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ANAPT: Additive noise analysis for persistence thresholding

  • * Corresponding author: Audun D. Myers

    * Corresponding author: Audun D. Myers 

The first author is supported by NSF grants CMMI-1759823 and DMS-1759824

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  • We introduce a novel method for Additive Noise Analysis for Persistence Thresholding (ANAPT) which separates significant features in the sublevel set persistence diagram of a time series based on a statistics analysis of the persistence of a noise distribution. Specifically, we consider an additive noise model and leverage the statistical analysis to provide a noise cutoff or confidence interval in the persistence diagram for the observed time series. This analysis is done for several common noise models including Gaussian, uniform, exponential, and Rayleigh distributions. ANAPT is computationally efficient, does not require any signal pre-filtering, is widely applicable, and has open-source software available. We demonstrate the functionality of ANAPT with both numerically simulated examples and an experimental data set. Additionally, we provide an efficient $ \Theta(n\log(n)) $ algorithm for calculating the zero-dimensional sublevel set persistence homology.

    Mathematics Subject Classification: Primary: 55-08.

    Citation:

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  • Figure 1.  Persistence diagram summarizing the sublevel sets of a function $ f(t) $ over finite domain $ t \in[t_a, t_b] $. This function has two local minima (blue squares) and two local maxima (red circles

    Figure 2.  Sublevel set persistence applied to $ x(t) $ of a single variable function or time series with and without additive noise $ \epsilon $ from $ \mathcal{N} $, shown in red and blue, respectively. This demonstrates the stability of persistent homology with the time series (left) with and without additive noise and the small effect on the resulting persistence diagrams (right). In addition, the light red region separates the significant features from those associated to additive noise

    Figure 3.  Histograms $ h(*) $ of the zero mean normal distriubtion $ \mathcal{N}(0,\sigma^2 =1) $ and the resulting birth times $ B $ and death times $ D $, which are compared to the density distributions from Equation(5)

    Figure 4.  Additive noise probability distributions $ f(z) $ for the four models realized in this work: uniform, Gaussian, Rayleigh, and exponential

    Figure 5.  Example time series showing sample $ \delta_i $

    Figure 6.  Numeric function fitting of Equationk__ge (52) to the mean of the median lifetime $ \tilde{L} $ of $ f_i(t) $ for $ i \in [1,3] $ where $ \mathcal{N} $ is unit variance Gaussian additive noise with $ \delta \in [0,2] $ being incremented to understand the effects of signal on the median lifetime

    Figure 7.  Demonstration of distribution parameter $ \sigma $ estimation of Gaussian additive noise in $ x(t) = A\sin(\pi t) + \mathcal{N} $ using the median lifetime with and without signal compensation as $ \sigma $ and $ \sigma^* $, respectively

    Figure 8.  Comparison between the persistent homology, bootstrap resampling, and ANAPT methods for separating noise from topological signal in the sublevel set persistence diagram. The example signal is generated from a chaotic Lorenz system

    Figure 9.  Comparison of distribution parameter estimation techniques (low-Pass filter residuals, cubic spline residuals, and sublevel set persistence) for estimating $ \sigma $ of the additive Gaussian noise contaminating the $ x(t) $ solution of the Lorenz system in Eq.55

    Figure 10.  Numerical simulations with $ 10^5 $ data points for each of the four investigated noise models with probability densities on the top row and lifetime probability densities on the bottom row with their associated cutoff $ C_\alpha $. The distribution parameters were set as $ \sigma = \Delta = \lambda = \sigma = 1 $ and were estimated as $ \sigma \approx 0.996 $, $ \Delta \approx 1.013 $, $ \lambda \approx 1.009 $, and $ \sigma \approx 0.998 $ for the Gaussian, uniform, Rayleigh, and exponential distributions, respectively

    Figure 11.  Example cutoff calculation for time series $ x(t) = A(\sin(\pi t) + \sin(t)) + \epsilon $, where $ \epsilon $ is additive noise from a Gaussian distribution with zero mean and four different standard deviations as $ \sigma \in [0.2, 1, 2, 4] $. The resulting sublevel set persistence diagrams with cutoff $ C_\alpha^* $ are shown to the right

    Figure 12.  Experimental time series from free drop response of single pendulum with corresponding cutoffs in the persistence diagram and time-order lifetimes plot

    Figure 13.  Histogram of the highlighted section of signal in Fig. 12 with fitted Rayleigh distribution

    Table 1.  Ratios $ \rho = \bar{L} / \tilde{L} $ for estimating sample mean from the sample median with uncertainty as three standard deviations

    Distribution Gussian Uniform Rayleigh Exponential
    $ \rho = \bar{L}/\tilde{L} $ $ 1.154 \pm 0.012 $ $ 1.000 \pm 0.010 $ $ 1.136 \pm 0.013 $ $ 1.265 \pm 0.016 $
     | Show Table
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    Table 2.  Constants of Equation (52) for each distribution type investigated in this work with associated uncertainty from ten trials

    Distribution Gussian Uniform Rayleigh Exponential
    $ c_1 $ $ 0.845 \pm 0.029 $ $ 0.880 \pm 0.017 $ $ 0.726 \pm 0.026 $ $ 0.436 \pm 0.036 $
    $ c_2 $ $ 0.809 \pm 0.061 $ $ 0.639 \pm 0.026 $ $ 0.605 \pm 0.054 $ $ 0.393 \pm 0.075 $
     | Show Table
    DownLoad: CSV

    Table Algorithm 1.  Zero-Dimensional Persistence Algorithm

    Data: Array A of n Real Numbers
    Result: Persistence Diagram
    $\begin{array}{l} {\bf{begin}}\\ \;\;\left| \begin{array}{l} \;\;\;\;M = {\rm{list}}\;{\rm{of}}\;{\rm{local}}\;{\rm{extrema}}\;{\rm{in}}\;A\left( {{\rm{non - endpoint}}\;{\rm{maxima}}\;{\rm{and}}\;{\rm{minima}}} \right);\\ \;\;\;\;Q = {\rm{priority}}\;{\rm{queue}}\;{\rm{of}}\;{\rm{pointers}}\;{\rm{to}}\;{\rm{elements}}\;{\rm{of}}\;M;\\ \;\;\;\;{\bf{while}}\;\left| M \right| > 3\;{\bf{do}}\\ \;\;\;\;\;\;\left| \begin{array}{l} m \leftarrow Q.pop;\\ b \leftarrow {\rm{min}}\left\{ {m.height,m.next.height} \right\};\\ d \leftarrow {\rm{max}}\left\{ {m.height,m.next.height} \right\};\\ {\rm{Add}}\;(b,d){\rm{ to}}\;pairs;\\ {\rm{Update}}\;Q\;{\rm{and}}\;M; \end{array} \right.\\ \;\;\;\;{\bf{end}}\\ \;\;\;\;{\bf{return}}\;pairs \end{array} \right.\\ {\bf{end}} \end{array}$
     | Show Table
    DownLoad: CSV
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