(top right) The P(| X| ≥ x) as a function of x for the same distributions. (top left) The probability density function of the noise of an AR(3) process modeling the daily NAO index (blue) compared to the best-fitted densities of three non-Gaussian distributions, namely, the generalized hyperbolic secant distribution (green), the α-stable distribution (red), and the nonstandardized t distribution (black). (right) A Q–Q plot of the noise distribution compared to a normal sample. (center) The P(| X| ≥ x) as a function of x for the same distributions. (left) The probability density function of the noise in the AR(3) model (blue) compared to a Gaussian density (red) with the same mean and variation as the noise. The reference line (red) equals F( s) = s 1/2.
#Non gaussian noise series
(right) The fluctuation function F( s) as a function of s for the time series of the daily NAO index (blue) and for the noise in the AR( p) processes modeling the time series (the green line corresponds to p = 1 and the black line to p = 3). The variances are normalized by division by the sample standard deviation of the entire time series and the reference line (red) is n −1 (red). (left) The sample variance as a function of the sample size for the time series of the daily NAO index (blue) and for the noise in the AR( p) processes modeling the time series (the green line corresponds to p = 1 and the black line to p = 3). For the AR(3) process, the median (red × symbols) and 5%–95% range (red dashed line) of 1000 realizations is displayed. Sample ACF of the daily NAO index (blue) compared to the sample ACF of an AR(3) process forced with nonstandardized t-distributed noise. The size of the statistical error is indicated by the red lines. Sample PACF as a function of p for the daily NAO index.
The 95% confidence bounds for the ACF are on the order of 1.96 n −1/2 = 8.6 × 10 −3. (top left) The sample ACF for the daily NAO index in (bottom left) standard scale and (bottom right) logarithmic scale. (top right) A Q–Q plot of the daily NAO index compared to a normal sample. (top left) The probability density function of the daily NAO index (blue) and normal distributions with the same mean and variance (red). (bottom) A 3-yr running mean of the observed daily NAO index during the years 1872–2014. The opposite phase is associated with a weaker pressure gradient and weaker westerlies following a more southward trajectory, resulting in wetter conditions over the Mediterranean region and harsh winters over northern Europe. The positive phase of the NAO is characterized by a strong pressure gradient between Iceland and the Azores, yielding stronger westerlies and, in turn, leading to wetter and milder conditions over western and northern Europe and drier conditions over the Mediterranean region. (2001), shown are typical weather conditions under the positive and negative phases of the NAO.
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