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detailed analysis of prominence definitions

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  • Adam Helman
    Hello folks, Many of you will be merely intrigued by this work - http://www.cohp.org/prominence/definitions/prominence_definitions.html - while some will
    Message 1 of 3 , Apr 29, 2012
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      Hello folks,

      Many of you will be merely intrigued by this work -

      http://www.cohp.org/prominence/definitions/prominence_definitions.html

      - while some will actually read it all because it features much new
      material
      addressing the various "flavors" of prominence: mean ("interpolated"),
      clean and optimistic from a purely theoretical view.

      How well they fare, and WHY, in selecting "gray zone" peaks at the
      bottom of a prominence list is a major topic.

      I recommend reading all BUT the Appendix. It contains some serious
      mathematics required to evaluate certain error properties for these
      measures.

      Still, if you want to see how Jacobians, orthogonal transformations
      and double integrals actually work in the "real world" then, by all means,
      read the Appendix as well!

      ****************************************

      There is a new measure related to prominence just as intimately as
      peaks and saddles are related to one another. I denote it "q" in the
      derivations,
      having failed to conjure a decent English name for it.
      "Continentality" comes to mind but that's too long and has too many
      syllables.
      Your recommendations are welcome for naming this measure which seems to
      select-out peaks that are high and yet NOT prominent.

      It would make a terrible list-builder except for those "heightists" who
      love
      bashing prominence. In fact, "q" could very well be their poster-child.

      This work consumed the better part of one week.
      So I hope that somebody takes note of its contents and concludes
      it might be of value.

      Sincerely,
      Adam Helman
    • Adam Helman
      OK, Dave, I will make it simple for you: 1. Interpolated prominence has overwhelmingly superior mathematical properties compared against clean and optimistic
      Message 2 of 3 , Apr 30, 2012
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        OK, Dave, I will make it simple for you:

        1. Interpolated prominence has overwhelmingly superior mathematical properties compared against clean and optimistic prominence.

        a. It more frequently correctly assigns and expels peaks from appearing at the bottom of a prominence-based peak list.
        EVERY combination of {prominence type, list cutoff value, map contour spacing} is exhaustively considered.

        b. It is an unbiased estimator for the unknowable true prominence when the latter is only available as a range of values,
        i.e. the average error (calculated - true prominence) is 0 for interpolated ("mean") prominence. The other prominence definitions
        are biased, but can also recover the same 0 mean error by simple linear transformations:

        clean prominence -> clean prominence - half contour interval
        and
        optimistic prom -> optimistic prom + half contour interval

        2. The q-value, defined as (peak elevation + saddle elevation) (although more properly as (peak elevation + saddle elevation) / sqrt (2) ).
        From a purely theoretical perspective it is the equal of prominence. However from a practical aspect it suffers....

        a. Living in Colorado you should note that every tiny bump just under the top of all 14ers are climbworthy: for they, along with
        Point Success on Mount Rainier, will lead the list of CONUS peaks ordered by their "continentality" q - the antithesis of prominence.

        In effect, "q" is Colorado's answer to Washington's prominence challenge 8-).

        b. When you climb Denali this Spring be certain to also do Archdeacons Tower at 19,500 feet.
        In theory it leads North America in q-value unless there's a minor bump higher-still between it and the top.


        I'll add a "Summary" expounding on these critical points.

        ****************************************************************************************************************************

        Non-USA residents should note that a P610 meter list will have a larger fraction of correct gray zone peak assignments than a P600 meter list.
        Please read the relevant text to understand WHY this is so.

        Sincerely,
        Adam H.

        PS Good luck at Denali. WHEN do you fly?
        ----- Original Message -----
        From: Dave_Covill@...
        To: Adam Helman
        Cc: davidwmolson@... ; dmg@... ; esquared@... ; gregslayden@... ; jkirk_14@... ; john.hamann@... ; mark Adrian ; oldadit@... ; Petter.Bjorstad@... ; prominence@yahoogroups.com ; rfbolton@... ; rhwxyz@... ; robert.packard@... ; Roy Schweiker ; surgent@...
        Sent: Monday, April 30, 2012 9:17 AM
        Subject: Re: detailed analysis of prominence definitions


        Yikes !!!!!!!!!! I have a headache.....


        ;)





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      • Adam Helman
        Reverse the signs to obtain these (corrected) prescriptions - clean prominence - clean prominence + half contour interval and optimistic prom - optimistic
        Message 3 of 3 , Apr 30, 2012
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          Reverse the signs to obtain these (corrected) prescriptions -

          clean prominence -> clean prominence + half contour interval
          and
          optimistic prom -> optimistic prom - half contour interval


          > b. It is an unbiased estimator for the unknowable true prominence when the latter is only available as a range of values,
          > i.e. the average error (calculated - true prominence) is 0 for interpolated ("mean") prominence. The other prominence definitions
          > are biased, but can also recover the same 0 mean error by simple linear transformations:

          > clean prominence -> clean prominence - half contour interval
          > and
          > optimistic prom -> optimistic prom + half contour interval


          *******************************************************************

          Fortunately the online article already had the correct versions as Equations (4a,b).

          - Adam H.


          [Non-text portions of this message have been removed]
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