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## Abstract

The Madden–Julian oscillation (MJO) is the leading mode of tropical variability on subseasonal time scales and has predictable impacts in the extratropics. Whether or not the MJO has a discernible influence on U.S. tornado occurrence has important implications for the feasibility of extended-range forecasting of tornado activity. Interpretation and comparison of previous studies is difficult because of differing data periods, methods, and tornado activity metrics. Here, a previously described modulation of the frequency of violent tornado outbreaks (days with six or more tornadoes reported rated EF2 or greater) by the MJO is shown to be fairly robust to the addition or removal of years to the analysis period and to changes in the number of tornadoes used to define outbreak days, but is less robust to the choice of MJO index. Earlier findings of a statistically significant MJO signal in the frequency of days with at least one tornado report are shown to be incorrect. The reduction of the frequency of days with tornadoes rated EF1 and greater when MJO convection is present in the Maritime Continent and western Pacific is statistically significant in April and robust across varying thresholds of reliably reported tornado numbers and MJO indices.

## Abstract

The Madden–Julian oscillation (MJO) is the leading mode of tropical variability on subseasonal time scales and has predictable impacts in the extratropics. Whether or not the MJO has a discernible influence on U.S. tornado occurrence has important implications for the feasibility of extended-range forecasting of tornado activity. Interpretation and comparison of previous studies is difficult because of differing data periods, methods, and tornado activity metrics. Here, a previously described modulation of the frequency of violent tornado outbreaks (days with six or more tornadoes reported rated EF2 or greater) by the MJO is shown to be fairly robust to the addition or removal of years to the analysis period and to changes in the number of tornadoes used to define outbreak days, but is less robust to the choice of MJO index. Earlier findings of a statistically significant MJO signal in the frequency of days with at least one tornado report are shown to be incorrect. The reduction of the frequency of days with tornadoes rated EF1 and greater when MJO convection is present in the Maritime Continent and western Pacific is statistically significant in April and robust across varying thresholds of reliably reported tornado numbers and MJO indices.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## Abstract

This paper shows that if a measure of predictability is invariant to affine transformations and monotonically related to forecast uncertainty, then the component that maximizes this measure for normally distributed variables is independent of the detailed form of the measure. This result explains why different measures of predictability such as anomaly correlation, signal-to-noise ratio, predictive information, and the Mahalanobis error are each maximized by the same components. These components can be determined by applying principal component analysis to a transformed forecast ensemble, a procedure called *predictable component analysis* (PrCA). The resulting vectors define a complete set of components that can be ordered such that the first maximizes predictability, the second maximizes predictability subject to being uncorrelated of the first, and so on. The transformation in question, called the whitening transformation, can be interpreted as changing the norm in principal component analysis. The resulting norm renders noise variance analysis equivalent to signal variance analysis, whereas these two analyses lead to inconsistent results if other norms are chosen to define variance. Predictable components also can be determined by applying singular value decomposition to a whitened propagator in linear models. The whitening transformation is tantamount to changing the initial and final norms in the singular vector calculation. The norm for measuring forecast uncertainty has not appeared in prior predictability studies. Nevertheless, the norms that emerge from this framework have several attractive properties that make their use compelling. This framework generalizes singular vector methods to models with both stochastic forcing and initial condition error. These and other components of interest to predictability are illustrated with an empirical model for sea surface temperature.

## Abstract

This paper shows that if a measure of predictability is invariant to affine transformations and monotonically related to forecast uncertainty, then the component that maximizes this measure for normally distributed variables is independent of the detailed form of the measure. This result explains why different measures of predictability such as anomaly correlation, signal-to-noise ratio, predictive information, and the Mahalanobis error are each maximized by the same components. These components can be determined by applying principal component analysis to a transformed forecast ensemble, a procedure called *predictable component analysis* (PrCA). The resulting vectors define a complete set of components that can be ordered such that the first maximizes predictability, the second maximizes predictability subject to being uncorrelated of the first, and so on. The transformation in question, called the whitening transformation, can be interpreted as changing the norm in principal component analysis. The resulting norm renders noise variance analysis equivalent to signal variance analysis, whereas these two analyses lead to inconsistent results if other norms are chosen to define variance. Predictable components also can be determined by applying singular value decomposition to a whitened propagator in linear models. The whitening transformation is tantamount to changing the initial and final norms in the singular vector calculation. The norm for measuring forecast uncertainty has not appeared in prior predictability studies. Nevertheless, the norms that emerge from this framework have several attractive properties that make their use compelling. This framework generalizes singular vector methods to models with both stochastic forcing and initial condition error. These and other components of interest to predictability are illustrated with an empirical model for sea surface temperature.

## 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.

## 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.

## Abstract

This paper introduces the average predictability time (APT) for characterizing the overall predictability of a system. APT is the integral of a predictability measure over all lead times. The underlying predictability measure is based on the Mahalanobis metric, which is invariant to linear transformation of the prediction variables and hence gives results that are independent of the (arbitrary) basis set used to represent the state. The APT is superior to some integral time scales used to characterize the time scale of a random process because the latter vanishes in situations when it should not, whereas the APT converges to reasonable values. The APT also can be written in terms of the power spectrum, thereby clarifying the connection between predictability and the power spectrum. In essence, predictability is related to the width of spectral peaks, with strong, narrow peaks associated with high predictability and nearly flat spectra associated with low predictability. Closed form expressions for the APT for linear stochastic models are derived. For a given dynamical operator, the stochastic forcing that minimizes APT is one that allows transformation of the original stochastic model into a set of uncoupled, independent stochastic models. Loosely speaking, coupling enhances predictability. A rigorous upper bound on the predictability of linear stochastic models is derived, which clarifies the connection between predictability at short and long lead times, as well as the choice of norm for measuring error growth. Surprisingly, APT can itself be interpreted as the “total variance” of an alternative stochastic model, which means that generalized stability theory and dynamical systems theory can be used to understand APT. The APT can be decomposed into an uncorrelated set of components that maximize predictability time, analogous to the way principle component analysis decomposes variance. Part II of this paper develops a practical method for performing this decomposition and applies it to meteorological data.

## Abstract

This paper introduces the average predictability time (APT) for characterizing the overall predictability of a system. APT is the integral of a predictability measure over all lead times. The underlying predictability measure is based on the Mahalanobis metric, which is invariant to linear transformation of the prediction variables and hence gives results that are independent of the (arbitrary) basis set used to represent the state. The APT is superior to some integral time scales used to characterize the time scale of a random process because the latter vanishes in situations when it should not, whereas the APT converges to reasonable values. The APT also can be written in terms of the power spectrum, thereby clarifying the connection between predictability and the power spectrum. In essence, predictability is related to the width of spectral peaks, with strong, narrow peaks associated with high predictability and nearly flat spectra associated with low predictability. Closed form expressions for the APT for linear stochastic models are derived. For a given dynamical operator, the stochastic forcing that minimizes APT is one that allows transformation of the original stochastic model into a set of uncoupled, independent stochastic models. Loosely speaking, coupling enhances predictability. A rigorous upper bound on the predictability of linear stochastic models is derived, which clarifies the connection between predictability at short and long lead times, as well as the choice of norm for measuring error growth. Surprisingly, APT can itself be interpreted as the “total variance” of an alternative stochastic model, which means that generalized stability theory and dynamical systems theory can be used to understand APT. The APT can be decomposed into an uncorrelated set of components that maximize predictability time, analogous to the way principle component analysis decomposes variance. Part II of this paper develops a practical method for performing this decomposition and applies it to meteorological data.

## 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.

## 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.