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Graphs of Functions01:30

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Graphs of functions provide a visual representation of how output values change in response to varying inputs. Each point on the graph corresponds to an ordered pair, where the x-coordinate (independent variable) determines the horizontal position and the y-coordinate (dependent variable) determines the vertical position. Linear functions like y = x give a straight line, indicating a constant rate of change.Nonlinear functions display more complex behaviors. Even power functions generate...
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Trigonometric functions exhibit periodic and symmetrical behavior, deeply rooted in the unit circle. The sine and cosine functions correspond to the vertical and horizontal projections, respectively, of a point rotating counterclockwise around the circle. These functions trace smooth, repeating waveforms with identical periods and bounded ranges. The tangent function is defined as the ratio of sine to cosine and produces an unbounded curve that repeats every units, with vertical asymptotes...
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The term "bootstrap" originated in the 19th century as a metaphor for self-improvement or achieving something independently, without external assistance. This concept extends to statistical bootstrapping, a self-contained method for estimating population parameters through resampling, even though it can be computationally intensive. Developed by the American statistician Dr. Bradley Efron in 1979, bootstrapping provides a robust way to perform inference when the original sample size is...
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Related Experiment Video

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Probing C84-embedded Si Substrate Using Scanning Probe Microscopy and Molecular Dynamics
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High-veracity functional imaging in scanning probe microscopy via Graph-Bootstrapping.

Xin Li1,2,3,4, Liam Collins1,2, Keisuke Miyazawa5

  • 1Institute for Functional Imaging of Materials, Oak Ridge National Laboratory, Oak Ridge, TN, 37831, USA.

Nature Communications
|June 23, 2018
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Summary
This summary is machine-generated.

We developed Graph-Bootstrapping, a data-driven method for analyzing complex nanoscale data from scanning probe microscopy (SPM). This approach enhances visualization and analysis of high-dimensional datasets, improving material characterization.

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Area of Science:

  • Materials Science
  • Nanotechnology
  • Analytical Chemistry

Background:

  • Scanning probe microscopy (SPM) is crucial for nanoscale surface analysis.
  • Increasingly complex SPM data (multiple channels, high dimensions) challenges visualization and analysis, especially with poorly understood dynamics.
  • Existing methods struggle with high-dimensional, information-rich SPM datasets.

Purpose of the Study:

  • To introduce a novel data-driven approach, Graph-Bootstrapping, for analyzing complex SPM data.
  • To address the challenges of visualizing and analyzing high-dimensional SPM datasets.
  • To improve the accuracy and efficiency of nanoscale material characterization using SPM.

Main Methods:

  • A data-driven approach utilizing low-dimensional manifold learning on full SPM spectra.
  • Application of Graph-Bootstrapping to analyze complex, high-dimensional SPM datasets.
  • Demonstration on mixed polymer thin films and calcite hydration structures.

Main Results:

  • Successful high-veracity mechanical mapping of a mixed polymer thin film.
  • Resolution of irregular hydration structures in calcite at atomic resolution.
  • Efficient revelation and hierarchical representation of salient material features with rich local details.

Conclusions:

  • Graph-Bootstrapping offers an effective solution for analyzing complex SPM data.
  • The methodology enables enhanced denoising, classification, and high-resolution functional imaging.
  • This approach advances the capabilities of SPM for detailed nanoscale material analysis.