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- Riemannian Geometry I.
William Wylie wwylie syr. Program Reference Code s :. Text Only Version. Whereas the unbounded case represented by the Fisher Information Matrix is embedded in the geometric framework as vanishing Christoffel Symbols, non-vanishing constant Christoffel Symbols prove to define prototype non-linear models featuring boundedness and flat parameter directions of the log-likelihood. Finally, we investigate if those models could allow to approximate and replace computationally expensive objective functions originating from non-linear models by a surrogate objective function in parameter estimation problems.
This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Data Availability: All relevant data are within the manuscript and its Supporting Information files.
The funders had no role in study design, data collection and analysis, decision to publish or preparation of the manuscript. Competing interests: The authors have declared that no competing interests exist. Parameter estimation by the maximum-likelihood method has numerous applications in different fields of physics, engineering, and other quantitative sciences. In systems biology, e.
Parameter estimation in these non-linear models can easily become time-consuming. Solving the ODEs and computing model sensitivities for numerical optimization is computationally demanding. The difficulty is further increased if many experimental conditions contribute to the evaluation of the likelihood function because the model ODE needs to be solved independently for each condition. Upon successful parameter estimation, thorough investigation of the log-likelihood around the optimum frequently reveals that some parameters, although having a unique optimum, cannot be constrained to finite confidence intervals.
This situation is denoted as practical non-identifiability [ 3 ]. The reason for practical non-identifiability is the non-linear relationship between model parameters and model predictions. The non-linearity culminates in the boundedness of model predictions for all possible combinations of parameters and, consequently, in upper limits of the negative log-likelihood that are not exceeded along certain paths.
Based on the likelihood-ratio test statistics, log-likelihood thresholds relative to the value at the optimum can be derived [ 4 ] that, when being exceeded by the choice of parameters, allow to reject the model specification. Conversely, if the derived thresholds are above the upper limit of the negative log-likelihood, the model cannot be rejected over an infinite range of parameter values. In this work we discuss that for certain models, practical non-identifiability can already be detected from local information, i.
Semi Riemannian Geometry
The construction is based on a differential geometric point of view on least squares estimation as laid out in [ 5 , 6 ]. The geometry of least squares estimation has already previously been discussed, e. Also the usage of second order model sensitivities to derive equations for parameter transformations providing the log-likelihood with a more quadratic shape around the optimum has been suggested in earlier statistical works, see [ 8 , 9 ].
However, these previous attempts have been too general to be either solved analytically or be feasible numerically. In contrast, by sticking to a local approximation of Christoffel Symbols, i. The boundary value problem underlying the parameter transformation can be solved efficiently by numerical methods [ 11 ]. The result is that despite being based on purely local second order sensitivity information, the log-likelihood function constructed in this way reflects a fundamental property of the original log-likelihood: its boundedness.
It is thereby possible to capture not only the parameter estimates but also their correlation structure locally as well as in the limit of practical non-identifiability. Given a mathematical model to describe a set of M data points, one is interested in the N parameters such that the model fits the data best. This boundedness has implications for parameter estimation and confidence interval determination [ 3 ].
All residuals can be combined into a vector r which is an element of the M -dimensional data space. Fig 1A shows an example of an extrinsically flat, one-dimensional model manifold in a two-dimensional data space.
A The tangent at the optimum is perpendicular to its residual vector. Boundaries of the model manifold, shown in orange, are marked by black segments. As a defining property of geodesics, the absolute value of the velocity stays constant along a geodesic. For bounded model manifolds, the RNC are bounded in their domain, since the boundary can be reached by a geodesic with finite initial velocity.
We emphasize that Eq 12 is exact if and only if the model manifold is extrinsically flat. For non-linear models, the extrinsic curvature generally is non-zero and Eq 12 only holds locally, since in this case the assumption in Eq 6 is violated when moving further away from the optimum and the geodesic is not a straight path in. In this work, we do not account for this deviation.
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It has been noted in [ 6 ] that extrinsic curvature of model manifolds can often be neglected. To perform the coordinate change from the original parameters to the RNC, the geodesic equation has to be solved as a two-point boundary value problem. The first point is the point around which the RNC are constructed, in our case. Since the geodesic equation is a non-linear ordinary differential equation, in most cases a closed-form solution does not exist and approximations are made to solve the geodesic equation. A popular approach in literature e. Our approach approximates only the Christoffel Symbols by their values at and inserts them in the otherwise unmodified geodesic equation: 13 This approximated geodesic equation can be numerically solved without repeated model evaluation.
By construction, the resulting curves approximate the true geodesics in a neighborhood of. This can be understood from the fact that the solution of a quadratic ordinary differential equation can diverge in finite time. We discuss various features of the approximation by means of three models with increasing complexity. The first model is an example for which the approximation is exact. Consider the model 17 with. The sign of the parameter k 1 determines whether the model describes exponential decay or exponential growth.
Clearly, the model manifold is bounded in one direction: In a one-dimensional data space, it covers the positive real numbers shifted negatively by the value of the data point. In this rare case, the geodesic equation can be solved exactly detailed steps are presented in section E in S1 Text : 18 The integration constants are chosen as 19 We therefore conclude that boundedness of a model along a parameter axis relates to a restricted co-domain of the model: The amount of the decaying substance can never drop below zero, regardless of the rate constant.
This boundedness is represented appropriately by a model with constant Christoffel Symbol. We now modify Model 1 at two stages: On the one hand, we introduce a parameter A 0 for the initial amount: 23 On the other hand, we restrict both the parameter k 1 and A 0 to values greater or equal to zero. In Fig 2A , the contours of the original objective function are shown as solid line.
The colored lines represent paths that are optimal with respect to the parameter k 1 for any given value of parameter A 0. The paths computed for A are shown in the new coordinates as colored lines. Thresholds for different confidence levels are depicted in gray. However, also the third source of boundedness can be observed, the coupling of parameters such that a flat canyon is formed which means that a change of one parameter is compensated by the other parameter. The dashed contour lines represent the approximated objective function.
In comparison to the original objective function, the asymptotic behaviour for cross sections at constant A 0 does not appear to be bounded, but for cross sections at constant k 1. However, the parameter coupling between A 0 and k 1 is matched very well, a behavior which an approximation by a Taylor expansion could never exhibit. It is noticeable that the path of the parameter coupling is straight as opposed to the curved path of the original objective function. However, we note that usually the paths of parameter coupling tend to straighten out asymptotically. They each follow the paths of the parameter coupling, which, though different in parameter space, appear to have very similar objective function values along their path, which is shown in Fig 2C.
The same paths and contours are shown in Fig 2B in the new coordinates v. Next, the approximation is tested on an enzymatic reaction modeled by mass-action kinetics. In this model, an enzyme E and its substrate S first form a complex C which can either dissociate back into E and S , or form a product P , in which case P and E are released. The corresponding ODEs are given by The enzyme model typically exhibits two time-scales: the binding and dissociation of E and S are usually much faster than the product formation, i. This can lead to non-identifiable parameters k 1 and k 2.
For large values of k 1 and k 2 , the complex quickly reaches a quasi-equilibrium in which only the ratio of k 1 and k 2 can be determined. Because the visualization of higher-dimensional parameter spaces by contour lines is not feasible, we have evaluated the original and approximated objective function along profile-likelihood paths for different parameters.