Delineating Parameter Unidentifiabilities in Complex Models

Scientists use mathematical modelling to understand and predict the properties of complex physical systems. In highly parameterised models there often exist relationships between parameters over which model predictions are identical, or nearly so. These are known as structural or practical unidentifiabilities, respectively. They are hard to diagnose and make reliable parameter estimation from data impossible. They furthermore imply the existence of an underlying model simplification. We describe a scalable method for detecting unidentifiabilities, and the functional relations defining them, for generic models. This allows for model simplification, and appreciation of which parameters (or functions thereof) cannot be estimated from data. Our algorithm can identify features such as redundant mechanisms and fast timescale subsystems, as well as the regimes in which such approximations are valid. We base our algorithm on a novel quantification of regional parametric sensitivity: multiscale sloppiness. Traditionally, the link between parametric sensitivity and the conditioning of the parameter estimation problem is made locally, through the Fisher Information Matrix. This is valid in the regime of infinitesimal measurement uncertainty. We demonstrate the duality between multiscale sloppiness and the geometry of confidence regions surrounding parameter estimates made where measurement uncertainty is non-negligible. Further theoretical relationships are provided linking multiscale sloppiness to the Likelihood-ratio test. From this, we show that a local sensitivity analysis (as typically done) is insufficient for determining the reliability of parameter estimation, even with simple (non)linear systems. Our algorithm provides a tractable alternative. We finally apply our methods to a large-scale, benchmark Systems Biology model of NF-$\kappa$B, uncovering previously unknown unidentifiabilities

Significance Regression: A Statistical Approach to Biased Linear Regression and Partial Least Squares

This paper first examines the properties of biased regressors that proceed by restricting the search for the optimal regressor to a subspace. These properties suggest features such biased regression methods should incorporate. Motivated by these observations, this work proposes a new formulation for biased regression derived from the principle of statistical significance. This new formulation, significance regression (SR), leads to partial least squares (PLS) under certain model assumptions and to more general methods under various other model kumptions. For models with multiple outputs, SR will be shown to have certain advantages over PLS. Using the new formulation a significance test is advanced for determining the number of directions to be used; for PLS, cross-validation has been the primary method for determining this quantity. The prediction and estimation properties of SR are discussed. A brief numerical example illustrates the relationship between SR and PLS

Image patch analysis of sunspots and active regions. II. Clustering via matrix factorization

Separating active regions that are quiet from potentially eruptive ones is a key issue in Space Weather applications. Traditional classification schemes such as Mount Wilson and McIntosh have been effective in relating an active region large scale magnetic configuration to its ability to produce eruptive events. However, their qualitative nature prevents systematic studies of an active region's evolution for example. We introduce a new clustering of active regions that is based on the local geometry observed in Line of Sight magnetogram and continuum images. We use a reduced-dimension representation of an active region that is obtained by factoring the corresponding data matrix comprised of local image patches. Two factorizations can be compared via the definition of appropriate metrics on the resulting factors. The distances obtained from these metrics are then used to cluster the active regions. We find that these metrics result in natural clusterings of active regions. The clusterings are related to large scale descriptors of an active region such as its size, its local magnetic field distribution, and its complexity as measured by the Mount Wilson classification scheme. We also find that including data focused on the neutral line of an active region can result in an increased correspondence between our clustering results and other active region descriptors such as the Mount Wilson classifications and the $R$ value. We provide some recommendations for which metrics, matrix factorization techniques, and regions of interest to use to study active regions.Comment: Accepted for publication in the Journal of Space Weather and Space Climate (SWSC). 33 pages, 12 figure

The two-dimensional random-bond Ising model, free fermions and the network model

We develop a recently-proposed mapping of the two-dimensional Ising model with random exchange (RBIM), via the transfer matrix, to a network model for a disordered system of non-interacting fermions. The RBIM transforms in this way to a localisation problem belonging to one of a set of non-standard symmetry classes, known as class D; the transition between paramagnet and ferromagnet is equivalent to a delocalisation transition between an insulator and a quantum Hall conductor. We establish the mapping as an exact and efficient tool for numerical analysis: using it, the computational effort required to study a system of width $M$ is proportional to $M^{3}$, and not exponential in $M$ as with conventional algorithms. We show how the approach may be used to calculate for the RBIM: the free energy; typical correlation lengths in quasi-one dimension for both the spin and the disorder operators; even powers of spin-spin correlation functions and their disorder-averages. We examine in detail the square-lattice, nearest-neighbour $\pm J$ RBIM, in which bonds are independently antiferromagnetic with probability $p$, and ferromagnetic with probability $1-p$. Studying temperatures $T\geq 0.4J$, we obtain precise coordinates in the $p-T$ plane for points on the phase boundary between ferromagnet and paramagnet, and for the multicritical (Nishimori) point. We demonstrate scaling flow towards the pure Ising fixed point at small $p$, and determine critical exponents at the multicritical point.Comment: 20 pages, 25 figures, figures correcte