dimensional analysis

Examples of dimensional analysis in the following topics:

• Dimensional Analysis

  • Dimensional analysis is the practice of checking relations between physical quantities by identifying their dimensions.
  • Dimensional analysis is based on the fact that physical law must be independent of the units used to measure the physical variables.

• Converting from One Unit to Another

  • Converting units using dimensional analysis makes working with large and small measurements more convenient.
  • Converting between metric units is called unit analysis or dimensional analysis.
  • Unit analysis is a form of proportional reasoning where a given measurement can be multiplied by a known proportion or ratio to give a result having a different unit or dimension.
  • If you had a sample of a substance with a mass of 0.0034 grams, and you wanted to express that mass in mg, you could use the following dimensional analysis.

• Crystallographic Analysis

  • Crystallographic analysis reveals the arrangement of atoms in solids that help build the three-dimensional model of molecules.
  • Nuclear magnetic resonance spectroscopy and x-ray crystallography have become the methods of choice for understanding three-dimensional protein structures.
  • Studies of protein crystallography help determine the three dimensional structure of proteins and analyze their function alone or within multimolecular assemblies.
  • The structure-function analysis is completed by biochemical and biophysical studies in solution.
  • The protocol for completing a successful crystallographic analysis requires production of proteins (cloning, mutagenesis, bacterial culture, etc.), purification of recombinant proteins (such as chromatography of affinity and gel filtration), enzymatic tests and inhibition measurement (spectrophotometry), crystallization, x-rays crystallography and structural analysis, interactions determination (microcalorimetry, fluorescence, BIAcore), conformational analyses (circular dichroism, ultracentrifugation, light scattering), modifications analysis (mass spectrometry), bioinformatics, and molecular modelisation.

• Two-mode factor analysis

  • The method used by factor analysis to identify the dimensions differs from SVD.
  • Figure 17.10 shows the eigenvalues (by principle components) calculated by Tools>2-Mode Scaling>Factor Analysis.
  • This solution, although different from SVD, also suggests considerable dimensional complexity in the joint variance of actors and events.
  • The factor analysis method does produce somewhat lower complexity than SVD.
  • With the caveat of pretty poor fit of a low-dimensional solution in mind, let's examine the scaling of actors on the first three factors (figure 17.11).

• Multi-dimensional scaling tools

  • Usually our goal in equivalence analysis is to identify and visualize "classes" or clusters of cases.
  • In using cluster analysis, we are implicitly assuming that the similarity or distance among cases reflects as single underlying dimension.
  • Factor or components analysis could be applied to correlations or covariances among cases.
  • Clustering and scaling tools can be useful in many kinds of network analysis.
  • Two-dimensional map of non-metric MDS of Knoke information adjacency

• Two-mode correspondence analysis

  • For binary data, the use of factor analysis and SVD is not recommended.
  • As an alternative for binary actor-by-event scaling, the method of correspondence analysis (Tools>2-Mode Scaling>Correspondence) can be used.
  • Correspondence analysis (rather like Latent Class Analysis) operates on multi-variate binary cross-tabulations, and its distributional assumptions are better suited to binary data.
  • Figure 17.13 shows the location of events (initiatives) along three dimensions of the joint actor-event space identified by the correspondence analysis method.
  • Figure 17.15 show the plot of the actors and events in the first two dimensions of the joint correspondence analysis space.

• Intorduction to qualitative analysis

  • This is because the various dimensional methods operate on similarity/distance matrices, and measures like correlations (as used in two-mode factor analysis) can be misleading with binary data.
  • Even correspondence analysis, which is more friendly to binary data, can be troublesome when data are sparse.
  • This approach doesn't involve any of the distributional assumptions that are made in scaling analysis.

• X-Ray Diffraction Analysis

  • The analysis consists of indexing, merging, and phasing variations in electron density.
  • Further analysis involves structure refinement and quantitative phase using the general structure analysis system (GSAS), which ultimately leads to the identification of the amorphous or crystalline phase of a matter and helps construct its three dimensional atomic model .
  • X-ray diffraction analysis workflow.
  • The two-dimensional images taken at different rotations are converted into a three-dimensional model of the density of electrons within the crystal using the mathematical method of Fourier transforms, combined with chemical data known for the sample.
  • Summarize the methods used for x-ray diffraction analysis and the contributions they have made to science

• Regression Analysis for Forecast Improvement

  • Regression Analysis is a causal / econometric forecasting method that is widely used for prediction and forecasting improvement.
  • Regression Analysis is a causal / econometric forecasting method.
  • A large body of techniques for carrying out regression analysis has been developed.
  • Nonparametric regression refers to techniques that allow the regression function to lie in a specified set of functions, which may be infinite-dimensional.
  • Regression analysis shows the relationship between a dependent variable and one or more independent variables.

• Basic Techniques in Protein Analysis

  • The basic technique for protein analysis, analogous to DNA sequencing, is mass spectrometry.
  • Proteins are naturally-unstable molecules, which makes proteomic analysis much more difficult than genomic analysis.
  • X-ray crystallography enables scientists to determine the three-dimensional structure of a protein crystal at atomic resolution.
  • The result is a three-dimensional digital image of the molecule.
  • Mass spectrometry can be used in protein analysis.