Package xal.extension.wirescan.profile

Class ProfileDataStatistics

java.lang.Object
xal.extension.wirescan.profile.ProfileDataStatistics
public class ProfileDataStatistics extends Object

Computes statistical properties of profile data.

Define the weighted, central summation Sn(μ) as

  Sn(μ) = Σk(k - μ)nfk

where {fk} is the set of discrete samples for a projection view (that is, the discrete function representing the projection). Then the quantities provided by this class are typically some ratio of the Sn(μ) for a combination of parameters (n,μ). For example, the nth central moment moment, denoted <(x - μ)n>, is defined as

   <(x - μ)n> = Sn(μ)/S0(0)

Note that the units of the moments will be in terms of samples (see below). For example, the value returned by the method getCenter(ProfileData.Angle) is the index location of the center of mass. Note further that although the indices are integer valued, the statistic values in units of samples need not be.

NOTE:
- The quantities provided by this class are for discrete system. When considering these systems as approximations for continuous systems (sampled data systems) then we must "unnormalized" the results by the sampling interval h. For example, to convert <(x - μ)n> to the continuous approximation the value must be multiplied by hn.
- The above conversions are not provided by methods in this class. There are, however, methods for computing the sampling length h for the various axes and actuator positions. The rationale for not providing this service at present is the profile data sets might not contain the axis positions, as these are version-sensitive data. Thus, such methods have the potential for returning erroneous results.
- Higher-order moments are highly sensitive to signal noise.
- Strongly peaked distributions create numerically unstable computations. In general, for accurate numerical results the distribution should have a standard deviation σ > 2. Standard deviations smaller than this value imply that the sampling interval is too large.

Author:
Christopher K. Allen

  • Constructor Details

    • ProfileDataStatistics

      public ProfileDataStatistics(ProfileData pdoData)
      Create a new ProfileDataStatistics object and attach it to the given profile data set.

  • Method Details

    • getMass

      public double getMass(ProfileData.Angle view)

      Return the total mass (i.e., the first integral, or zeroth moment) of the given projection.

      NOTE:
      - The returned value is for a discrete function and must be treated as a "normalized" quantity when considering sampled data from a continuous system. To convert to the continuous approximation the returned value must be multiplied by h where h is the sampling interval.

      Parameters:
      view - projection data
      Returns:
      the value S0(0)

    • getCenter

      public double getCenter(ProfileData.Angle view)

      Return the (normalized) center of mass (i.e., the first moment) for the given projection.

      NOTE:
      - The returned value is for a discrete function and must be treated as a "normalized" quantity when considering sampled data from a continuous system. To convert to the continuous approximation for <x> the returned value must be multiplied by h where h is the sampling interval.
      - To convert to the center of the projection axis indicated by view, the returned value must be multiply by h then offset by x0, the left-hand axis end-point.

      Parameters:
      view - projection data
      Returns:
      the value S1(0)/S0(0) in

    • getActuatorOffset

      public double getActuatorOffset()
      Return the initial actuator position, that is the actuator location x0 of the first sample.
      Returns:
      initial position of the actuator

    • getAxisOffset

      public double getAxisOffset(ProfileData.Angle view)
      Return the initial axis position for the given projection angle, that is the axis location x0 of the first sample.
      Returns:
      initial axis position for the given viewing angle

    • compAveActuatorStepSize

      public double compAveActuatorStepSize()

      Compute and return the average step size between actuator positions for the entire sample projection sample set. This value is given by

         (xN-1 - x0)/(N - 1)

      where N is the number of sample points and {xk} are the actuator positions.

      NOTE:
      - The above result is the same as computing the steps size hk = xk - xk-1 between each sample then averaging the set.

      Returns:
      average step length between actuator positions

    • compAveAxisStepSize

      public double compAveAxisStepSize(ProfileData.Angle view)

      Compute and return the average step size between axis sample positions for the given projection view. This value is given by

         (xN-1 - x0)/(N - 1)

      where N is the number of sample points and {xk} are the axis positions.

      NOTE:
      - The above result is the same as computing the steps size hk = xk - xk-1 between each sample then averaging the set.

      Returns:
      average step length between axis sample positions

    • compStdDev

      public double compStdDev(ProfileData.Angle view)

      Return the (normalized) standard deviation σ of the given projection. The standard deviation is defined as

         σ = <(x - μ)2>½

      NOTE:
      - The returned value is for a discrete function and must be treated as a "normalized" quantity when considering sampled data from a continuous system. To convert to the continuous approximation for σ the returned value must be multiplied by h where h is the sampling interval.
      - Because of signal noise and numerical rounding it is possible to compute a value for <(x - μ)2> which is less than zero. Such an occurrence indicates either that sampling interval h is too large, or that the signal-to-noise ratio is too small to permit meaningful results. A zero value is returned in this situation.

      Parameters:
      view - project data to analyze
      Returns:
      standard deviation of given view, or zero if undefined

    • computeMoment

      public double computeMoment(int intOrder, ProfileData.Angle view)

      Compute and return the indicated moment for the given projection data. The value of the returned moment <xn> is defined as

         <xn> = Sn(0)/S0(0)

      NOTE:
      - The returned value is for a discrete function and must be treated as a "normalized" quantity when considering sampled data from a continuous system. To convert to the continuous approximation for <xn> the returned value must be multiplied by hn where h is the sampling interval.

      Parameters:
      intOrder - moment order
      view - project data to analyze
      Returns:
      moment of given view

    • computeCentralMoment

      public double computeCentralMoment(int intOrder, ProfileData.Angle view)

      Compute and return the indicated central moment for the given projection data. The value of the returned moment <(x - μ)n> is defined as

         <(x - μ)n> = Sn(μ)/S0(0)

      NOTE:
      - The returned value is for a discrete function and must be treated as a "normalized" quantity when considering sampled data from a continuous system. To convert to the continuous approximation for <(x - μ)n> the returned value must be multiplied by hn where h is the sampling interval.

      Parameters:
      intOrder - moment order
      view - project data to analyze
      Returns:
      moment of given view

    • computeWeightedSum

      public double computeWeightedSum(int intOrder, double dblCntr, ProfileData.Angle view)

      Compute and return the weighted central summation Sn(μ) which is defined

        Sn(μ) = Σk(k - μ)nfk

      Parameters:
      intOrder - order of the summation weight (i.e., n)
      dblCntr - center of the summation weight (i.e., μ)
      view - projection view angle (i.e., the set {fk})
      Returns:
      the value Sn(μ) as defined above