a. Hypotheses

The aim is to understand any differences between the probability distributions of two variables, such as daily maximum temperatures at two sites. Simple characterizations of the possibly complex differences are often able to capture the main features and aid understanding. Two such characterizations are of particular interest: differences in location or scale, for which the distributions are related by a constant translation or scaling, respectively.

Let X and Y denote the two variables, and let their distribution functions be F(x) = P(X ≤ x) and G(y) = P(Y ≤ y), where P(A) denotes the probability of an event A. See von Storch and Zwiers (2001) or Wilks (1995) for basic introductions to probability distributions. The following hypotheses are of interest for understanding changing distributions:

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for all −∞ < z < ∞ and unknown constants μ_X, μ_Y, σ_X > 0, and σ_Y > 0. Hypothesis H₀ claims no difference between F and G, H_S a difference only in scale, H_L a difference only in location, and H_LS a difference only in location and scale. The relative impacts of location and scale changes have been discussed using parametric approaches by Mearns et al. (1984) and Katz and Brown (1992) among others.

In the remainder of this section, functions of quantiles that summarize the differences between F and G are defined, and simple, informative plots are described that support an informal assessment of the legitimacy of hypotheses (1). Methods are presented for computing confidence intervals to represent the variability of the quantile estimators, and formal testing of the hypotheses is discussed.

b. Quantiles

The p quantile (100p percentile) of a continuous distribution is the value below which a proportion p of the probability mass falls. For example, the p quantile, x_p, of F satisfies F(x_p) = p. If {X₁, . . . , X_m} and {Y₁, . . . , Y_n} are samples from distributions F and G, and X₍₁₎ ≤ . . . ≤ X_(m) and Y₍₁₎ ≤ . . . ≤ Y_(n) are the order statistics (samples arranged in increasing order), then estimators for the p quantiles of F and G are

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where denotes the integer part of z. Different quantile estimators might be preferred if the sample sizes are small (Parrish 1990). See Bonsal et al. (2001) for an application examining changes in temperature quantiles and Wilcox (1997) for more material on such nonparametric statistics.

Three useful statistics for summarizing a distribution are the median, interquartile range, and Yule–Kendall skewness measure, which are computed from just three quantiles:

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These statistics are resistant measures of the location, scale, and shape (asymmetry) of F, and can be compared with the corresponding statistics, m_Y, s_Y, and a_Y, for G. See Lanzante (1996) for examples demonstrating the benefits of using such measures.

For illustration, these statistics are now computed from all of the winter daily minimum temperatures simulated by HIRHAM4 at a single grid point (46.4805°N, 7.9761°E) in both the control (X) and scenario (Y) 30-yr integrations. Winter is defined to cover December–January–February (DJF), yielding sample sizes m = n = 2700. In the control, m_X = −12.2°C, s_X = 10.1°C, and a_X = −0.23; in the scenario, m_Y = −8.2°C, s_Y = 7.3°C, and a_Y = −0.06. Histograms and boxplots of the gridpoint temperatures are reproduced in Fig. 1. Note that the skewness measures compare the relative heights of the lower and upper boxes in the boxplots. These statistics and plots indicate a general warming together with a reduction in scale and a change in shape of the distribution: the long, colder tail evident in the control becomes shorter, resulting in a more symmetric distribution in the scenario. Similar behavior has been noted in observations such as the Central England Temperature series (Antoniadou et al. 2001).

Another informative comparison is made by the quantile–quantile plot of ŷ_p against x̂_p for p = 1/N, 2/N, . . . , 1, where N = min(m, n). The hypotheses (1) correspond to different linear relationships between the two sets of quantiles:

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for all 0 < p < 1. The right-hand sides of equalities H_S, H_L, and H_LS are the quantiles for the distribution obtained by adjusting F to have, respectively, the same scale, location, and location and scale as G.

The quantile–quantile plot of the scenario versus control gridpoint temperatures is reproduced in Fig. 2a. Estimates of the linear relationships (3) for hypotheses H₀, H_L, and H_LS are superimposed, where the location parameters, μ_X and μ_Y, are estimated by the medians, m_X and m_Y, and the scale parameters, σ_X and σ_Y, by the interquartile ranges, s_X and s_Y. The estimated location-scale model for H_LS (dotted line) is reasonably close to the quantile–quantile plot except for some discrepancies at low and high temperatures. This indicates that, while the changes to the body of the temperature distribution are well described by a change in location and scale, a more complex model is required to describe the changes in the extreme tails of the distribution.

c. Exploring changes in gridded fields

The quantile–quantile plot is practicable if only a small number of pairwise comparisons are able to be made. With gridded data, comparing distributions between two time periods at each of several thousand grid points is often of interest. In this case, it is more valuable to plot maps showing the change in a single quantile, such as ŷ_p − x̂_p, at each grid point. Values of p can be chosen to cover different parts of the distribution: for the center, p = 0.25, 0.5, and 0.75 might suffice; for the lower or upper tail, p = 0.01, 0.05, and 0.1 or p = 0.9, 0.95, and 0.99 could be used. All nine of these quantiles will be examined in our application. The 0.1 and 0.9 quantiles correspond to the Watson and Core Writing Team (2001) definition of an extreme event; the rarer quantiles provide more information about the tails. Sample size will dictate how far into the tails quantile estimators are acceptably precise.

If there is no difference between F and G (hypothesis H₀), then from (3) ŷ_p − x̂_p is expected to be approximately zero for each p. If maps of these quantile differences show nonzero values, then a simple explanation could be a change in location. If hypothesis H_L and the corresponding relationship (3) hold, then the estimators

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for the location-adjusted quantile differences are expected to be zero, and maps of these differences are useful for diagnosing a location shift. If significant patterns still remain, then it is possible to look for an additional change in scale. If hypothesis H_LS and the corresponding relationship (3) hold, then the estimators

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for the location- and scale-adjusted quantile differences are expected to be zero, and maps of these differences are useful for diagnosing location and scale shifts. Careful scrutiny of such maps can lead to a good understanding of how distributions differ at each grid point, and how these differences vary geographically.

For illustration, the quantile differences between the control and scenario temperatures at the example grid point are shown in Fig. 2b. The location- and scale-adjusted differences confirm the earlier finding that model H_LS provides a reasonable fit except in the tails. See Beniston and Stephenson (2004) and McGregor et al. (2005) for other applications of this technique.

d. Confidence intervals

Sample estimates d̂_p = ŷ_p − x̂_p differ from the true differences d_p = y_p − x_p due to sampling uncertainty. This can be quantified by calculating confidence intervals for d_p. The interval [L_p, U_p] is a pointwise (1 − α) confidence interval for d_p if

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The probability that the interval contains the true quantile difference is 1 − α. See von Storch and Zwiers (2001) for background on confidence intervals and other aspects of statistical inference.

There are many ways to construct confidence intervals. The approach employed here is based on bootstrap resampling, a popular and effective technique that can be adapted to account for dependence within and between the samples. Dunn (2001), for example, uses the bootstrap to estimate pointwise confidence intervals for rainfall quantiles from a single sample; see also Wilks (1995, 1997). A discussion of bootstrap confidence intervals and their implementation in the current setting is deferred to the appendixes. Pointwise confidence intervals for the quantile differences between the control and scenario temperatures at the example grid point are shown in Fig. 3, where it can be seen that the uncertainty is greater in the relatively long, colder tail.

The hypotheses (1) refer not to single quantiles but to entire distributions: if hypothesis H₀ holds, for example, then d_p = 0 for all p. Suppose that limits L′_p and U ′_p are available for each of M values, p₁, . . . , p_M, of p such that

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If the pointwise limits L_p and U_p satisfying expression (4) are used to define these simultaneous confidence intervals, then the coverage probability 1 − α is unlikely to be obtained in (5). For example, if the different quantiles were independent, then the pointwise limits would give

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Although independence is unrealistic, it remains true that simultaneous intervals are generally wider than pointwise intervals with the same coverage. The method used here to construct simultaneous intervals is described in appendix A.

e. Hypothesis tests

The simultaneous intervals (5) can be used to test H₀: the hypothesis is rejected at significance level α unless L′_p ≤ 0 ≤ U ′_p for all p. Rather than use all p₁ = 1/N, p₂ = 2/N, . . . , p_M = p_N = 1, attention can be restricted to a subset of quantiles, the choice of which involves a trade-off between statistical power and the strength of conclusions. Power reduces as more quantiles are considered because the simultaneous intervals widen, and a real change in distribution is less likely to be detected. On the other hand, a hypothesis test based on only a few quantiles ignores possible changes in other parts of the distribution. Our selected set of nine quantiles (p = 0.01, 0.05, 0.1, 0.25, 0.5, 0.75, 0.9, 0.95, and 0.99) form a compromise, but other choices may be more appropriate in other applications. A small simulation study of the power of tests based on these quantiles is summarized in appendix C.

Simultaneous confidence intervals for the quantile differences between the control and scenario temperatures at the example grid point are shown in Fig. 3. The line d_p = 0 lies outside the shaded simultaneous 90% confidence intervals, which therefore support rejection of H₀ at the 10% level of significance.

One might attempt to test H_L by constructing confidence intervals for y_p − {μ_Y + (x_p − μ_X)}, which is zero for all p if H_L holds, by substituting it for d_p in the previous section. This is successful only if μ_X and μ_Y are specified. For example, choosing μ_X = x_0.5 and μ_Y = y_0.5 produces confidence intervals for y_p − {y_0.5 + (x_p − x_0.5)}. Such intervals, however, do not support a test of H_L because H_L leaves μ_X and μ_Y unspecified: even if the confidence intervals for y_p − {y_0.5 + (x_p − x_0.5)} failed to contain zero for all p, other choices for μ_X and μ_Y might yield a different conclusion. Similar considerations apply to confidence intervals for y_p − σ_Yx_p/σ_X and y_p − {μ_Y + σ_Y(x_p − μ_X)/σ_X}.

Some authors (e.g., Sun et al. 2001) have proposed rejecting H_L if a horizontal line cannot pass completely through the confidence band for d_p, that is, if max_pL′_p > min_pU ′_p. Such a test is conservative, however: if H_L holds, then the band will contain the true value of μ_Y − μ_X with the appropriate probability, but the probability that it contains any constant line is greater, so H_L will be rejected too infrequently. Parametric models can be used to test for specific departures from H_L, H_S, or H_LS, but such tests can be sensitive to model assumptions, as mentioned in section 1.

Our preferred approach for testing hypotheses H₀, H_L, H_S, and H_LS would be first to test H_LS against the general alternative that there are some differences between the distributions that cannot be described by location and scale changes, then H_L or H_S against H_LS if the first test were passed, and finally H₀ against H_L or H_S if the second test were passed. Several nonparametric procedures exist for the latter two tests when there is no serial dependence: see Conover et al. (1981) and Lehmann (1975, p. 95) for example. We have failed to find any published nonparametric tests for H_LS against the general alternative, so we are investigating elsewhere the use of minimized distance measures, such as the Kolmogorov–Smirnov distance and the quantile distance considered by Zhang and Yu (2002), as test statistics.

a. Temperature

The summary statistics (2) are displayed in Fig. 4. In the control, cooler median temperatures are found over northeastern Europe, regions that also exhibit greater variability, and there is widespread negative skewness. The uniform increase in scenario median temperatures is greater in cooler regions, the general decrease in variability is greatest in the continental interior, and skewness moves closer to zero except, most noticeably, for a band in the east that may correspond to snow retreat (Kjellström 2004).

That the probability distribution of scenario temperatures is not everywhere merely a location shift of the control distribution is evident in Fig. 5, where the greater increase in cold quantiles, particularly in the continental interior, is clear. Figure 6, showing the location-adjusted quantile differences, indicates that the distributional changes over much of the seas and western Europe can be described by a shift in location. Figure 7 indicates that additional changes in scale describe some of the remaining changes in northern and eastern Europe, but that some regions exhibit a more complex distributional change.

b. Precipitation

Location shifts may be inappropriate descriptions of changes in distributions of nonnegative variables such as precipitation. A positive location shift would exclude values near zero; a negative shift would admit negative values. Although common measures of location such as the median may still change, such effects are better described by shifts in scale and shape. Distinguishing between wet and dry days, the former denoting days on which the precipitation strictly exceeds an amount u, may also be important.

Let F and G denote the distribution functions for excess wet-day precipitation above u = 1 mm in the control and scenario integrations. Let also x̂_p and ŷ_p be the estimators for the p quantiles of F and G. The scale- and shape-change hypothesis is formulated as

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for all z > 0 and unknown, positive constants σ_X, σ_Y, α_X, and α_Y. This hypothesis corresponds to the nonlinear relationship

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Taking logarithms yields

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where x*_p = logx_p and y*_p = logy_p. This has the same form as the final equality (3) and shows that a scale- shape change of wet-day precipitation excess is equivalent to a location-scale change of log-transformed excess. Adjusted quantile differences can therefore bedefined as in section 2c. For example, if hypothesis H_SS holds, then the estimators

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for the location- and scale-adjusted transformed quantile differences are expected to be zero, where m*_X = x̂*_0.5, s*_X = x̂*_0.75 − x̂*_0.25, and m*_Y and s*_Y are defined similarly. Inverting the transformation yields scale- and shape-adjusted quantile ratios:

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Maps of these scale- and shape-adjusted quantities are useful for diagnosing scale and shape changes. Setting s*_X = s*_Y yields quantities appropriate for diagnosing a pure scale change.

These methods are illustrated by application to winter daily total precipitation from the control and scenario integrations. Summary statistics are reproduced in Fig. 8. In the control, greater median precipitation amounts are found windward of steep altitude gradients, regions that also exhibit greater variability, and there is widespread positive skewness. Scenario median precipitation decreases only in the Mediterranean and over the Scandinavian mountains, a pattern that is replicated for scale, but the spatially complex changes in skewness are difficult to summarize. The change in proportion of wet days is shown in Fig. 9, revealing a decrease in the Mediterranean and in the far north, and an increase in the intervening latitudes that is strongest around the North and Baltic Seas.

The changes in six quantiles are investigated in Fig. 10. (Low quantiles are of less interest so they are excluded.) All quantiles decrease in the Mediterranean and over the Scandinavian mountains. Largest increases are found elsewhere in Scandinavia and in eastern Europe. The scale-adjusted quantile ratios in Fig. 11 indicate that changes in scale of the distributions can explain many of the differences over northern Europe, the Alps, and around the Adriatic. Accounting for additional changes in shape (not shown) explained little of the remaining differences.