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AI-GEOSTATS: Re: mixtures of populations

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  • Isobel Clark
    Hello All The common Normal Score transform assumes one population. Transformations such as rank or logarithm do not assume one population. The best way to
    Message 1 of 9 , Mar 9, 2004
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      Hello All

      The common 'Normal Score' transform assumes one
      population. Transformations such as rank or logarithm
      do not assume one population.

      The best way to identify likely mixtures is with
      programs such as Peter MacDonald's Mix (cited in
      Ruben's email I think):

      http://www.math.mcmaster.ca/peter/mix/mix31.html

      or with probability plots. Many software packages have
      these and mixtures are easily identifiable by
      break-points or points of inflexion in the plot.

      For those (like myself) without easy access to
      libraries, there are a couple of papers which describe
      (geological) applications and using a combination of
      indicator and ordinary kriging to solve some problems.


      Papers can be found at
      http://uk.geocities.com/drisobelclark/resume
      follow the publications link. Look for my 1974 paper,
      now available in pdf format, the 1993 IMGC paper and
      the 1992 Troia paper with Jonathan Vieler.

      We have had good experience with this approach for 30
      years in fields as diverse as mineral resource
      estimation and seabird preservation.

      Isobel Clark
      http://geoecosse.bizland.com/courses.htm







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    • Chaosheng Zhang
      Dear Isobel and all, This is an interesting topic. I appreciate the ideas of the software MIX , but have not tried it yet. If somebody finds it really
      Message 2 of 9 , Mar 9, 2004
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        Dear Isobel and all,



        This is an interesting topic. I appreciate the ideas of the software "MIX", but have not tried it yet. If somebody finds it "really" useful, I may have a try. My main concern is that one needs to define subpopulations prior to "separate" mixed populations, but these subpopulations are in fact what we are looking for and thus they may not be pre-defined in many cases. This may be a "chicken and egg" dilemma in this case.



        For the method of "probability plot", I have some concerns. It is quite often declared that sub-populations can be "separated" based on the "break points" or "kinks/inflexions". I feel this is hard to prove. Whichever threshold values (e.g., break points) you choose to separate a mixture, you can always say the low values are background and the high ones are polluted, and you can always map them separately and explain them with your geological/environmental knowledge in a very good way. Bearing my doubt in mind, some time ago, I tried to use normal score values to produce two perfect normal populations, say one with mean=0 and stddev=1 (n=10000, min=-3.8361, max=3.8361), and the other with mean=8, and stddev=1 (n=10000, min=4.1639, max=11.8361). The value "4" should be the separating point for the two "sub-populations". The probability plot (specific method used was "Normal Q-Q plot" which should be equivalent to others) of the mixed population showed that the separating point of 4 was located in the middle of a "plateau", not on the kinks. Reducing the mean value of the second "sub-population" just created several kinks on the plot, and the expected separating points never appeared on a kink. I also tried to use 3 normal subpopulations, and couldn't get the "expected" results. Therefore, I don't think points of kinks can be used to separate mixed populations into several subpopulations. Well, that was just an experiment, but it did prove that my concern is real.



        I'm looking for some constructive comments.:)



        Cheers,



        Chaosheng





        ----- Original Message -----
        From: "Isobel Clark" <drisobelclark@...>
        To: <ai-geostats@...>
        Sent: Tuesday, March 09, 2004 2:18 PM
        Subject: AI-GEOSTATS: Re: mixtures of populations


        > Hello All
        >
        > The common 'Normal Score' transform assumes one
        > population. Transformations such as rank or logarithm
        > do not assume one population.
        >
        > The best way to identify likely mixtures is with
        > programs such as Peter MacDonald's Mix (cited in
        > Ruben's email I think):
        >
        > http://www.math.mcmaster.ca/peter/mix/mix31.html
        >
        > or with probability plots. Many software packages have
        > these and mixtures are easily identifiable by
        > break-points or points of inflexion in the plot.
        >
        > For those (like myself) without easy access to
        > libraries, there are a couple of papers which describe
        > (geological) applications and using a combination of
        > indicator and ordinary kriging to solve some problems.
        >
        >
        > Papers can be found at
        > http://uk.geocities.com/drisobelclark/resume
        > follow the publications link. Look for my 1974 paper,
        > now available in pdf format, the 1993 IMGC paper and
        > the 1992 Troia paper with Jonathan Vieler.
        >
        > We have had good experience with this approach for 30
        > years in fields as diverse as mineral resource
        > estimation and seabird preservation.
        >
        > Isobel Clark
        > http://geoecosse.bizland.com/courses.htm
        >
        >
        >
        >
        >
        >
        >
        > ___________________________________________________________
        > Yahoo! Messenger - Communicate instantly..."Ping"
        > your friends today! Download Messenger Now
        > http://uk.messenger.yahoo.com/download/index.html
        >
        > --
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        >

        [Non-text portions of this message have been removed]
      • Isobel Clark
        AH me, the English language slips away from me again. I said that the PRESENCE {pardon the capitals, no way to italicise email} of more than one population is
        Message 3 of 9 , Mar 9, 2004
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          AH me, the English language slips away from me again.

          I said that the PRESENCE {pardon the capitals, no way
          to italicise email} of more than one population is
          indicated by the points of inflexion on the
          probability plot. Not that these were breakpoints
          between populations.

          Normal (or lognormal) populations overlap. The break
          point in the probability plot allows us to distinguish
          between data which are skewed and multiple
          populations. Skewed data give curved probability
          plots. Mixtures of populations give plots with abrupt
          changes in slope. These are very rarely equivalent to
          'equal probability' points - that is, statistical
          break points between the population. But, they are a
          good place to start looking ;-)

          Once you have deduced that multiple populations are
          present there are lots of things you can do, including
          simple stuff like post-plotting an indicator transform
          of the data at various threshholds just to see if
          there is any spatial pattern obvious to the naked eye.

          In many cases, ordinary kriging can proceed even with
          a mixture, since it only requires second-order
          stationarity not the existance of one single
          population.

          In 34 years of searching, I have never seen a
          probability plot with breakpoint(s) which did not have
          a matching multiple population explanation. The number
          of times I have argued with a 'customer' about this is
          legion. In some cases, we have found more populations
          than expected (witness my 1993 IGMC paper).

          In environmental studies, as in many geological
          situations, one would normally expect a broad
          background population of readings with the 'pollution'
          showing as a more cohesive, generally higher valued
          overlying one. Where both exist in the same locality,
          it is often difficult to separate them in the data set
          because you need both to characterise that area. This
          is the case where you would co-krige an indicator and
          two populations to get one estimate.

          Peter MacDonald's work is pretty definitive in North
          America and his MIX program for separating a histogram
          out into components has been around for 30 years, to
          my knowledge (I met him in 1976 at a Biometrics
          Congress!).

          There is a great monograph by Alistair Sinclair called
          "Application of Probability Plots in Mineral
          Exploration" which costs around $10 from the
          Association of Exploration Geochemists and was first
          published about 30 years ago. The task of identifying
          mineral targets is very like that of identifying
          pollution sources or other types of 'secondary'
          populations.

          It is much better to identify multiple populations
          from other knwledge of the site, but this is not
          always possible. If you don't know whether or not you
          have a mixture, statistical plots are one way of
          checking - and very quick and easy to produce
          nowadays. I am open to any other suggestions on how
          to identify multiple populations when all you have is
          the sample data.

          Isobel Clark
          http://uk.geocities.com/drisobelclark





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        • Monica Palaseanu-Lovejoy
          Hi, I have to recognize that i am very interested in this topic. I did QQ norm plots on my data, and it is curved - but it is not an ... arc ... or how can i
          Message 4 of 9 , Mar 10, 2004
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            Hi,

            I have to recognize that i am very interested in this topic.

            I did QQ norm plots on my data, and it is curved - but it is not an ...
            arc ... or how can i explain it? the lower part of the graph is almost
            a line for about 2/3 or the data and after it is a curve - kind of large -
            and after the rest of the data is almost on another line. Because
            the curvature is large it is hard to decide where to "split" the data
            with the naked eye. So i used box-plot to identify "outliers" and see
            if a QQ norm plot of these points represents a line or not. Well, it is
            not a perfect line - but it is quite close. So i was happy with that.
            But i plot in a QQ graph the rest of the "non-outliers" - and guess
            what??? It is a kind of curve, the nice line i had before in the first
            graph is now "diluted" (if i can say that) ....

            I know from the data semi-variogram that i don't have spatial
            autocorrelation - and i should. If i eliminate the "outliers" i do have
            spatial autocorrelation, with a decent global Moran's I. Now, if i
            reintroduce some of the previous identified "outliers", the global
            Moran's I increases. I would interpret this saying that box-plot over-
            estimated in this case the number of outliers.

            I have prior knowledge about the site i am working on, and since
            the contamination i am speaking about is PAH16, even so called
            background contamination (which is way below the environmental
            threshold) is anthropogenic in origin. Of course i am much more
            interested in "outliers" because there i have the "pollution" values.
            These values mainly cluster in 3 areas, and i consider those the
            "point source pollution" which actually correlate well with the
            industrial activity on the site.

            I think this is a nice and quite believable story. But the situation is
            totally different with heavy metals, or better put - not so obvious. So
            i am trying now to apply some MIX procedures (P. Mcdonald) and
            see if this will shed more light. Now my fear is that the spatial
            autocorrelation for heavy metals is at a much smaller scale than
            the actual sampling i have for the site.Unfortunately it is out of
            question to do more sampling.

            Monica

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          • Monica Palaseanu-Lovejoy
            Hi, I am attaching a pdf file which is a DD plot (mahalanobis distances versus robust distances) for a heavy metal multivariate data. I am not very sure about
            Message 5 of 9 , Mar 10, 2004
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              Hi,

              I am attaching a pdf file which is a DD plot (mahalanobis distances
              versus robust distances) for a heavy metal multivariate data. I am
              not very sure about how i should interpret this particular graph.

              Any help is greatly appreciated.

              Monica


              ----------

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              [Non-text portions of this message have been removed]
            • Ruben Roa Ureta
              ... Hi Chaosheng First, the MIX program is a good one and based on solid statistical theory. The original paper was published in a fisheries journal. This is
              Message 6 of 9 , Mar 10, 2004
              • 0 Attachment
                > Dear Isobel and all,
                >
                >
                >
                > This is an interesting topic. I appreciate the ideas of the software
                > "MIX", but have not tried it yet. If somebody finds it "really" useful, I
                > may have a try. My main concern is that one needs to define subpopulations
                > prior to "separate" mixed populations, but these subpopulations are in
                > fact what we are looking for and thus they may not be pre-defined in many
                > cases. This may be a "chicken and egg" dilemma in this case.

                Hi Chaosheng

                First, the MIX program is a good one and based on solid statistical
                theory. The original paper was published in a fisheries journal. This is
                the reference,

                Macdonald and Pitcher. 1979. Journal of the Fisheries Research Board of
                Canada (now known as Canadian Journal of Fisheries and Aquatic Sciences)
                36:987.

                The program shows a menu-based DOS window with graphic capabilities (at
                least the version that i have, which is, v 3.0).

                Second, regarding the number of components in the mixture, normally you
                don't know it, but there are ways to estimate it as a parameter/test it as
                a hypothesis. For the mixtures-of-normal-distributions case, see:
                Lo et al. 2001. Biometrika 88:767
                http://www3.oup.co.uk/biomet/hdb/Volume_88/Issue_03/880767.sgm.abs.html
                Another alternative is to use Akaike Information Criteria for several
                models (say with r-1, r, r+1, etc, components).
                And a still simpler alternative is to try a visual inspection of the data,
                fit say r components as suggested by the data or separate information, get
                the chi-square goodness-of-fit statistics (provided by the program), and
                then do the fitting with say r-1, and r+1 components and compare results.
                It usually happens that only very few alternatives are allowed since the
                algorithm will not converge when an absurd number of components (from the
                'point of view' of the data and the model) is introduced.
                Cheers
                Ruben


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              • Monica Palaseanu-Lovejoy
                Hi, Sorry for cross posting. It seems that my attachment with the pdf file DD plot failed so - i am trying again to attach it - as a jpg file this time. I
                Message 7 of 9 , Mar 10, 2004
                • 0 Attachment
                  Hi,

                  Sorry for cross posting. It seems that my attachment with the pdf
                  file DD plot failed so - i am trying again to attach it - as a jpg file
                  this time.

                  I would like to see what is your interpretation of this plot.

                  Thank you so much,

                  Monica

                  ----------

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                  [Non-text portions of this message have been removed]
                • Din Chen
                  To add to this discussion, one of Peter MacDonald s students changed MIX to Rmix using the Free R-package with nlm for the mle optimization. So it is free(but
                  Message 8 of 9 , Mar 10, 2004
                  • 0 Attachment
                    To add to this discussion, one of Peter MacDonald's students changed MIX to
                    Rmix using the Free R-package with nlm for the mle optimization. So it is
                    free(but it is good for Peter if you want to spend some money to buy MIX)
                    and excellent for mixture population from my view.

                    As for plateau/kinks/breakpoints, it depends on the proportions of each
                    subpopulation in the mixture. Furthermore, the goodness-of-fit statistics in
                    Rmix will give you some directions on how many subpopulations in the
                    mixture.

                    Din



                    -----Original Message-----
                    From: Isobel Clark [mailto:drisobelclark@...]
                    Sent: Tuesday, March 09, 2004 3:34 PM
                    To: Chaosheng Zhang
                    Cc: ai-geostats@...
                    Subject: AI-GEOSTATS: Re: mixtures of populations


                    AH me, the English language slips away from me again.

                    I said that the PRESENCE {pardon the capitals, no way
                    to italicise email} of more than one population is
                    indicated by the points of inflexion on the
                    probability plot. Not that these were breakpoints
                    between populations.

                    Normal (or lognormal) populations overlap. The break
                    point in the probability plot allows us to distinguish
                    between data which are skewed and multiple
                    populations. Skewed data give curved probability
                    plots. Mixtures of populations give plots with abrupt
                    changes in slope. These are very rarely equivalent to
                    'equal probability' points - that is, statistical
                    break points between the population. But, they are a
                    good place to start looking ;-)

                    Once you have deduced that multiple populations are
                    present there are lots of things you can do, including
                    simple stuff like post-plotting an indicator transform
                    of the data at various threshholds just to see if
                    there is any spatial pattern obvious to the naked eye.

                    In many cases, ordinary kriging can proceed even with
                    a mixture, since it only requires second-order
                    stationarity not the existance of one single
                    population.

                    In 34 years of searching, I have never seen a
                    probability plot with breakpoint(s) which did not have
                    a matching multiple population explanation. The number
                    of times I have argued with a 'customer' about this is
                    legion. In some cases, we have found more populations
                    than expected (witness my 1993 IGMC paper).

                    In environmental studies, as in many geological
                    situations, one would normally expect a broad
                    background population of readings with the 'pollution'
                    showing as a more cohesive, generally higher valued
                    overlying one. Where both exist in the same locality,
                    it is often difficult to separate them in the data set
                    because you need both to characterise that area. This
                    is the case where you would co-krige an indicator and
                    two populations to get one estimate.

                    Peter MacDonald's work is pretty definitive in North
                    America and his MIX program for separating a histogram
                    out into components has been around for 30 years, to
                    my knowledge (I met him in 1976 at a Biometrics
                    Congress!).

                    There is a great monograph by Alistair Sinclair called
                    "Application of Probability Plots in Mineral
                    Exploration" which costs around $10 from the
                    Association of Exploration Geochemists and was first
                    published about 30 years ago. The task of identifying
                    mineral targets is very like that of identifying
                    pollution sources or other types of 'secondary'
                    populations.

                    It is much better to identify multiple populations
                    from other knwledge of the site, but this is not
                    always possible. If you don't know whether or not you
                    have a mixture, statistical plots are one way of
                    checking - and very quick and easy to produce
                    nowadays. I am open to any other suggestions on how
                    to identify multiple populations when all you have is
                    the sample data.

                    Isobel Clark
                    http://uk.geocities.com/drisobelclark





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                  • Chaosheng Zhang
                    Dear Isobel and all, Thanks for your clarification and the replies from several others. Well, I didn t mean to question what you said (or wrote, -:)), but just
                    Message 9 of 9 , Mar 11, 2004
                    • 0 Attachment
                      Dear Isobel and all,

                      Thanks for your clarification and the replies from several others. Well, I didn't mean to question what you said (or wrote, -:)), but just wanted to discuss with you and make this issue clearer as I saw so many messages in the list talking about "probability plot".

                      Glad to know that I am not alone with such a concern. However, perhaps the appropriate way is to say that a probability plot is still very useful, but care should be taken when explain it. It may be still ok to visually identify outliers, even to explain different sections on the plot with scientific knowledge, but we cannot go too far. Since it is a "graphical" method, we may regard it as a "practical" way.

                      Cheers,

                      Chaosheng
                      --------------------------------------------------------------------------
                      Dr. Chaosheng Zhang
                      Lecturer in GIS
                      Department of Geography
                      National University of Ireland, Galway
                      IRELAND
                      Tel: +353-91-524411 x 2375
                      Fax: +353-91-525700
                      E-mail: Chaosheng.Zhang@...
                      Web 1: www.nuigalway.ie/geography/zhang.html
                      Web 2: www.nuigalway.ie/geography/gis/index.htm
                      ----------------------------------------------------------------------------

                      ----- Original Message -----
                      From: "Isobel Clark" <drisobelclark@...>
                      To: "Chaosheng Zhang" <Chaosheng.Zhang@...>
                      Cc: <ai-geostats@...>
                      Sent: Tuesday, March 09, 2004 11:34 PM
                      Subject: AI-GEOSTATS: Re: mixtures of populations


                      > AH me, the English language slips away from me again.
                      >
                      > I said that the PRESENCE {pardon the capitals, no way
                      > to italicise email} of more than one population is
                      > indicated by the points of inflexion on the
                      > probability plot. Not that these were breakpoints
                      > between populations.
                      >
                      > Normal (or lognormal) populations overlap. The break
                      > point in the probability plot allows us to distinguish
                      > between data which are skewed and multiple
                      > populations. Skewed data give curved probability
                      > plots. Mixtures of populations give plots with abrupt
                      > changes in slope. These are very rarely equivalent to
                      > 'equal probability' points - that is, statistical
                      > break points between the population. But, they are a
                      > good place to start looking ;-)
                      >
                      > Once you have deduced that multiple populations are
                      > present there are lots of things you can do, including
                      > simple stuff like post-plotting an indicator transform
                      > of the data at various threshholds just to see if
                      > there is any spatial pattern obvious to the naked eye.
                      >
                      > In many cases, ordinary kriging can proceed even with
                      > a mixture, since it only requires second-order
                      > stationarity not the existance of one single
                      > population.
                      >
                      > In 34 years of searching, I have never seen a
                      > probability plot with breakpoint(s) which did not have
                      > a matching multiple population explanation. The number
                      > of times I have argued with a 'customer' about this is
                      > legion. In some cases, we have found more populations
                      > than expected (witness my 1993 IGMC paper).
                      >
                      > In environmental studies, as in many geological
                      > situations, one would normally expect a broad
                      > background population of readings with the 'pollution'
                      > showing as a more cohesive, generally higher valued
                      > overlying one. Where both exist in the same locality,
                      > it is often difficult to separate them in the data set
                      > because you need both to characterise that area. This
                      > is the case where you would co-krige an indicator and
                      > two populations to get one estimate.
                      >
                      > Peter MacDonald's work is pretty definitive in North
                      > America and his MIX program for separating a histogram
                      > out into components has been around for 30 years, to
                      > my knowledge (I met him in 1976 at a Biometrics
                      > Congress!).
                      >
                      > There is a great monograph by Alistair Sinclair called
                      > "Application of Probability Plots in Mineral
                      > Exploration" which costs around $10 from the
                      > Association of Exploration Geochemists and was first
                      > published about 30 years ago. The task of identifying
                      > mineral targets is very like that of identifying
                      > pollution sources or other types of 'secondary'
                      > populations.
                      >
                      > It is much better to identify multiple populations
                      > from other knwledge of the site, but this is not
                      > always possible. If you don't know whether or not you
                      > have a mixture, statistical plots are one way of
                      > checking - and very quick and easy to produce
                      > nowadays. I am open to any other suggestions on how
                      > to identify multiple populations when all you have is
                      > the sample data.
                      >
                      > Isobel Clark
                      > http://uk.geocities.com/drisobelclark
                      >
                      >
                      >
                      >
                      >
                      > ___________________________________________________________
                      > Yahoo! Messenger - Communicate instantly..."Ping"
                      > your friends today! Download Messenger Now
                      > http://uk.messenger.yahoo.com/download/index.html
                      >
                      > --
                      > * To post a message to the list, send it to ai-geostats@...
                      > * As a general service to the users, please remember to post a summary of any useful responses to your questions.
                      > * To unsubscribe, send an email to majordomo@... with no subject and "unsubscribe ai-geostats" followed by "end" on the next line in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list
                      > * Support to the list is provided at http://www.ai-geostats.org
                      >

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