A quick look at the field of probabilistic forecasting (background and theory) for applications in renewable energy prediction.

1. Probabilistic Forecasting:
How and Why?
By Kostas Hatalis
hatalis@gmail.com
Dept. of Electrical & Computer Engineering
Lehigh University, Bethlehem, PA
2019
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2. Motivation - Application Standpoint
Demand for power, and usage of renewables is both increasing rapidly.
Source: EIA Annual Energy Outlook 2018
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3. Motivation - Application Standpoint
Several challenges can be identified with a higher penetration of
renewables energy into the smart grid:
1) managing variability and uncertainty
2) balancing supply and demand
3) security
All will require the use of forecasting. Applied motivation of my
research.
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4. Motivation - Methodology Standpoint
Goal:develop novel forecasting frameworks that can answer most of
these questions.
1 How do we make robust nonparametric probabilistic forecasts?
2 Can a forecasting model be domain independent?
3 How can nonlinearity and nonstationary be captured in data?
4 Can forecasts be made multi-step?
5 Can the model scale to big data sets?
6 Can we autonomously engineer features?
7 Can we conduct easy model selection (have few parameters)?
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5. Motivation - Methodology Standpoint
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6. Forecasting Setup
Forecasting involves taking models fit on historical data and using them to
predict future observations.
ˆyi
output prediction
=
forecast model
f (xi ) where xi = xi,1, xi,2, ..., xi,k
input features
A forecast model is fit on a set of training samples in the form of (xi , yi )
where xi is the predictor variable, yi is the target variable, and i = 1...N.
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7. Forecasting Questions
When forecasting, it is important to understand your goal:
1 How much data do we have available and are we able to gather it all
together?
2 What is the time horizon of predictions that is required? Short,
medium, or long term?
3 Can forecasts be updated frequently over time or must they be made
once and remain static?
4 At what temporal frequency are forecasts required?
5 Are we forecasting single or multi step predictions?
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8. Point Forecasting
Traditional form of statistical numeric prediction.
The goal is to forecast a single expected value in the future ˆy = E[y|x].
Many methods exist:
Regression: ordinary least squares, ridge, lasso, polynomial, etc.
Time Series: autoregression, ARMA, exponential smoothing, etc.
Note: Time series analysis and regression analysis share many models,
and both are used for forecasting, but they are theoretically different!
Time series methods account for autocorrelation, regression methods
assume iid in features and no serial correlation.
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9. Types of Uncertainty Forecasting
1 Probabilistic Forecasting: predicts future densities. Most popular in
wind power.
2 Risk Indexes: finds the expected level of forecasting error.
3 Scenario Forecasting: creates several point forecasts (scenarios) for
a future period to find spatial-temporal correlations.
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10. Parametric vs Nonparametric
From ”Stochastic predictive control of battery energy storage for wind farm dispatching: Using probabilistic wind power
forecasts.” Renewable Energy (2015).
Parametric - assume a predefined shape of the density.
Nonparametric - we don’t assume any shape.
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11. Types of Forecasts
Types of probabilistic forecasts:
Quantiles: dividing the range of a probability distribution into
intervals.
Prediction Intervals: a range of specified coverage probability under
that distribution.
Predictive Densities: the full distribution.
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12. Quantiles
If we define Ft as the cumulative distribution function (CDF) of the
random variable xt and Ft is a strictly increasing function, the quantile qτ
t
with proportion α ∈ [0, 1] of xt is uniquely defined as the value x such that:
P(Pt < x) = τ
or equivalently as:
qτ
t = F−1
t (τ)
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13. Prediction Intervals
A prediction interval ˆIβ
t+k produced at time t for future time t + k is
defined by its lower and upper bounds, which are the quantile forecasts:
ˆIβ
t+k = ˆqαl
t+k, ˆqαu
t+k
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14. Prediction Density
If predicting quantiles we smooth between them to get the CDF. Else we
can use density estimation techniques.
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15. Forecasting Paradigms
How are the intervals centered?
Centered on the median
Centered on the mean
What type of data do we use for prediction?
Renewable Power
Numerical Weather Forecasts (NWP)
Past Time Series Data
Forecasting done using either current NWP or a combination of
past power and NWP data.
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16. Probabilistic Forecast Evaluation
Reliability Score - measures if PIs cover the observations.
Prediction Interval Coverage Probability (PICP):
PICP =
1
Ntest
Ntest
i=1
ci where ci =
1, ti ∈ Iτ
t (xi )
0, ti /∈ Iτ
t (xi )
Average coverage Error (ACE):
ACE = |PICP − 100 × (1 − τ)|
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17. Probabilistic Forecast Evaluation
Sharpness Score - measures how wide PIs are.
Interval Score (IS):
IS =
1
Ntest
Ntest
i=1
(ˆq1−τ
t − ˆqτ
t )
Skill Score - a rule based score of the density.
Quantile Score (QS):
Q(qτ , y) =
τ
100 − 1 (y − qτ ) if y < qτ
τ
100(y − qτ ) if y ≥ qτ
where y is observed power and qτ is the quantile forecast.
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18. Approaches to Probabilistic Forecasting
Quantile Regression (QR): predict quantiles based on exogenous
inputs.
Kernel Density Estimation (KDE): smooths a histogram of predicted
Gaussian densities.
Machine Learning: use of supervised and unsurprised learning
methods.
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19. Quantile Regression
Uses the pin-ball loss function to apply asymmetric weights to prediction
errors to compute conditional quantiles.
ρτ (u) =
τu if u ≥ 0
(τ − 1)u if u < 0
, 0 < τ < 1 (1)
The pinball function is always positive. The lower the loss, the better the
quantile forecast.
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20. Quantile Regression
Given predictors xi , slope mi and intercept b coefficients in a linear
regression equation we get the conditional τ quantile ˆqτ :
ˆqτ (t) =
k
i=1
mi xi + b (2)
which is be estimated by minimizing the following error cost where y(t) is
the observed value:
Eτ =
1
N
N
t=1
ρτ (y(t) − ˆqτ (t)) (3)
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