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Timothy DelSole
and
Michael K. Tippett

Abstract

This paper proposes a new method for representing data in a general domain on a sphere. The method is based on the eigenfunctions of the Laplace operator, which form an orthogonal basis set that can be ordered by a measure of length scale. Representing data with Laplacian eigenfunctions is attractive if one wants to reduce the dimension of a dataset by filtering out small-scale variability. Although Laplacian eigenfunctions are ubiquitous in climate modeling, their use in arbitrary domains, such as over continents, is not common because of the numerical difficulties associated with irregular boundaries. Recent advances in machine learning and computational sciences are exploited to derive eigenfunctions of the Laplace operator over an arbitrary domain on a sphere. The eigenfunctions depend only on the geometry of the domain and hence require no training data from models or observations, a feature that is especially useful in small sample sizes. Another novel feature is that the method produces reasonable eigenfunctions even if the domain is disconnected, such as a land domain comprising isolated continents and islands. The eigenfunctions are illustrated by quantifying variability of monthly mean temperature and precipitation in climate models and observations. This analysis extends previous studies by showing that climate models have significant biases not only in global-scale spatial averages but also in global-scale dipoles and other physically important structures. MATLAB and R codes for deriving Laplacian eigenfunctions are available upon request.

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Timothy Hall
and
Michael K. Tippett

Abstract

A statistical model of northeastern Pacific Ocean tropical cyclones (TCs) is developed and used to estimate hurricane landfall rates along the coast of Mexico. Mean annual landfall rates for 1971–2014 are compared with mean rates for the extremely high northeastern Pacific sea surface temperature (SST) of 2015. Over the full coast, the mean rate and 5%–95% uncertainty range (in parentheses) for TCs that are category 1 and higher on the Saffir–Simpson scale (C1+ TCs) are 1.24 (1.05, 1.33) yr−1 for 1971–2014 and 1.69 (0.89, 2.08) yr−1 for 2015—a difference that is not significant. The increase for the most intense landfalls (category-5 TCs) is significant: 0.009 (0.006, 0.011) yr−1 for 1971–2014 and 0.031 (0.016, 0.036) yr−1 for 2015. The SST impact on the rate of category-5 TC landfalls is largest on the northern Mexican coast. The increased landfall rates for category-5 TCs are consistent with independent analysis showing that SST has its greatest impact on the formation rates of the most intense northeastern Pacific TCs. Landfall rates on Hawaii [0.033 (0.019, 0.045) yr−1 for C1+ TCs and 0.010 (0.005, 0.016) yr−1 for C3+ TCs for 1971–2014] show increases in the best estimates for 2015 conditions, but the changes are statistically insignificant.

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Michael K. Tippett
and
Alessandra Giannini

Abstract

An ensemble of general circulation model (GCM) integrations forced by observed sea surface temperature (SST) represents the climate response to SST forcing as well as internal variability or “noise.” Signal-to-noise analysis is used to identify the most reproducible GCM patterns of African summer precipitation related to the SST forcing. Two of these potentially predictable components are associated with the precipitation of the Guinea Coast and Sahel regions and correlate well with observations. The GCM predictable component associated with rainfall in the Sahel region reproduces observed temporal variability on both interannual and decadal time scales, though with reduced amplitude.

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Timothy DelSole
and
Michael K. Tippett

Abstract

A basic question in forecasting is whether one prediction system is more skillful than another. Some commonly used statistical significance tests cannot answer this question correctly if the skills are computed on a common period or using a common set of observations, because these tests do not account for correlations between sample skill estimates. Furthermore, the results of these tests are biased toward indicating no difference in skill, a fact that has important consequences for forecast improvement. This paper shows that the magnitude of bias is characterized by a few parameters such as sample size and correlation between forecasts and their errors, which, surprisingly, can be estimated from data. The bias is substantial for typical seasonal forecasts, implying that familiar tests may wrongly judge that differences in seasonal forecast skill are insignificant. Four tests that are appropriate for assessing differences in skill over a common period are reviewed. These tests are based on the sign test, the Wilcoxon signed-rank test, the Morgan–Granger–Newbold test, and a permutation test. These techniques are applied to ENSO hindcasts from the North American Multimodel Ensemble and reveal that the Climate Forecast System, version 2, and the Canadian Climate Model, version 3 (CanCM3), outperform other models in the sense that their squared error is less than that of other single models more frequently. It should be recognized that while certain models may be superior in a certain sense for a particular period and variable, combinations of forecasts are often significantly more skillful than a single model alone. In fact, the multimodel mean significantly outperforms all single models.

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Michael K. Tippett
and
Timothy DelSole

Abstract

The constructed analog procedure produces a statistical forecast that is a linear combination of past predictand values. The weights used to form the linear combination depend on the current predictor value and are chosen so that the linear combination of past predictor values approximates the current predictor value. The properties of the constructed analog method have previously been described as being distinct from those of linear regression. However, here the authors show that standard implementations of the constructed analog method give forecasts that are identical to linear regression forecasts. A consequence of this equivalence is that constructed analog forecasts based on many predictors tend to suffer from overfitting just as in linear regression. Differences between linear regression and constructed analog forecasts only result from implementation choices, especially ones related to the preparation and truncation of data. Two particular constructed analog implementations are shown to correspond to principal component regression and ridge regression. The equality of linear regression and constructed analog forecasts is illustrated in a Niño-3.4 prediction example, which also shows that increasing the number of predictors results in low-skill, high-variance forecasts, even at long leads, behavior typical of overfitting. Alternative definitions of the analog weights lead naturally to nonlinear extensions of linear regression such as local linear regression.

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Timothy DelSole
and
Michael K. Tippett

Abstract

This paper proposes a procedure based on random walks for testing and visualizing differences in forecast skill. The test is formally equivalent to the sign test and has numerous attractive statistical properties, including being independent of distributional assumptions about the forecast errors and being applicable to a wide class of measures of forecast quality. While the test is best suited for independent outcomes, it provides useful information even when serial correlation exists. The procedure is applied to deterministic ENSO forecasts from the North American Multimodel Ensemble and yields several revealing results, including 1) the Canadian models are the most skillful dynamical models, even when compared to the multimodel mean; 2) a regression model is significantly more skillful than all but one dynamical model (to which it is equally skillful); and 3) in some cases, there are significant differences in skill between ensemble members from the same model, potentially reflecting differences in initialization. The method requires only a few years of data to detect significant differences in the skill of models with known errors/biases, suggesting that the procedure may be useful for model development and monitoring of real-time forecasts.

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Chiara Lepore
and
Michael K. Tippett

Abstract

The scaling of U.S. tornado frequency with enhanced Fujita (EF)-rated intensity is examined for the range EF1–EF3. Previous work has found that tornado frequency decreases exponentially with increasing EF rating and that many regions around the world show the same exponential rate of decrease despite having quite different overall tornado frequencies. This scaling is important because it relates the frequency of the most intense tornadoes to the overall tornado frequency. Here we find that U.S. tornado frequency decreases more sharply with increasing intensity during summer than during other times of the year. One implication of this finding is that, despite their rarity, when tornadoes do occur during the cool season, the relative likelihood of more intense tornadoes is higher than during summer. The environmental driver of this scaling variability is explored through new EF-dependent tornado environmental indices (TEI-EF) that are fitted to each EF class. We find that the sensitivity of TEI-EF to storm relative helicity (SRH) increases with increasing EF class. This increasing sensitivity to SRH means that TEI-EF predicts a slower decrease in frequency with increasing intensity for larger values of SRH (e.g., cool season) and a sharper decrease in tornado frequency in summer when wind shear plays a less dominant role. This explanation is also consistent with the fact that the fraction of supercell tornadoes is smaller during summer.

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Balachandrudu Narapusetty
,
Timothy DelSole
, and
Michael K. Tippett

Abstract

This paper shows theoretically and with examples that climatological means derived from spectral methods predict independent data with less error than climatological means derived from simple averaging. Herein, “spectral methods” indicates a least squares fit to a sum of a small number of sines and cosines that are periodic on annual or diurnal periods, and “simple averaging” refers to mean averages computed while holding the phase of the annual or diurnal cycle constant. The fact that spectral methods are superior to simple averaging can be understood as a straightforward consequence of overfitting, provided that one recognizes that simple averaging is a special case of the spectral method. To illustrate these results, the two methods are compared in the context of estimating the climatological mean of sea surface temperature (SST). Cross-validation experiments indicate that about four harmonics of the annual cycle are adequate, which requires estimation of nine independent parameters. In contrast, simple averaging of daily SST requires estimation of 366 parameters—one for each day of the year, which is a factor of 40 more parameters. Consistent with the greater number of parameters, simple averaging poorly predicts samples that were not included in the estimation of the climatological mean, compared to the spectral method. In addition to being more accurate, the spectral method also accommodates leap years and missing data simply, results in a greater degree of data compression, and automatically produces smooth time series.

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Li Zhang
,
Ping Chang
, and
Michael K. Tippett

Abstract

A novel noise filter is used to effectively reduce internal atmospheric variability in the air–sea fluxes of a coupled model. This procedure allows for a test of the impact of the internal atmospheric variability on ENSO through its effect on the Pacific meridional mode (MM). Three 100-yr coupled experiments are conducted, where the filter is utilized to suppress internal atmospheric variability in 1) both the surface wind stress and the heat flux (fully filtered run), 2) only the surface heat flux (filtered-flux run), and 3) only the surface wind stress (filtered-wind run). The fully filtered run indicates that suppressing internal atmospheric variability weakens the MM, which in turn results in substantially reduced ENSO variability. ENSO is no longer phase locked to the boreal winter. The filtered-flux and filtered-wind experiments reveal that different types of noise affect ENSO in different ways. The noise in the wind stress does not have a significant impact on the MM and its relationship to ENSO. This type of noise, however, tends to broaden the spectral peak of ENSO while shifting it toward lower frequencies. The noise in the heat flux, on the other hand, has a direct impact on the strength of the MM and consequently its ability to influence ENSO. Reducing the effect of heat flux noise yields substantially weakened MM activity and a weakened relationship to ENSO, which leads to altered seasonal phase-locking characteristics.

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Liwei Jia
,
Timothy DelSole
, and
Michael K. Tippett

Abstract

An optimal projection for improving the skill of dynamical model forecasts is proposed. The proposed method uses statistical optimization techniques to identify the most skillful or most predictable patterns, and then projects forecasts onto these patterns. Applying the method to seasonal mean 2-m temperature from the Ensemble-Based Predictions of Climate Changes and Their Impacts (ENSEMBLES) multimodel hindcast dataset reveals that the method improves skill only in South America and Africa, suggesting that the benefit of optimal projection is limited to certain regions, but can be substantial. Further investigation reveals that the improvement in skill comes not from optimal projection itself, but from the EOF prefiltering that is done to reduce the dimension of the optimization space. Thus, much of the improvement attributable to optimal projection can be achieved by suitable EOF filtering. Interestingly, models are found to generate patterns that project only weakly on observational datasets but are strongly correlated between models. An important by-product of the method is a concise summary of the skillful or predictable structures in a given forecast. For the ENSEMBLES dataset, the method convincingly demonstrates that most of the seasonal prediction skill over continents comes from two components, ENSO and the global warming trend. In addition, the method can be used to determine whether a pattern exists that is well predicted by one model but not by another model (complementary skill).

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