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Re: AI-GEOSTATS: Akaike's information criterion (AIC)

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  • Brian R Gray
    suspect Ruben would note that, under a normal assumption, OLS and ML coincide. also, I suspect that Ruben s comments also apply to REML results--altho in that
    Message 1 of 7 , Dec 18, 2002
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      suspect Ruben would note that, under a normal assumption, OLS and ML
      coincide. also, I suspect that Ruben's comments also apply to REML
      results--altho in that case you may need to restrict inference to random
      components. brian

      ****************************************************************
      Brian Gray
      USGS Upper Midwest Environmental Sciences Center
      575 Lester Avenue, Onalaska, WI 54650
      ph 608-783-7550 ext 19, FAX 608-783-8058
      brgray@...
      *****************************************************************



      "Ruben Roa"
      <rroa@...> To: vanessa stelzenmüller <vstelzenmueller@...>
      Sent by: cc: ai-geostats@...
      ai-geostats-list@ Subject: Re: AI-GEOSTATS: Akaike's information criterion (AIC)
      unil.ch


      12/18/2002 10:26
      AM
      Please respond to
      "Ruben Roa"






      >Dear all,
      >
      >The AIC is used to select the "best" model from a list
      >of theoretical functions. I wonder if its necessary
      >the models need to be fitted by the same method ?

      Yes. The model must be fitted my maximum likelihood.

      >Would it be possible to stress the AIC to select the
      >"best" model from models which were fitted for example
      >by OLS,WLS, REML etc. This means to use AIC to choose
      >the model and the fitting method ?

      The algebraic expression for the AIC results from the bias in the maximum
      log-likelihood of a model as estimator of the mean expected log-likelihood,
      this bias being a function of the number of free parameters in the model.
      So it only covers those models fitted by maximum likelihood.

      Rubén
      http://webmail.udec.cl

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    • Ruben Roa
      ... coincide. True, though i d say that OLS is a particular case of MLE iff the process being modelled is additive and the additive stochastic component is
      Message 2 of 7 , Dec 18, 2002
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        >suspect Ruben would note that, under a normal assumption, OLS and ML
        coincide.

        True, though i'd say that OLS is a particular case of MLE iff the process
        being modelled is additive and the additive stochastic component is normal.

        >also, I suspect that Ruben's comments also apply to REML
        >results--altho in that case you may need to restrict inference to random
        components. brian

        There are so many acronyms that i got lost with REML. Is it Random Effects
        etc...?
        Rubén
        http://webmail.udec.cl

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      • Ruben Roa
        ... log-likelihood of a model as estimator of the mean expected log-likelihood, this bias being a function of the number of free parameters in the model. So it
        Message 3 of 7 , Dec 18, 2002
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          >>The algebraic expression for the AIC results from the bias in the maximum
          log-likelihood of a model as estimator of the mean expected log-likelihood,
          this bias being a function of the number of free parameters in the model.
          So it only covers those models fitted by maximum likelihood.
          >
          >Please, let me know. I'm interested in the AIC.

          Check out 'Akaike Information Criterion Statistics', 1986, by Sakamoto,
          Ishiguro, and Kitagawa (who are working associates to Akaike himself). KTK
          Scientific Publishers, Tokyo. There is an English translation distributed
          by Kluwer.

          >If I have 3 models each one fitted with a least square method, are them
          suitable for AIC application?

          Yes if the models have different number of free parameters, they have an
          additive stochastic component, and this component distributes normally.

          >Are their SSRs the correct ones to use in the AIC?

          Not quite. Compute the log likelihood under the normal assumption for each
          model and use that in the AIC. If both the mean and variance of the normal
          stochastic component are unknown, the log likelihood is

          L(mu,sigma^2)=
          -(n/2)ln(2*pi*sigma^2)-(1/2sigma^2)SUM_n(x_i-mu)^2

          By taking the partial derivative of the log likelihood with respect to mu
          and sigma^2, making it zero, solving for the MLE of mu and sigma^2, and
          replacing these solutions into L, you get the maximum log likelihood of
          each model,

          L(mu_hat,sigma^2_hat)=-(n/2)ln(2*pi*sigma^2_hat)-n/2
          =-(n/2)ln[(2*pi/n)SUM_n(x_i-mu_hat)^2]-n/2

          Note that mu_hat would be each one of your models.
          Cheers
          Rubén
          http://webmail.udec.cl

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        • Brian R Gray
          REML = variously, restricted or residual ML. the kicker is that, under REML, a function of the outcomes are estimated, such that the function contains none of
          Message 4 of 7 , Dec 18, 2002
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            REML = variously, restricted or residual ML. the kicker is that, under
            REML, a function of the outcomes are estimated, such that the function
            contains none of the fixed effects present/suspected in the original
            outcomes. brian

            ****************************************************************
            Brian Gray
            USGS Upper Midwest Environmental Sciences Center
            575 Lester Avenue, Onalaska, WI 54650
            ph 608-783-7550 ext 19, FAX 608-783-8058
            brgray@...
            *****************************************************************



            "Ruben Roa"
            <rroa@...> To: "Brian R Gray" <brgray@...>
            Sent by: cc: ai-geostats@...
            rroa@... Subject: Re: AI-GEOSTATS: Akaike's information criterion (AIC)


            12/18/2002 12:45
            PM
            Please respond to
            rroa






            >suspect Ruben would note that, under a normal assumption, OLS and ML
            coincide.

            True, though i'd say that OLS is a particular case of MLE iff the process
            being modelled is additive and the additive stochastic component is normal.

            >also, I suspect that Ruben's comments also apply to REML
            >results--altho in that case you may need to restrict inference to random
            components. brian

            There are so many acronyms that i got lost with REML. Is it Random Effects
            etc...?
            Rubén
            http://webmail.udec.cl






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