Stochastic modelling of rainfall patterns across the United Kingdom

METEOROLOGICAL APPLICATIONS Meteorol. Appl. 24: 5 595 (2017) Published online 7 July 2017 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.2/met.1658 Stochastic modelling of rainfall patterns across the United Kingdom Terence C. Mills* School of Business and Economics, Loughborough University, UK ABSTRACT: In the last 2 years several regions of the United Kingdom have experienced extended periods of heavy rainfall, most notably the south of England over the winter of 2013 2014 and northern England and much of Scotland during the winter of 2015 2016. Since extensive monthly regional rainfall data are available, this study develops statistical models for these regional rainfall series, estimates the seasonal patterns implied by them, assesses whether rainfall has become more unpredictable over time and uses these seasonal patterns to assess how unusually wet these recent months have actually been in terms of departures from the predictions of the models. KEY WORDS rainfall; statistical models; seasonal patterns; prediction Received 2 June 2016; Revised 19 December 2016; Accepted 19 December 2016 1. Introduction In the last 2 years several regions of the United Kingdom have experienced extended periods of heavy rainfall, most notably the south of England over the winter of 2013 2014 and northern England and much of Scotland during the winter of 2015 2016. The Met Office reported that the earlier of these winters was exceptionally wet: for England and Wales this was one of, if not the most, exceptional periods of winter rainfall in at least 248 years, with the southeast and central southern England region experiencing the wettest 2 month period in a data series beginning in 1910 (Met Office, 2014, p. 2). Two years later Storm Desmond produced heavy rainfall from Friday 4 to Sunday 6 December [which] led to widespread flooding in Cumbria and across other parts of northern Britain. Many parts of north-west Britain had already recorded more than twice the monthly average rainfall during November. Across north-west England and North Wales, November 2015 was the second wettest November in a series from 1910 (http:// www.metoffice.gov.uk/climate/interesting/december2015). Following in quick succession, Storms Eva and Frank brought a sustained period of exceptionally wet weather from mid-december to New Year [which] resulted in severe and extensive flooding across the north of the U.K. so that December 2015 was the wettest calendar month for the U.K. in a series from 1910, while November and December combined was the wettest any two-month period (http://www.metoffice.gov.uk/climate/ interesting/december2015_further). Since extensive monthly rainfall data are available from the Met Office s Hadley Centre for five regions of England, three regions of Scotland and for Northern Ireland, an interesting exercise is to develop statistical models for these regional rainfall series, to estimate the seasonal patterns implied by them and to use these patterns to assess how unusually wet these recent months have actually been in terms of departures from the predictions of the models. * Correspondence: T. C. Mills, School of Business and Economics, Loughborough University, Ashby Road, Loughborough LE11 3TU, UK. E-mail: t.c.mills@lboro.ac.uk To this end, Section 2 extends the modelling approach of Mills (2005, 2015) to construct and estimate models of monthly regional rainfall. In doing so, two interesting statistical problems need to be addressed: how to transform naturally skewed data to syetry and, it is hoped, normality, and whether seasonal rainfall patterns evolve, either deterministically or stochastically, over time. Section 3 focuses on the annual series for each month and finds that a very simple noise model adequately characterizes these series. Section 4 investigates whether rainfall patterns across the United Kingdom have varied in unpredictability over time by considering the evolution of moving residual standard deviations from the models fitted in Section 2. The question posed in the previous paragraph is addressed in Section 5, which uses the standardized residuals and forecast errors from the two sets of models to assess how unusual these recent months of excessive rainfall have actually been compared to predictions based on the past record of rainfall. Section 6 offers a suary and conclusions. 2. A model for monthly rainfall Following Mills (2005, 2015), a basic model for a monthly rainfall series observed from time t = 1 to time t = T has been found to be: x (λ) t = 12 i=1 ( αi s i,t + β i s i,t t ) + u t t = 1, 2,, T (1) In Equation (1), rainfall x t is transformed by the Box and Cox (1964) power transformation x (λ) t = xλ t 1 λ x (0) t = log x t to ameliorate the skewness found in the raw data, a consequence of x t being bounded below at zero and possibly having a long right (positive) tail. Such a transformation will aid in inducing normality, linearity and constancy of variance in the model. The s i,t, t = 1, 2,, T, are duy variables defined to take the value 1 in month i and 0 elsewhere (where i = 1 signifies January etc.). Their inclusion allows a deterministic monthly pattern to be 2017 Royal Meteorological Society

Rainfall patterns across the UK 581 0 Central England 0.014 North East England 0.016 0.012 0.010 0.012 6 2 0 20 1 1 1 0 20 1 1 1 200 0.012 North West England and Wales 0.014 South East England 0.010 0.012 0.010 6 6 2 2 0 20 1 1 1 200 220 2 2 2 0 20 1 1 1 200 220 (e) 0.010 South West England and Wales 6 2 0 20 1 1 1 200 220 2 2 2 Figure 1. (e) Histograms and kernel densities (measured in relative frequencies) for English regional rainfall. [Colour figure can be viewed at wileyonlinelibrary.com]. modelled. The presence of the s i,t t interaction variables allows for the possibility of different monthly linear time trends. The α i and β i parameters measure the intercept and slope of these trends, so that if β i 0 then the seasonal pattern for month i evolves linearly over time. The error u t can in general follow a seasonal autoregressivemoving average process (see, for example, Mills, 2014): where φ (B) Φ ( B 12) u t = θ (B) Θ ( B 12) a t (2) φ (B) = 1 φ 1 B φ p B p θ (B) = 1 + θ 1 B + +θ q B q are non-seasonal polynomials of orders p and q in the lag operator B, defined such that B j a t a t j. a t is zero mean white noise (E(a t ) = 0, E(a t a t j ) = 0 for all j t) with variance E ( a 2 t ) = σ 2 a. The seasonal polynomials: Φ ( B 12) = 1 Φ 1 B 12 Θ ( B 12) = 1 +Θ 1 B 12 + Φ P B 12P +Θ Q B 12Q are of orders P and Q and their presence allows the error to be autocorrelated at seasonal lags, such as 12, 24,,aswellas being autocorrelated at non-seasonal lags. 2.1. Deterministic and stochastic trends and seasonality More general models result if unit roots are allowed in the φ(b)andφ(b 12 ) polynomials. If the non-seasonal autoregressive polynomial contains a unit root, i.e. the characteristic equation

582 T. C. Mills 0.016 Eastern Scotland Northern Scotland 0.014 7 0.012 6 0.010 5 6 3 2 2 1 0 20 1 1 1 200 220 0 1 200 2 2 320 3 9 Southern Scotland 0.012 Northern Ireland 7 0.010 6 5 6 3 2 1 2 0 1 200 2 2 320 0 20 1 1 1 200 220 Figure 2. Histograms and kernel densities (measured in relative frequencies) for Scottish regional and Northern Ireland rainfall. [Colour figure can be viewed at wileyonlinelibrary.com]. Table 1. Distributional statistics for regional rainfall. β 1 β 2 J-B λ 95% CI Central 0.7 3.48 155.3 [] 0.49 [0.42, 0.56] Northeast 0.7 3.57 182.7 [] 0.45 [0.38, 0.52] Northwest 0.6 3.33 124.7 [] 0.55 [0.48, 0.62] Southeast 0.9 3.82 257.6 [] 0.49 [0.43, 0.55] Southwest 0.7 3.28 148.3 [] 0.54 [0.48, 0.] East Scotland 0.7 3.65 111.1 [] 0.43 [0.32, 0.54] North Scotland 0.7 3.08 81.5 [] 0.52 [0.43, 0.61] South Scotland 0.7 3.12 82.5 [] 0.54 [0.46, 0.62] Northern Ireland 0.3 2. 22.9 [] 0. [0., 0.] β 1, moment measure of skewness; β 2, moment measure of kurtosis; J-B, Jarque and Bera (19) test for normality (β 1 = 0, β 2 = 3); J-B χ 2 (2), probability value shown in square brackets. associated with φ(b) contains a root of unity, then φ(b) can be factorized as: ) φ (B) = (1 B) (1 φ 1 B φ p 1 Bp 1 = (1 B) φ (B) where φ*(b) is a polynomial of order p 1. Equation (1) then becomes, with =1 B signifying the first-difference operator and u t = u t : x (λ) t = 12 i=1 ( αi s i,t + β i s i,t t ) + u t φ (B) Φ ( B 12) u t = θ (B) Θ ( B 12) a t (3) Noting that s i,t = s i,t s i + 1,t and s i,t t = (12 + i)(s i,t s i + 1,t ), where it is taken that s 12,t = s 12,t s 1,t + 1, Equation (3) in turn becomes: x (λ) t = 12 i=1 { αi + (12 + i) β i }( si,t s i+1,t ) + u t (4) Noting that x (λ) t = α + u would depict a random walk with t drift α, Equation (4) may be interpreted as implying that x (λ) t contains a stochastic, random walk, trend with differing seasonal drifts, i.e. each month evolves as a random walk with its own drift. Alternatively, suppose that the seasonal autoregressive polynomial contains a unit root: Φ ( B 12) = ( 1 B 12)( 1 Φ 1 B12 Φ P 1 B12(P 1)) = ( 1 B 12) Φ ( B 12) Equation (1) then becomes, with 12 = 1 B 12 and u t = 12 u t : 12 x (λ) t = 12 ( αi 12 s i,t + β i 12 s i,t t ) + u t φ (B) Φ ( B 12) u t i=1 = θ (B) Θ ( B 12) a t (5) Now 12 s i,t = 0 and 12 s i,t t = 12s i,t, so that Equation (5) becomes: 12 12 x (λ) t = 12 β i s i,t + u t (6) i=1

Rainfall patterns across the UK 583 0.12 Central England: Box-Cox transformed 0.09 North West England and Wales: Box-Cox transformed 0.10 0.08 0.07 0.08 0.06 0.06 0.05 0.03 0.01 0 2 4 6 8 10 12 14 16 18 20 22 24 0 4 8 12 16 20 24 28 32 0.09 South West England and Wales: Box-Cox transformed 0.12 North East England: Box-Cox transformed 0.08 0.07 0.10 0.06 0.08 0.05 0.06 0.03 0.01 0 4 8 12 16 20 24 28 32 0 2 4 6 8 10 12 14 16 18 20 22 24 26 (e) 0.10 South East England: Box-Cox transformed 0.08 0.06 0 2 4 6 8 10 12 14 16 18 20 22 24 26 Figure 3. (e) Histograms and kernel densities (measured in relative frequencies) for transformed English regional rainfall (horizontal units are Box-Cox transformed.). [Colour figure can be viewed at wileyonlinelibrary.com]. and x (λ) t now contains a stochastic seasonal random walk with differing seasonal drifts. If Φ(B 12 ) =Θ(B 12 ) then there is only deterministic seasonality (for similar set-ups and additional analysis see Pierce (1978) and Mills and Mills (1992). 2.2. Estimating and testing the model For the English regions the sample period is January 1873 to December 2015, a total of T = 1716 observations; for the Scottish regions and Northern Ireland the sample period begins in January 1931, so that there are only 1020 observations. The data were taken from the UK Met Office (Hadley Centre) website at http://www.metoffice.gov.uk/hadobs/hadukp/data/monthly/ HadEWP_monthly_qg.txt Figures 1 and 2 provide histograms and empirical kernel densities for each of the regions and accompanying distributional statistics are given in Table 1. All regions are confirmed to be right skewed (β 1 > 0) and all, apart from Northern Ireland, are excessively kurtotic (β 2 > 3); consequently, the Jarque and Bera (19) statistics for testing the null hypothesis of normality (β 1 = 0, β 2 = 3) all conclusively reject this hypothesis by very wide margins. Also shown in Table 1 are the maximum likelihood estimates of the Box Cox transformation parameter in

584 T. C. Mills 0.12 Eastern Scotland: Box-Coxtransformed 0.08 Northern Scotland: Box-Coxtransformed 0.10 0.07 0.06 0.08 0.05 0.06 0.03 0.01 0 2 4 6 8 10 12 14 16 18 20 22 24 26 0 4 8 12 16 20 24 28 32 36 0.09 Southern Scotland: Box-Cox transformed 0.05 Northern Ireland: Box-Cox transformed 0.08 0.07 0.06 0.05 0.03 0.03 0.01 0.01 0 4 8 12 16 20 24 28 32 36 0 5 10 15 20 25 30 35 45 55 Figure 4. Histograms and kernel densities (measured in relative frequencies) for transformed Scottish regional and Northern Ireland rainfall (horizontal units are Box-Cox transformed.). [Colour figure can be viewed at wileyonlinelibrary.com]. Equation (1), accompanied by 95% confidence intervals. For the English and Scottish regions all estimates are close to and insignificantly different from 0.5 while the estimate for Northern Ireland is 0.7. These values were thus chosen as the values of λ with which to transform the rainfall series. Figures 3 and 4 therefore provide histograms and empirical kernel densities for each of the transformed regional series, with accompanying distributional statistics being given in Table 2. The Box Cox transformation (essentially a square root transformation for all regions apart from Northern Ireland) is seen to do an excellent job in transforming these series to syetry and, indeed, normality. To determine the most appropriate form of the combined model (1) and (2), initial analysis using the information from the sample Table 2. Distributional statistics for transformed regional rainfall. λ β 1 β 2 J-B Central 0.5 0.0 2.83 2.1 [0.35] Northeast 0.5 0.1 2.82 3.7 [0.16] Northwest 0.5 0.0 2.83 2.4 [0.31] Southeast 0.5 0.0 2. 1.2 [0.54] Southwest 0.5 0.0 2.82 2.4 [0.31] East Scotland 0.5 0.1 2.78 4.0 [0.13] North Scotland 0.5 0.1 2.81 2.2 [0.34] South Scotland 0.5 0.0 2.81 1.9 [0.39] Northern Ireland 0.7 0.0 2.63 5.9 [0.05] As for Table 1. Table 3. Stochastic parameter estimates for Equation (7). φ Φ Θ σ a R 2 Central 0.051 (4) 0.7 (0.102) 0.776 (0.102) 3.557 0.09 Northeast 0.018 (4) 0.8 (0.0) 0.844 (0.064) 3.773 0.11 Northwest 0.013 (4) 0.925 (0.010) 0.983 (3) 4.081 0.20 Southeast 0.033 (4) 0.916 (0.011) 0.986 (3) 4.171 0.13 Southwest 0.015 (4) 0.2 (0.076) 0.814 (0.075) 4.530 0.21 East Scotland 1 (0.032) 0.741 (0.087) 0.761 (0.085) 3.528 0.15 North Scotland 0.105 (0.032) 0.876 (0.030) 0.912 (7) 4.697 0.34 South Scotland 0.089 (0.054) 0.891 (7) 0.5 (7) 4.625 0.29 Northern Ireland 0.191 (0.057) 0.954 (0.010) 0.964 (9) 8.977 0.21 Standard errors are shown in parentheses.

Rainfall patterns across the UK 585 Table 4. Test statistics for Equation (7). φ = 0 Φ+Θ= 0 Seasonal trends L-B Probability value Probability value Probability value Probability value Central 0. 0.76 0.68 Northeast 0.97 0.24 0.14 0.66 Northwest 0.84 0.35 Southeast 0.17 0.55 Southwest 0.54 0.42 0.14 0.52 East Scotland 0.97 0.33 0.82 0. North Scotland 0.01 0.06 0.14 South Scotland 0.10 0.21 0.39 Northern Ireland 0.14 0.22 0.61 L-B, Ljung and Box (1978) test for 10 th order autocorrelation; L-B χ 2 (10). 65 55 45 35 110 75 65 55 45 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Central England seasonal pattern North East England seasonal pattern 85 75 65 55 45 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec South West England and Wales seasonal pattern Eastern Scotland seasonal pattern (e) 110 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Northern Ireland seasonal pattern Deterministic seasonal patterns Figure 5. (e) Regions with deterministic seasonal patterns: the shaded area depicts one-standard-error bands. [Colour figure can be viewed at wileyonlinelibrary.com].

586 T. C. Mills 130 1874-18 130 19-1944 110 110 1875 18 1885 18 1895 10 15 1910 1915 1920 1925 1930 1935 19 130 110 1945-19 1981-2016 130 110 1945 19 1955 19 1965 19 1975 19 1985 19 1995 2000 2005 2010 2015 Figure 6. Northwest England and Wales: fitted seasonal pattern to monthly rainfall. [Colour figure can be viewed at wileyonlinelibrary.com]. 1874-18 19-1944 30 1875 18 1885 18 1895 10 15 30 1910 1915 1920 1925 1930 1935 19 1945-19 1981-2016 30 1945 19 1955 19 1965 19 1975 19 30 1985 19 1995 2000 2005 2010 2015 Figure 7. Southeast England: fitted seasonal pattern to monthly rainfall. [Colour figure can be viewed at wileyonlinelibrary.com].

Rainfall patterns across the UK 587 220 1934-1959 200 1934-1959 200 1 1 1 1 1 1 1935 19 1945 19 1955 1935 19 1945 19 1955 220 19-1987 200 19-1987 200 1 1 1 1 1 1 19 1965 19 1975 19 1985 19 1965 19 1975 19 1985 220 1988-2016 200 1988-2016 200 1 1 1 1 1 1 19 1995 2000 2005 2010 2015 19 1995 2000 2005 2010 2015 Figure 8. Northern Scotland: fitted seasonal pattern to monthly rainfall. [Colour figure can be viewed at wileyonlinelibrary.com]. Figure 9. Southern Scotland: fitted seasonal pattern to monthly rainfall. [Colour figure can be viewed at wileyonlinelibrary.com]. autocorrelation and partial autocorrelation functions, along with residual diagnostic checks from fitted models, established that the polynomial orders could typically be set at p = P = Q = 1and q = 0, leading to the model: x (λ) t = 12 i=1 ( αi s i,t + β i s i,t t ) + 1 +ΘB 12 (1 φb) ( 1 ΦB 12) a t (7) for all regions except southern Scotland and Northern Ireland, where in both cases a setting of p = 3 was required so that for these two regions φ(b) = 1 φ 1 B φ 2 B 2 φ 3 B 3 appears as the non-seasonal autoregressive polynomial in Equation (7). The estimates of this model for each region are reported in Tables 3 and 4. For there to be a random walk trend in rainfall, φ (or φ 1 + φ 2 + φ 3 ) would have to be unity (the unit root condition). The estimates show clearly the absence of such a trend since, for every region, φ <1 by a considerable margin (for southern Scotland and Northern Ireland the sum φ 1 + φ 2 + φ 3 with accompanying standard error is provided). For central England, northern Scotland, Northern Ireland and, marginally, southern Scotland, there is evidence of positive (φ>0) non-seasonal autocorrelation. The estimates of Φ are less than unity for all regions, thus indicating the absence of seasonal non-stationarity and implying that seasonal amplitudes have not increased or declined continually throughout the sample period. For three regions, northwest England and Wales, southeast England and northern Scotland, the hypothesis of no stochastic seasonality (Φ+Θ=0)

588 T. C. Mills 10 Central 10 North East 0 0 0 0 0 0 1875 10 1925 19 1975 2000 1875 10 1925 19 1975 2000 10 North West and Wale 10 South East 0 0 0 0 0 0 1875 10 1925 19 1975 2000 1875 10 1925 19 1975 2000 (e) 10 South West and Wales 0 0 0 1875 10 1925 19 1975 2000 Figure 10. (e) Annual rainfall for English and Welsh regions. [Colour figure can be viewed at wileyonlinelibrary.com]. 20 East 20 North 2000 2000 10 10 0 0 0 0 0 19 19 19 19 19 19 2000 2010 0 19 19 19 19 19 19 2000 2010 20 South 20 Northern Ireland 2000 2000 10 10 0 0 0 0 0 19 19 19 19 19 19 2000 2010 0 19 19 19 19 19 19 2000 2010 Figure 11. Annual rainfall for Scottish regions and Northern Ireland. [Colour figure can be viewed at wileyonlinelibrary.com]. can be conclusively rejected. These three regions also saw the rejection of the hypothesis of no deterministic seasonal trends (β 1 = β 2 = =β 12 = 0), so their seasonal patterns evolved both linearly and stochastically throughout the sample period. The residuals from this model exhibit no autocorrelation for any region. The question of whether the deterministic seasonal model might contain a nonlinear component was addressed by including additional quadratic trends, taking the form s i,t t 2,but these were found to be insignificant for all regions. The overall fit of the model varies considerably across regions, with R 2 statistics ranging from 0.09 for central England to 0.34 for northern Scotland, thus demonstrating that a good deal of the variation

Rainfall patterns across the UK 589 4.0 4.4 3.8 4.2 3.6 4.0 3.4 3.8 3.6 3.2 3.4 3.0 3.2 2.8 10 1925 19 1975 2000 3.0 10 1925 19 1975 2000 Central England North East England 4.6 4.8 4.4 4.6 4.2 4.4 4.0 4.2 3.8 4.0 3.6 3.8 3.4 3.6 3.2 10 1925 19 1975 2000 North West England and Wales 3.4 10 1925 19 1975 2000 South East England (e) 5.2 5.0 4.8 4.6 4.4 4.2 4.0 3.8 10 1925 19 1975 2000 South West Englandand Wales Figure 12. (e) England and Wales: 10-year moving residual standard deviations. [Colour figure can be viewed at wileyonlinelibrary.com]. in rainfall remains unexplained and, indeed, unpredictable with this model. The deterministic seasonal patterns for the five regions for which this specification is appropriate are shown in Figure 5. The driest months are from February to June, the wettest during autumn, although each region has its own idiosyncrasies. Figures 6 9 plot the evolving stochastic seasonal patterns of the remaining four regions. For northwest England and Wales the seasonal amplitude is smallest between 10 and 19, whereupon it increases to become similar to the pattern in the 19 th Century. The pattern looks to have stabilized since 1995. Southeast England has seen distinct shifts in the pattern of seasonality, with the smallest amplitude occurring between 19 and 19 and with the pattern stabilizing during the 2000s. Northern Scotland has a fairly regular pattern throughout with the largest amplitude occurring during the 1930s. In contrast, southern Scotland has a distinctive evolving pattern with amplitude increasing since 19. 3. Modelling the annual rainfall data The data may also be modelled on an annual basis, i.e. by constructing 12 annual series by month. Obviously such annual

5 T. C. Mills 4.0 5.6 3.8 5.2 3.6 4.8 3.4 4.4 3.2 4.0 3.0 3.6 19 19 19 19 19 2000 2010 19 19 19 19 19 2000 2010 5.6 Eastern Scotland 10.5 Northern Scotland 5.2 10.0 9.5 4.8 9.0 4.4 8.5 4.0 8.0 19 19 19 19 19 2000 2010 7.5 19 19 19 19 19 2000 2010 Southern Scotland Northern Ireland Figure 13. Scotland and Northern Ireland: 10-year moving residual standard deviations. [Colour figure can be viewed at wileyonlinelibrary.com]. Table 5. Standardized residuals for selected months: figures in parentheses give Gaussian marginal probabilities; figures in square brackets give marginal probabilities calculated from the empirical distribution. 2014 2015 January February November December Central Monthly 1.76 0.75 0. 0.41 (0.039) (0.228) (0.346) (0.339) Annual 2.22 ab 0.82 0.33 0.35 (0.013) (0.206) (0.369) (0.363) [1] [0.210] [0.315] [0.293] Northeast Monthly 1.42 0. 1.54 2.36 a (0.078) (0.185) (0.061) (9) Annual 1.65 0.87 1.62 2.86 ab (9) (0.192) (0.053) (2) [0.056] [0.203] [0.063] [0.014] Northwest Monthly 1.33 1.65 1.76 2. a (0.091) (0.0) (0.039) (2) Annual 1.62 1.49 2.46 ab 3.74 ab (0.053) (0.068) (7) () [9] [0.084] [7] [] Southeast Monthly 2.13 a 2.05 a 0.06 0.09 (0.016) (0) (0.524) (0.463) Annual 3.39 ab 2.65 ab 0.08 0.09 () () (0.533) (0.537) [] [7] [0.454] [0.510] Southwest Monthly 2.26 a 2.33 a 0.55 0.88 (0.012) (0.010) (0.292) (0.188) Annual 2.67 ab 2.38 ab 0.45 0.82 () (9) (0.327) (0.206) [] [7] [0.273] [0.203]

Rainfall patterns across the UK 591 Table 5. continued 2014 2015 January February November December Eastern Scotland Monthly 0.97 1.01 0.24 1.73 (0.166) (0.157) (0.7) (2) Annual 1.11 1.07 0.62 2.22 ab (0.133) (0.142) (0.267) (0.013) [0.118] [0.141] [0.235] [0.012] Monthly 0.33 1.22 0.95 1.63 (0.369) (0.111) (0.172) (0.052) Northern Scotland Annual 0.31 0.97 0.68 1.63 (0.3) (0.166) (0.248) (0.052) [0.412] [0.153] [0.247] [0.059] Monthly 1.49 1.61 1.94 2.00 (0.068) (0.054) (6) (3) Southern Scotland Annual 1.27 1.92 2.54 ab 2. ab (0.101) (8) (6) (5) [0.118] [0.035] [] [] Monthly 1.15 1.66 1.47 2.05 a (0.126) (9) (0.071) (0) Northern Ireland Annual 1.31 1.66 1.46 2.45 a, b (0.072) (9) (0.072) (7) [0.082] [7] [0.082] [0.012] a A Gaussian marginal probability less than 5. b An empirical marginal probability less than 5. Table 6. Standardized forecast errors for January and February 2016: figures in parentheses give Gaussian marginal probabilities. January 2016 February Central Monthly 0.78 0.20 (0.218) (0.420) Annual 0.87 0.08 (0.193) (0.469) Northeast Monthly 2.26 a 0.10 (0.012) (0.462) Annual 2. a 0.03 (2) (0.514) Northwest Monthly 1.59 1.22 (0.056) (0.111) Annual 1.86 0.96 (0.031) (0.168) Southeast Monthly 0.98 0.10 (0.164) (0.461) Annual 1.39 0.07 (0.083) (0.053) Southwest Monthly 1. 0.89 (0.081) (0.188) Annual 1.48 0.66 (0.0) (0.255) Eastern Scotland Monthly 2.52 a 0.38 (6) (0.351) Annual 3.51 a 0.30 () (0.383) Northern Scotland Monthly 0.85 1.02 (0.197) (0.154) Annual 0.93 0. (0.175) (0.184) Southern Scotland Monthly 1.39 1.27 (0.082) (0.102) Annual 2.04 a 1.29 (1) (0.098) Northern Ireland Monthly 1.34 1.09 (0.0) (0.137) Annual 1. 0.75 (0.054) (0.225) a A marginal probability less than 5. series display no seasonality and, in fact, each can typically be modelled as a simple white noise process: y t = α + a t (8) where y t is the annual series of observations for a particular month. Suary statistics from fitting this model to the annual monthly series for each region are presented in the Appendix, Table A1, parts (i), accompanied by explanatory coents. As a consequence, the annual rainfall for each month of every region can be regarded as being essentially white noise. However, there is an interesting departure from this for the March data for northwest England and Wales and the three Scottish regions, which all show runs of low order positive, although not numerically particularly large, autocorrelations. This implies that there are sequences of unusually wet, or indeed dry, Marches for these contiguous regions. To investigate these annual rainfall patterns further, each region was aggregated to an annual frequency, the suary statistics of these annual aggregates being shown in the Appendix, Table A2, and the aggregates themselves being plotted in Figures 10 and 11. These plots clearly show the absence of any trend in rainfall and all, except for southern Scotland and Northern Ireland which show small positive autocorrelation, are essentially normally distributed white noise around widely differing means, ranging from 654 for central England to 1637 for northern Scotland with widely differing standard deviations, these ranging from 89 for eastern Scotland to 320 for southern Scotland. 4. Has rainfall pattern unpredictability altered through time? To assess whether the patterns of rainfall have altered in predictability over time relative to the fitted models, moving residual standard deviations were computed. The n-period moving residual standard deviation at time t, σ a,t,n, is defined from: ( n 1 ) σ 2 a,t,n = (n 1) t = n, n + 1,, T â 2 t i i=0

592 T. C. Mills so that the conventional residual standard deviation σ a reported in Table 3 implicitly sets the moving window size n equal to the sample size T. Figures 12 and 13 plot these moving residual standard deviations for n =, i.e. for a 10-year (decadal) moving window, for the English and Scottish (and Northern Ireland) regions respectively. For the English regions the σ a,t, series oscillate around their corresponding σ a, with minimum values occurring in the early part of the 20 th Century. Only for northeast England have values during the 2000s persistently exceeded σ a. For the Scottish regions and Northern Ireland an oscillatory pattern is again observed (note, however, that the series of standard deviations do not begin until 1941). However, for none of the Scottish regions, and only just for Northern Ireland, do any values exceed σ a. Thus, while there are certainly periods for which rainfall is more or less unpredictable relative to the fitted model, there is no indication, apart from some weak evidence for northeast England, that rainfall patterns have generally become more unpredictable in recent years. 5. How unusual was early 2014 and the winter of 2015 2016? Table 5 focuses on the extent to which rainfall in January and February 2014 and November and December 2015 was unusually large, in the sense that rainfall levels were much greater than predicted by the monthly and annual models. Standardized residuals were computed from Equations (7) and (8), i.e. â t σ a, and marginal probabilities (i.e. the probability of obtaining a value larger than that actually observed) were calculated from a standard normal (Gaussian) distribution and also from the empirical distribution for the residuals from Equation (8), given that these distributions are often skewed. Table 6 presents standardized forecast errors for January and February 2016, the two out-of-sample months available at the time of writing, from the two models, along with Gaussian marginal probabilities. If a marginal probability level of 5 is chosen as the criterion for an unusually wet month, then the southeast and southwest regions of England met this criterion in both January and February 2014, with there being weak evidence of the central region being unusually wet in the January. The winter months of 2015 2016 saw unusually heavy rainfall concentrated in the northern regions of England, southern and eastern, but not northern, Scotland and, for December 2015 only, Northern Ireland. There are some striking differences, however, in the implied frequency of such unusual rainfall from the two models. A choice of 5 as the marginal probability level for designating a month as unusually wet implies that such an event should be expected every 1/5 = years using the annual model (8), and many of these unusual months had probabilities much lower than this: for example, the largest standardized residual, 3.74, was for the northwest region in December 2015, which implies a one in 1/ 092 10 000 year event, while the standardized residuals for southeast and southwest England were 3.39 and 2.67, implying one in approximately 3000 and 2 year events respectively! A once in years event for the monthly model (7), however, would imply a marginal probability of 1/4 = 2, which is the probability attached to the corresponding standardized residual for the northwest in December 2015 of 2.. Most of the unusually large rainfall months found by the monthly model had marginal probabilities in the range 0.01, i.e. months (approximately 4 8 years): certainly unusual but not strikingly so. Thus, unsurprisingly, if only, say, Decembers are considered then the 2015 rainfall was indeed an extremely unusual event for the northwest region, but it was a much less unlikely event when the entire rainfall history of the region is taken into account. 6. Suary and conclusions A relatively simple model has been found to fit the regional monthly rainfall series for the United Kingdom adequately. Box Cox transformed rainfall, with a typical parameter of 0.5 (essentially a square root transformation), may generally be modelled as a set of deterministic seasonal linear trends with stationary stochastic components modelling short-run autocorrelation and seasonality, although simpler forms of this model are found to be appropriate for several regions. If just the annual series for each month are analysed then an even simpler model results, that of normal white noise deviations around a constant mean, with the mean and standard deviation of the distribution varying widely across regions. By computing moving residual standard deviations, it was found that rainfall patterns had not become more unpredictable over time. These models were then used to provide an assessment of how unusual, in terms of departures from the fitted model, were the high rainfalls of January and February 2014 and the winter of 2015 2016 observed in several of the regions. Using the annual models, the first 2 months of 2014 were found to be very unusually wet in southern England and similarly for the northern regions of England and southern and eastern Scotland in December 2015 and January 2016. When the monthly models were used, however, the unusualness of these rainfall events was much lower, since the entire rainfall history of the regions, not just a particular month, is then taken into account. The use of statistical models to analyse rainfall patterns would thus seem to be a useful complement to the simulation approach to assessing extreme rainfall events recently provided by Otto et al. (2015), Pearson et al. (2015) and Rhodes et al. (2015). Appendix: Suary statistics for the annual monthly rainfall series Although the typical annual series is right skewed, the data are analysed without transformation as the extent of non-normality, in most cases, was found not to be excessive from the reported Jarque Bera normality statistics in Tables A1 and A2. Nevertheless, standard errors attached to the sample standard deviation, the estimate of σ a in Equation (8), were adjusted for kurtosis in the manner described in the notes to the tables. The Ljung and Box (1978) statistics for autocorrelation in the residuals from Equation (8) were occasionally significant (13 of the 108 residual series were found to be autocorrelated at the 10% level, just seven at the 5% level), in which case the sample standard deviations were replaced by heteroscedasticity and autocorrelation consistent (HAC) estimates of the Newey and West (1994) variety, which adjust these statistics for autocorrelation and, indeed, for any heteroscedasticity that might also be present.

Rainfall patterns across the UK 593 Table A1. Suary statistics for rainfall by month for each region. y s β 1 β 2 J-B L-B Central England January 54.10 ± 1.93 23.10 ± 1.28 0.5 2.76 5.5 [0.06] 12.2 [0.27] February 43.16 ± 1.86 22.20 ± 1.32 0.6 3.01 8.3 [] 14.3 [0.16] March 43.51 ± 1.84 21.98 ± 1.37 0.6 3.22 8.3 [] 11.3 [0.33] April 44.85 ± 1.91 22.84 ± 1. 0.7 3.82 16.9 [] 11.8 [0.24] May 51. ± 2.06 24.59 ± 1.64 0.9 3.55 20.2 [] 15.4 [0.12] June 51.96 ± 2.24 26.74 ± 1.73 0.7 3. 12.9 [] 7.9 [0.64] July 61.83 ± 2.45 29.27 ± 2.00 0.7 3.66 14.2 [] 9.5 [0.48] August 64.95 ± 2.39 28.59 ± 2.06 0.6 3.97 14.6 [] 10.7 [0.38] September 54.01 ± 2.36 28.23 ± 1.46 0.5 2.52 8.2 [] 10.0 [0.44] October 63.38 ± 2.46 29.45 ± 1.77 0.5 3.06 7.1 [0.03] 10.1 [0.43] November 62.27 ± 2.08 24.88 ± 1.55 0.7 3.21 12.4 [] 5.7 [0.84] December 58.51 ± 2.24 25.65 ± 1. 0.6 3.03 9.1 [0.01] 16.4*[0.09] Northeast England January 68.76 ± 2.43 28.74 ± 1. 0.6 3.44 8.6 [0.01] 18.5*[0.05] February 55.54 ± 2.28 27.29 ± 1.42 0.5 2.54 6.6 [] 14.1 [0.17] March 55.08 ± 2.15 25.65 ± 1.53 0.5 3.04 6.6 [] 9.7 [0.45] April 51.63 ± 2.22 26.59 ± 2.14 1.0 4.69 42.8 [] 6.8 [0.74] May 57.25 ± 2.09 24.96 ± 1.49 0.6 3.03 8.7 [0.01] 15.5 [0.12] June 57.74 ± 2.59 31.00 ± 2.25 1.0 4.01 29.7 [] 11.7 [0.38] July 69.13 ± 2.59 30.94 ± 1.77 0.5 2.88 5.9 [0.05] 12.1 [0.28] August 78. ± 2.92 34.87 ± 2.29 0.6 3.47 10.8 [] 7.2 [0.71] September 62.86 ± 2.71 32. ± 2.01 0.8 3.20 15.6 [] 9.8 [0.45] October 77.75 ± 2.87 32.46 ± 2.07 0.5 3.08 6.3 [] 4.7 [0.91] November 78.91 ± 2.86 34.17 ± 1.84 0.5 2.66 6.1 [0.05] 11.4 [0.32] December 74.53 ± 2.68 32.11 ± 2.61 1.0 4.77 41.5 [] 6.1 [0.81] Northwest England and Wales January 98.59 ± 3.33 39.79 ± 2.36 0.3 3.02 2.0 [0.37] 12.6 [0.24] February.74 ± 3.18 38.05 ± 1.81 0.3 2.30 5.5 [0.06] 6.3 [0.79] March 69.74 ± 2.62 34.35 ± 1.86 0.7 3.02 10.5 [0.01] 26.4*[] April 58.42 ± 1.97 23.55 ± 1.28 0.1 2.69 1.1 [0.59] 7.7 [0.66] May 63.84 ± 2.19 26.22 ± 1. 0.5 2.87 5.6 [0.06] 3.5 [0.97] June 69.15 ± 2.71 32.43 ± 2.20 0.8 3.64 16.5 [] 12.5 [0.25] July 84.95 ± 2.95 35.23 ± 1.82 0.4 2.53 5.4 [0.07] 15.3 [0.12] August.30 ± 3.35.10 ± 2.45 0.2 3.13 1.1 [0.57] 9.8 [0.46] September.71 ± 3.45 41.28 ± 2.61 0.4 3.28 3.4 [0.18] 5.3 [0.87] October 108. ± 3.83 45.81 ± 2.65 0.4 2.92 4.6 [0.10] 7.3 [0.] November 102.82 ± 3.53 42.23 ± 2.30 0.4 2.69 5.1 [0.08] 8.5 [0.48] December 106.72 ± 3.69 44.07 ± 2.82 0.5 3.35 7.1 [0.03] 2.3 [0.99] Southwest England and Wales January 109. ± 3.93 46.96 ± 2.36 0.0 2.44 1.9 [0.39] 10.6 [0.39] February 82.27 ± 3.92 46.86 ± 2.48 0.4 2. 4.9 [0.09] 10.5 [0.] March 75.57 ± 3.20 38.23 ± 2.79 0.8 4.04 21.0 [] 6.7 [0.75] April 63.95 ± 2.58 30.91 ± 1.99 0.5 3.36 6.0 [0.05] 7.6 [0.67] May 63. ± 2.45 29.25 ± 1.62 0.4 2.76 4.4 [0.11] 10.1 [0.43] June 61.08 ± 2.59 30.97 ± 2.01 0.7 3.41 12.8 [] 4.6 [0.91] July 72.11 ± 2.89 34.53 ± 1.97 0.6 2.87 7.6 [] 4.9 [0.] August 85.87 ± 3.24 38.77 ± 2. 0.4 3.19 3.9 [0.14] 6.8 [0.75] September 82.01 ± 3. 41.81 ± 2.38 0.6 2.85 9.0 [0.01] 11.2 [0.34] October 111.28 ± 3.91 46. ± 2.35 0.3 2.44 3.6 [0.17] 5.5 [0.86] November 114.31 ± 4.14 49.55 ± 2.86 0.6 2. 8.9 [0.01] 9.5 [0.48] December 119.97 ± 4.23 44.13 ± 2.75 0.4 2.69 4.5 [0.10] 19.1*[] (e) Southeast England January 68.21 ± 2. 33.52 ± 1. 0.4 2.84 3.8 [0.15] 9.5 [0.48] February 51.30 ± 2.65 31.68 ± 1. 0.6 2.64 9.6 [0.01] 16.7*[0.08] March 49.79 ± 2. 28.72 ± 1.89 0.8 3.47 16.2 [] 8.0 [0.63] April 48.83 ± 2.10 25.07 ± 1.81 0.6 3.99 13.1 [] 12.3 [0.27] May 51. ± 2.10 25.07 ± 1.43 0.5 2.86 6.9 [0.03] 13.2 [0.21] June 51.47 ± 2.28 27.22 ± 1.62 0.6 3.03 9.7 [0.01] 5.9 [0.83] July 58.17 ± 2.33 27.89 ± 1.38 0.4 2.41 6.4 [] 5.7 [0.84] August 62.69 ± 2.58 30.81 ± 1. 0.5 2.75 5.7 [0.06] 7.9 [0.64] September 59.97 ± 2.85 34.03 ± 2.26 0.9 3.52 20.0 [] 12.6 [0.25] October 78.82 ± 3.56 42.59 ± 2.51 0.6 2.98 10.0 [0.01] 6.9 [0.74] November 78.52 ± 3.35.09 ± 2.45 0.8 3.14 14.0 [] 14.0 [0.17] December 73. ± 3.09 34.93 ± 2.33 0.7 3.28 10.6 [] 16.8*[0.08]

594 T. C. Mills Table A1. Continued y s β 1 β 2 J-B L-B (f) Northern Scotland January 179.33 ± 7.65.61 ± 4.58 0.1 2.43 1.2 [0.54] 8.7 [0.56] February 129.56 ± 6.95 64.09 ± 6.01 0.7 3.99 11.2 [] 14.2 [0.16] March 129.34 ±.74 79.24 ± 7.24 0.9 3.84 12.8 [] 31.6*[] April 94.10 ± 3.99 36. ± 3. 0.6 4.07 9.8 [0.01] 10.0 [0.44] May 84.06 ± 4.03 37.17 ± 3.42 0.7 3.84 8.9 [0.01] 10.0 [0.44] June.76 ± 3.62 33.36 ± 2.74 0.5 3.29 3.6 [0.17] 8.5 [0.58] July 108.79 ± 4.65 37.35 ± 2.99 0.5 3.18 3.5 [0.18] 9.2 [0.51] August 124.71 ± 5.46.30 ± 4.30 0.4 3.49 3.0 [0.23] 8.0 [0.63] September 151.27 ± 5.88 54.19 ± 3.62 0.0 2.52 0.8 [0.66] 10.4 [0.] October 185.12 ± 7.25 66.83 ± 4.44 0.0 2. 0.9 [0.64] 3.7 [0.96] November 1.97 ± 6.78 62.51 ± 4.46 0.2 2.73 0.9 [0.65] 4.1 [0.94] December 188.93 ± 7..99 ± 4.44 0.1 2.33 1.8 [0.41] 10.2 [0.42] (g) Eastern Scotland January 73.08 ± 2.86 26.41 ± 1.83 0.2 2.64 1.1 [0.58] 15.9 [0.10] February 53.33 ± 2.59 23.91 ± 1.61 0.2 2.54 1.3 [0.51] 6.8 [0.75] March 52.01 ± 3.18 24.48 ± 1.58 0.1 2.41 1.6 [0.46] 19.3*[] April 48.07 ± 2.51 23.14 ± 2.56 1.2 5.17 38.3 [] 7.8 [0.64] May 56.64 ± 2.49 22.75 ± 1.75 0.5 3.01 4.2 [0.12] 17.6*[0.06] June 56.87 ± 2. 25.83 ± 2.12 0.9 3.28 11.3 [] 8.6 [0.57] July 72.64 ± 3.64 33. ± 2. 0.5 3.03 4.1 [0.13] 8.5 [0.58] August 74.12 ± 3.85 35. ± 2.39 0.3 2.54 2.4 [0.31] 13.5 [0.20] September 66. ± 3. 32.27 ± 4.01 1.3 6.25 61.3 [] 5.2 [0.87] October 81. ± 3.81 35.13 ± 2.22 0.0 2.36 1.5 [0.48] 7.5 [0.68] November 75. ± 3.53 32.55 ± 3.08 0.8 4.04 12.4 [] 7.7 [0.66] December 73.69 ± 2.87 26.47 ± 1.82 0.4 2.61 2.5 [0.28] 12.0 [0.29] (h) Southern Scotland January 151.92 ± 6.39 58.95 ± 4.13 0.1 2.67 0.7 [0.71] 10.7 [0.38] February 104.73 ± 6.04 55.67 ± 4.94 0.7 3.67 9.5 [0.01] 8.4 [0.59] March 104.75 ± 8.37 62.79 ± 4.92 0.6 3.09 4.7 [0.10] 39.9*[] April.27 ± 3. 33.15 ± 2.97 0.5 3.72 5.8 [0.05] 14.2 [0.16] May 79.36 ± 3.92 36.17 ± 2.75 0.4 2.97 2.3 [0.31] 14.9 [0.13] June 85.16 ± 3.59 33.11 ± 2.99 0.9 3.78 12.8 [] 12.1 [0.28] July.45 ± 3.77 34. ± 2.59 0.4 2.89 1.8 [0.41] 8.4 [0.59] August 111.24 ± 5.67 52.32 ± 4.32 0.5 3.32 4.1 [0.13] 3.4 [0.97] September 128.83 ± 5.92 54. ± 4.15 0.2 2.96 0.7 [0.69] 13.2 [0.21] October 155.22 ± 6.89 63.51 ± 3.82 0.1 2.23 2.2 [0.33] 4.4 [0.93] November 148.62 ± 6. 62.74 ± 4.05 0.1 2.42 1.2 [0.54] 4.2 [0.94] December 156.75 ± 6.87 63.31 ± 4.13 0.3 2.45 2.0 [0.36] 8.5 [0.58] (i) Northern Ireland January 104.88 ± 3.53 32.51 ± 2.31 0.2 2.72 0.9 [0.64] 13.7 [0.19] February 73.58 ± 3.73 34.43 ± 2.49 0.3 2.78 1.8 [0.41] 10.8 [0.38] March 76.12 ± 3.20 29.46 ± 1.92 0.3 2.45 2.8 [0.35] 10.8 [0.38] April 65.61 ± 2.85 26.23 ± 1.74 0.2 2.49 1.3 [0.52] 2.1 [0.99] May 66. ± 3.18 27.73 ± 1.63 0.3 2.17 3.5 [0.17] 16.3*[0.09] June 73.87 ± 3.33 30.73 ± 2.54 0.7 3.32 6.8 [0.03] 12.2 [0.27] July 84.64 ± 3.43 36.74 ± 2.64 0.3 2.75 1.4 [0.49] 22.0*[0.01] August 91.29 ± 4.24 39.09 ± 2.76 0.2 2. 0.7 [0.72] 14.5 [0.15] September 92.20 ± 4.43.12 ± 2.91 0.2 2.79 0.7 [0.] 16.3*[0.09] October 107.00 ± 4.39.51 ± 2.58 0.2 2.38 1.9 [0.39] 4.9 [0.] November 103.94 ± 4.11 37.89 ± 2.23 0.1 2.18 2.5 [0.28] 8.7 [0.56] December 107.58 ± 3.81 35.15 ± 2.68 0.1 2.98 0.2 [0.91] 8.7 [0.56] y, sample mean ± one standard error; s, sample standard deviation ± one standard error; β 1, moment measure of skewness; β 2, moment measure of kurtosis; J-B, Jarque and Bera (19) test for normality; J-B χ 2 (2), probability value shown in square brackets; L-B, Ljung and Box (1978) test for 10 th order autocorrelation; L-B χ 2 (10), probability value shown in square brackets. Standard error of x is s/ n, where the sample size n = 143 for English regions and n = 85 for Scottish regions and Northern Ireland. s is replaced by the Newey West (1991) HAC long-run standard deviation when the accompanying L-B statistic has a probability value less than 0.10 (denoted by *). {1 ( Standard error of s is (s 2n) + β2 3 ) 2 } (Yule and Kendall, 19, 21.12).

Rainfall patterns across the UK 595 Table A2. Suary statistics for annual regional rainfall. y s β 1 β 2 J-B L-B Central 653.9 ± 8.0 95.3 ± 4.8 0.0 2.44 1.8 [0.] 2.6 [0.99] Northeast 787.8 ± 8.7 104.4± 6.6 0.4 3.28 3.5 [0.17] 4.3 [0.93] Northwest 1024.5 ± 11.0 130.6 ± 7.9 0.2 3.09 1.1 [0.58] 11.5 [0.32] Southeast 733.1 ± 9.3 111.0 ± 7.0 0.3 3.25 2.2 [0.32] 8.3 [0.] Southwest 1041.9 ± 11.7 138.4 ± 7.5 0.1 2.89 0.2 [0.92] 5.2 [0.88] East Scotland 784.6 ± 9.6 88.9 ± 6.2 0.5 2.63 3.6 [0.16] 10.7 [0.38] North Scotland 1637.0 ± 24.8 228.5 ± 20.8 0.6 3.81 7.0 [0.03] 15.4 [0.12] South Scotland 1354.6 ± 19.4 319.9 ± 21.0 0.2 2.46 1.6 [0.44] 24.7*[0.01] Northern Ireland 1047.5 ± 11.6 127.3 ± 9.0 0.3 2.71 1.4 [0.49] 18.0*[0.06] y, sample mean ± one standard error; s, sample standard deviation ± one standard error; β 1, moment measure of skewness; β 2, moment measure of kurtosis; J-B, Jarque and Bera (19) test for normality; J-B χ 2 (2), probability value shown in square brackets; L-B, Ljung and Box (1978) test for 10 th order autocorrelation; L-B χ 2 (10), probability value shown in square brackets. Standard error of x is s/ n, where the sample size n = 143 for English regions and n = 85 for Scottish regions and Northern Ireland. s is replaced by the Newey West (1991) HAC long-run standard deviation when the accompanying L-B statistic has a probability value less than 0.10 (denoted by *). {1 ( Standard error of s is (s 2n) + β2 3 ) 2 } (Yule and Kendall, 19, 21.12). References Box GEP, Cox DR. 1964. An analysis of transformations. J. R. Stat. Soc. Ser. B 26: 211 246. Jarque CM, Bera AK. 19. Efficient tests for normality, homoscedasticity and serial dependence in regression residuals. Econ. Lett. 6: 255 259. Ljung GM, Box GEP. 1978. On a measure of lack of fit in time series models. Biometrika 65: 297 303. Met Office. 2014. The Recent Storms and Floods in the UK. MetOffice and Centre for Ecology and Hydrology, NERC: Exeter, UK. Mills TC. 2005. Modelling precipitation trends in England and Wales. Meteorol. Appl. 12: 169 176. Mills TC. 2014. Time series modelling of temperatures: an example from Kefalonia. Meteorol. Appl. 21: 578 584. Mills TC. 2015. Modelling rainfall trends in England and Wales. Cogent GeoSci. OA 1: 1133218. Mills TC, Mills AG. 1992. Modelling the seasonal patterns in UK macroeconomic time series. J. R. Stat. Soc. Ser. A 155: 61 75. Newey WK, West KD. 1994. Automatic lag length selection in covariance matrix estimation. Rev. Econ. Stud. 61: 631 653. Otto FEL, Rosier SM, Allen MR, Massey NR, Rye CJ, Quintana JI. 2015. Attribution analysis of high precipitation events in suer in England and Wales over the last decade. Clim. Change 132: 77 91. Pearson KJ, Shaffrey LC, Methven J, Hodges KI. 2015. Can a climate model reproduce extreme regional precipitation events over England and Wales? Q. J. R. Meteorol. Soc. 141: 1466 1472. Pierce DA. 1978. Seasonal adjustment when both deterministic and stochastic seasonality are present. In Seasonal Analysis of Economic Time Series, Zellner A (ed). US Department of Coerce: Washington, DC; 242 269. Rhodes RI, Shaffrey LC, Gray SL. 2015. Can reanalyses represent extreme precipitation over England and Wales? Q. J. R. Meteorol. Soc. 141: 1114 1. Yule GU, Kendall MG. 19. An Introduction to the Theory of Statistics, 12th edn. Griffin & Company: London.