SpringerBriefs in the Mathematics of Planet Earth shows topics of current relevance to the Mathematics of Planet Earth. CN is greatly obliged to the University of Reading for supporting Open Access publication of this book.
Forecasting and Challenges
These are reasonably simple and transparent and thus quite favorable in the electrical load forecasting community. On the one hand, peaks in data can be perceived as local extremes.
Data
Irish Smart Meter Data
Similarly, the upper right graph in Figure 1.3 shows the total daily consumption for each day over a 7-week period. It is also valuable to see how the top left profile in Figure 1.3 changes for each day of the week.
Thames Valley Vision Data
The average profile also appears to be less smooth as expected, than in the case of the Irish smart meter data, as there are fewer households. Of course these values depend on the time of year and the day of the week as shown in the upper right and lower left plots of Fig.1.10.
Outline and Objectives
Haben, S., Giasemidis, G., Ziel, F., Arora, S.: Short-term load forecasting and the effect of temperature at the low voltage level. The growth of literature on short-term load forecasting at the individual level began with the wider access to higher resolution data in the last two decades, and is still developing.
Forecasts
- Linear Regression
- Time Series Based Algorithms
- Permutation Based Algorithms
- Machine Learning Based Algorithms
First, the unconditional density is estimated using historical observations of the variable to be predicted. The LSTM network that was implemented in Kong et al. i) the energy demand from K past time steps, (ii) the time of day for each of the past K energy demands which is one of 48 to reflect the half hours in the day, (iii) the day of the week which is one of 7 , (iv) a binary vector of length K indicating whether the day is a holiday or not. The normalization of the last three inputs is done using a hot coder.9 The LSTM network is designed with two hidden layers and 20 nodes in each hidden layer.
Forecast Errors
Point Error Measures
Thus, from the reviewed literature we conclude that, while SVR is a promising aggregate load forecasting algorithm, data volatility reduces its effectiveness when it comes to forecasting individual smart meter data. In this book, we adopt a common arrangement that allows the data points to be zero and define the MAPE to be. 19], we get p=4 as it allows larger errors to be penalized more and smaller errors to be penalized less.
Time Shifted Error Measures
The error measure assumes that an observation is predicted well enough (in time) if both the prediction and the observation are within a certain time windowω. 19], the minimization is done using the Hungarian algorithm, but faster implementations are possible with Dijkstra's shortest path algorithm or topological sorting as discussed by Charlton et al. 38], a distance based on the adjusted p-norm error, the local permutation invariant (LPI) is formalized.
Discussion
In the literature, EVI is also referred to as the shape parameter for GEV. Pareto, Generalized Pareto, and Inverse Gamma distributions are examples of distributions in the attraction domain of the Fréchet distribution. Finally, for γ<0, F is said to be in the domain of attraction of the Weibull distribution with d.f.−1/γ.
We now proceed to discuss the asymptotic distribution for each order statistic. Considering the problem of statistical selection of the regions of attraction postulated in Eq. Therefore, we will continue with the evaluation of the final upper limit in the POT framework.
Neves, C., Picek, J., Alves, M.F.: The contribution of the maximum to the sum of residuals for the test of maximum domains of attraction.
Basic De fi nitions
Maximum of a Random Sample
The parameter γ∈, the so-called extreme value index (EVI), controls the shape of the GEV distribution. The right side of the same plot shows how quickly the normalized maxima go to the Gumbel distribution; the exponential distribution belongs to the Gumbel domain of attraction. Thus, without knowledge of the appropriate normalizing constants, the von Mises conditions allow us to find the domain of attraction of F.
Exceedances and Order Statistics
Exceedances
For large sample sizes, the performance of the block maxima method and the peaks over threshold method are comparable. However, when the sample is not large, there may be some estimates where the Peaks over threshold (POT) model is more efficient [21]. As before, we will start with definitions and then proceed to establish the generalized Pareto as the limiting distribution.
Asymptotic Distribution of Certain Order Statistics
Note that result3.8 in Theorem 3.8 relates to intermediate order statistics, while result3.8 relates to central order statistics. We can consider the asymptotic distribution of the extreme order statistic (also known as the upper order statistic), which no longer exhibits asymptotic normality. Similarly, Theorems 3.9 and 3.10 can be used to choose appropriate normalization for the relevant order statistics.
Extended Regular Variation
The next theorem establishes the form of H and gives some properties of the auxiliary functions. The semiparametric framework, where inference takes place in the domains of attraction rather than through the prescribed limiting distribution to the data—. Figure 4.2 is a scatter plot representation of the actual observations in the form of household number.
Block Maxima and Peaks over Threshold Methods
Proposition 4.1 is clearly valid and therefore the Burr distribution satisfies the second-order extreme value condition (Eq.4.3) with γ=1/(γτ)inρ˜= max(−1/λ,−1)ifτ =1. It is clear that U does not satisfy the second-order condition (Eq.4.2) in a simple way, but we will show that the corresponding V(t)=U. Therefore, GPD confirms Proposition 4.1 if it goes through the consideration that BDP satisfies (Eq.4.2) with ρ=.
Maximum Lq-Likelihood Estimation with the BM Method
Upper Endpoint Estimation
72 4 Statistics of Extreme Values Since the adopted estimators do not seem to target a value, it is not easy to agree between them. Furthermore, since we are dealing with small sample sizes (we take the maximum over 7 weeks), the skewed version, i.e. the MLq estimator, should be considered. Therefore, we find it reasonable to take as an estimate for the upper bound the midpoint of the region where MLq changes direction and travels to the simple ML estimator with increasingq.
Estimating and Testing with the POT Method
Selection of the Max-Domain of Attraction
According to the asymptotic results given in [27], the null hypothesis F ∈D(G0) is rejected against the two-sided alternative F ∈D(Gγ)γ=0 at the asymptotic level α∈(0,1)ifTn∗( k) < gα/2orTn∗(k) >g1−α/2, where gε denotes the ε-quantile of the Gumble distribution, i.e. gε= −log(−logε). The critical region for a two-tailed test of nominal size α is given by |Tn∗(k)|>z1−α/2, where zε denotes the ε-quantile of the standard normal distribution and where T should conveniently be replaced by RorW. While the R ∗ -based test barely detects small negative values of γ , Hasofer and Wang's test is the most powerful one studied with respect to alternatives in the Weibull attraction domain.
Testing for a Finite Upper Endpoint
When these critical limits are crossed, the null hypothesis of the Gumbel domain is rejected in favor of the Weibull domain. The test problem statement at the top point is slightly different as it involves a different partition of the Gumbel domain. The application of the HW test appears to do justice to this Irish smart meter data set and finds sufficient evidence to reject the null hypothesis of the Gumbel domain in favor of fun estimation procedures appropriate for the Weibull domain.
Upper Endpoint Estimation
For the asymptotic properties of the POT maximum likelihood estimator for EVI in the semiparametric approach, see e.g. However, there is one estimator for the upper endpoint xF that does not depend on the EVIγ estimator developed in [18], which is designed for distributions with a finite upper endpoint enclosed in a Gumble region of attraction. A suitable maximum likelihood estimator of Lq is found with the distortion parameter q set to 0.88, with values of k at which the Lq likelihood method had trouble converging on the EVI estimate now omitted.
Non-identically Distributed Observations — Scedasis Function
82 4 Statistics of extreme values The maximum temperature obtained during a heat wave will be part of the same weather system and then depend and also less representative of wild nature. We will put scedasis assessment into practice in Chapter 5 using Thames Valley Vision data. Castillo, E., Hadi, A., Balakrishnan, N., Sarabia, J.M.: Extreme Value and Related Models with Applications in Engineering and Science.
Predicting Electricity Peaks on a Low Voltage Network
Short Term Load Forecasts
The parameters were obtained by performing localized search based on the Akaike Information Criterion (AIC) for each household.3 An example showing some success in the prediction of peaks can be seen in Fig.5.6. 92 5 Case Study We used MLRegressor from the Python scikit-learn library [5], using 'relu', the rectified linear unit function, f(x)=max(0,x) as the activation function for the hidden layers. An example is shown in Fig. 5.10, where the timing of regular peaks is mostly captured, but amplitudes are underestimated.
Forecast Uncertainty
An optimizer in the family of quasi-Newton methods, 'lbfgs', was used as the solver for the weight optimization. The learning rate for weight updates was set to "adaptive", i.e. was held constant at 0.01 as long as the training loss continued to decrease. In the week commencing 31 August 2015, there is the double challenge of the summer bank holiday (31 August) and the start of the school year, while the summer weeks before that in the UK are generally characterized by lower consumption.
Heteroscedasticity in Forecasts
Both PM4 and SD4 appear to have better predictive power on later days of the week, as they show a decreasing trend in the probability of major errors. The images or other third-party material in this chapter are included in the chapter's Creative Commons license, unless otherwise indicated in a credit line for the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulations or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
