of, for example, standard deviation , the estimation of a quantile depends upon whether one is operating with a statistical population or with a sample drawn from it. For a population, of discrete values or for a continuous population density, the k -th q -quantile is the data value where
602-434: The range of a probability distribution into continuous intervals with equal probabilities, or dividing the observations in a sample in the same way. There is one fewer quantile than the number of groups created. Common quantiles have special names, such as quartiles (four groups), deciles (ten groups), and percentiles (100 groups). The groups created are termed halves, thirds, quarters, etc., though sometimes
645-416: The " p -quantile" is based on a real number p with 0 < p < 1 then p replaces k / q in the above formulas. This broader terminology is used when quantiles are used to parameterize continuous probability distributions . Moreover, some software programs (including Microsoft Excel ) regard the minimum and maximum as the 0th and 100th percentile, respectively. However, this broader terminology
688-930: The QRA method is based on quantile regression, not least squares , it still suffers from the same problems: the exogenous variables should not be correlated strongly and the number of variables included in the model has to be relatively small in order for the method to be computationally efficient. The main difficulty associated with applying QRA comes from the fact that only individual models that perform well and (preferably) are distinct should be used. However, there may be many well performing models or many different specifications of each model (with or without exogenous variables, with all or only selected lags, etc.) and it may not be optimal to include all of them in Quantile Regression Averaging. In Factor Quantile Regression Averaging (FQRA) , instead of selecting individual models
731-412: The appropriate index; the corresponding data value is the k -th q -quantile. On the other hand, if I p is an integer then any number from the data value at that index to the data value of the next index can be taken as the quantile, and it is conventional (though arbitrary) to take the average of those two values (see Estimating quantiles from a sample ). If, instead of using integers k and q ,
774-430: The case for the median (2-quantile) of a uniform probability distribution on a set of even size. Quantiles can also be applied to continuous distributions, providing a way to generalize rank statistics to continuous variables (see percentile rank ). When the cumulative distribution function of a random variable is known, the q -quantiles are the application of the quantile function (the inverse function of
817-492: The corresponding observed target variable as the dependent variable in a standard quantile regression setting. The Quantile Regression Averaging method yields an interval forecast of the target variable, but does not use the prediction intervals of the individual methods. One of the reasons for using point forecasts (and not interval forecasts) is their availability. For years, forecasters have focused on obtaining accurate point predictions. Computing probabilistic forecasts , on
860-409: The cumulative distribution function crosses k / q . That is, x is a k -th q -quantile for a variable X if and For a finite population of N equally probable values indexed 1, …, N from lowest to highest, the k -th q -quantile of this population can equivalently be computed via the value of I p = N k / q . If I p is not an integer, then round up to the next integer to get
903-770: The data, in essence the data are simply numbers or more generally, a set of items that can be ordered. These algorithms are computer science derived methods. Another class of algorithms exist which assume that the data are realizations of a random process. These are statistics derived methods, sequential nonparametric estimation algorithms in particular. There are a number of such algorithms such as those based on stochastic approximation or Hermite series estimators. These statistics based algorithms typically have constant update time and space complexity, but have different error bound guarantees compared to computer science type methods and make more assumptions. The statistics based algorithms do present certain advantages however, particularly in
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#1732851620852946-441: The dataset [3, 6, 7, 8, 8, 10, 13, 15, 16, 20] are [7, 9, 15]. If also required, the zeroth quartile is 3 and the fourth quartile is 20. Consider an ordered population of 11 data values [3, 6, 7, 8, 8, 9, 10, 13, 15, 16, 20]. What are the 4-quantiles (the "quartiles") of this dataset? So the first, second and third 4-quantiles (the "quartiles") of the dataset [3, 6, 7, 8, 8, 9, 10, 13, 15, 16, 20] are [7, 9, 15]. If also required,
989-431: The default. The standard error of a quantile estimate can in general be estimated via the bootstrap . The Maritz–Jarrett method can also be used. The sample median is the most examined one amongst quantiles, being an alternative to estimate a location parameter, when the expected value of the distribution does not exist, and hence the sample mean is not a meaningful estimator of a population characteristic. Moreover,
1032-445: The distribution is discrete, then the distribution of the sample median and the other quantiles fails to be Normal (see examples in https://stats.stackexchange.com/a/86638/28746 ). A solution to this problem is to use an alternative definition of sample quantiles through the concept of the "mid-distribution" function, which is defined as The definition of sample quantiles through the concept of mid-distribution function can be seen as
1075-517: The estimate for the p -quantile (the k -th q -quantile, where p = k / q ) from a sample of size N by computing a real valued index h . When h is an integer, the h -th smallest of the N values, x h , is the quantile estimate. Otherwise a rounding or interpolation scheme is used to compute the quantile estimate from h , x ⌊ h ⌋ , and x ⌈ h ⌉ . (For notation, see floor and ceiling functions ). The first three are piecewise constant, changing abruptly at each data point, while
1118-444: The family of data sketches that are subsets of Streaming Algorithms with useful properties: t-digest or KLL sketches can be combined. Computing the sketch for a very large vector of values can be split into trivially parallel processes where sketches are computed for partitions of the vector in parallel and merged later. The algorithms described so far directly approximate the empirical quantiles without any particular assumptions on
1161-514: The forecasting models directly. QRA may be viewed as an extension of combining point forecasts. The well-known ordinary least squares (OLS) averaging uses linear regression to estimate weights of the point forecasts of individual models. Replacing the quadratic loss function with the absolute loss function leads to quantile regression for the median, or in other words, least absolute deviation (LAD) regression . Quantile In statistics and probability , quantiles are cut points dividing
1204-415: The intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=QRA&oldid=1174520227 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Quantile regression averaging The individual point forecasts are used as independent variables and
1247-413: The last six use linear interpolation between data points, and differ only in how the index h used to choose the point along the piecewise linear interpolation curve, is chosen. Mathematica , Matlab , R and GNU Octave programming languages support all nine sample quantile methods. SAS includes five sample quantile methods, SciPy and Maple both include eight, EViews and Julia include
1290-418: The letter used in place of Q in some Inuit orthographies, also spelled Qra Radio Q code for "What is the name of your vessel (or station)?" or "The name of my vessel (or station) is ____." Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title QRA . If an internal link led you here, you may wish to change the link to point directly to
1333-416: The mean, then quantiles may be more useful descriptive statistics than means and other moment-related statistics. Closely related is the subject of least absolute deviations , a method of regression that is more robust to outliers than is least squares, in which the sum of the absolute value of the observed errors is used in place of the squared error. The connection is that the mean is the single estimate of
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1376-430: The mean. This is because the exponential distribution has a long tail for positive values but is zero for negative numbers. Quantiles are useful measures because they are less susceptible than means to long-tailed distributions and outliers. Empirically, if the data being analyzed are not actually distributed according to an assumed distribution, or if there are other potential sources for outliers that are far removed from
1419-463: The non-stationary streaming setting i.e. time-varying data. The algorithms of both classes, along with some respective advantages and disadvantages have been recently surveyed. Standardized test results are commonly reported as a student scoring "in the 80th percentile", for example. This uses an alternative meaning of the word percentile as the interval between (in this case) the 80th and the 81st scalar percentile. This separate meaning of percentile
1462-651: The other hand, is generally a much more complex task and has not been discussed in the literature nor developed by practitioners so extensively. Therefore, QRA may be found particularly attractive from a practical point of view as it allows to leverage existing development of point forecasting. The quantile regression problem can be written as follows: Q y ( q | X t ) = X t β q {\displaystyle Q_{y}(q|X_{t})=X_{t}\beta _{q}} , where Q y ( q | ⋅ ) {\displaystyle Q_{y}(q|\cdot )}
1505-545: The sample median is a more robust estimator than the sample mean. One peculiarity of the sample median is its asymptotic distribution: when the sample comes from a continuous distribution, then the sample median has the anticipated Normal asymptotic distribution, This extends to the other quantiles, where f ( x p ) is the value of the distribution density at the p -th population quantile ( x p = F − 1 ( p ) {\displaystyle x_{p}=F^{-1}(p)} ). But when
1548-422: The six piecewise linear functions, Stata includes two, Python includes two, and Microsoft Excel includes two. Mathematica, SciPy and Julia support arbitrary parameters for methods which allow for other, non-standard, methods. The estimate types and interpolation schemes used include: Notes: Of the techniques, Hyndman and Fan recommend R-8, but most statistical software packages have chosen R-6 or R-7 as
1591-420: The stream is made of a repetition of 100 times v1 and 100 times v2, there is no reason to keep a sorted list of 200 elements, it is enough to keep two elements and two counts to be able to recover the quantiles. With more values, these algorithms maintain a trade-off between the number of unique values stored and the precision of the resulting quantiles. Some values may be discarded from the stream and contribute to
1634-404: The terms for the quantile are used for the groups created, rather than for the cut points. q - quantiles are values that partition a finite set of values into q subsets of (nearly) equal sizes. There are q − 1 partitions of the q -quantiles, one for each integer k satisfying 0 < k < q . In some cases the value of a quantile may not be uniquely determined, as can be
1677-452: The value μ + zσ for z = −3 will never exceed Q ( p = 0.1) , the first decile. One problem which frequently arises is estimating a quantile of a (very large or infinite) population based on a finite sample of size N . Modern statistical packages rely on a number of techniques to estimate the quantiles. Hyndman and Fan compiled a taxonomy of nine algorithms used by various software packages. All methods compute Q p ,
1720-409: The value that is z standard deviations above the mean has a lower bound μ + z σ ≥ Q ( z 2 1 + z 2 ) , f o r z ≥ 0. {\displaystyle \mu +z\sigma \geq Q\left({\frac {z^{2}}{1+z^{2}}}\right)\,,\mathrm {~for~} z\geq 0.} For example,
1763-631: The value that is z = 1 standard deviation above the mean is always greater than or equal to Q ( p = 0.5) , the median, and the value that is z = 2 standard deviations above the mean is always greater than or equal to Q ( p = 0.8) , the fourth quintile. When z ≤ 0 , there is instead an upper bound μ + z σ ≤ Q ( 1 1 + z 2 ) , f o r z ≤ 0. {\displaystyle \mu +z\sigma \leq Q\left({\frac {1}{1+z^{2}}}\right)\,,\mathrm {~for~} z\leq 0.} For example,
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1806-432: The weight of a nearby value without changing the quantile results too much. The t-digest maintains a data structure of bounded size using an approach motivated by k -means clustering to group similar values. The KLL algorithm uses a more sophisticated "compactor" method that leads to better control of the error bounds at the cost of requiring an unbounded size if errors must be bounded relative to p . Both methods belong to
1849-608: The zeroth quartile is 3 and the fourth quartile is 20. For any population probability distribution on finitely many values, and generally for any probability distribution with a mean and variance, it is the case that μ − σ ⋅ 1 − p p ≤ Q ( p ) ≤ μ + σ ⋅ p 1 − p , {\displaystyle \mu -\sigma \cdot {\sqrt {\frac {1-p}{p}}}\leq Q(p)\leq \mu +\sigma \cdot {\sqrt {\frac {p}{1-p}}}\,,} where Q(p)
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