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Maximum likelihood estimation in constrained parameter spaces for mixtures of factor analyzers

# Maximum likelihood estimation in constrained parameter spaces for mixtures of factor analyzers

Francesca Greselin Francesca Greselin Department of Statistics and Quantitative Methods
Milano-Bicocca University
Via Bicocca degliArcimboldi 8 - 20126 Milano (Italy). 22email: francesca.greselin@unimib.itSalvatore IngrassiaDepartment of Economics and Business
University of Catania
Corso Italia 55, - Catania (Italy). 44email: s.ingrassia@unict.it
Salvatore Ingrassia Francesca Greselin Department of Statistics and Quantitative Methods
Milano-Bicocca University
Via Bicocca degliArcimboldi 8 - 20126 Milano (Italy). 22email: francesca.greselin@unimib.itSalvatore IngrassiaDepartment of Economics and Business
University of Catania
Corso Italia 55, - Catania (Italy). 44email: s.ingrassia@unict.it
###### Abstract

Mixtures of factor analyzers are becoming more and more popular in the area of model based clustering of high-dimensional data. According to the likelihood approach in data modeling, it is well known that the unconstrained log-likelihood function may present spurious maxima and singularities and this is due to specific patterns of the estimated covariance structure, when their determinant approaches 0. To reduce such drawbacks, in this paper we introduce a procedure for the parameter estimation of mixtures of factor analyzers, which maximizes the likelihood function in a constrained parameter space. We then analyze and measure its performance, compared to the usual non-constrained approach, via some simulations and applications to real data sets.

###### Keywords:
Constrained estimation Factor Analyzers Modeling Mixture Models Model-Based Clustering.

## 1 Introduction and motivation

Finite mixture distributions have been receiving a growing interest in statistical modeling. Their central role is mainly due to their double nature: they combine the flexibility of non-parametric models with the strong and useful mathematical properties of parametric models. According to this approach, when we know that a sample of observations has been drawn from different populations, we assume a specific distributional form in each of the underlying populations. The purpose is to decompose the sample into its mixture components, which, for quantitative data, are usually modeled as a multivariate Gaussian distribution, and to estimate parameters. The assumption of underlying normality, besides the elegant analytic properties, allows also to employ the EM algorithm for the ML estimation of the parameters. On the other side, when considering a large number of observed variables, Gaussian mixture models can provide an over-parameterized solution as, besides the mixing weights, it is required to estimate the mean vector and the covariance matrix for each component (Peel and McLachlan, 2000). As a consequence, we observe at the same time an undue load of computationally intensive procedures for the estimation.

This is the reason why a number of strategies have been introduced in the literature to avoid over-parameterized solutions. Among the various proposal, some authors developed methodologies for variable selection (see, f.i., Liu et al. (2003) and Hoff (2005) in the Bayesian framework, Pan and Shen (2007) and Raftery and Dean (2006) in the frequentist one). They further motivate their approach from the observation that the presence of non-informative variables can be strongly misleading for some clustering methods. With the same purpose of parsimony, but a completely different approach, Banfield and Raftery (1993) devised a methodology to identify common patterns among the component-covariance matrices; their proposal arose a great attention in the literature. Along a slightly different line of thinking, Ghahramani and Hilton (1997) and McLachlan et al. (2003) proposed to employ latent variables to perform dimensional reduction in each component, starting from the consideration that in many phenomena some few unobserved features could be explained by the many observed ones.

In this paper we address mixtures of factor analyzers by assuming that the data have been generated by a linear factor model with latent variables modeled as Gaussian mixtures. Our purpose is to improve the performances of the EM algorithm, by facing with some of its issues and giving practical recipes to overcome them. It is well known that the EM algorithm generates a sequence of estimates, starting from an initial guess, so that the corresponding sequence of the log-likelihood values is not decreasing. However, the convergence toward the MLE is not guaranteed, because the log-likelihood is unbounded and presents local maxima. Another critical point is that the parameter estimates as well as the convergence of the whole estimation process may be affected by the starting values (see, f.i., McLachlan and Krishnan (2007)) so that the final estimate crucially depends on the initial guess. This issue has been investigated by many authors, starting from the seminal paper of Redner and Walker (1984). Along the lines of (Ingrassia, 2004), in this paper we introduce and implement a procedure for the parameters estimation of mixtures of factor analyzers, which maximizes the likelihood function in a constrained parameter space, having no singularities and a reduced number of spurious local maxima. We then analyze and measure its performance, compared to the usual non-constrained approach.

We have organized the rest of the paper as follows. In Section 2 we summarize main ideas about Gaussian Mixtures of Factor Analyzer model; in Section 3 we provide fairly extensive notes concerning the likelihood function and the AECM algorithm. Some well known considerations (Hathaway, 1985) related to spurious maximizers and singularities in the EM algorithm are recalled in Section 4, and motivate our proposal to introduce constraints on factor analyzers. Further, we give a detailed methodology to implement such constraints into the EM algorithm. In Section 6 we show and discuss the improved performance of our procedure, on the ground of some numerical results based on both simulated and real data. Section 7 contains concluding notes and provides ideas for future research.

## 2 The Gaussian Mixture of Factor analyzers

Within the Gaussian Mixture (GM) model-based approach to density estimation and clustering, the density of the -dimensional random variable of interest is modelled as a mixture of a number, say , of multivariate normal densities in some unknown proportions . That is, each data point is taken to be a realization of the mixture probability density function,

 f(x;\boldmathθ)=G∑g=1πgϕd(x;\boldmathμg,\boldmathΣg) (1)

where denotes the -variate normal density function with mean and covariance matrix . Here the vector of unknown parameters consists of the mixing proportions , the elements of the component means , and the distinct elements of the component-covariance matrices . Therefore, the -component normal mixture model (1) with unrestricted component-covariance matrices is a highly parametrized model. We crucially need some method for parsimonious parametrization of the matrices , because they requires parameters. Among the various proposals for dimensionality reduction, we are interested here in considering Mixtures of Gaussian Factor Analyzers (MGFA), which allows to explain data by explicitly modeling correlations between variables in multivariate observations. We postulate a finite mixture of linear sub-models for the distribution of the full observation vector , given the (unobservable) factors . That is we can provide a local dimensionality reduction method by assuming that the distribution of the observation can be given as

 Xi=\boldmathμg+\boldmathΛgUig+eigwith probability πg(g=1,…,G)fori=1,…,n, (2)

where is a matrix of factor loadings, the factors are distributed independently of the errors , which are independently distributed, and is a diagonal matrix . We suppose that , which means that unobservable factors are jointly explaining the observable features of the statistical units. Under these assumptions, the mixture of factor analyzers model is given by (1), where the -th component-covariance matrix has the form

 \boldmathΣg=\boldmathΛg\boldmathΛ′g+\boldmathΨg(g=1,…,G). (3)

The parameter vector now consists of the elements of the component means , the , and the , along with the mixing proportions , on putting . Note that in the case of , there is an infinity of choices for , since model (2) is still satisfied if we replace by , where is any orthogonal matrix of order . As constraints are needed for to be uniquely defined, the number of free parameters, for each component of the mixture, is

 dq+d−12q(q−1).

Comparing the two approaches and willing now to measure the gained parsimony when we use mixtures of factor analyzers, with respect to the more usual gaussian mixtures, and denoting by and , the number of the estimated parameters for the covariance matrices in the GM and MGFA models, respectively, we have to choose values of such that the following quantity is positive

 P=|\boldmathθCovGM(d,G)|−|\boldmathθCovMGFA(d,q,G)|=G2d(d+1)−G[dq−d+12q(q−1)]

i.e.:

 P=G2[(d−q)2−(d+q)].

This is the only requirement for parsimony. Now, we can express the relative reduction given by

 RR(d,q) =|\boldmathθCovGM(d,G)|−|\boldmathθCovGMFA(d,q,G)||\boldmathθCovGM(d,G)|=(d−q)2−(d+q)d(d+1).

In Table 1 we report the relative reduction, in term of lower number of estimated parameters for the covariance matrices in the MGFA models, with respect to the GM models.

The relative reduction represents the extent to which the factor model offers a simpler interpretation for the behaviour of than the alternative assumption given by the gaussian mixture model.

## 3 The likelihood function and the EM algorithm for MGFA

In this section we summarize the main steps of the EM algorithm for mixtures of Factor analyzers, see e.g. McLachlan and Peel (2000) for details.

Let be a sample of size from density (1), and let () denotes the realization of in (2). For given data , parameters in (1) can be estimated according to the likelihood approach via the EM algorithm, where the likelihood function is given by:

 L(\boldmathθ;X∼) =n∏i=1{G∑g=1ϕd(xi;\boldmathμg,\boldmathΣg)πg}=n∏i=1{G∑g=1ϕd(xi;\boldmathμg,\boldmathΛg,\boldmathΨg)πg},

where we set (). Consider the augmented data , where , with if comes from the -th population and otherwise. Then, the complete-data likelihood function can be written in the form:

 Lc(\boldmathθ;X∼)=n∏i=1G∏g=1[ϕd(xi|ui;% \boldmathμg,Λg,Ψg)ϕq(uig)πg]zig. (4)

In particular, due to the factor structure of the model, see Meng and van Dyk (1997), we have to consider the alternating expectation-conditional maximization (AECM) algorithm. Such a procedure is an extension of the EM algorithm that uses different specifications of missing data at each stage. The idea is to partition in such a way that is easy to maximize for given and vice versa. Then, we can iterate between these two conditional maximizations until convergence. In this case where the missing data are the unobserved group labels , and the second part of the parameters vector is given by where the missing data are the group labels and the unobserved latent factors . Hence, the application of the AECM algorithm consists of two cycles, and there is one E-step and one CM-step alternatively considering and in each pair of cycles.

#### First Cycle.

Here it is where the missing data are the unobserved group labels . The complete data likelihood is

 Lc1(\boldmathθ1) =n∏i=1G∏g=1[ϕd(xi;\boldmathμg,\boldmathΣg)πg]zig. (5)

The E-step on the first cycle on the -th iteration requires the calculation of which is the expected complete-data log-likelihood given the data and using the current estimate for . In practice it requires calculating and usual computations show that this step is achieved by replacing each by its current conditional expectation given the observed data , that is we replace by , where

 (6)

On the M-step, the maximization of this complete-data log-likelihood yields

 π(k+1)g =∑ni=1z(k+1)ign \boldmathμ(k+1)g =1ngn∑i=1z(k+1)igxi

where . According to notation in McLachlan and Peel (2000), we set .

#### Second Cycle.

Here it is , where the missing data are the unobserved group labels and the latent factors . Therefore, the complete data likelihood is

 Lc2(\boldmathθ2) =n∏i=1G∏g=1[ϕd(xi|uig;\boldmathμ(k+1)g,\boldmathΣg)ϕq(uig)π(k+1)g]zig =n∏i=1G∏g=1[ϕd(xi|uig;\boldmathμ(k+1)g,\boldmathΛg,\boldmathΨg)ϕq(uig)π(k+1)g]zig, (7)

where

 ϕd(xi|uig;\boldmathμ% (k+1)g,\boldmathΛg,\boldmathΨg) =1|2π\boldmathΨg|1/2exp{−12(xi−\boldmathμ(k+1)g−\boldmathΛguig)′\boldmathΨ−1g(xi−\boldmathμ(k+1)g−\boldmathΛguig)}. ϕq(uig) =1(2π)q/2exp{−12u′iguig}.

Now the complete data log-likelihood is given by

 Lc2(\boldmathθ2) =−nd2ln2π+G∑g=1nglnπg+12n∑i=1G∑g=1zigln|\boldmathΨ−1g| −12n∑i=1G∑g=1zigtr{(xi−\boldmathμ(k+1)g−\boldmathΛguig)(xi−\boldmathμ(k+1)g−\boldmathΛguig)′\boldmathΨ−1g}. (8)

Some algebras lead to the following estimate of , :

 ^\boldmathΛg =S(k+1)g\boldmathγ(k)′g[\boldmathΘ(k)g]−1 ^\boldmathΨg =diag{S(k+1)g−^\boldmathΛg\boldmathγ(k)gS(k+1)g}.

where we set

 S(k+1)g =(1/n(k+1)g)n∑i=1z(k+1)ig(xi−\boldmathμ(k+1)g)(xi−\boldmathμ(k+1)g)′ \boldmathγ(k)g =\boldmathΛ(k)′g(\boldmathΛ(k)g\boldmathΛ(k)′g+% \boldmathΨ(k)g)−1 \boldmathΘ(k)ig =Iq−\boldmathγ(k)g\boldmath% Λ(k)g+\boldmathγ(k)g(xi−% \boldmathμg)(xi−\boldmathμg)′% \boldmathγ(k)′g.

Hence the maximum likelihood estimates and for and can be obtained by alternatively computing the update estimates and , by

 \boldmathΛ+g =S(k+1)g\boldmathγ(k)′g[\boldmathΘ(k)g]−1and% \boldmathΨ+g=diag{S(k+1)g−% \boldmathΛ+g\boldmathγ(k)gS(k+1)g}, (9)

and, from the latter, evaluating the update estimates and by

 \boldmathγ+g=\boldmathΛ′g(\boldmathΛg\boldmathΛ′g+\boldmathΨg)−1and\boldmath% Θ+g=Iq−\boldmathγg\boldmathΛg+\boldmathγgS(k+1)g% \boldmathγ′g, (10)

iterating these two steps until convergence on and , so giving and .

In summary, the procedure can be described as follows. For a given initial random clustering , on the iteration, the algorithm carries out the following steps, for :

1. Compute and consequently obtain , , and ;

2. Set a starting value for and from ;

3. Repeat the following steps, until convergence on and :

1. Compute and from (10);

2. Set and ;

3. Compute and ;

4. Set and ;

To completely describe the algorithm, here we give more details on how to specify the starting values for and from , as it is needed in Step 2.

Starting from the eigen-decomposition of , say , computed on the base of , the main idea is that has to synthesize the ”more important” relations between the observed features, see McNicholas and Murphy (2008). Then, looking at the equality , the initial values of were set as

 λij=√djaij (11)

where is the th largest eigenvalue of and is the th element of the corresponding eigenvector (the th column in ), for and . Finally the matrices can be initialized by the position .

## 4 Likelihood maximization in constrained parametric spaces

Properties of maximum likelihood estimation for normal mixture models have been deeply investigated. It is well known that is unbounded on and may present many local maxima. Day (1969) was perhaps the first noting that any small number of sample points, grouped sufficiently close together, can give raise to spurious maximizers, corresponding to parameters points with greatly differing component standard deviation. To overcome this issue and to prevent from singularities, Hathaway (1985) proposed a constrained maximum likelihood formulation for mixtures of univariate normal distributions, suggesting a natural extension to the multivariate case. Let , then the following constraints

 min1≤h≠j≤kλ(\boldmathΣh\boldmathΣ−1j)≥c (12)

on the eigenvalues of leads to properly defined, scale-equivariant, consistent ML-estimators for the mixture-of-normal case, see Hennig (2004). It is easy to show that a sufficient condition for (12) is

 a≤λig≤b,i=1,…,d;g=1,…,G (13)

where denotes the th eigenvalue of i.e. , and for such that , see Ingrassia (2004). Differently from (12), condition (13) can be easily implemented in any optimization algorithm. Let us consider the constrained parameter space of :

 \boldmathΘc= {(π1,…,πG,\boldmathμ1,…,% \boldmathμG,\boldmathΣ1,…,\boldmathΣG)∈Rk[1+d+(d2+d)/2]: πg≥0,π1+⋯+πG=1,a≤λig≤b,g=1,…,Gi=1,…,d}. (14)

Due to the structure of the covariance matrix given in (3), bound in (13) yields

 λmin(\boldmathΛg\boldmathΛ′g+\boldmathΨg)≥aandλmax(\boldmathΛg\boldmathΛ′g+\boldmathΨg)≤b,g=1,…,G (15)

where and denote the smallest and the largest eigenvalue of respectively. Since and are symmetric and positive definite, then it results:

 λmin(\boldmathΛg\boldmathΛ′g+\boldmathΨg) ≥λmin(\boldmathΛg% \boldmathΛ′g)+λmin(\boldmathΨg)≥a (16) λmax(\boldmathΛg\boldmathΛ′g+\boldmathΨg) ≤λmax(\boldmathΛg% \boldmathΛ′g)+λmax(\boldmathΨg)≤b, (17)

see Lütkepohl (1996). Moreover, being a diagonal matrix, then

 λmin(\boldmathΨg) =miniψig andλmax(\boldmathΨg) =maxiψig, (18)

where denotes the -th diagonal entry of the matrix .

Concerning the square matrix (), we can get its eigenvalue decomposition, i.e. we can find and such that

 \boldmathΛg\boldmathΛ′g=% \boldmathΓg\boldmathΔg\boldmathΓ′g (19)

where is the orthonormal matrix whose rows are the eigenvectors of and is the diagonal matrix of the eigenvalues of , sorted in non increasing order, i.e. , and .

Now, we can apply the singular value decomposition to the rectangular matrix , so giving , where is a unitary matrix (i.e., such that ) and is a rectangular diagonal matrix with nonnegative real numbers on the diagonal, known as singular values, and is a unitary matrix. The columns of and the columns of are called the left singular vectors and right singular vectors of , respectively. Now we have that

 \boldmathΛg\boldmathΛ′g=(UgDgV′g)(VgD′gU′g)=UgDgIqD′gU′g=UgDgD′gU′g (20)

and equating (19) and (20) we get and , that is

 diag(δ1g,…,δqg)=diag(d21g,…,d2qg). (21)

with . In particular, it is known that only the first values of are non negative, and the remaining terms are null. Thus it results

 λmax(\boldmathΛg\boldmathΛ′g)=d21g. (22)

Supposing now to choose a value for the upper bound in such a way that for and , then constraints (16) and (17) are satisfied when

 d2ig+ψig ≥a i=1,…,d (23) dig ≤√b−\boldmathΨig i=1,…,q (24) ψig ≤b i=q+1,…,d (25)

for . In particular, we remark that condition (23) reduces to for .

## 5 Constraints on the covariance matrix for factor analyzers

The two-fold (eigenvalue and singular value) decomposition of the presented above, suggests how to modify the EM algorithm in such a way that the eigenvalues of the covariances (for ) are confined into suitable ranges. To this aim we have to implement constraints (23), (24) and (25).

We proceed as follows on the th iteration:

1. Decompose according to the singular value decomposition as ;

2. Compute the squared singular values of ;

3. Create a copy of and a copy of ;

4. For to , if , then if set else into ;

5. For to , if then set into ;

6. For to , if , then if set into else into ;

7. For to , if then set into ;

8. Set ;

9. Set .

10. Stop.

It is important to remark that the resulting EM algorithm is monotone, once the initial guess, say , satisfies the constraints. Further, as shown in the case of gaussian mixtures in Ingrassia and Rocci (2007), the maximization of the complete loglikelihood is guaranteed. From the other side, it is apparent that the above recipes require some a priori information on the covariance structure of the mixture, throughout the bounds and .

## 6 Numerical studies

In this section we present numerical studies, based on both simulated and real data sets, in order to show the performance of the constrained EM algorithm with respect to unconstrained approaches.

### 6.1 Artificial data

We consider here three mixtures of components of -variate normal distributions, for different values of the parameter . First, we point out that the point of local maximum corresponding to the consistent estimator , has been chosen to be the limit of the EM algorithm using the true parameter as initial estimate, i.e. considering the true classification. In other words, we set if the th unit comes from the th component and otherwise. In the following, such estimate will be referred to as the right maximum of the likelihood function.

To begin with, we generate a set of 100 different random initial clusterings to initialize the algorithm at each run. To this aim, for a fixed number of components of the mixture, we draw each time a set of random starting values for the from the multinomial distribution with values in with parameters . Then we run a hundred times both the unconstrained and the constrained AECM algorithms (for different values of the constraints ) using the same set of initial clusterings in both cases. The initial values for the elements of and can be obtained as described at the end of Section 3 from the eigen-decomposition of , and the algorithms run until convergence or it reaches the fixed maximum number of iterations.

The stopping criterion is based on the Aitken acceleration procedure (Aitken, 1926), to estimate the asymptotic maximum of the log-likelihood at each iteration of the EM algorithm (in such a way, a decision can be made regarding whether or not the algorithm reaches convergence; that is, whether or not the log-likelihood is sufficiently close to its estimated asymptotic value). The Aitken acceleration at iteration is given by

 a(k)=L(k+1)−L(k)L(k)−L(k−1),

where , , and are the log-likelihood values from iterations , , and , respectively. Then, the asymptotic estimate of the log-likelihood at iteration is given by

 L(k+1)∞=L(k)+11−a(k)(L(k+1)−L(k)),

see Böhning et al. (1994). In our analyses, the algorithms stop when , with . Programs have been written in the R language; the different cases and the obtained results are described below.

#### Mixture 1: G=3, d=6, q=2, N=150.

The sample has been generated with weights according to the following parameters:

 \boldmathμ1 =(0,0,0,0,0,0)′ \boldmathΨ1 =diag(0.1,0.1,0.1,0.1,0.1,0.1) \boldmathμ2 =(5,5,5,5,5,5)′ \boldmathΨ2 =diag(0.4,0.4,0.4,0.4,0.4,0.4) \boldmathμ3 =(10,10,10,10,10,10)′ \boldmathΨ3 =diag(0.2,0.2,0.2,0.2,0.2,0.2)

 \boldmathΛ1=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝0.501.001.000.450.05−0.50−0.600.500.500.101.00−0.15⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠\boldmathΛ2=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝0.100.200.200.501.00−1.00−0.200.501.000.701.20−0.30⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠\boldmathΛ3=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝0.100.200.200.001.000.00−0.200.001.000.000.00−1.30⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠.

Hence, the covariance matrices () have the following eigenvalues:

 λ(\boldmathΣ1) =(3.17,1.63,0.10,0.10,0.10,0.10)′ λ(\boldmathΣ2) =(4.18,2.27,0.40,0.40,0.40,0.40)′ λ(\boldmathΣ3) =(2.29,1.93,0.20,0.20,0.20,0.20)′,

whose largest value is given by

First we run the unconstrained algorithm: the right solution has been attained in 24% of cases, without incurring in singularities. Summary statistics (minimum, first quartile , median , third quartile and maximum) about the distribution of the misclassification error over the 100 runs are reported in Table 2. Due to the choice on parameters, we rarely expect too small eigenvalues in the estimated covariance matrices: we set to protect from them; conversely, as local maxima are quite often due to large estimated eigenvalues, we consider setting also a constraint from above, taking into account some values for , the upper bound. To compare how the choice of the bounds and influences the performance of the constrained EM, we experimented with different pairs of values, and in Table 3 we report the more interesting cases. Further results are reported in Figure 1, which provides the boxplots of the distribution of the misclassification errors obtained in the sequence of runs, showing the poor performance of the unconstrained algorithm compared with the good behaviour of its constrained version. For all values of the upper bound , the third quartile of the misclassification error is steadily equal to . Indeed, for and 15 we had no misclassification error, while we observed very low and rare misclassification errors only for and (respectively 3 and 11 not null values, over 100 runs). Moreover, the robustness of the results with respect to the choice of the upper constraint is apparent.


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## Calculus (3rd Edition)

The series $\sum_{n=2}^{\infty} \frac{\cos n \pi}{(\ln n)^{2}}$ converges conditionally.
Since $\cos (n\ pi)$ is either $1$ or $-1$ (depending on whether $n$ is even or odd), then we can write the series $\sum_{n=2}^{\infty} \frac{\cos n \pi}{(\ln n)^{2}}$ in the form $\sum_{n=2}^{\infty} \frac{(-1)^n}{(\ln n)^{2}}$. Now, we study the positive series $\sum_{n=2}^{\infty} \frac{1}{(\ln n)^{2}}$. Using the property that $\ln n \lt n^a, \ \ a\gt 0$, then we have $\ln n \lt n^{1/2}$ then $(\ln n)^{2}\lt n$, now we have $$\frac{1}{(\ln n)^{2}}\geq\frac{ 1}{n}$$ Since $\Sigma_{n=2}^{\infty}\frac{ 1}{n}$ is a divergent p-series ( $p=1$) then the series $\Sigma_{n=2}^{\infty} \frac{1}{(\ln n)^{2}}$ diverges. Also, $$\lim_{n\to\infty}b_n=\lim_{n\to\infty} \frac{1}{(\ln n)^{2}}=0.$$ So, by using the alternating series test, the series $\sum_{n=2}^{\infty} \frac{\cos n \pi}{(\ln n)^{2}}$ converges conditionally.

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# Analysis problem

1. Nov 12, 2008

Let x_{0} element of N. Then there is a sequence (x_{n}) that converges at x_{0} but has no terms that are nutural numbers.

Is that true?

Thank you

2. Nov 12, 2008

### Pere Callahan

Welcome tp PF.

What do you mean by a sequence to converge at some point?

If you ask if a sequence of real numbers all of whose term are not natural numbers can converge to some natural number, the answer is yes. I assume, you prefer trying to prove it yourself to being given a hint right away.

3. Nov 12, 2008

### HallsofIvy

Staff Emeritus
Can you think of a sequence that converges to 0? Does that meet the requirements?


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# How to turn linear regression into logistical regression

I followed these articles to implement logistic regression.

I'm confused however because after training the model and getting the weights for my variables I don't now how to use the one-hot vector to turn this into confidence scores for the different classes.

I've got the formula: y' = x1W1 + x2W2 + x3W3 + b

I've got values for all Ws and b.

I've got my one-hot vector: [[1, 0, 0], [0, 1, 0], [0, 0, 1]]

How do I combine all this into confidence for each class?

You should use softmax to convert your output in probabilities. For only two classes, you have the formula $$P(x \in class 1) = \frac{\exp(y_{\text{class1}})}{\exp(y_{\text{class1}}) + \exp(y_{\text{class2}})}$$. It mentioned in your article.


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Reply To: Pre 12/2007 EZBridge/EZServe 2017-12-21T19:48:54-07:00

HOME Forums Gateways EZSrve Pre 12/2007 EZBridge/EZServe Reply To: Pre 12/2007 EZBridge/EZServe

Anonymous
Post count: 20

Okay, I haven’t had all my questions answered (like which firmware revision initiated the need for a hardware fix), but I did get my device upgraded.

In short: I have a 1st generation (or so) EZBridge unit, which I got in early/mid 2007. In late summer 2007, it was sent back for some kind of hardware upgrade to make it into an EZServe (though still a separate unit from the PLM, unlike the current EZServe product).

Apparently, there was still some need for hardware tweaking (not sure how) even though I’d had the EZServe upgrade. Also, they upgraded the PLM somehow (actually sending that out to Smartlabs). So though it was a pain (this makes the 3rd time I’ve had to RMA this damned thing! 🙂 ), at least now I know I’m at the absolutely current revision for everything.

It’s no longer a free process (unlike when we beta users got upgraded to EZServe), which I’m not sure how I feel about that but I suppose I can’t really fault them since I got a PLM upgrade out of the deal (which I’d expect to be Smartlabs’ problem, not SimpleHomeNet’s).

And thanks to Al and everyone for getting it fixed QUICKLY and sent back to me fast. I really do appreciate that.

Now I just have to figure out everything I’ve missed over the past year. I’ll save that for other threads…


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### Tropical storms

"Hurricane" redirects here. For other uses, see Hurricane (disambiguation).

A tropical cyclone is a rapidly-rotating storm system characterized by a low-pressure centre, strong winds, and a spiral arrangement of thunderstorms that produce heavy rain. Tropical cyclones typically form over large bodies of relatively warm water. They derive their energy from the evaporation of water from the ocean surface, which ultimately recondenses into clouds and rain when moist air rises and cools to saturation. This energy source differs from that of mid-latitude cyclonic storms, such as nor'easters and European windstorms, which are fueled primarily by horizontal temperature contrasts. The strong rotating winds of a tropical cyclone are a result of the (partial) conservation of angular momentum imparted by the Earth's rotation as air flows inwards toward the axis of rotation. As a result, they rarely form within 5° of the equator.[1] Tropical cyclones are typically between 100 and 4,000 km (62 and 2,485 mi) in diameter.

The term "tropical" refers to the geographical origin of these systems, which usually form over the tropical oceans. The term "cyclone" refers to their cyclonic nature, with wind blowing counterclockwise in the Northern Hemisphere and clockwise in the Southern Hemisphere. The opposite direction of circulation is due to the Coriolis force. Depending on its location and strength, a tropical cyclone is referred to by names such as hurricane ( or ), typhoon , tropical storm, cyclonic storm, tropical depression, and simply cyclone.

In addition to strong winds and rain, tropical cyclones are capable of generating high waves, damaging storm surge, and tornadoes. They typically weaken rapidly over land where they are cut off from their primary energy source. For this reason, coastal regions are particularly vulnerable to damage from a tropical cyclone as compared to inland regions. Heavy rains, however, can cause significant flooding inland, and storm surges can produce extensive coastal flooding up to 40 kilometres (25 mi) from the coastline. Though their effects on human populations are often devastating, tropical cyclones can relieve drought conditions. They also carry heat energy away from the tropics and transport it toward temperate latitudes, which may play an important role in modulating regional and global climate.

## Physical structure

Tropical cyclones are areas of relatively low pressure in the troposphere, with the largest pressure perturbations occurring at low altitudes near the surface. On Earth, the pressures recorded at the centres of tropical cyclones are among the lowest ever observed at sea level.[2] The environment near the center of tropical cyclones is warmer than the surroundings at all altitudes, thus they are characterized as "warm core" systems.[3]

### Wind field

The near-surface wind field of a tropical cyclone is characterised by air rotating rapidly around a centre of circulation while also flowing radially inwards. At the outer edge of the storm, air may be nearly calm; however, due to the Earth's rotation, the air has non-zero angular momentum. As air flows radially inward, it begins to rotate cyclonically (counter-clockwise in the Northern Hemisphere, and clockwise in the Southern Hemisphere) in order to conserve angular momentum. At an inner radius, air begins to ascend to the top of the troposphere. This radius is typically coincident with the inner radius of the eyewall, and has the strongest near-surface winds of the storm; consequently, it is known as the radius of maximum winds.[4] Once aloft, air flows away from the storm's center, producing a shield of cirrus clouds.[5]

The previously-mentioned processes result in a wind field that is nearly axisymmetric: Wind speeds are low at the centre, increase rapidly moving outwards to the radius of maximum winds, and then decay more gradually with radius to large radii. However, the wind field often exhibits additional spatial and temporal variability due to the effects of localized processes, such as thunderstorm activity and horizontal flow instabilities. In the vertical direction, winds are strongest near the surface and decay with height within the troposphere.[6]

### Eye and center

At the center of a mature tropical cyclone, air sinks rather than rises. For a sufficiently strong storm, air may sink over a layer deep enough to suppress cloud formation, thereby creating a clear "eye". Weather in the eye is normally calm and free of clouds, although the sea may be extremely violent.[7] The eye is normally circular in shape, and is typically 30–65 km (19–40 mi) in diameter, though eyes as small as 3 km (1.9 mi) and as large as 370 km (230 mi) have been observed.[8][9]

The cloudy outer edge of the eye is called the "eyewall". The eyewall typically expands outward with height, resembling an arena football stadium; this phenomenon is sometimes referred to as the stadium effect.[10] The eyewall is where the greatest wind speeds are found, air rises most rapidly, clouds reach to their highest altitude, and precipitation is the heaviest. The heaviest wind damage occurs where a tropical cyclone's eyewall passes over land.[7]

In a weaker storm, the eye may be obscured by the central dense overcast, which is the upper-level cirrus shield that is associated with a concentrated area of strong thunderstorm activity near the center of a tropical cyclone.[11]

The eyewall may vary over time in the form of eyewall replacement cycles, particularly in intense tropical cyclones. Outer rainbands can organize into an outer ring of thunderstorms that slowly moves inward, which is believed to rob the primary eyewall of moisture and angular momentum. When the primary eyewall weakens, the tropical cyclone weakens temporarily. The outer eyewall eventually replaces the primary one at the end of the cycle, at which time the storm may return to its original intensity.[12]

### Intensity

Storm "intensity" is defined as the maximum wind speed in the storm. This speed is taken as either a 1-minute or a 10-minute average at the standard reference height of 10 meters. The choice of averaging period, as well as the naming convention for classifying storms, differs across forecast centers and ocean basins.

### Size

Size descriptions of tropical cyclones
ROCI Type
Less than 2 degrees latitude Very small/midget
2 to 3 degrees of latitude Small
3 to 6 degrees of latitude Medium/Average
6 to 8 degrees of latitude Large
Over 8 degrees of latitude Very large[13]

There are a variety of metrics commonly used to measure storm size. The most common metrics include the radius of maximum wind, the radius of 34-knot wind (i.e. gale force), the radius of outermost closed isobar (ROCI), and the radius of vanishing wind.[14][15] An additional metric is the radius at which the cyclone's relative vorticity field decreases to 1×10−5 s−1.[16]

On Earth, tropical cyclones span a large range of sizes, from 100–2000 km as measured by the radius of vanishing wind. They are largest on average in the northwest Pacific Ocean basin and smallest in the eastern Pacific Ocean basin. If the radius of outermost closed isobar is less than two degrees of latitude (222 km (138 mi)), then the cyclone is "very small" or a "midget". A radius of 3–6 latitude degrees (333–670 km (207–416 mi)) is considered "average sized". "Very large" tropical cyclones have a radius of greater than 8 degrees (888 km (552 mi)).[13] Observations indicate that size is only weakly correlated to variables such as storm intensity (i.e. maximum wind speed), radius of maximum wind, latitude, and maximum potential intensity.[15][17]

Size plays an important role in modulating damage caused by a storm. All else equal, a larger storm will impact a larger area for a longer period of time. Additionally, a larger near-surface wind field can generate higher storm surge due to the combination of longer wind fetch, longer duration, and enhanced wave setup.[18] For example, Hurricane Sandy, which struck the eastern U.S. in 2012, barely attained hurricane intensity prior to landfall yet was one of the costliest landfalling hurricanes in U.S. history because of its extremely large size.

The upper circulation of strong hurricanes extends into the tropopause of the atmosphere, which at low latitudes is 50,000-60,000 ft.[19]

## Physics and energetics

The three-dimensional wind field in a tropical cyclone can be separated into two components: a "primary circulation" and a "secondary circulation". The primary circulation is the rotational part of the flow; it is purely circular. The secondary circulation is the overturning (in-up-out-down) part of the flow; it is in the radial and vertical directions. The primary circulation has the strongest winds and is responsible for the majority of the damage a storm causes, while the secondary circulation is slower but governs the energetics of the storm.

### Secondary circulation: A Carnot heat engine

A tropical cyclone's primary energy source is the evaporation of water from the ocean surface, which ultimately recondenses into clouds and rain when moist air rises and cools to saturation. The energetics of the system may be idealized as an atmospheric Carnot heat engine.[21] First, inflowing air near the surface acquires heat primarily via evaporation of water (i.e. latent heat) at the temperature of the warm ocean surface. Second, air rises and cools within the eyewall while conserving total heat content (latent heat is simply converted to sensible heat during condensation). Third, air outflows and loses heat via infrared radiation to space at the temperature of the cold tropopause. Finally, air subsides and warms at the outer edge of the storm while conserving total heat content. The first and third legs are nearly isothermal, while the second and fourth legs are nearly isentropic. This in-up-out-down overturning flow is known as the secondary circulation. The Carnot perspective provides an upper bound on the maximum wind speed that a storm can attain.

Scientists estimate that a tropical cyclone releases heat energy at the rate of 50 to 200 exajoules (1018 J) per day,[22] equivalent to about 1 PW (1015 watt). This rate of energy release is equivalent to 70 times the world energy consumption of humans and 200 times the worldwide electrical generating capacity, or to exploding a 10-megaton nuclear bomb every 20 minutes.[22][23]

### Primary circulation

The primary rotating flow in a tropical cyclone results from the conservation of angular momentum by the secondary circulation. Absolute angular momentum on a rotating planet $M$ is given by

$M = \frac\left\{1\right\}\left\{2\right\}fr^2 + vr$

where $f$ is the Coriolis parameter, $v$ is the azimuthal (i.e. rotating) wind speed, and $r$ is the radius to the axis of rotation. The first term on the right hand side is the component of planetary angular momentum that projects onto the local vertical (i.e. the axis of rotation). The second term on the right hand side is the relative angular momentum of the circulation itself with respect to the axis of rotation. Because the planetary angular momentum term vanishes at the equator (where $f=0$ ), tropical cyclones rarely form within 5° of the equator.[1][24]

As air flows radially inward at low levels, it begins to rotate cyclonically in order to conserve angular momentum. Similarly, as rapidly-rotating air flows radially outward near the tropopause, its cyclonic rotation decreases and ultimately changes sign at large enough radius, resulting in an upper-level anti-cyclone. The result is a vertical structure characterized by a strong cyclone at low levels and a strong anti-cyclone near the tropopause; from thermal wind balance, this corresponds to a system that is warmer at its center than in the surrounding environment at all altitudes (i.e. "warm-core"). From hydrostatic balance, the warm core translates to lower pressure at the center at all altitudes, with the maximum pressure drop located at the surface.[6]

### Maximum Potential Intensity

Due to surface friction, the inflow only partially conserves angular momentum. Thus, the sea surface lower boundary acts as both a source (evaporation) and sink (friction) of energy for the system. This fact leads to the existence of a theoretical upper bound on the strongest wind speed that a tropical cyclone can attain. Because evaporation increases linearly with wind speed (just as climbing out of a pool feels much colder on a windy day), there is a positive feedback on energy input into the system known as the Wind-Induced Surface Heat Exchange (WISHE) feedback.[21] This feedback is offset when frictional dissipation, which increases with the cube of the wind speed, becomes sufficiently large. For wind speeds above this upper bound, energy loss due to frictional dissipation is larger than energy input due to evaporation, and the storm weakens.

This upper bound is called the "Maximum Potential Intensity", $v_p$, and is given by

$v_p = \sqrt\left\{\frac\left\{C_k\right\}\left\{C_d\right\}\frac\left\{T_s - T_o\right\}\left\{T_o\right\}\left(k^*_s-k\right)\right\}$

where $T_s$ is the temperature of the sea surface, $T_o$ is the temperature of the outflow ([K]), $k^*_s$ is the saturation enthalpy of the sea surface ([J]), $k$ is the enthalpy of boundary layer air overlying the surface ([J]), and $C_k$ and $C_d$ are the surface exchange coefficients ([dimensionless]) of enthalpy and momentum, respectively.[25] Thus, $k^*_s-k$ is the enthalpy disequilibrium between the surface and overlying air.

The maximum potential intensity is predominantly a function of the background environment alone (i.e. without a tropical cyclone). This quantity can be used to determine which regions on Earth can support tropical cyclones of a given intensity, and how these regions may evolve in time.[26][27]

#### Derivation

The maximum potential intensity is calculated from the balance between the rate of net energy production in the system and the rate of energy loss due to frictional dissipation at the surface. The rate of energy loss per unit area from surface friction, $W_\left\{out\right\}$, is given by

$W_\left\{out\right\} = C_d \rho |\mathbf\left\{u\right\}|^3$

where $\rho$ is the density of near-surface air ([kg/m3]) and $|\mathbf\left\{u\right\}|$ is the near surface wind speed ([m/s]). The rate of net energy production per unit surface area, $W_\left\{in\right\}$, for a Carnot heat engine is given by

$W_\left\{in\right\} = \epsilon Q_\left\{in\right\}$

where $\epsilon$ is the Carnot efficiency and $Q_\left\{in\right\}$ is the total rate of heat input into the system per unit surface area. The Carnot efficiency is given by

$\epsilon = \frac\left\{T_s-T_o\right\}\left\{T_s\right\}$

There are two sources of heat input. The dominant source is the input of heat at the surface, primarily due to evaporation. The bulk aerodynamic formula for the rate of heat input per unit area at the surface, $Q_\left\{in:k\right\}$, is given by

$Q_\left\{in:k\right\} = C_k \rho |\mathbf\left\{u\right\}|\left(k^*_s-k\right)$

The second source is the internal sensible heat generated from frictional dissipation (equal to $W_\left\{out\right\}$), which occurs near the surface within the tropical cyclone and is recycled to the system.

$Q_\left\{in:friction\right\} = C_d \rho |\mathbf\left\{u\right\}|^3$

Thus, the total rate of net energy production per unit surface area is given by

$W_\left\{in\right\} = \frac\left\{T_s-T_o\right\}\left\{T_s\right\}\left\left(C_k \rho |\mathbf\left\{u\right\}|\left(k^*_s-k\right) + C_d \rho |\mathbf\left\{u\right\}|^3\right\right)$

Setting $W_\left\{in\right\} = W_\left\{out\right\}$ and taking $|\mathbf\left\{u\right\}| \approx v$ (i.e. the rotational wind speed is dominant) leads to the solution for $v_p$ given above. This derivation assumes that total energy input and loss within the system can be approximated by their values at the radius of maximum wind. The inclusion of $Q_\left\{in:friction\right\}$ acts to multiply the total heat input rate by the factor $\frac\left\{T_s\right\}\left\{T_o\right\}$. Mathematically, this has the effect of replacing $T_s$ with $T_o$ in the denominator of the Carnot efficiency.

An alternative definition for the maximum potential intensity, which is mathematically-equivalent to the above formulation, is

$v_p = \sqrt\left\{\frac\left\{T_s\right\}\left\{T_o\right\}\frac\left\{C_k\right\}\left\{C_d\right\}\left(CAPE^*_s-CAPE_b\right)|_m\right\}$

where CAPE stands for the Convective Available Potential Energy, $CAPE^*_s$ is the CAPE of an air parcel lifted from saturation at sea level in reference to the environmental sounding, $CAPE_b$ is the CAPE of the boundary layer air, and both quantities are calculated at the radius of maximum wind.[28]

#### Characteristic values on Earth

On Earth, a characteristic temperature for $T_s$ is 300 K and for $T_o$ is 200 K, corresponding to a Carnot efficiency of $\epsilon = 1/3$. The ratio of the surface exchange coefficients, $C_k/C_d$, is typically taken to be 1. However, observations suggest that the drag coefficient $C_d$ varies with wind speed and may decrease at high wind speeds within the boundary layer of a mature hurricane.[29] Additionally, $C_k$ may vary at high wind speeds due to the effect of sea spray on evaporation within the boundary layer.[30]

A characteristic value of the maximum potential intensity, $v_p$, is 80 m/s. However, this quantity varies significantly across space and time, particularly within the seasonal cycle, spanning a range of 0–100 m/s.[28] This variability is primarily due to variabliity in the surface enthalpy disequilibrium ( $k^*_s - k$ ) as well as in the thermodynamic structure of the troposphere, which are controlled by the large-scale dynamics of the tropical climate. These processes are modulated by factors including the sea surface temperature (and underlying ocean dynamics), background near-surface wind speed, and the vertical structure of atmospheric radiative heating.[31] The nature of this modulation is complex, particularly on climate time-scales (decades or longer). On shorter time-scales, variability in the maximum potential intensity is commonly linked to sea surface temperature perturbations from the tropical mean, as regions with relatively warm water have thermodynamic states much more capable of sustaining a tropical cyclone than regions with relatively cold water.[32] However, this relationship is indirect via the large-scale dynamics of the tropics; the direct influence of the absolute sea surface temperature on $v_p$ is relatively weak.

### Interaction with the upper ocean

The passage of a tropical cyclone over the ocean causes the upper layers of the ocean to cool substantially, which can influence subsequent cyclone development. This cooling is primarily caused by wind-driven mixing of cold water from deeper in the ocean with the warm surface waters. This effect results in a negative feedback process that can inhibit further development or lead to weakening. Additional cooling may come in the form of cold water from falling raindrops (this is because the atmosphere is cooler at higher altitudes). Cloud cover may also play a role in cooling the ocean, by shielding the ocean surface from direct sunlight before and slightly after the storm passage. All these effects can combine to produce a dramatic drop in sea surface temperature over a large area in just a few days.[33]

## Major basins and related warning centers

Basins and WMO Monitoring Institutions[34]
Basin Responsible RSMCs and TCWCs
North Atlantic National Hurricane Center (United States)
North-East Pacific National Hurricane Center (United States)
North-Central Pacific Central Pacific Hurricane Center (United States)
North-West Pacific Japan Meteorological Agency
North Indian Ocean India Meteorological Department
South-West Indian Ocean Météo-France
Australian region Bureau of Meteorology (Australia)
Indonesian Meteorological and Geophysical Agency
Papua New Guinea National Weather Service
Southern Pacific Fiji Meteorological Service
Meteorological Service of New Zealand
: Indicates a Tropical Cyclone Warning Center

On 26 March 2004, Cyclone Catarina became the first recorded South Atlantic cyclone and subsequently struck southern Brazil with winds equivalent to Category 2 on the Saffir-Simpson Hurricane Scale. As the cyclone formed outside the authority of another warning center, Brazilian meteorologists initially treated the system as an extratropical cyclone, but later on classified it as tropical.[39]

## Formation

Main article: Tropical cyclogenesis

Worldwide, tropical cyclone activity peaks in late summer, when the difference between temperatures aloft and sea surface temperatures is the greatest. However, each particular basin has its own seasonal patterns. On a worldwide scale, May is the least active month, while September is the most active month. November is the only month in which all the tropical cyclone basins are active.[40]

### Times

In the Northern Atlantic Ocean, a distinct cyclone season occurs from June 1 to November 30, sharply peaking from late August through September.[40] The statistical peak of the Atlantic hurricane season is 10 September. The Northeast Pacific Ocean has a broader period of activity, but in a similar time frame to the Atlantic.[41] The Northwest Pacific sees tropical cyclones year-round, with a minimum in February and March and a peak in early September. In the North Indian basin, storms are most common from April to December, with peaks in May and November.[40] In the Southern Hemisphere, the tropical cyclone year begins on July 1 and runs all year-round encompassing the tropical cyclone seasons, which run from November 1 until the end of April, with peaks in mid-February to early March.[40][42]

Template:TC Season Stats

### Factors

The formation of tropical cyclones is the topic of extensive ongoing research and is still not fully understood.[43] While six factors appear to be generally necessary, tropical cyclones may occasionally form without meeting all of the following conditions. In most situations, water temperatures of at least 26.5 °C (79.7 °F) are needed down to a depth of at least 50 m (160 ft);[44] waters of this temperature cause the overlying atmosphere to be unstable enough to sustain convection and thunderstorms.[45] Another factor is rapid cooling with height, which allows the release of the heat of condensation that powers a tropical cyclone.[44] High humidity is needed, especially in the lower-to-mid troposphere; when there is a great deal of moisture in the atmosphere, conditions are more favorable for disturbances to develop.[44] Low amounts of wind shear are needed, as high shear is disruptive to the storm's circulation.[44] Tropical cyclones generally need to form more than 555 km (345 mi) or five degrees of latitude away from the equator, allowing the Coriolis effect to deflect winds blowing towards the low pressure center and creating a circulation.[44] Lastly, a formative tropical cyclone needs a pre-existing system of disturbed weather. Tropical cyclones will not form spontaneously.[44] Low-latitude and low-level westerly wind bursts associated with the Madden-Julian oscillation can create favorable conditions for tropical cyclogenesis by initiating tropical disturbances.[46]

### Locations

Most tropical cyclones form in a worldwide band of thunderstorm activity near the equator, referred to as the Intertropical Front (ITF), the Intertropical Convergence Zone (ITCZ), or the monsoon trough.[47][48][49] Another important source of atmospheric instability is found in tropical waves, which contribute to the development of about 85% of intense tropical cyclones in the Atlantic ocean and become most of the tropical cyclones in the Eastern Pacific.[50][51][52] The majority forms between 10 and 30 degrees of latitude away of the equator, and 87% forms no farther away than 20 degrees north or south.[53][54] Because the Coriolis effect initiates and maintains their rotation, tropical cyclones rarely form or move within 5 degrees of the equator, where the effect is weakest.[53] However, it is still possible for tropical systems to form within this boundary as Cyclone Agni and Tropical Storm Vamei did in 2004 and 2010, respectively.[55][56]

## Movement and track

### Steering winds

Although tropical cyclones are large systems with great amounts of intrinsic energy, their movements over the Earth's surface are controlled by large-scale winds—the streams in the Earth's atmosphere. The path of motion is referred to as a tropical cyclone's track and has been compared by Dr. Neil Frank, former director of the National Hurricane Center, to "leaves carried along by a stream".[57]

Tropical systems, while generally located within 20 degrees of the equator, are steered primarily westward by the east-to-west trade winds on the equatorward side of the subtropical ridge—a persistent high-pressure area over the world's oceans.[57] In the tropical North Atlantic and Northeast Pacific oceans, the trade winds steer tropical waves westward from the African coast toward the Caribbean Sea, North America, and ultimately into the central Pacific ocean before the waves dampen out.[51] These waves are the precursors to many tropical cyclones within this region.[50] In the Indian Ocean and Western Pacific, both spanning north and south of the equator, tropical cyclogenesis is strongly influenced by the seasonal movement of the Intertropical Convergence Zone, or the monsoon trough, rather than by easterly waves.[58] Tropical cyclones can also be steered by other weather systems, such as other low-pressure systems, high-pressure systems, warm fronts, and cold fronts.

### Coriolis effect

The Earth's rotation imparts an acceleration known as the Coriolis effect, Coriolis acceleration, or colloquially, Coriolis force. This acceleration causes cyclonic systems to turn towards the poles in the absence of strong steering currents.[59] The poleward portion of a tropical cyclone contains easterly winds, and the Coriolis effect pulls them slightly more poleward. The westerly winds on the equatorward portion of the cyclone pull slightly towards the equator, but, because the Coriolis effect weakens toward the equator, the net drag on the cyclone is poleward. Thus, tropical cyclones in the Northern Hemisphere usually turn north (before being blown east), and tropical cyclones in the Southern Hemisphere usually turn south (before being blown east) when no other effects counteract the Coriolis effect.[60]

The Coriolis effect also initiates cyclonic rotation, but unlike the heat of condensation it is not a driving force in bringing this rotation to high speeds.[22]

### Interaction with the mid-latitude westerlies

When a tropical cyclone crosses the subtropical ridge axis, its general track around the high-pressure area is deflected significantly by winds moving towards the general low-pressure area to its north. When the cyclone track becomes strongly poleward with an easterly component, the cyclone has begun recurvature.[61] A typhoon moving through the Pacific Ocean towards Asia, for example, will recurve offshore of Japan to the north, and then to the northeast, if the typhoon encounters southwesterly winds (blowing northeastward) around a low-pressure system passing over China or Siberia. Many tropical cyclones are eventually forced toward the northeast by extratropical cyclones in this manner, which move from west to east to the north of the subtropical ridge. An example of a tropical cyclone in recurvature was Typhoon Ioke in 2006, which took a similar trajectory.[62]

### Landfall

Officially, landfall is when a storm's center (the center of its circulation, not its edge) crosses the coastline.[4] Storm conditions may be experienced on the coast and inland hours before landfall; in fact, a tropical cyclone can launch its strongest winds over land, yet not make landfall; if this occurs, then it is said that the storm made a direct hit on the coast.[4] As a result of the narrowness of this definition, the landfall area experiences half of a land-bound storm by the time the actual landfall occurs. For emergency preparedness, actions should be timed from when a certain wind speed or intensity of rainfall will reach land, not from when landfall will occur.[4]

### Multiple storm interaction

Main article: Fujiwhara effect

When two cyclones approach one another, their centers will begin orbiting cyclonically about a point between the two systems. The two vortices will be attracted to each other, and eventually spiral into the center point and merge. When the two vortices are of unequal size, the larger vortex will tend to dominate the interaction, and the smaller vortex will orbit around it. This phenomenon is called the Fujiwhara effect, after Sakuhei Fujiwhara.[63]

## Dissipation

### Factors

A tropical cyclone can cease to have tropical characteristics in several different ways. One such way is if it moves over land, thus depriving it of the warm water it needs to power itself, quickly losing strength.[64] Most strong storms lose their strength very rapidly after landfall and become disorganized areas of low pressure within a day or two, or evolve into extratropical cyclones. There is a chance a tropical cyclone could regenerate if it managed to get back over open warm water, such as with Hurricane Ivan. If it remains over mountains for even a short time, weakening will accelerate.[65] Many storm fatalities occur in mountainous terrain, when diminishing cyclones unleash their moisture as torrential rainfall.[66] This may lead to deadly floods and mudslides, as was the case with Hurricane Mitch in 1998.[67] Without warm surface water, the storm cannot survive.[68]

A tropical cyclone can dissipate when it moves over waters significantly below 26.5 °C (79.7 °F). This will cause the storm to lose its tropical characteristics, such as a warm core with thunderstorms near the center, and become a remnant low-pressure area. These remnant systems may persist for up to several days before losing their identity. This dissipation mechanism is most common in the eastern North Pacific.[69] Weakening or dissipation can occur if it experiences vertical wind shear, causing the convection and heat engine to move away from the center; this normally ceases development of a tropical cyclone.[70] In addition, its interaction with the main belt of the Westerlies, by means of merging with a nearby frontal zone, can cause tropical cyclones to evolve into extratropical cyclones. This transition can take 1–3 days.[71] Even after a tropical cyclone is said to be extratropical or dissipated, it can still have tropical storm force (or occasionally hurricane/typhoon force) winds and drop several inches of rainfall. In the Pacific Ocean and Atlantic Ocean, such tropical-derived cyclones of higher latitudes can be violent and may occasionally remain at hurricane or typhoon-force wind speeds when they reach the west coast of North America. These phenomena can also affect Europe, where they are known as European windstorms; Hurricane Iris's extratropical remnants are an example of such a windstorm from 1995.[72] A cyclone can also merge with another area of low pressure, becoming a larger area of low pressure. This can strengthen the resultant system, although it may no longer be a tropical cyclone.[70] Studies in the 2000s have given rise to the hypothesis that large amounts of dust reduce the strength of tropical cyclones.[73]

### Artificial dissipation

In the 1960s and 1970s, the United States government attempted to weaken hurricanes through Project Stormfury by seeding selected storms with silver iodide. It was thought that the seeding would cause supercooled water in the outer rainbands to freeze, causing the inner eyewall to collapse and thus reducing the winds.[74] The winds of Hurricane Debbie—a hurricane seeded in Project Stormfury—dropped as much as 31%, but Debbie regained its strength after each of two seeding forays.[75] In an earlier episode in 1947, disaster struck when a hurricane east of Jacksonville, Florida promptly changed its course after being seeded, and smashed into Savannah, Georgia.[76] Because there was so much uncertainty about the behavior of these storms, the federal government would not approve seeding operations unless the hurricane had a less than 10% chance of making landfall within 48 hours, greatly reducing the number of possible test storms. The project was dropped after it was discovered that eyewall replacement cycles occur naturally in strong hurricanes, casting doubt on the result of the earlier attempts. Today, it is known that silver iodide seeding is not likely to have an effect because the amount of supercooled water in the rainbands of a tropical cyclone is too low.[77]

Other approaches have been suggested over time, including cooling the water under a tropical cyclone by towing icebergs into the tropical oceans.[78] Other ideas range from covering the ocean in a substance that inhibits evaporation,[79] dropping large quantities of ice into the eye at very early stages of development (so that the latent heat is absorbed by the ice, instead of being converted to kinetic energy that would feed the positive feedback loop),[78] or blasting the cyclone apart with nuclear weapons.[80] Project Cirrus even involved throwing dry ice on a cyclone.[81] These approaches all suffer from one flaw above many others: tropical cyclones are simply too large and short-lived for any of the weakening techniques to be practical.[82]

## Effects

Tropical cyclones out at sea cause large waves, heavy rain and high winds, disrupting international shipping and, at times, causing shipwrecks.[83] Tropical cyclones stir up water, leaving a cool wake behind them, which causes the region to be less favorable for subsequent tropical cyclones.[33] On land, strong winds can damage or destroy vehicles, buildings, bridges, and other outside objects, turning loose debris into deadly flying projectiles. The storm surge, or the increase in sea level due to the cyclone, is typically the worst effect from landfalling tropical cyclones, historically resulting in 90% of tropical cyclone deaths.[84] The broad rotation of a landfalling tropical cyclone, and vertical wind shear at its periphery, spawns tornadoes. Tornadoes can also be spawned as a result of eyewall mesovortices, which persist until landfall.[85]

Over the past two centuries, tropical cyclones have been responsible for the deaths of about 1.9 million people worldwide. Large areas of standing water caused by flooding lead to infection, as well as contributing to mosquito-borne illnesses. Crowded evacuees in shelters increase the risk of disease propagation.[86] Tropical cyclones significantly interrupt infrastructure, leading to power outages, bridge destruction, and the hampering of reconstruction efforts.[86][87] On average, the Gulf and east coasts of the United States suffer approximately US $5 billion (1995 US$) in cyclone damage every year. Majority (83%) of tropical cyclone damage is caused by severe hurricanes, category 3 or greater. However, category 3 or greater hurricanes only account for about one-fifth of cyclones that make land every year.[88]

Although cyclones take an enormous toll in lives and personal property, they may be important factors in the precipitation regimes of places they impact, as they may bring much-needed precipitation to otherwise dry regions.[89] Tropical cyclones also help maintain the global heat balance by moving warm, moist tropical air to the middle latitudes and polar regions,[90] and by regulating the thermohaline circulation through upwelling.[91] The storm surge and winds of hurricanes may be destructive to human-made structures, but they also stir up the waters of coastal estuaries, which are typically important fish breeding locales. Tropical cyclone destruction spurs redevelopment, greatly increasing local property values.[92]

## Observation and forecasting

### Observation

Intense tropical cyclones pose a particular observation challenge, as they are a dangerous oceanic phenomenon, and weather stations, being relatively sparse, are rarely available on the site of the storm itself. In general, surface observations are available only if the storm is passing over an island or a coastal area, or if there is a nearby ship. Real-time measurements are usually taken in the periphery of the cyclone, where conditions are less catastrophic and its true strength cannot be evaluated. For this reason, there are teams of meteorologists that move into the path of tropical cyclones to help evaluate their strength at the point of landfall.[93]

Tropical cyclones far from land are tracked by weather satellites capturing visible and infrared images from space, usually at half-hour to quarter-hour intervals. As a storm approaches land, it can be observed by land-based Doppler weather radar. Radar plays a crucial role around landfall by showing a storm's location and intensity every several minutes.[94]

In situ measurements, in real-time, can be taken by sending specially equipped reconnaissance flights into the cyclone. In the Atlantic basin, these flights are regularly flown by United States government hurricane hunters.[95] The aircraft used are WC-130 Hercules and WP-3D Orions, both four-engine turboprop cargo aircraft. These aircraft fly directly into the cyclone and take direct and remote-sensing measurements. The aircraft also launch GPS dropsondes inside the cyclone. These sondes measure temperature, humidity, pressure, and especially winds between flight level and the ocean's surface. A new era in hurricane observation began when a remotely piloted Aerosonde, a small drone aircraft, was flown through Tropical Storm Ophelia as it passed Virginia's Eastern Shore during the 2005 hurricane season. A similar mission was also completed successfully in the western Pacific ocean. This demonstrated a new way to probe the storms at low altitudes that human pilots seldom dare.[96]

### Forecasting

Because of the forces that affect tropical cyclone tracks, accurate track predictions depend on determining the position and strength of high- and low-pressure areas, and predicting how those areas will change during the life of a tropical system. The deep layer mean flow, or average wind through the depth of the troposphere, is considered the best tool in determining track direction and speed. If storms are significantly sheared, use of wind speed measurements at a lower altitude, such as at the 70 kPa pressure surface (3,000 metres or 9,800 feet above sea level) will produce better predictions. Tropical forecasters also consider smoothing out short-term wobbles of the storm as it allows them to determine a more accurate long-term trajectory.[97] High-speed computers and sophisticated simulation software allow forecasters to produce computer models that predict tropical cyclone tracks based on the future position and strength of high- and low-pressure systems. Combining forecast models with increased understanding of the forces that act on tropical cyclones, as well as with a wealth of data from Earth-orbiting satellites and other sensors, scientists have increased the accuracy of track forecasts over recent decades.[98] However, scientists are not as skillful at predicting the intensity of tropical cyclones.[99] The lack of improvement in intensity forecasting is attributed to the complexity of tropical systems and an incomplete understanding of factors that affect their development.

## Classifications, terminology, and naming

### Intensity classifications

Tropical cyclones are classified into three main groups, based on intensity: tropical depressions, tropical storms, and a third group of more intense storms, whose name depends on the region. For example, if a tropical storm in the Northwestern Pacific reaches hurricane-strength winds on the Beaufort scale, it is referred to as a typhoon; if a tropical storm passes the same benchmark in the Northeast Pacific Basin, or in the North Atlantic, it is called a hurricane.[4] Neither "hurricane" nor "typhoon" is used in either the Southern Hemisphere or the Indian Ocean. In these basins, storms of tropical nature are referred to simply as "cyclones".

As indicated in the table below, each basin uses a separate system of terminology, making comparisons between different basins difficult. In the Pacific Ocean, hurricanes from the Central North Pacific sometimes cross the 180th meridian into the Northwest Pacific, becoming typhoons (such as Hurricane/Typhoon Ioke in 2006); on rare occasions, the reverse will occur.[100] It should also be noted that typhoons with sustained winds greater than 67 metres per second ([convert: unit mismatch]) or 150 miles per hour (240 km/h) are called Super Typhoons by the Joint Typhoon Warning Center.[101]

#### Tropical depression

A tropical depression is an organized system of clouds and thunderstorms with a defined, closed surface circulation and maximum sustained winds of less than 34 knots (39 mph). It has no eye and does not typically have the organization or the spiral shape of more powerful storms. However, it is already a low-pressure system, hence the name "depression".[102] In the Philippines, the practice is to name tropical depressions from their own naming convention when the depressions are within the country's area of responsibility.[103]

#### Tropical storm

A tropical storm is an organized system of strong thunderstorms with a defined surface circulation and maximum sustained winds between 34 knots (39 mph) and 64 knots (74 mph). At this point, the distinctive cyclonic shape starts to develop, although an eye is not usually present. Government weather services first assign names to systems that reach this intensity (thus the term named storm).[102]

#### Hurricane or Typhoon

A hurricane or typhoon (sometimes simply referred to as a tropical cyclone, as opposed to a depression or storm) is a system with sustained winds of at least 34 metres per second (66 kn) or 74 miles per hour (119 km/h).[102] A cyclone of this intensity tends to develop an eye, an area of relative calm (and lowest atmospheric pressure) at the center of circulation. The eye is often visible in satellite images as a small, circular, cloud-free spot. Surrounding the eye is the eyewall, an area about 16 kilometres (9.9 mi) to 80 kilometres (50 mi) wide in which the strongest thunderstorms and winds circulate around the storm's center. Maximum sustained winds in the strongest tropical cyclones have been estimated at about 85 metres per second (165 kn) or 195 miles per hour (314 km/h).[104]

Tropical Cyclone Classifications [105][106]
The
Beaufort
scale
1-minute sustained winds 10-minute sustained winds NE Pacific &
N Atlantic
NHC/CPHC
NW Pacific
JTWC
NW Pacific
JMA
N Indian Ocean
IMD
SW Indian Ocean
MF
Australia & S Pacific
BOM/FMS[107]
0–7 <32 knots (37 mph; 59 km/h) <28 knots (32 mph; 52 km/h) Tropical Depression Tropical Depression Tropical Depression Depression Zone of Disturbed Weather Tropical Disturbance
Tropical Depression
Tropical Low
7 33 knots (38 mph; 61 km/h) 28–29 knots (32–33 mph; 52–54 km/h) Deep Depression Tropical Disturbance
8 34–37 knots (39–43 mph; 63–69 km/h) 30–33 knots (35–38 mph; 56–61 km/h) Tropical Storm Tropical Storm Tropical Depression
9-10 38–54 knots (44–62 mph; 70–100 km/h) 34–47 knots (39–54 mph; 63–87 km/h) Tropical Storm Cyclonic Storm Moderate Tropical Storm Category 1
tropical cyclone
11 55–63 knots (63–72 mph; 102–117 km/h) 48–55 knots (55–63 mph; 89–102 km/h) Severe Tropical Storm Severe Cyclonic Storm Severe Tropical Storm Category 2
tropical cyclone
12+ 64–71 knots (74–82 mph; 119–131 km/h) 56–63 knots (64–72 mph; 104–117 km/h) Category 1 hurricane Typhoon
72–82 knots (83–94 mph; 133–152 km/h) 64–72 knots (74–83 mph; 119–133 km/h) Typhoon Very Severe
Cyclonic Storm
Tropical Cyclone Category 3 severe
tropical cyclone
83–95 knots (96–109 mph; 154–176 km/h) 73–83 knots (84–96 mph; 135–154 km/h) Category 2 hurricane
96–97 knots (110–112 mph; 178–180 km/h) 84–85 knots (97–98 mph; 156–157 km/h) Category 3 Hurricane
98–112 knots (113–129 mph; 181–207 km/h) 86–98 knots (99–113 mph; 159–181 km/h) Intense Tropical Cyclone Category 4 severe
tropical cyclone
113–122 knots (130–140 mph; 209–226 km/h) 99–107 knots (114–123 mph; 183–198 km/h) Category 4 hurricane
123–129 knots (142–148 mph; 228–239 km/h) 108–113 knots (124–130 mph; 200–209 km/h) Category 5 severe
tropical cyclone
130–136 knots (150–157 mph; 241–252 km/h) 114–119 knots (131–137 mph; 211–220 km/h) Super Typhoon Super Cyclonic Storm Very Intense Tropical Cyclone
>137 knots (158 mph; 254 km/h) >120 knots (140 mph; 220 km/h) Category 5 hurricane

### Origin of storm terms

The word typhoon, which is used today in the Northwest Pacific, may be derived from Arabic ţūfān (طوفان) (similar in Hindi/Urdu and Persian), which in turn originates from Greek Typhon (Τυφών), a monster from Greek mythology associated with storms.[108] The related Portuguese word tufão, used in Portuguese for typhoons, is also derived from Typhon.[109] The word is also similar to Chinese "táifēng" (Simplified Chinese: 台风) (fēng = wind), "toifung" in Cantonese (Traditional Chinese: 颱風), "taepoong" (태풍) in Korean, and "taifu" (台風) in Japanese.

The word hurricane, used in the North Atlantic and Northeast Pacific, is derived from huracán, the Spanish word for the Carib/Taino storm god, Juracán. This god is believed by scholars to have been at least partially derived from the Mayan creator god, Huracan. Huracan was believed by the Maya to have created dry land out of the turbulent waters. The god was also credited with later destroying the "wooden people", the precursors to the "maize people", with an immense storm and flood.[110][111] Huracan is also the source of the word orcan, another word for a particularly strong European windstorm.[111]

### Naming

Storms reaching tropical storm strength were initially given names to eliminate confusion when there are multiple systems in any individual basin at the same time, which assists in warning people of the coming storm.[112] In 1953, the United States abandoned a plan to name hurricanes using a phonetic alphabet, and began using women's names. The practice of using women's names exclusively came to an end in 1978, when both men's and women's names were added to the Eastern North Pacific storm lists. The lists for the Atlantic and Gulf of Mexico followed in 1979.[113] In most cases, a tropical cyclone retains its name throughout its life; however, under special circumstances, tropical cyclones may be renamed while active. These names are taken from lists that vary from region to region and are usually drafted a few years ahead of time. The lists are decided on, depending on the regions, either by committees of the World Meteorological Organization (called primarily to discuss many other issues) or by national weather offices involved in the forecasting of the storms. Within all basins except for the southwestern Indian Ocean, the names of particularly destructive or deadly storms may be retired and replaced with a new name.

## Notable tropical cyclones

Tropical cyclones that cause extreme destruction are rare, although when they occur, they can cause great amounts of damage or thousands of fatalities. The 1970 Bhola cyclone is the deadliest tropical cyclone on record, killing more than 300,000 people[114] and potentially as many as 1 million[115] after striking the densely populated Ganges Delta region of Bangladesh on 13 November 1970. Its powerful storm surge was responsible for the high death toll.[114] The North Indian cyclone basin has historically been the deadliest basin.[86][116] Elsewhere, Typhoon Nina killed nearly 100,000 in China in 1975 due to a 100-year flood that caused 62 dams including the Banqiao Dam to fail.[117] The Great Hurricane of 1780 is the deadliest North Atlantic hurricane on record, killing about 22,000 people in the Lesser Antilles.[118] A tropical cyclone does need not be particularly strong to cause memorable damage, primarily if the deaths are from rainfall or mudslides. Tropical Storm Thelma in November 1991 killed thousands in the Philippines,[119] while in 1982, the unnamed tropical depression that eventually became Hurricane Paul killed around 1,000 people in Central America.[120]

Hurricane Katrina is estimated as the costliest tropical cyclone worldwide,[121] causing $81.2 billion in property damage (2008 USD)[122] with overall damage estimates exceeding$100 billion (2005 USD).[121] Katrina killed at least 1,836 people after striking Louisiana and Mississippi as a major hurricane in August 2005.[122] Hurricane Sandy is the second most destructive tropical cyclone in U.S history, with damages totaling $68 billion (2012 USD), and with damage costs at$37.5 billion (2012 USD), Hurricane Ike is the third most destructive tropical cyclone in U.S history. The Galveston Hurricane of 1900 is the deadliest natural disaster in the United States, killing an estimated 6,000 to 12,000 people in Galveston, Texas.[123] Hurricane Mitch caused more than 10,000 fatalities in Latin America. Hurricane Iniki in 1992 was the most powerful storm to strike Hawaii in recorded history, hitting Kauai as a Category 4 hurricane, killing six people, and causing U.S. $3 billion in damage.[124] Kauai was also struck by Hurricanes Dot (1959) and Iwa (1982) (see List of Hawaii hurricanes). Other destructive Eastern Pacific hurricanes include Pauline and Kenna, both causing severe damage after striking Mexico as major hurricanes.[125][126] In March 2004, Cyclone Gafilo struck northeastern Madagascar as a powerful cyclone, killing 74, affecting more than 200,000, and becoming the worst cyclone to affect the nation for more than 20 years.[127] Hurricane Sandy which hit the United States east coast in late October 2012, caused unprecedented damage, flooded subways, closed down all major airports, resulted in cancellation of over 20,000 flights,[128][129] costing the aviation industry nearly 200 million dollars.[130] It claimed more than one-hundred lives.[131][132] Eleven states and the District of Columbia were declared disaster states.[133] Estimates for the damage cost vary from 50–70 billion dollars.[134][135] The most intense storm on record was Typhoon Tip in the northwestern Pacific Ocean in 1979, which reached a minimum pressure of 870 mbar (652.5 mmHg) and maximum sustained wind speeds of 165 knots (85 m/s) or 190 miles per hour (310 km/h).[136] Tip, however, does not solely hold the record for fastest sustained winds in a cyclone. Typhoon Keith in the Pacific and Hurricanes Camille and Allen in the North Atlantic currently share this record with Tip.[137] Camille was the only storm to actually strike land while at that intensity, making it, with 165 knots (85 m/s) or 190 miles per hour (310 km/h) sustained winds and 183 knots (94 m/s) or 210 miles per hour (340 km/h) gusts, the strongest tropical cyclone on record at landfall.[138] Typhoon Nancy in 1961 had recorded wind speeds of 185 knots (95 m/s) or 215 miles per hour (346 km/h), but recent research indicates that wind speeds from the 1940s to the 1960s were gauged too high, and this is no longer considered the storm with the highest wind speeds on record.[104] Likewise, a surface-level gust caused by Typhoon Paka on Guam was recorded at 205 knots (105 m/s) or 235 miles per hour (378 km/h). Had it been confirmed, it would be the strongest non-tornadic wind ever recorded on the Earth's surface, but the reading had to be discarded since the anemometer was damaged by the storm.[139] In addition to being the most intense tropical cyclone on record, Tip was the largest cyclone on record, with tropical storm-force winds 2,170 kilometres (1,350 mi) in diameter. The smallest storm on record, Tropical Storm Marco, formed during October 2008, and made landfall in Veracruz. Marco generated tropical storm-force winds only 37 kilometres (23 mi) in diameter.[140] Hurricane John is the longest-lasting tropical cyclone on record, lasting 31 days in 1994. Before the advent of satellite imagery in 1961, however, many tropical cyclones were underestimated in their durations.[141] John is also the longest-tracked tropical cyclone in the Northern Hemisphere on record, which had a path of 8,250 mi (13,280 km).[142] Cyclone Rewa of the 1993-94 South Pacific and Australian region cyclone seasons had one of the longest tracks observed within the Southern Hemisphere, travelling a distance of over 5,545 mi (8,920 km) during December 1993 and January 1994.[142] ## Changes caused by El Niño-Southern Oscillation Most tropical cyclones form on the side of the subtropical ridge closer to the equator, then move poleward past the ridge axis before recurving into the main belt of the Westerlies.[143] When the subtropical ridge position shifts due to El Niño, so will the preferred tropical cyclone tracks. Areas west of Japan and Korea tend to experience much fewer September–November tropical cyclone impacts during El Niño and neutral years. During El Niño years, the break in the subtropical ridge tends to lie near 130°E which would favor the Japanese archipelago.[144] During El Niño years, Guam's chance of a tropical cyclone impact is one-third of the long term average.[145] The tropical Atlantic ocean experiences depressed activity due to increased vertical wind shear across the region during El Niño years.[146] During La Niña years, the formation of tropical cyclones, along with the subtropical ridge position, shifts westward across the western Pacific ocean, which increases the landfall threat to China and much greater intensity in the Philippines.[144] ## Long-term activity trends While the number of storms in the Atlantic has increased since 1995, there is no obvious global trend; the annual number of tropical cyclones worldwide remains about 87 ± 10 (Between 77 and 97 tropical cyclones annually). However, the ability of climatologists to make long-term data analysis in certain basins is limited by the lack of reliable historical data in some basins, primarily in the Southern Hemisphere.[147] In spite of that, there is some evidence that the intensity of hurricanes is increasing. Kerry Emanuel stated, "Records of hurricane activity worldwide show an upswing of both the maximum wind speed in and the duration of hurricanes. The energy released by the average hurricane (again considering all hurricanes worldwide) seems to have increased by around 70% in the past 30 years or so, corresponding to about a 15% increase in the maximum wind speed and a 60% increase in storm lifetime."[148] Atlantic storms are becoming more destructive financially, as evidenced by the fact that five of the ten most expensive storms in United States history have occurred since 1990. According to the World Meteorological Organization, "recent increase in societal impact from tropical cyclones has been caused largely by rising concentrations of population and infrastructure in coastal regions."[149] Pielke et al. (2008) normalized mainland U.S. hurricane damage from 1900–2005 to 2005 values and found no remaining trend of increasing absolute damage. The 1970s and 1980s were notable because of the extremely low amounts of damage compared to other decades. The decade 1996–2005 was the second most damaging among the past 11 decades, with only the decade 1926–1935 surpassing its costs. The most damaging single storm is the 1926 Miami hurricane, with$157 billion of normalized damage.[150]

Often in part because of the threat of hurricanes, many coastal regions had sparse population between major ports until the advent of automobile tourism; therefore, the most severe portions of hurricanes striking the coast may have gone unmeasured in some instances. The combined effects of ship destruction and remote landfall severely limit the number of intense hurricanes in the official record before the era of hurricane reconnaissance aircraft and satellite meteorology. Although the record shows a distinct increase in the number and strength of intense hurricanes, therefore, experts regard the early data as suspect.[151]

The number and strength of Atlantic hurricanes may undergo a 50–70 year cycle, also known as the Atlantic Multidecadal Oscillation. Nyberg et al. reconstructed Atlantic major hurricane activity back to the early 18th century and found five periods averaging 3–5 major hurricanes per year and lasting 40–60 years, and six other averaging 1.5–2.5 major hurricanes per year and lasting 10–20 years. These periods are associated with the Atlantic multidecadal oscillation. Throughout, a decadal oscillation related to solar irradiance was responsible for enhancing/dampening the number of major hurricanes by 1–2 per year.[152]

Although more common since 1995, few above-normal hurricane seasons occurred during 1970–94.[153] Destructive hurricanes struck frequently from 1926–60, including many major New England hurricanes. Twenty-one Atlantic tropical storms formed in 1933, a record only recently exceeded in 2005, which saw 28 storms. Tropical hurricanes occurred infrequently during the seasons of 1900–25; however, many intense storms formed during 1870–99. During the 1887 season, 19 tropical storms formed, of which a record 4 occurred after 1 November and 11 strengthened into hurricanes. Few hurricanes occurred in the 1840s to 1860s; however, many struck in the early 19th century, including an 1821 storm that made a direct hit on New York City. Some historical weather experts say these storms may have been as high as Category 4 in strength.[154]

These active hurricane seasons predated satellite coverage of the Atlantic basin. Before the satellite era began in 1960, tropical storms or hurricanes went undetected unless a reconnaissance aircraft encountered one, a ship reported a voyage through the storm, or a storm hit land in a populated area.[151]

Proxy records based on paleotempestological research have revealed that major hurricane activity along the Gulf of Mexico coast varies on timescales of centuries to millennia.[155][156] Few major hurricanes struck the Gulf coast during 3000–1400 BC and again during the most recent millennium. These quiescent intervals were separated by a hyperactive period during 1400 BC and 1000 AD, when the Gulf coast was struck frequently by catastrophic hurricanes and their landfall probabilities increased by 3–5 times. This millennial-scale variability has been attributed to long-term shifts in the position of the Azores High,[156] which may also be linked to changes in the strength of the North Atlantic Oscillation.[157]

According to the Azores High hypothesis, an anti-phase pattern is expected to exist between the Gulf of Mexico coast and the Atlantic coast. During the quiescent periods, a more northeasterly position of the Azores High would result in more hurricanes being steered towards the Atlantic coast. During the hyperactive period, more hurricanes were steered towards the Gulf coast as the Azores High was shifted to a more southwesterly position near the Caribbean. Such a displacement of the Azores High is consistent with paleoclimatic evidence that shows an abrupt onset of a drier climate in Haiti around 3200 14C years BP,[158] and a change towards more humid conditions in the Great Plains during the late-Holocene as more moisture was pumped up the Mississippi Valley through the Gulf coast. Preliminary data from the northern Atlantic coast seem to support the Azores High hypothesis. A 3000-year proxy record from a coastal lake in Cape Cod suggests that hurricane activity increased significantly during the past 500–1000 years, just as the Gulf coast was amid a quiescent period of the last millennium.

## Global warming

According to IPCC SREX 2012, "attribution of single extreme events to anthropogenic climate change is challenging".[159] On one hand, the report said that there is "medium evidence" that long-term trends in normalized losses have not been attributed to tropical and extratropical [winter] storms. On the other hand, the report also noted that much more research is needed in part due to "confounding factors" that might have increased losses, such as increased population and development in at-risk areas, and those that might have decreased losses, such as better forecasting, emergency alert systems, emergency management, building codes, and near-instantaneous media coverage of weather emergencies.[160] The report also noted that storm surges are already worse due to sea level rise, and that windspeed of tropical cyclones will likely increase, as will total rainfall in extreme events.[160][page needed]

Some experts who agree that we can not yet detect any increase in frequency or intensity of tropical cyclones include Thomas Knutson,[161] and Roger Pielke Jr..[162]

Others say that there is evidence for a causal connection. The U.S. National Oceanic and Atmospheric Administration Geophysical Fluid Dynamics Laboratory performed a simulation to determine if there is a statistical trend in the frequency or strength of tropical cyclones over time. The simulation concluded "the strongest hurricanes in the present climate may be upstaged by even more intense hurricanes over the next century as the earth's climate is warmed by increasing levels of greenhouse gases in the atmosphere".[163]

Costliest U.S. Atlantic hurricanes 1900–2010
Total estimated property damage, adjusted for wealth normalization[150][164]
Rank Hurricane Season Cost (2010 USD)
1 "Miami" 1926 $164.8 billion 2 Katrina 2005$113.4 billion
3 "Galveston" 1900 $104.3 billion 4 "Galveston" 1915$ 71.3 billion
5 Andrew 1992 $58.5 billion 6 "New England" 1938$ 41.1 billion
7 "Cuba–Florida" 1944 $40.6 billion 8 "Okeechobee" 1928$ 35.2 billion
9 Ike 2008 $29.5 billion 10 Donna 1960$ 28.1 billion
Main article: List of costliest Atlantic hurricanes

In an article in Nature, meteorology professor Kerry Emanuel stated that potential hurricane destructiveness, a measure combining hurricane strength, duration, and frequency, "is highly correlated with tropical sea surface temperature, reflecting well-documented climate signals, including multidecadal oscillations in the North Atlantic and North Pacific, and global warming". Emanuel predicted "a substantial increase in hurricane-related losses in the twenty-first century".[165] In more recent work, Emanuel states that new climate modeling data indicates "global warming should reduce the global frequency of hurricanes."[166] According to the Houston Chronicle, the new work suggests that, even in a dramatically warming world, hurricane frequency and intensity may not substantially rise during the next two centuries.[167]

P.J. Webster and others published an article in Science examining the "changes in tropical cyclone number, duration, and intensity" over the past 35 years, the period when satellite data has been available. Their main finding was although the number of cyclones decreased throughout the planet excluding the north Atlantic Ocean, there was a great increase in the number and proportion of very strong cyclones.[168]

The strength of the reported effect is surprising in light of modeling studies[169] that predict only a one-half category increase in storm intensity as a result of a ~2 °C (3.6 °F) global warming. Such a response would have predicted only a ~10% increase in Emanuel's potential destructiveness index during the 20th century rather than the ~75–120% increase he reported.[165] Second, after adjusting for changes in population and inflation, and despite a more than 100% increase in Emanuel's potential destructiveness index, no statistically significant increase in the monetary damages resulting from Atlantic hurricanes has been found.[150][170]

Sufficiently warm sea surface temperatures are considered vital to the development of tropical cyclones.[171] Although neither study can directly link hurricanes with global warming, the increase in sea surface temperatures is believed to be due to both global warming and natural variability, e.g. the hypothesized Atlantic Multidecadal Oscillation (AMO), although an exact attribution has not been defined.[172] However, recent temperatures are the warmest ever observed for many ocean basins.[165]

In February 2007, the United Nations Intergovernmental Panel on Climate Change released its fourth assessment report on climate change. The report noted many observed changes in the climate, including atmospheric composition, global average temperatures, ocean conditions, and others. The report concluded the observed increase in tropical cyclone intensity is larger than climate models predict. In addition, the report considered that it is likely that storm intensity will continue to increase through the 21st century, and declared it more likely than not that there has been some human contribution to the increases in tropical cyclone intensity.[173] However, there is no universal agreement about the magnitude of the effects anthropogenic global warming has on tropical cyclone formation, track, and intensity. For example, critics such as Chris Landsea assert that man-made effects would be "quite tiny compared to the observed large natural hurricane variability".[174] A statement by the American Meteorological Society on February 1, 2007 stated that trends in tropical cyclone records offer "evidence both for and against the existence of a detectable anthropogenic signal" in tropical cyclogenesis.[175] Although many aspects of a link between tropical cyclones and global warming are still being "hotly debated",[176] a point of agreement is that the strength of destructiveness no individual tropical cyclone or season can be attributed entirely to global warming.[172][176] Research reported in the September 3, 2008 issue of Nature found that the strongest tropical cyclones are getting stronger, in particular over the North Atlantic and Indian oceans. Wind speeds for the strongest tropical storms increased from an average of 140 miles per hour (230 km/h) in 1981 to 156 miles per hour (251 km/h) in 2006, while the ocean temperature, averaged globally over all the regions where tropical cyclones form, increased from 28.2 °C (82.8 °F) to 28.5 °C (83.3 °F) during this period.[177][178]

## Related cyclone types

In addition to tropical cyclones, there are two other classes of cyclones within the spectrum of cyclone types. These kinds of cyclones, known as extratropical cyclones and subtropical cyclones, can be stages a tropical cyclone passes through during its formation or dissipation.[179] An extratropical cyclone is a storm that derives energy from horizontal temperature differences, which are typical in higher latitudes. A tropical cyclone can become extratropical as it moves toward higher latitudes if its energy source changes from heat released by condensation to differences in temperature between air masses; although not as frequently, an extratropical cyclone can transform into a subtropical storm, and from there into a tropical cyclone.[180] From space, extratropical storms have a characteristic "comma-shaped" cloud pattern.[181] Extratropical cyclones can also be dangerous when their low-pressure centers cause powerful winds and high seas.[182]

A subtropical cyclone is a weather system that has some characteristics of a tropical cyclone and some characteristics of an extratropical cyclone. They can form in a wide band of latitudes, from the equator to 50°. Although subtropical storms rarely have hurricane-force winds, they may become tropical in nature as their cores warm.[183] From an operational standpoint, a tropical cyclone is usually not considered to become subtropical during its extratropical transition.[184]

## In popular culture

In popular culture, tropical cyclones have made several appearances in different types of media, including films, books, television, music, and electronic games. These media often portray tropical cyclones that are either entirely fictional or based on real events.[185] For example, George Rippey Stewart's Storm, a best-seller published in 1941, is thought to have influenced meteorologists on their decision to assign female names to Pacific tropical cyclones.[186] Another example is the hurricane in The Perfect Storm, which describes the sinking of the Andrea Gail by the 1991 Perfect Storm.[187] Hypothetical hurricanes have been featured in parts of the plots of series such as The Simpsons, Invasion, Family Guy, Seinfeld, Dawson's Creek, Burn Notice and CSI: Miami.[185][188][189][190][191][192] The 2004 film The Day After Tomorrow includes several mentions of actual tropical cyclones and features fantastical "hurricane-like", albeit non-tropical, Arctic storms.[193][194]