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

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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 1 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 2 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 3 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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