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GEOSTATS: spatial sampling

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  • arthur getis
    With regard to spatial sampling, one must keep in mind that the assumptions embodied in the theory of the test statistics that are to be used must be taken
    Message 1 of 5 , Oct 29, 1998
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      With regard to spatial sampling, one must keep in mind that the assumptions
      embodied
      in the theory of the test statistics that are to be used must be taken into
      consideration. If the
      test statistics require independent observations, then sites must be chosen
      in such a way
      that near neighbor sampled sites are at distance from each other beyond the
      range of the
      semivariogram, otherwise observations will be spatially autocorrelated and
      therefore dependent.
      Of course, some statistics are designed to identify spatial dependence by
      rejecting hypotheses
      of independence.

      Art Getis
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    • Vera Pawlowsky-Glahn
      Independence of samples is defined as absence of correlation among them and thus zero covariance, which is intrinsically a contradiction with spatial
      Message 2 of 5 , Oct 30, 1998
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        Independence of samples is defined as absence of correlation among
        them and thus zero covariance, which is intrinsically a contradiction
        with spatial dependence.

        Best regards,

        Vera

        Yetta Jager escribi�:

        > In response to:"...sites must be chosen in such a way that near neighbor sampled sites are at distance from each other beyond the range of the semivariogram, otherwise observations will be spatially autocorrelated and therefore dependent."
        >
        > If the objective is to characterize spatial autocorrelation, then clearly
        > it is necessary to focus on shorter distances.
        >
        > Let me ask this though. I hear this often, that using data that may be spatially autocorrelated
        > violates the independence assumption. Maybe I'm wrong, but my understanding is that the
        > correlation structure of the sample has no bearing on the independence assumption of the
        > sampling. The main purpose of the independence assumption is to ensure that the sample
        > is representative of the population that it will be used to make inferences about. Therefore,
        > the only requirement is that the sample be drawn in a random manner or according to some
        > design where the inclusion probabilities are known. There can be all kinds of dependencies and
        > relationships among the actual sample points, but as long as these represent dependencies and
        > relationships that are also properties of the population, its not a problem. If this view
        > is in error, I'd appreciate a clarification.
        >
        > ------------------------------------------------------
        > Yetta Jager
        > Environmental Sciences Division
        > Oak Ridge National Laboratory
        > P.O. Box 2008, MS 6036
        > Oak Ridge, TN 37831-6036
        > OFFICE: 423/574-8143
        > FAX: 423/576-8543
        > Work email: jagerhi@...
        > Home email: hjager@...
        > WEBpage: http://www.esd.ornl.gov/~zij/
        > -----------------------------------------------------
        >
        > "One man's mean is another man's Poisson" J.W. Haeffner
        > --
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        --
        *************************************************

        Vera Pawlowsky Glahn
        Universitat Polit�cnica de Catalunya - ETSECCPB
        Departament de Matem�tica Aplicada III
        UPC - Campus Nord (edificio C2)
        Jordi Girona Salgado 1-3
        E-08034 Barcelona (Espa�a)

        Tel.: 34-93-401 69 17
        Fax.: 34-93-401 65 04
        e-mail: pawlowsky@...

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



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      • Vera Pawlowsky-Glahn
        Sorry, I made a formulation mistake: Independence of samples, like that of any random vector, IMPLIES absence of correlation. The definition is that the joint
        Message 3 of 5 , Oct 30, 1998
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          Sorry, I made a formulation mistake:
          Independence of samples, like that of any random vector, IMPLIES
          absence of correlation. The definition is that the joint distribution
          function factorizes into the product of the marginals.

          Best regards,

          Vera


          Vera Pawlowsky-Glahn escribi�:

          > Independence of samples is defined as absence of correlation among
          > them and thus zero covariance, which is intrinsically a contradiction
          > with spatial dependence.
          >
          > Best regards,
          >
          > Vera
          >
          > Yetta Jager escribi�:
          >
          > > In response to:"...sites must be chosen in such a way that near neighbor sampled sites are at distance from each other beyond the range of the semivariogram, otherwise observations will be spatially autocorrelated and therefore dependent."
          > >
          > > If the objective is to characterize spatial autocorrelation, then clearly
          > > it is necessary to focus on shorter distances.
          > >
          > > Let me ask this though. I hear this often, that using data that may be spatially autocorrelated
          > > violates the independence assumption. Maybe I'm wrong, but my understanding is that the
          > > correlation structure of the sample has no bearing on the independence assumption of the
          > > sampling. The main purpose of the independence assumption is to ensure that the sample
          > > is representative of the population that it will be used to make inferences about. Therefore,
          > > the only requirement is that the sample be drawn in a random manner or according to some
          > > design where the inclusion probabilities are known. There can be all kinds of dependencies and
          > > relationships among the actual sample points, but as long as these represent dependencies and
          > > relationships that are also properties of the population, its not a problem. If this view
          > > is in error, I'd appreciate a clarification.
          > >
          > > ------------------------------------------------------
          > > Yetta Jager
          > > Environmental Sciences Division
          > > Oak Ridge National Laboratory
          > > P.O. Box 2008, MS 6036
          > > Oak Ridge, TN 37831-6036
          > > OFFICE: 423/574-8143
          > > FAX: 423/576-8543
          > > Work email: jagerhi@...
          > > Home email: hjager@...
          > > WEBpage: http://www.esd.ornl.gov/~zij/
          > > -----------------------------------------------------
          > >
          > > "One man's mean is another man's Poisson" J.W. Haeffner
          > > --
          > > *To post a message to the list, send it to ai-geostats@....
          > > *As a general service to list users, please remember to post a summary
          > > of any useful responses to your questions.
          > > *To unsubscribe, send email to majordomo@... with no subject and
          > > "unsubscribe ai-geostats" in the message body.
          > > DO NOT SEND Subscribe/Unsubscribe requests to the list!
          >
          > --
          > *************************************************
          >
          > Vera Pawlowsky Glahn
          > Universitat Polit�cnica de Catalunya - ETSECCPB
          > Departament de Matem�tica Aplicada III
          > UPC - Campus Nord (edificio C2)
          > Jordi Girona Salgado 1-3
          > E-08034 Barcelona (Espa�a)
          >
          > Tel.: 34-93-401 69 17
          > Fax.: 34-93-401 65 04
          > e-mail: pawlowsky@...
          >
          > *************************************************
          >
          > --
          > *To post a message to the list, send it to ai-geostats@....
          > *As a general service to list users, please remember to post a summary
          > of any useful responses to your questions.
          > *To unsubscribe, send email to majordomo@... with no subject and
          > "unsubscribe ai-geostats" in the message body.
          > DO NOT SEND Subscribe/Unsubscribe requests to the list!



          --
          *************************************************

          Vera Pawlowsky Glahn
          Universitat Polit�cnica de Catalunya - ETSECCPB
          Departament de Matem�tica Aplicada III
          UPC - Campus Nord (edificio C2)
          Jordi Girona Salgado 1-3
          E-08034 Barcelona (Espa�a)

          Tel.: 34-93-401 69 17
          Fax.: 34-93-401 65 04
          e-mail: pawlowsky@...

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



          --
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        • Yetta Jager
          In response to: ...sites must be chosen in such a way that near neighbor sampled sites are at distance from each other beyond the range of the semivariogram,
          Message 4 of 5 , Oct 30, 1998
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            In response to:"...sites must be chosen in such a way that near neighbor sampled sites are at distance from each other beyond the range of the semivariogram, otherwise observations will be spatially autocorrelated and therefore dependent."

            If the objective is to characterize spatial autocorrelation, then clearly
            it is necessary to focus on shorter distances.

            Let me ask this though. I hear this often, that using data that may be spatially autocorrelated
            violates the independence assumption. Maybe I'm wrong, but my understanding is that the
            correlation structure of the sample has no bearing on the independence assumption of the
            sampling. The main purpose of the independence assumption is to ensure that the sample
            is representative of the population that it will be used to make inferences about. Therefore,
            the only requirement is that the sample be drawn in a random manner or according to some
            design where the inclusion probabilities are known. There can be all kinds of dependencies and
            relationships among the actual sample points, but as long as these represent dependencies and
            relationships that are also properties of the population, its not a problem. If this view
            is in error, I'd appreciate a clarification.



            ------------------------------------------------------
            Yetta Jager
            Environmental Sciences Division
            Oak Ridge National Laboratory
            P.O. Box 2008, MS 6036
            Oak Ridge, TN 37831-6036
            OFFICE: 423/574-8143
            FAX: 423/576-8543
            Work email: jagerhi@...
            Home email: hjager@...
            WEBpage: http://www.esd.ornl.gov/~zij/
            -----------------------------------------------------

            "One man's mean is another man's Poisson" J.W. Haeffner
            --
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          • Philippe Aubry
            Dear colleagues (I apologize for my poor grammar and vocabulary) ... Simple Random Sampling theory applies whatever the underlying population structure, and
            Message 5 of 5 , Nov 2, 1998
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              Dear colleagues (I apologize for my poor grammar and vocabulary)

              In response to Yetta:

              >Let me ask this though. I hear this often, that using data that may be
              >spatially autocorrelated
              >violates the independence assumption. Maybe I'm wrong, but my
              >understanding is that the
              >correlation structure of the sample has no bearing on the independence
              >assumption of the
              >sampling. The main purpose of the independence assumption is to ensure
              >that the sample
              >is representative of the population that it will be used to make
              >inferences about. Therefore,
              >the only requirement is that the sample be drawn in a random manner or
              >according to some
              >design where the inclusion probabilities are known. There can be all
              >kinds of dependencies and
              >relationships among the actual sample points, but as long as these
              >represent dependencies and
              >relationships that are also properties of the population, its not a problem.

              -------------------

              Simple Random Sampling theory applies whatever the underlying population
              structure, and therefore, SRS estimator for the variance of the spatial
              mean holds, even with spatially autocorrelated populations. Moreover, the
              sampling distribution of the spatial mean is Gaussian (central limit
              theorem). Consequently, if sampling is SRS, calculation of the confidence
              interval of the spatial mean is straightforward.

              For every sampling design (defined by the set of all possible samples,
              first-order inclusion probabilities and second-order inclusion
              probabilities) with second-order inclusion probabilities greater than zero,
              the Horvitz-Thompson estimator for the variance of the mean is unbiased,
              without regards to the spatial autocorrelation structure.

              Of course, with purposive sampling (non probabilistic sampling), or even
              systematic sampling (many second order inclusion probabilities equal to
              zero), using the SRS estimator for the variance of the mean can lead to an
              inclusion bias which magnitude depends on the underlying spatial
              autocorrelation. In such a case, one should turn to model-based inference
              since design-based inference is either impossible (purposive sampling) or
              problematic (systematic sampling).

              -----------------------

              Speaking about "stochastic independence" or "statistical independence" for
              the data is a non-sense since the data alone do not give a set of random
              variables. Random variables are introduced by a stochastic mecanism.
              Drawing SRS is a way of producing stochastically independent random
              variables (the first RV is for all the first values we draw by repeating
              the samplign scheme, the second RV is for all the second values and so on
              ... the nth RV is for all the nth values we draw). Assuming a
              superpopulation model (geostatistics use random functions as
              superpopulation models) from which the population is one realization is
              another way to introduce stochasticity in order to perform statistical
              inference. But now the RV must be statisically dependent (in the model) if
              the data are spatially dependent (in the reality) or statistical inference
              about the population will be very poor.

              ------------------------

              For classical statistical tests it is required that the data are spatially
              independent. Spatial independence (= no spatial autcorrelation, for all
              lags = pure nugget) is similar to experimental independence for
              biostatistics. With spatially autocorrelated data, it is necessary to take
              into account the spatial dependence when assessing the p-value of any
              statistic (i.e. the Pearson correlation coefficient between two
              regionalized variables).



              Hope this help


              Best regards


              Philippe AUBRY

              -----------------------------------------
              Laboratoire de Biometrie
              UMR CNRS 5558
              Universite Claude Bernard - Lyon 1
              43 bd. du 11 Novembre 1918
              69622 VILLEURBANNE Cedex
              FRANCE
              -----------------------------------------
              private fax number : 04.72.74.47.46
              -----------------------------------------
              e-mail : paubry@...-lyon1.fr
              -----------------------------------------


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