Linear observation function. Importance of using the Mahalanobis distance. We want to demonstrate the utility of using the Mahalanobis distance compared to using the typical Euclidean distance. Errors in covariance estimation for the linear example.
The average is computed over 10 data points and using 50 sample points to estimate each covariance. Errors in covariance estimation. For the example in 6. Recovery of the fast variable. Note that, for the example in 6. Therefore, for the two-dimensional data considered here, the fast variable will not necessarily appear as the second nontrivial eigenvector. As the time-scale separation increases, the relative importance of the slow and fast directions will also increase.
This implies that the eigenvalue corresponding to the eigenvector which parametrizes the fast direction will decrease, and the number of harmonics of the slow mode which appear before the fast mode will increase. This implies that the apparent time-scale separation is smaller, the attenuation of the fast variable is less pronounced relative to the slow variable, and the fast variable is recovered in an earlier eigenvector in our ordering of the spectrum.
Nonlinear observation function. In the second example, our data from section 6. Row 1: Data gray and representative burst red used to estimate the local covariance. The eigenvalues corresponding to the coordinates for the slow and fast modes are indicated by red circles.
The data from Figure 4, transformed by f in 6. Mahalanobis distance to obtain a parametrization that is consistent with the underlying fast- slow dynamics.
Errors in Mahalanobis distance. Therefore, for the example in 6. This is shown in Figure 9 b , and the deviation from quadratic behavior is consistent with the intersection of the analytical expressions plotted in Figure 9 a. Errors in the Mahalanobis distance and the covariance estimation for the nonlinear example in 6. From 5. These results are shown by plotting C in Figure 9 d. Furthermore, we showed how to estimate the covariances required for this Maha- lanobis distance computation directly from data.
A key point in our approach is that the embedding coordinates we compute are not only insensitive to the fast variables, but are also approximately invariant to nonlinear observation functions. In the examples presented, the initial data came from a single trajectory of a dynamical system, and the local covariance at each point in the trajectory was estimated using brief simulation bursts. However, the initial data need not be collected from a single trajectory, and other sampling schemes could be employed.
In our examples, we controlled the time scale of sampling; we could therefore set the time scale over which to estimate the noise covariance and could set the simulation time step to be arbitrarily small. Such intermediates are more complex statistical functions than simple averages and may be able to capture additional structure within the data.
However, constructing such intermediates often requires additional a priori knowledge about the system dynamics and noise structure, and it was not pursued here.
The method presented here provides a bridge between traditional data mining and a class of multi-time-scale dynamical systems. Belkin and P. Niyogi, Laplacian eigenmaps for dimensionality reduction and data representation, Neural Comput. Berry, J. Cressman, Z. Brey, R. Zwanzig, and J. Dorfman, Nonlinear transport equations in statistical mechanics, Phys.
A, , pp. Chorin, O. Hald, and R. Kupferman, Optimal prediction and the Mori—Zwanzig represen- tation of irreversible processes, Proc. USA, 97 , pp. Coifman and S. Coifman, S. Lafon, A. Lee, M. Maggioni, B. Nadler, F. Warner, and S. Contou-Carrere, V. Sotiropoulos, Y. Kaznessis, and P. Dong, L. Jakobowski, M. Iafolla, and D. Dsilva, R. Talmon, R. Coifman, and I. Kevrekidis, Parsimonious representation of nonlinear dynamical systems through manifold learning: A chemotaxis case study, Appl.
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The person requires more assistance with activities of daily living, such as bathing, toileting, and eating. Patients in this stage may develop delusions, hallucinations, or obsessions. Patients show increased anxiety and may become violent. The person in this stage begins to sleep during the day and stay awake at night. Stage 6 is severe dementia. NCD Alzheimer and related type: Noticeable deficits in job and demanding situations. Mild NCD: Requires assistance in complicated tasks such as handling finances, planning parties, etc.
Moderately severe NCD: Requires assistance dressing, bathing, and toileting. Experiences urinary and fecal incontinence.
Severe NCD: Speech ability declines to about a half-dozen intelligible words. Progressive losses of abilities to sit up, smile, and hold head up.
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