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

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  • vanessa stelzenmüller
    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
    Message 1 of 7 , Dec 18, 2002
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      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 ?
      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 ?

      Best wishes
      Vanessa

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    • Ruben Roa
      ... Yes. The model must be fitted my maximum likelihood. ... The algebraic expression for the AIC results from the bias in the maximum log-likelihood of a
      Message 2 of 7 , Dec 18, 2002
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        >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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      • Claudio Cocheo
        Dear all, ... Please, let me know. I m interested in the AIC. If I have 3 models each one fitted with a least square method, are them suitable for AIC
        Message 3 of 7 , Dec 18, 2002
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          Dear all,

          >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.
          If I have 3 models each one fitted with a least square method, are them
          suitable for AIC application?
          Are their SSRs the correct ones to use in the AIC?

          Many thanks to all
          Claudio Cocheo


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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 4 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 5 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 6 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 7 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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