Correlations and Clustering in Wholesale Electricity Markets

We study the structure of locational marginal prices in day-ahead and real-time wholesale electricity markets. In particular, we consider the case of two North American markets and show that the price correlations contain information on the locational structure of the grid. We study various clustering methods and introduce a type of correlation function based on event synchronization for spiky time series, and another based on string correlations of location names provided by the markets. This allows us to reconstruct aspects of the locational structure of the grid.Comment: 30 pages, several picture

Perspectives on the Formalism of Quantum Theory

Quantum theory has the distinction among physical theories of currently underpinning most of modern physics, while remaining essentially mysterious, with no general agreement about the nature of its principles or the underlying reality. Recently, the rise of quantum information science has shown that thinking in operational or information-theoretic terms can be extremely enlightening, and that a fruitful direction for understanding quantum theory is to study it in the context of more general probabilistic theories. The framework for such theories will be reviewed in the Chapter Two. In Chapter Three we will study a property of quantum theory called self-duality, which is a correspondence between states and observables. In particular, we will show that self-duality follows from a computational primitive called bit symmetry, which states that every logical bit can be mapped to any other logical bit by a reversible transformation. In Chapter Four we will study a notion of probabilistic interference based on a hierarchy of interference-type experiments involving multiple slits. We characterize theories which do not exhibit interference in experiments with k slits, and give a simple operational interpretation. We also prove a connection between bit symmetric theories which possess certain natural transformations, and those which exhibit at most two-slit interference. In Chapter Five we will focus on reconstructing the algebraic structures of quantum theory. We will show that the closest cousins to standard quantum theory, namely the finite-dimensional Jordan-algebraic theories, can be characterized by three simple principles: (1) a generalized spectral decomposition, (2) a high degree of symmetry, and (3) a generalization of the von Neumann-Luders projection postulate. Finally, we also show that the absence of three-slit interference may be used as an alternative to the third principle. In Chapter Six, we focus on quantum statistical mechanics and the problem of understanding how its characteristic features can be derived from an exact treatment of the underlying quantum system. Our central assumptions are sufficiently complex dynamics encoded as a condition on the complexity of the eigenvectors of the Hamiltonian, and an information theoretic restriction on measurement resources. We show that for almost all Hamiltonian systems measurement outcome probabilities are indistinguishable from the uniform distribution

Higher-order interference and single-system postulates characterizing quantum theory

We present a new characterization of quantum theory in terms of simple physical principles that is different from previous ones in two important respects: first, it only refers to properties of single systems without any assumptions on the composition of many systems; and second, it is closer to experiment by having absence of higher-order interference as a postulate, which is currently the subject of experimental investigation. We give three postulates -- no higher-order interference, classical decomposability of states, and strong symmetry -- and prove that the only non-classical operational probabilistic theories satisfying them are real, complex, and quaternionic quantum theory, together with 3-level octonionic quantum theory and ball state spaces of arbitrary dimension. Then we show that adding observability of energy as a fourth postulate yields complex quantum theory as the unique solution, relating the emergence of the complex numbers to the possibility of Hamiltonian dynamics. We also show that there may be interesting non-quantum theories satisfying only the first two of our postulates, which would allow for higher-order interference in experiments while still respecting the contextuality analogue of the local orthogonality principle.Comment: 21 + 6 pages, 1 figure. v4: published version (includes several minor corrections

Self-duality and Jordan structure of quantum theory follow from homogeneity and pure transitivity

Among the many important geometric properties of quantum state space are: transitivity of the group of symmetries of the cone of unnormalized states on its interior (homogeneity), identification of this cone with its dual cone of effects via an inner product (self-duality), and transitivity of the group of symmetries of the normalized state space on the pure normalized states (pure transitivity). Koecher and Vinberg showed that homogeneity and self-duality characterize Jordan-algebraic state spaces: real, complex and quaternionic quantum theory, spin factors, 3-dimensional octonionic quantum state space and direct sums of these irreducible spaces. We show that self-duality follows from homogeneity and pure transitivity. These properties have a more direct physical and information-processing significance than self-duality. We show for instance (extending results of Barnum, Gaebeler, and Wilce) that homogeneity is closely related to the ability to steer quantum states. Our alternative to the Koecher-Vinberg theorem characterizes nearly the same set of state spaces: direct sums of isomorphic Jordan-algebraic ones, which may be viewed as composites of a classical system with an irreducible Jordan-algebraic one. There are various physically and informationally natural additional postulates that are known to single out complex quantum theory from among these Jordan-algebraic possibilities. We give various such reconstructions based on the additional property of local tomography

The structure of reversible computation determines the self-duality of quantum theory

Predictions for measurement outcomes in physical theories are usually computed by combining two distinct notions: a state, describing the physical system, and an observable, describing the measurement which is performed. In quantum theory, however, both notions are in some sense identical: outcome probabilities are given by the overlap between two state vectors - quantum theory is self-dual. In this paper, we show that this notion of self-duality can be understood from a dynamical point of view. We prove that self-duality follows from a computational primitive called bit symmetry: every logical bit can be mapped to any other logical bit by a reversible transformation. Specifically, we consider probabilistic theories more general than quantum theory, and prove that every bit-symmetric theory must necessarily be self-dual. We also show that bit symmetry yields stronger restrictions on the set of allowed bipartite states than the no-signalling principle alone, suggesting reversible time evolution as a possible reason for limitations of non-locality.Comment: 4 pages, 1 figure. v2: published version. Title slightly changed, interpretation of Theorem 2 correcte

Information-theoretic equilibration: the appearance of irreversibility under complex quantum dynamics

The question of how irreversibility can emerge as a generic phenomena when the underlying mechanical theory is reversible has been a long-standing fundamental problem for both classical and quantum mechanics. We describe a mechanism for the appearance of irreversibility that applies to coherent, isolated systems in a pure quantum state. This equilibration mechanism requires only an assumption of sufficiently complex internal dynamics and natural information-theoretic constraints arising from the infeasibility of collecting an astronomical amount of measurement data. Remarkably, we are able to prove that irreversibility can be understood as typical without assuming decoherence or restricting to coarse-grained observables, and hence occurs under distinct conditions and time-scales than those implied by the usual decoherence point of view. We illustrate the effect numerically in several model systems and prove that the effect is typical under the standard random-matrix conjecture for complex quantum systems.Comment: 15 pages, 7 figures. Discussion has been clarified and additional numerical evidence for information theoretic equilibration is provided for a variant of the Heisenberg model as well as one and two-dimensional random local Hamiltonian

PJM and MISO electricity markets price data

<p>Hourly price data both day-ahead and real time for PJM and MISO electricity markets, going from 00:00 of 1st January 2014 to 23:00 of 9th March. The data has been collected, cleaned and provided by Invenia Technical Computing Corporation; when a price was not available, it has been replaced with a NaN value.</p> <p>The file contains a dataset in Matlab 2012 .mat file extension and a readme text file.</p> <p>The dataset contains day-ahead price data for the MCC, MEC and LMP time series for both markets. Each time series is in a matrix format - Node x Time - with variable names da_(mcc/mec/lmp)_(miso/pjm). </p> <p> </p> <p> </p