Task 6. Akaike Information Criterion

In HEC-SSP, the Akaike Information Criterion is computed using the following equation (Burnham and Anderson 1998):

$//<![CDATA[ \begin{array}{l}AIC_c = \frac{2Nk}{N-k-1} - 2ln(\hat{L})\end{array} //]]>$

where $//<![CDATA[ \begin{array}{l}k\end{array} //]]>$ is the number of parameters estimated by the probability model, $//<![CDATA[ \begin{array}{l}N\end{array} //]]>$ is the number of data points, and $//<![CDATA[ \begin{array}{l}\hat{L}\end{array} //]]>$ is the maximized value of the likelihood function of the probability model i.e. $//<![CDATA[ \begin{array}{l}\hat{L} = p(x|\hat{\theta}, M)\end{array} //]]>$ where $//<![CDATA[ \begin{array}{l}\hat{\theta}\end{array} //]]>$ are the parameter values that maximize the likelihood function and $//<![CDATA[ \begin{array}{l}M\end{array} //]]>$ is the probability model. 

The above form of AIC is used for small samples sizes ($//<![CDATA[ \begin{array}{l}N/k\end{array} //]]>$ < 40). As the sample size increases with respect to the number of parameters, this equation converges to the standard form of the AIC:

$//<![CDATA[ \begin{array}{l}AIC = 2k - 2ln(\hat{L})\end{array} //]]>$

• Copy the sheet titled Data and name the new sheet AIC.
• Compute the PDF, $//<![CDATA[ \begin{array}{l}f_Q\end{array} //]]>$, of Q where random variable Q (discharge) follows a log10-normal distribution. The PDF of discharge Q is given in Task 5. Bayesian Information Criterion.
• Compute the natural log of the likelihood function.
• Compute the AIC using the equation provided above for $//<![CDATA[ \begin{array}{l}AIC_c\end{array} //]]>$

Question: What is the computed Akaike Information Criterion, AIC?

Download the final Excel spreadsheet here: Point_of_Rocks_Goodness_of_Fit_Tests.xlsx