# Last edited on 2022-10-20 18:07:23 by stolfi

TEST RESULTS

  120_distance
  
    The distance squared between two corners {x,y} of the canonical tetrahedron 
    is {z = SUM{(x[c] - y[c])^2 : c \in 0..2}}, that is 
    {z = SUM{ x[c]^2 - 2*x[c]*y[c] + y[c]^2 : c \in 0..2}}.
    Since the coordinates are all {±1}, that is 
    
      {z(x,y) = 6 - 2*dot(x,y)}.
      
    The value of {dot(x,y)} when {x} and {y} are corners of the tetrahedron
    is either {3} (same corner) of {-1} (different corners).  So 
    the value of {z} is either 0 or 8.
  
    The goal is to find a formula that takes /filtered/ datums {X[i][0..2]}
    and {Y[i][0..2]} and gives the best possible estimate for the 
    distance squared of the original (unfiltered) datums 
    {z[j] = |x[j] - y[j]|^2 = SUM{(x[j][c] - y[j][c])^2 : c \in 0..2}}
    where {x[j],y[j]} are the datums of the original sequences 
    that correspond to the filtered datums {X[i],Y[i]}.

    Tested with datum filtering weight table of 11 entries
 
      {FILTER_WEIGHTS   := 2  18 105 366 774 994 774 366 105  18   2}
      
    Result: the best estimator for {z[j] = |x[j] - y[j]|^2} is about
    
      {U1(X,Y) = 6 - 4*dot(X,Y)}
      
    Since {dot} is bilinear, and {X,Y} (with non-negative smoothing weights)
    are convex combinations of original datum vectors, the value 
    of {dot(X,Y)} has the same range as {dot(x,y)}, namely {-1} to {3}.
    Thus {U1(X,Y)} varies from 
      
    Also computed the best estimator for the smoothed version {Z[i]}
    of the {z} signal, that is, the RMS local mean discrepancy between
    the two original sequences.  This value too ranges between 0 and 8.
    
    Using weights
      
      DIFF_SQR_WEIGHTS := 0   0  11 133 599 988 599 133  11   0   0
      
    to smooth the {z} signal, the best estimator for {Z} from {X,Y} 
    turned out to be
    
      V1(X,Y) = 6 - 2.833*dot(X,Y)
      
    and 2.833 is about 17/6.  This estimate ranges from 

    When the {DIFF_SQR_WEIGHTS} weights were the same as the datum
    filter weights, the best estimate for {Z} mas
    
      V2(X,Y) = 6 - 2.31*dot(X,Y)

  125_dm_lav_to_cdv
  127_dm_chop_cands
  128_dm_scramble_cands
  130_dm_analyse_cdv
  150_scoring
  160_dm_show_alignments
  180_discrim
  190_candfilter
  220_refining
  250_extract
  270_weights
  330_dm_cand_view
  500_simple


Running dm_test_120_distance