# AIRLINE_AIC - Akaike's Information Criterion (AIC) of an Airline Model

Calculates the Akaike's information criterion (AIC) of the given airline model (with correction to small sample sizes)

## Syntax

AIRLINE_AIC(X, Order, mean, sigma, s, theta, theta2)

X is the univariate time series data (one dimensional array of cells (e.g. rows or columns)).

Order is the time order in the data series (i.e. the first data point's corresponding date (earliest date=1 (default), latest date=0)).

Order Description
1 ascending (the first data point corresponds to the earliest date) (default)
0 descending (the first data point corresponds to the latest date)

mean is the model mean (i.e. mu).

sigma is the standard deviation of the model's residuals/innovations.

s is the length of seasonality (expressed in terms of lags, where s > 1).

theta is the coefficient of first-lagged innovation (see model description).

theta2 is the coefficient of s-lagged innovation (see model description).

## Remarks

1. The underlying model is described here.
2. Akaike's Information Criterion (AIC) is described here.
3. Warning: AIRLINE_AIC() function is deprecated as of version 1.63: use AIRLINE_GOF function instead.
4. The time series is homogeneous or equally spaced.
5. The time series may include missing values (e.g. #N/A) at either end.
6. The airline model with order $s$ has 4 parameters: $\mu\,,\sigma\,\,,\theta\,,\mathit{and} \: \Theta$
7. The Airline model is a special case of multiplicative seasonal ARIMA model, and it assumes independent and normally distributed residuals with constant variance.

## Examples

Example 1:

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
A B C D
Date Data
1/1/2008 -0.300 AIRLINE
1/2/2008 -1.278 Mean 0.0481
1/3/2008 0.244 theta(1) 0.14
1/4/2008 1.276 theta(2) 0.30
1/6/2008 1.733 Sigma 2.74127
1/7/2008 -2.184 s 2
1/8/2008 -0.234
1/9/2008 1.095
1/10/2008 -1.087
1/11/2008 -0.690
1/12/2008 -1.690
1/13/2008 -1.847
1/14/2008 -0.978
1/15/2008 -0.774

Formula Description (Result)
=AIRLINE_AIC(Sheet1!$B$2:$B$15,1,$D$3,$D$6,$D$7,$D$4,$D$5) 65.6 Akaike's information criterion (AIC)
=AIRLINE_LLF(Sheet1!$B$2:$B$15,1,$D$3,$D$6,$D$7,$D$4,$D$5) -25.47 Log-Likelihood Function
=AIRLINE_CHECK($D$3,$D$6,$D$7,$D$4,$D$5) 1 Is the AIRLINE model stable?

## References

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