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Sara van de Geer
Compressed sensing, sparsity and p-values
One of the problems in compressed sensing can be phrased as follows:
suppose \(\beta^0 \in \mathbb{R}^p\) is a unknown vector we want to recover
from \(n \ll p\) measurements \(y\) where \(y = X \beta^0\) and \(X\)
is a \(n\times p\) matrix. The \(\ell_1\)-approach is to minimize \(\| \beta \|_1\)
subject to \(X \beta = y\). This "works" if "most" of the entries of \(\beta^0\)
are exactly zero (sparseness) and \(X\) satisfies the so-called
null-space property
on the support of \(\beta^0\). The statistical variant of the compressed
sensing problem sketched above generally faces a view complications. First of all, we usually will
have noisy measurements \(y\), i.e. we observe \(Y = X \beta^0 + \epsilon\) with \(\epsilon\)
unobservable noise. Secondly, we usually do not believe that
\(\beta^0\) is sparse in the strong sense (many zeroes), but rather in the weak sense,
where there are many non-zeroes but most of them are very small.
Then in applications we often cannot "design" the matrix \(X\) but it is just given to us.
The linear model could not be appropriate (for example when the
entries in \(Y\) are binary). Finally, recovering \(\beta^0\) exactly is
not possible in the noisy situation, instead we aim at interval estimates or p-values.
In our talk, we will present sparsity-regularized estimators
which are shown to trade-off approximation error and estimation error (an example will be the
so-called square-root Lasso). Our bounds have a learning-type of
flavour in the sense that if the model is wrong one is as good as the best approximation
within the model plus "small" error. Finally, we show a technique for establishing p-values.
Download the slides
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Gernot Müller
Estimation methods for an electricity spot price model based on a stable CARMA process
In recent years, electricity markets throughout the world have undergone massive changes due to deregulations. Extreme price volatility has forced producers
and wholesale consumers to hedge not only against volume risk but also against price movements. Consequently, statistical modeling and estimation of electricity
prices are an important issue for the risk management of electricity markets. We consider a model for the electricity spot price dynamics, which is able to capture
seasonality, low-frequency dynamics and the extreme spikes in the market. Instead of the usual purely deterministic trend we introduce a non-stationary
independent increments process for the low-frequency dynamics, and model the large fluctuations by a non-Gaussian stable CARMA process. We suggest a first
estimation procedure, where we fit the components of the model step by step. Then we look at a Bayesian approach to fit the model to data. Finally, we apply the
procedures to base load and peak load data from the German electricity exchange EEX.
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