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S.1: Half Normal Distribution
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If x is a normal distribution, N (0, σ2), and u = absolute (x) then u follows half
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normal distribution whose probability density function, mean, and variance is given
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below:
æ u2 ö
exp ç - 2 ÷
s p
è 2s ø
2
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f (u;s ) =
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mean(u) =
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æ 2ö
var iance(u) = s 2 ç1- ÷
è pø
s 2
p
Eq. S.1
Eq. S.2
Eq. S.3
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For σ = 1.0, mean (u) = 0.80, variance (u) = 0.36, standard deviation (u) = 0.60.
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Source: Wikipedia, 2013. http://en.wikipedia.org/wiki/Half-normal_distribution
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S.2: Hypothesis’ Testing Design
With reference to formulae (1) and (2) given in main text, and notations discussed
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in sub-section S.1: u denotes AADm,l,y or ADm,l,y. The Null Hypothesis is – observation is
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randomly distributed about the ensemble mean (or vice-versa, N(0, 1), white-noise).
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Since numerator for AAD or AD calculation (Eqs. 1 and 2) is the difference between the
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ensemble mean and the observation, the mean value for this difference should be zero (if
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we do not consider the absolute values). The denominator is the normalization of
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difference by the ensemble standard deviation; hence the variance of the random normal
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is one. Since AAD or AD calculation considers absolute difference between ensemble
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mean and the observation standardized by the ensemble standard deviation (Eqs. 1 and
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2), hence u follows the half normal distribution. Hence, following the standard hypothesis
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testing procedure for difference in mean (Miller and Miller, 2004), the 95% confidence
1
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interval range for rejecting the null hypothesis can be constructed as discussed in Eq. 3 of
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the main text.
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S.3: f-ratio Test
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The ‘f-ratio’ test is used to test the null hypothesis that the forecast variance (σf2)
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is same as the inter-annual climate variability (σc2) at verification lead-time/month. For
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testing the null hypothesis against two-sided alternative σf2≠σc2 (Miller and Miller, 2004):
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s 2f
³ fa /2,nf -1,nc-1 if s 2f ³ sc2
Eq. S.4
sc2
³ fa /2,nc-1,nf -1 if s 2f < s 2c
2
sf
Eq. S.5
s
2
c
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Where s denotes standard deviation of forecast (sf) and inter-annual climate variability
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(sc); nf is sample size for forecast (=24) and nc is sample size for climatology (=17, 1982
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to 1998); α/2 = 0.025; and fa /2,nf -1,nc-1 and fa /2,nc-1,nf -1 are F-distribution with given
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parameters α/2, nf-1, and nc-1.
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Please note that we have only considered ratio of forecast to climatological
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æ s 2f ö
variance çç 2 ÷÷ in our calculations (Figure 5). For the cases when s 2f < s 2c , we took the
è sc ø
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values of F-distribution corresponding to fa /2,nc-1,nf -1 and then reversed these values (
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æ s2 ö
fa /2,nc-1,nf -1 ^-1) and calculated statistical significance if çç 2f ÷÷ < ( fa /2,nc-1,nf -1 ^-1). In other
è sc ø
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words, for s 2f ³ sc2 the F-distribution parameters are f0.025,23,16 and for s 2f < s 2c the F-
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distribution parameters are f0.025,16,23 . Since, forecast variance comes in numerator, we
2
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used threshold of S f ≥0.01 in this part of the study (e.g., Fig. 5) as opposed to S f ≥0.25 in
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first part of the study (e.g., Fig. 3).
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Figure S.1: Same as Figure 3(a) except using 0.01°C threshold for the ensemble spread.
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3
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Figure S.2: Same as Figure 3(a) except using 0.1°C threshold for the ensemble spread.
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4