Independent Components in the Northern Hemisphere Winter: Is the

Journal of the Meteorological Society of Japan, Vol. 85, No. 6, pp. 825--846, 2007

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Independent Components in the Northern Hemisphere Winter: Is the Arctic Oscillation Independent?

Hisanori ITOH Department of Earth and Planetary Sciences, Kyushu University, Fukuoka, Japan

Atsushi MORI Department of Natural Science, J. F. Oberlin University, Tokyo, Japan

and Seiji YUKIMOTO Meteorological Research Institute, Tsukuba, Japan (Manuscript received 12 January 2007, in final form 8 August 2007)

Abstract This paper examines true oscillations in the Northern Hemisphere winter by using independent component analysis (ICA). ICA can distinguish between true and apparent oscillations under the assumption that the true oscillations are mutually independent. Particular attention is paid to the Arctic Oscillation (AO). For this purpose, the NCEP-NCAR reanalysis data (NCEP-NCAR data) and the data of the present climate experiment of the Meteorological Research Institute (MRI data) are used. There may be certain issues in ICA applied to meteorology: the selection of data periods, treatment of noise, and relationship between the number of dimensions in phase space and the number of independent components. We make several considerations and proposals about these issues. ICA should be performed for periods for which the variance is almost the same. Since independent components after whitening are not necessarily uncorrelated with each other under the existence of noise, and the relation between the dimension of phase space and the number of independent components cannot be predetermined, we propose the method for seeking independent components by kurtosis as the most relevant in meteorology. On the basis of the above considerations and proposals, ICA is performed on the NCEP-NCAR data. Independent components are found for the sea level pressure (SLP) and 500 hPa height (Z500) fields. They are the North Atlantic Oscillation (NAO) and the Pacific-North American Oscillation (PNA). Thus, the AO is an apparent mode derived from them. However, since the period of the data is too short, statistical significance cannot be obtained. Then, ICA is performed on the MRI data. Also in this data, the NAO and the PNA-like oscillation are independent for both the SLP and Z500, where the PNA-like oscillation implies that its pattern is somewhat different from the observed PNA pattern. It can be concluded again that the AO is not independent, but this time with statistical significance. Corresponding author: Hisanori Itoh, Department of Earth and Planetary Sciences, Kyushu University, 6-10-1, Hakozaki, Higashi-ku, Fukuoka 8128581, Japan. E-mail: [email protected] ( 2007, Meteorological Society of Japan

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Journal of the Meteorological Society of Japan

Terminology

First, we define the terminology used in this paper. The term ‘‘oscillation,’’ as in ‘‘the North Atlantic Oscillation (NAO)’’ or ‘‘the PacificNorth American Oscillation (PNA),’’ indicates both a spatial pattern and time variation of amplitude. When a spatial pattern is meant exclusively, the word ‘‘pattern’’ is always attached, as in ‘‘the NAO pattern’’ or ‘‘the PNA pattern.’’ A time series of amplitude is generally called a ‘‘signal.’’ Thus, when a signal and the basis vector of a pattern in phase space are denoted as aðtÞ (t: time) and e, respectively, an oscillation can be expressed as aðtÞe. ‘‘Mode’’ and ‘‘component’’ have a similar meaning to ‘‘oscillation.’’ ‘‘System,’’ as in ‘‘the NAO-PNA system,’’ indicates a combination of two oscillations. ‘‘Independence’’ is an important keyword in this paper, and means ‘‘statistical independence.’’ Suppose there are two random variables X1 and X2 (usually time series in this paper), and any information on the value of X1 yields no information on the value of X2 , and vice versa. Then, X1 and X2 are called ‘‘statistically independent.’’ ‘‘Independence’’ has a stronger meaning than ‘‘uncorrelatedness’’ because ‘‘uncorrelatedness’’ is a necessary condition for ‘‘independence’’ but not a sufficient condition. For instance, the two time series data, sinðtÞ and cosðtÞ, are uncorrelated but not independent, because sinðtÞ uniquely determines cosðtÞ, except its sign. Another typical example of uncorrelated but dependent time series is x1 ðtÞ ¼ r1 ðtÞ þ r2 ðtÞ and x2 ðtÞ ¼ r1 ðtÞ r2 ðtÞ, where r1 ðtÞ and r2 ðtÞ are two random numbers with zero mean and same variance, because both x1 and x2 do not hold large (positive or negative) values at the same time. Strictly speaking, ‘‘independence’’ is defined for signals under the above definitions, but expressions such as ‘‘two oscillations are independent’’ are often used in this paper for simplicity. Furthermore, the term ‘‘independent pattern’’ is also used in the sense of the basis vector for an independent signal, although this term appears only in figure captions. One major assumption is that true oscillations have independent signals and vice versa. In other words, apparent oscillations have dependent signals, i.e., a mixture of independent signals. This is a reasonable assumption.

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Introduction

Whether the Arctic Oscillation (AO, or annular mode, Thompson and Wallace 1998) is true or not is one of the major controversial issues in meteorology. Deser (2000), Ambaum et al. (2001) and Itoh (2002, hereafter referred to as I02) claim that the AO is apparent because it can be understood as a combination of the NAO and PNA; the basis vectors representing the NAO and PNA patterns are not orthogonal in phase space (the inner product of the two vectors is not zero), so that Empirical Orthogonal Function (EOF) analysis inescapably produces an apparent mode, i.e., the AO, as the first EOF mode. Further, an additional apparent EOF mode showing Atlantic-Pacific seesaw, which is referred to as the negative correlation mode (NCM) in this paper, simultaneously emerges. Wallace and Thompson (2002) have the following arguments against these studies; first, they present a schematic figure on the phase relationship among the four patterns, AO, NCM, NAO, and PNA, which are located on the same plane in phase space. They point out that this means that the AO-NCM and NAOPNA systems are equivalent in the sense that any points on the plane (any patterns with arbitrary amplitudes) can be expressed as linear combinations of either the AO-NCM system or the NAO-PNA system. In other words, it is impossible to distinguish between true and apparent oscillations; the two-dimensional observed data xðtÞ can be expressed by either of the following two mathematical equations, xðtÞ ¼ aN ðtÞeN þ aC ðtÞeC ;

ð1Þ

xðtÞ ¼ aA ðtÞeA þ aP ðtÞeP ;

ð2Þ

or

where aN , aC , aA , and aP (eN , eC , eA , and eP ) are signals (basis vectors) representing the AO, NCM, NAO, and PNA, respectively. Further, they ascribe no correlation between the Atlantic and Pacific regions to the coexistence of the AO and NCM, whose physical reality is obtained in a PNA having a wavetrain over the Euro-Atlantic region in the regression of the NCM on the 500 hPa height. Thus, they state that the AO-NCM view is an alternative to the NAO-PNA view. Similarly, Christiansen (2002)

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criticizes I02, claiming that the two equations representing the three-point seesaw system, which makes simple considerations about the AO-NCM and NAO-PNA systems in I02, are mathematically equivalent, and thereby, the two cannot be distinguished by observational analyses alone. However, as shown by Mori et al. (2006, referred to as M06) and Itoh (2007), it is now evident that we can determine which of the two systems (AO-NCM or NAO-PNA) is true, even though they are located on the same plane in phase space. It is also clear that the two equations of I02 are not equivalent. The basic idea of the discrimination is that true, i.e., independent oscillations have different probability density functions (PDFs) from those of apparent oscillations. The details of the difference in PDFs will be explained in Section 4. The most systematic method to detect independent signals, i.e., the difference in the PDFs, is independent component analysis (ICA). By using this analysis, true oscillations, i.e., independent components, can be distinguished from apparent oscillations, i.e., dependent components. Indeed, very recently, M06 examined independent components in the sea level pressure (referred to as SLP) using ICA. They concluded that the Aleutian-Icelandic seesaw (AIS) and the oscillation with the cold ocean-warm land pattern (COWL) are independent. However, their results seem to have difficulties with respect to data selection, which will be described in detail in Section 5. In ICA, it is critical to properly select data and to adequately interpret results. To show this is the first purpose of this study. Among several types of ICAs, basic ICA (Hyvarinen et al. 2001) is the simplest one; the number of source signals is equal to that of observed signals (the dimension of phase space and the number of independent components are equal) in the absence of noise. The theory and algorithm of basic ICA are almost complete. Thus, if basic ICA was applicable to meteorological problems, there would be no difficulty. However, we have two major difficulties applying basic ICA to meteorology, of which one is the existence of noise. When there are two independent oscillations a1 ðtÞe1 and a2 ðtÞe2 on the plane in phase space, where

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a1 ðtÞ and a2 ðtÞ represent two independent signals and e1 and e2 represent the basis vectors of two patterns, all trajectories on the plane can be expressed as linear combinations of the two oscillations, xðtÞ ¼ a1 ðtÞe1 þ a2 ðtÞe2 . It is easy to apply basic ICA to this situation; from xðtÞ, we can easily identify a1 and a2 as independent signals and e1 and e2 as associated patterns. However, when noise is introduced, the situation is completely changed; trajectories on the plane may be expressed as xðtÞ ¼ a1 ðtÞe1 þ a2 ðtÞe2 þ e3 ðtÞ, where e3 ðtÞ indicates noise. Basic ICA is not theorized for such situations, and we must use other methods. The second difficulty is the relationship between the dimension of phase space and the number of independent components. Basic ICA assumes them to be the same; however, in meteorology, the relation is unclear or may not even be unique. The second purpose of this study is to make considerations and propose solutions to these problems. Although considerable effort has been devoted in overcoming these difficulties from a generalized viewpoint, we will make further effort from the meteorological standpoint. The third purpose is to identify independent components, i.e., true oscillations, by using ICA on the basis of the abovementioned considerations and proposals. In order to obtain fruitful results from the controversy, it is necessary that a common basis is chosen, the points of contention are clarified, and convincing facts are revealed by using objective methods. The common basis could be that the AO-NCM, NAO-PNA, and AIS-COWL systems are located on the same plane in phase space. Further, we may agree that ICA can be used to estimate whether an oscillation is independent or true. Therefore, it is decisively important to identify independent components by an objective method, ICA. Studies to dynamically elucidate the AO are underway. Therefore, we require observational studies to ascertain whether the AO is true or not; if the AO is apparent, such dynamical studies themselves are unnecessary, and would result in major losses to the meteorological community. In this sense, this kind of observational study is important and pressing. Although a preliminary result of this study is presented in Itoh (2007), detailed consid-

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Fig. 1. (a) Regression of the SLP northward of 20 N on PC1 in the Atlantic sector in the NCEPNCAR 6-month data. (b) As in (a) but in the Pacific sector. The solid and dashed lines indicate positive and negative values, respectively. The contour interval is 1.0 hPa.

erations and results will be described in this paper. 3.

Data

Two datasets are used. One is the monthly mean of the the National Centers for Environmental Prediction-National Center for Atmospheric Research reanalysis (hereafter referred to as NCEP-NCAR data). In addition to the SLP, the 500 hPa height (referred to as Z500) is also used. They are the gridded data of a resolution with 2.5 longitude by 2.5 latitude and cover 53 years from 1948 to 2000. Anomalies are defined as deviations from the 53-year average of each calendar month, for which EOF analysis and ICA are performed. However, the period of this data is too short to obtain statistical significance. Therefore, we also use the monthly-mean data of the present climate experiment using a climate model (MRI-CGCM2) of the Meteorological Research Institute (referred to as MRI data). This experiment is integrated for 350 years with CO2 concentration of 348 ppm (Yukimoto et al. 2001). The SLP and Z500 are again used. The resolution is 2.8125 longitude by approximately 2.8 latitude (Gaussian latitude). Anomalies are also defined as deviations from the 350-year average of each calendar month. All analyses are performed for regions northward of 20 N. EOF analysis is carried out

based on covariance matrices on 10 longitude by 5 latitude grids for the NCEP-NCAR data, and on 5.625 longitude by the original latitude grids for the MRI data. Sectorial calculations, in which the hemisphere is divided into the Europe-Atlantic sector (hereafter referred to as the Atlantic sector) and the Pacific sector, are also carried out. The former is defined as 90 W–90 E including 0 , and the latter as 90 E–90 W including 180 . In this analysis, the length of the longitudinal grid for the NCEP-NCAR data is 5 . Hereafter, the spatial pattern of the n-th mode in the EOF analysis is called EOFn, while the time coefficient is named as PCn. Hemispherical EOFs 1 and 2 for the NCEPNCAR SLP 6-month (Nov.–Apr.) data are illustrated in Fig. 1 of I02. As previously mentioned, Figs. 1a and 1b of I02 are called the AO and NCM, respectively. Figure 1 in this paper shows hemispherical patterns regressed on PC1 of the Atlantic and Pacific sectors. Figure 1a clearly indicates the NAO pattern. Figure 1b illustrates an oscillating pattern of the Aleutian low; however, the PNA pattern appears in regression with Z500, so that we will call this pattern the PNA pattern. Note that this pattern has signals in the polar region. Leading EOF patterns for other periods in winter (e.g., Dec.–Feb. and so on) are almost the same as those for the 6-month data.

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Three-point seesaw system and the procedure of independent component analysis

In the following consideration, since it is simple to use the three-point seesaw model (I02), it will be described first. The AO has three centers of action, i.e., the north polar (N), the Atlantic (A), and the Pacific (P) regions. Furthermore, the NCM has two centers of action, A and P. Then, the time series at these three points (regions) are given as follows: xN ðtÞ ¼ 2re1 ðtÞ;

ð3aÞ

xA ðtÞ ¼ re1 ðtÞ þ re2 ðtÞ;

ð3bÞ

xP ðtÞ ¼ re1 ðtÞ re2 ðtÞ;

ð3cÞ

where re1 ðtÞ and re2 ðtÞ express the AO and NCM time series, respectively. For simplicity, it is assumed that all source signals (including those appearing later) have zero average and same variance, and that they are independent of each other. EOFs 1 and 2 in this model are ð2; 1; 1Þ t and ð0; 1; 1Þ t , respectively, where ð Þ t stands for transpose. These EOFs obviously correspond to the AO and NCM, which will be designated as ‘‘AO’’ and ‘‘NCM,’’ respectively, to discriminate the AO and NCM in this simple model from those in observations. Next, the following time series are considered, where rn ðtÞ is similar to ren ðtÞ above. yN ðtÞ ¼ r1 ðtÞ þ r2 ðtÞ;

ð4aÞ

yA ðtÞ ¼ r1 ðtÞ;

ð4bÞ

yP ðtÞ ¼ r2 ðtÞ:

ð4cÞ

This model expresses two seesaw oscillations between N and A (referred to as ‘‘NAO’’), and between N and P (referred to as ‘‘PNA’’). However, the EOF analysis results in the same EOF patterns as (3). This can be mathematically explained. That is, (4) can be transformed as follows: yN ðtÞ ¼ ½r1 ðtÞ þ r2 ðtÞ;

ð5aÞ

yA ðtÞ ¼ 0:5½r1 ðtÞ þ r2 ðtÞ 0:5½r1 ðtÞ r2 ðtÞ;

ð5bÞ

yP ðtÞ ¼ 0:5½r1 ðtÞ þ r2 ðtÞ þ 0:5½r1 ðtÞ r2 ðtÞ:

ð5cÞ

Thus, the apparent ‘‘AO’’ and ‘‘NCM’’ emerge

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from the ‘‘NAO’’ and ‘‘PNA’’ that vary independently. This is due to the fact that the ‘‘NAO’’ and ‘‘PNA’’ patterns are not orthogonal in phase space, but have a normalized inner product of 0.5. Figure 2a shows the positions of the four modes and a scatter plot based on (4). These positions are determined by the inner products among the four modes. For the scatter plot, the sum of two uniform random numbers is used, because this random number gives similar magnitudes of kurtosis to the real world (cf., Figs. 2c, 5, and 11). If uniform random number were used, the scatter plot area would indicate a rhombus. When the ‘‘AO’’ and ‘‘NCM’’ are independent, that is, (3) holds true, the scatter plot area using uniform random number would be rectangular. From this fact, (3) and (4) yield the same EOF, but these amplitude distributions of PC1 and PC2, i.e., PDFs, are different. In other words, PC1 and PC2 are independent in (3), while they are not in (4). This can be easily understood from (5); neglecting the normalization factor, PC1 ¼ r1 ðtÞ þ r2 ðtÞ and PC2 ¼ r1 ðtÞ r2 ðtÞ. Thus, they are uncorrelated but not independent. This characteristic enables us to principally distinguish between the ‘‘NAO-PNA’’ and ‘‘AONCM’’ systems. The most systematic method to distinguish different PDFs is ICA. Among several methods of ICA, we adopt a method using kurtosis as an estimating function. The relative superiority of this method in meteorology will be described in Section 6. Since the reader may not be familiar with ICA, it will be briefly described. The details are given in, e.g., Hyvarinen et al. (2001) and M06. The central limit theorem states that the distribution of a sum of independent random variables, under certain conditions, tends toward a normal (Gaussian) distribution. Thus a sum of two independent random variables usually has a distribution that is closer to Gaussian than the original random variables. The strict statement is found in Itoh (2007). Furthermore, M06 proved that it is a necessary condition for the independence of a certain component that it has the extremum of nonGaussianity. By this characteristic, we can distinguish between independent components and their combinations. There are various quantitative measures of non-Gaussianity; one

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Fig. 2. Procedure of independent component analysis. This example is based on the three-point seesaw system of Eq. (4), where the ‘‘NAO’’ and ‘‘PNA’’ are independent. (a) Locations of the four patterns on the two-dimensional phase space, and a scatter plot. Since the normalized inner product between the ‘‘NAO’’ and ‘‘PNA’’ patterns is 0.5, the two lines representing these patterns intersect at 60 . As the random number for the scatter plot, the sum of the two uniform random numbers is used. (b) Whitened scatter plot of (a). (c) Distribution of kurtosis calculated from (b). The abscissa and ordinate are angle and kurtosis, respectively. The 0 –180 and 180 –360 lines are the same since only the sign of patterns is different.

of popular is kurtosis defined as P the most 4 4 ðx mÞ /ðNs Þ 3, where xi , m, s, and N dei i note each sample, the average, standard deviation, and sample number, respectively. Since kurtosis is zero for the normal distribution, independent components are obtained as those whose kurtoses of PDFs are far from zero. We will explain how independent components are obtained more specifically, on the basis of Fig. 2. As already noted, Fig. 2a shows a scatter plot when the ‘‘NAO’’ and ‘‘PNA’’ are independent. However, we cannot use this configuration directly for ICA, because, for instance, when the maximum amplitude of the ‘‘AO’’ is projected on the ‘‘NAO,’’ it is not the true amplitude of the ‘‘NAO’’ but a larger amplitude. Therefore, whitening in which EOF1 and EOF2 have the same amplitude (Fig. 2b) is necessary. Now, we observe that the projected maximum amplitude of the ‘‘AO’’ on the

‘‘NAO’’ is the true amplitude of the ‘‘NAO.’’ Independent components can be investigated from this configuration. Figure 2c illustrates kurtosis as a function of angle (i.e., pattern) from the ‘‘AO’’ pattern. Since the sum of two uniform random numbers is used, all kurtoses are negative. Among them, the ‘‘NAO’’ and ‘‘PNA’’ comprise minimum kurtoses. Thus, the independent components can be successfully obtained. Here, we provide one comment to confirm our analysis. Prior to whitening, the ‘‘NAO’’ and ‘‘PNA’’ patterns that have spatial correlation are not orthogonal. This configuration is needed to give apparent oscillations. However, following the whitening, they are orthogonal (135 and 45 in Fig. 2c). The orthogonality after the whitening is a necessary condition for independence. Note that the orthogonality before and after the whitening is different.

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Fig. 3. (a) Normal distributions with the standard deviation of 1 (solid line) and 2 (dashed line). (b) Distribution of the sum of the two distributions in (a) (solid line). For comparison, a normal distribution with the same standard deviation is shown by the dashed line.

5.

Importance of data selection in independent component analysis

We must be cautious about the data selection for ICA. Otherwise, we unavoidably lead to misinterpretation. Suppose there is a dependent component (component A) whose PDF shows a normal distribution for every season. This means that component A is a sum of many independent components. Furthermore, for simplicity, we consider only two seasons—winter having a large amplitude and summer having a small amplitude. However, when these two sets (winter and summer sets) are added, a curious thing occurs; the PDF has positive kurtosis. The solid line in Fig. 3b represents the distribution constructed from the sum of two sets with normal distributions whose standard deviations are 1 and 2 (Fig. 3a). It obviously shows positive kurtosis, which is calculated as 1.0. Next, suppose there is an independent component (component B) whose amplitude PDF has negative kurtosis. Moreover, as in component A, it has large amplitudes in winter and small amplitudes in summer. Then, the distribution of all the season samples approaches the normal distribution. When we apply ICA on these samples ‘‘honestly,’’ it determines that component B is not independent, whereas component A is. We will further show this contradiction using the three-point seesaw system of (4). The am-

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Fig. 4. Kurtosis as a function of angle calculated from the artificial data comprising ‘‘winter’’ and ‘‘summer’’ whose amplitudes are 2 and 1, respectively. Others are the same as in Fig. 2c.

plitudes of 1 and 2 are given random number samples in ‘‘summer’’ and ‘‘winter,’’ respectively. Then, by making EOF analysis on the sum of the two sets, we calculate the kurtosis as a function of angle (Fig. 4). The line shape is similar to that in Fig. 2c, but the kurtoses are positive at all angles. Consequently, ICA judges that ‘‘NAO’’ and ‘‘PNA’’ are not independent since their kurtoses are close to zero, whereas ‘‘AO’’ and ‘‘NCM’’ are independent. This contradiction occurs because winter and summer samples are added, even though their means and variances are largely different, that is, they belong to different populations. If ICA is made on each population, we can obtain accurate results. However, if samples belonging to different populations are treated together, we cannot obtain accurate results. In extreme cases, contradictory results may even be obtained. The above example is typically unfavorable. We have assumed so far that samples in winter compose one population, while those in summer compose another. Practically, it was unknown as to which combinations of months are regarded as one population. However, the answer is simple; they are combinations of months whose variances are similar. Hence, the variances of the SLP are calculated for

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Table 1. Variances (hPa 2 ) of the SLP in each month for the NCEP-NCAR reanalysis data (second line) and the MRI data (third line). Month

Jan.

Feb.

Mar.

Apr.

May

June

July

Aug.

Sep.

Oct.

Nov.

Dec.

NCEP

21.4

22.9

16.2

9.1

6.1

4.8

4.3

5.3

6.6

8.2

13.6

17.4

MRI

19.7

21.1

18.1

11.8

6.9

4.9

3.7

3.8

5.7

8.6

13.5

17.1

Table 2. Kurtosis of the SLP for various periods for the NCEP-NCAR reanalysis data (second and fifth lines) and the MRI data (third and sixth lines). Month NCEP MRI Month

Dec.–Feb.

Jan.–Mar.

Dec.–Mar.

Nov.–Mar.

Nov.–Apr.

0.057

0.102

0.099

0.152

0.253

0.004

0.045

0.009

0.050

0.109

Sep.–May

all months

May–Oct.

June–Aug.

NCEP

0.609

0.954

0.341

0.051

MRI

0.478

0.889

0.483

0.143

each month. The result is shown in the upper part of Table 1. This table indicates that variances in winter are about 5 times larger than those in summer. It makes no sense that samples with amplitudes of such a wide range are treated together. When we consider periods in winter to be as long as possible, the periods of December–March or November–March are relatively relevant. We must ascertain whether average kurtosis is nearly zero when months whose variances are close to each other are selected. We choose several combinations of months, calculating the average kurtosis from the gridpoint data. For comparison, 6-month and all-month data are also chosen. Since there are large variations in amplitude from one gridpoint to another, the magnitude of the SLP anomaly is normalized by its standard deviation. Namely, when p denotes the pressure anomaly, i the gridpoint index, spi the standard deviation of pi , a PDF is constructed from pi /spi of all gridpoints of 5 latitude by 10 longitude northward of 20 N, where the frequencies are multiplied by a factor of cos f (f: latitude). The second and fifth lines of Table 2 show the result. It is found that the kurtoses of Dec.–Feb. or June–Aug. data are small; they can be regarded as one population. Kurtosis systematically increases

with the range of the variance. It should be avoidable to use all-month data. However, there may be cases in which we want to treat samples beyond one population, for instance, due to the paucity of the sample. In such cases, however, we should not conclude that components with kurtosis far from zero are automatically independent. Rather, by focussing on the continuity of results, i.e., by tracing the systematic displacements of kurtosis distributions with increasing sample numbers, independent components can be correctly identified. Because M06 performed ICA on all-month data, their results have the abovementioned difficulty. Independent components may not be correctly obtained. In this paper, ICA will be performed on properly selected data. In the case when the data are treated beyond one season, independent components will be systematically traced. 6.

Difficulties of independent component analysis in meteorology and some proposals

There may be two major difficulties in applying ICA to meteorological data. In order to overcome both the difficulties, considerable effort has been devoted. However, consensus has

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not been reached in the general approach. Here, we will make some considerations and proposals to these two problems from the meteorological standpoint. One difficulty is that the relationship between the dimension of phase space in analysis and the number of independent components is unclear. In basic ICA, it is assumed that they are the same. However, the relation between them is unclear, or may not be unique, in meteorology. Moreover, even though there exist independent components under ordinary circumstances, they might not be successfully identified due to large noise. Therefore, a method that does not assume a relationship between them is suitable in ICA used in meteorology. The second difficulty is that the discrimination of independent components from their combinations is not perfect. For example, even when the NAO and PNA are supposed to be independent, components on the plane spanned by the NAO and PNA do not always appear as their combinations. It is certainly possible for other patterns that cannot be regarded as their combinations, mainly noise, to appear on the plane. Further, with respect to the first difficulty, it is not guaranteed at all that any component can be expressed as a linear combination of independent components. A typical example is the case where the number of independent components is less than the dimension of phase space. The following equation system may be regarded to represent a model including noise: yN ðtÞ ¼ r1 ðtÞ þ r2 ðtÞ þ ar3 ðtÞ;

ð6aÞ

yA ðtÞ ¼ r1 ðtÞ þ ar4 ðtÞ;

ð6bÞ

yP ðtÞ ¼ r2 ðtÞ þ ar5 ðtÞ;

ð6cÞ

where yN , yA , yP , r1 , and r2 are the same as in (4). r3 , r4 , and r5 are other random numbers with unit variance, and a stands for their amplitude. This model expresses that, in addition to the ‘‘NAO’’ and ‘‘PNA,’’ other patterns randomly appear in the three-dimensional phase space. Making EOF analysis, we obtain 3 þ a 2 , 1 þ a 2 , and a 2 as eigenvalues, where we assume the variance of each random number as unity for simplicity. The associated eigenvectors are ð2; 1; 1Þ t , ð0; 1; 1Þ t , and ð1; 1; 1Þ t . Thus, the same EOFs 1 and 2 as (4) are ob-

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tained, even though a increases to a certain extent. However, this result is obtained under the assumption that the amplitude of all the components is a. In general cases, needless to say, EOFs in (6) are different from those in (4). When noise is included, whether independent components are well found on the plane spanned by EOF1 and EOF2 is an issue. If the variances of noise are known, the procedure for obtaining true independent components is simple; we can regard true independent components that add noise as fictitious ‘‘independent components,’’ obtaining them by the use of basic ICA. Then, noise can be removed from them (Hyvarinen et al. 2001). However, the variances of noise are generally unknown in meteorology, therefore this method cannot be applied. An alternative method must be devised. Before considering a different method, we should note an important characteristic in ICA including noise. The abovementioned method has the property that fictitious ‘‘independent components’’ are uncorrelated with each other. Therefore, patterns associated with true independent components without noise are not orthogonal to each other after whitening is conducted. This is the major difference between ICAs with and without noise. Next, we quantitatively consider departures from the orthogonality, using the ‘‘NAO’’ and ‘‘PNA’’ patterns. This is equivalent to the consideration on the departure from 45 of the ‘‘PNA’’ pattern after the whitening. The ‘‘PNA’’ pffiffi pattern is expressed as 23 ; 12 on the plane of EOFs 1 and 2, as shown in Fig. 2a. Multiplying them by the whitening factors (reciprocals pffiffiffiffiffiffiffiffiffiffiffiffiffiffi of the square root of eigenvalues), 1/ 3 þ a 2 and pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1/ 1 þ a , respectively, we obtain the position of the ‘‘PNA’’ pattern after the whitening. Then, the angle a from EOF1 can be calculated as pffiffiffi 3 1 1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi tan a ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 þ a2 2 3 þ a2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 þ a2 ¼ pffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð7Þ 3 1 þ a2 When a is 0.4, 0.6, 0.8, and 1.0, a ¼ 44:7 ; 43:2 ; 41:3 , and 39.2 , respectively. As a increases, the angle straddling EOF1 decreases slightly. This is because the noise level of

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EOF2 is relatively larger than that of EOF1. Since these values are theoretical ones (i.e., values when the sample number is infinite, and there is no computational error), actual values are considerably larger. Thus, when noise is included, patterns associated with independent components are placed with the angle straddling EOF1 less than 90 . Even in such cases, the angles of kurtosis extrema (minima when kurtosis is negative or maxima when it is positive) correctly indicate those of independent components under the assumption that the PDF of noise is the normal distribution; the kurtosis of the normal distribution is zero, so that noise has no influence on the angle of extrema. Thus, we can obtain independent components by kurtosis extrema even when noise is included. From the above considerations, a method for inspecting independent components by kurtosis is considered to be the best ICA in meteorology. In this method, the number of kurtosis extrema is not determined by the dimension of phase space in analysis. The orthogonality among the directions of kurtosis extrema is not certain. Deviations from orthogonality are dependent on two parts: the noise level and the random part resulting from the paucity of the sample. Although it is impossible to separate the two parts, the latter seems to be larger than the former, because the deviations due to the noise are very small unless the noise level is high. Estimating functions other than kurtosis are often used because kurtosis has the drawback that it is sensitive to outliers. However, the data used here are the reanalysis data and output of the numerical model. These data do not inherently include outliers. Thus, there is no drawback in using kurtosis in ICA. There is an additional difficulty, which is related to the first one. It is that we cannot predetermine the number of dimension of phase space required to obtain independent components. The higher the number of dimension is, the better the result seems. However, we cannot simply increase the dimension due to a noise problem associated with whitening; in the whitening, no matter how small the variance of a particular direction is, that direction is relatively increased to obtain the same magnitude as EOF1. This unavoidably causes large noises in that direction. Therefore, we have the

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maximum dimension to obtain independent components for which the influence of noise is relatively small. Then, we propose the following method to determine the maximum dimension: by increasing the space dimension one at a time, we seek independent components. Up to certain dimensions, the independent components gradually change, by maintaining the orthogonality between the patterns associated with the independent components (after the whitening). This is also valid for cases in which a new independent component is found. However, at certain dimensions, the angles between the patterns of ‘‘independent components’’ abruptly deviate from the right angle. This is considered to be due to strong effects of noise. The inspection should be terminated before this space dimension. As a result, we can obtain independent components for which the influence of noise is relatively small. Since the index of the deviation from orthogonality is objective, the method to obtain independent components by kurtosis proves to be superior in this respect as well. 7.

Independent components from the observed data

7.1 Independent components First, we consider independent components on the plane spanned by EOFs 1 and 2 for the NCEP-NCAR SLP data. Based on the consideration in Section 5 that the homogeneity of data is important, analyses are first made for the 3-month (Dec., Jan., and Feb.) data. The thick solid line in Fig. 5 shows the kurtosis as a function of angle from EOF1. It is clear that the overall values are negative. Since this data is almost homogeneous, components with kurtosis far from zero, i.e., kurtosis minima, should be independent. Minima are located at 64 and 167 . The difference of the two angles is 103 (77 ), which is almost a right angle. Thus, these two are independent on the two-dimensional plane. Note that the angle between the two minima straddling EOF1 is less than 90 . This is consistent with the prediction in Section 6. These patterns illustrated in Fig. 6 are similar to those in Fig. 1, corresponding to the NAO (b) and PNA (a) patterns. The spatial correlation (strictly speaking, normalized inner

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Fig. 5. Kurtosis as a function of angle on the plane spanned by EOF1 and EOF2 for the NCEP-NCAR SLP data. The abscissa is the angle from EOF1. The six lines show the difference in the data period: Dec.–Feb. (thick solid), Dec.– Mar. (thick dashed), Nov.–Mar. (thick dotted), Nov.–Apr. (thin solid), Sep.– May (thin dashed), and all months (thin dotted).

product) between this NAO pattern and Fig. 1a is as high as 0.99. Although the PNA pattern exhibits a more pronounced seesaw pattern with the Icelandic low than that in Fig. 1b, the

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spatial correlation is also as high as 0.98. In short, the NAO and PNA are independent, while the AO and NCM are expressed as linear combinations of the NAO and PNA. Since we have to determine whether the AO and NCM are independent or not, the answer that they are not is final. Thus, there is no room for further questions. However, it is uncertain whether the NAO and PNA are truly independent, because they might be projections of independent components located in the direction off the plane. Then, we further pursue independent components by increasing the space dimension. The inspection of independent components in threeor higher dimensional space cannot be carried out in the same manner as in the twodimensional space where the kurtosis is calculated as a function of angle. We adopt a method to seek kurtosis minima as a nonlinear optimization problem from various, say 20 000, initial angles obtained from random numbers. Results for the 3 month data are not stable because selected patterns and their numbers considerably vary depending on the space dimension. The sample number ð53 3 ¼ 159Þ may be too small. We then increase the sample number. However, the homogeneity of the data is lost with the increase in the analyzed months, so that we trace the change in the distribution of kurtosis. The result is shown in Fig. 5. Although

Fig. 6. Patterns corresponding to (a) 64 and (b) 167 in the thick solid line in Fig. 5. Kurtoses (percent variances) are 0.46 (16.5%) for (a) and 0.48 (20.7%) for (b). The contour interval is arbitrary, but values of the regression on PC1 and PC2 are used. (From Itoh 2007)

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Fig. 7. Independent patterns in the 4 dimensional phase space spanned by EOF1-EOF4 in the NCEP-NCAR SLP 6-month data. Angles between two patterns after the whitening are 85.7 between a and b, 78.4 between a and c, and 83.5 between b and c. Kurtoses (percent variances) are 0.55 (11.7%), 0.50 (10.8%), and 0.28 (17.7%) from the left.

EOFs are different for different data, EOFs 1 and 2, i.e., the AO and NCM patterns are almost the same. Thus, neglecting minor differences in EOFs, all distributions of kurtosis are presented in the same figure. Kurtosis systematically increases with the sample number, while the line shapes remain unchanged. From this characteristic, we can identify independent components even though the sample number increases, i.e., the homogeneity of the data is lost. They must be components with kurtosis minima, and not those with kurtosis far from zero. In particular, for the 9-month or all-month data, the components with values far from zero are well-mixed patterns, which are contrary to independent components. Even expanding data to at least 6 months, results do not essentially change. Then, the following analyses will be carried out for the 6-month data. Considering the 6-month data in Fig. 5, minima are located at 67 and 165 . The patterns with these minima are almost the same as those for the 3-month data, corresponding to the PNA and NAO patterns, respectively (not shown). We seek independent components by increasing the space dimension. Since the influence of noise is observed in 5 dimensions, the inspection is terminated in 4 dimensions. Three independent components are found, and their patterns are shown in Fig. 7. Although Fig. 7a indicates the PNA pattern, large amplitudes are seen in Eurasia, and the spatial correlation

with the PNA in 2 dimensions (2D) is only 0.78. Considering this result as well as results shown later, this mode may be somewhat contaminated by noise. Figure 7c can be identified as the NAO pattern having a spatial correlation of 0.87 with the NAO in 2D. Although Fig. 7b is not elucidated, we call it the AtlanticEurasian (AE) pattern for later convenience. Since the PNA and NAO can be continuously traced from the two-dimensional plane, these two are independent components located almost on the plane. Next, we analyze Z500. The inspection is terminated in 5 dimensions. The patterns of independent components are shown in Fig. 8. Figure 8a indicates the PNA pattern, while Fig. 8b can be identified as the NAO pattern, though it extends to Eurasia and the Pacific. Figure 8c corresponds to the AE pattern. Thus, similar patterns are obtained in the Z500 as in the SLP. The patterns regressed on the SLP time coefficients are evidently similar to Fig. 8. Correlations between the SLP and Z500 time coefficients are 0.64 for the PNA, 0.89 for the NAO, and 0.88 for the AE. Although the PNA correlation is not high, the correlation with PC1 of the SLP Pacific sector is as high as 0.80. These further confirm the result of the SLP field. Vertically-coupled independent components between the SLP and Z500 can be obtained by analyzing the normalized SLP and Z500 as one data, where the normalization factor is the total variance at each level. Six patterns (3 in-

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Fig. 8. Independent patterns in the 5 dimensional phase space spanned by EOF1-EOF5 in the NCEP-NCAR Z500 6-month data. Angles between two patterns after the whitening are 74.2 between a and b, 87.7 between a and c, and 79.1 between b and c. Kurtoses (percent variances) are 0.56 (11.5%), 0.55 (11.8%), and 0.45 (12.8%) from the left.

dependent components 2 levels) found in 5 dimensions, which is the relatively noise-free largest dimension, are highly correlated with corresponding patterns obtained at each level (Figs. 7 and 8). All the correlations exceed 0.97, except for the SLP PNA pattern with the correlation of 0.75. Since the vertically-coupled SLP PNA pattern has almost no amplitude over Eurasia (not shown), as in Figs. 1b and 6a, Fig. 7a may be influenced by noise, as already mentioned. 7.2 Significance test Two tests are required. One is a significance test for kurtosis minima themselves, that is, a test where the probability to accidentally select such minimum values is very less. The second is the significance of the directions (i.e., patterns) of kurtosis minima in phase space. In other words, we prove that the directions of kurtosis minima significantly differ from the directions of the AO or NCM patterns. In these tests, the difference in the variance from month to month should be reflected. We will explain the tests in 2 dimensions; however, extending to higher dimensions is easy. The first test may be designed as follows: Time series produced by the Markov process are given PC1 and PC2. Then, we make a scatter plot of PC1 and PC2, and calculate kurtosis at each angle, selecting the minimum value. We consider this value to be obtained by chance. Compared with this, when the kurtosis minima analyzed in the real data are small, the

minima are significant. In practice, we consider K M samples for each PC as follows: AðmÞrm; t

ðm ¼ 1; . . . ; M; t ¼ 1; . . . ; KÞ;

where K and M denote total years and months used in the analysis, respectively, and AðmÞ and rm; t are the amplitude of month m and time series produced by the Markov process, respectively. AðmÞ may be given by the square root of values in the second line in Table 1.1 The Markov process can be written as rm; t ¼ rrm1; t þ em; t ;

ð8Þ

where r is the lag-1 autocorrelation of PC (0.411 for the 6-month PC1) and em; t indicates normal random numbers with a variance of ð1 r 2 Þ. A time series having M samples is constructed for each t, and consequently K time series are generated. On repeating this way many times (B times), we make the PDF of kurtosis minima produced by chance. For instance, comparing a value at 5 percentile of this PDF with the analyzed kurtosis minima, we can estimate 95% significance. Here, we adopt 10 000 as B. The results do not reveal significance for the 3-month data as well as the 6-month and allmonth data. In our opinion, significant results 1

It may be more suitable that AðmÞ is given by amplitudes of PC1 and PC2 in each month and subsequently whitening is performed. However, since the amplitudes of PC1 and PC2 for each month are calculated from small sample numbers, it does not have high reliability.

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cannot be obtained from the observed data irrespective of analysis methods. For the second test, the bootstrap method is used. More specifically, the moving block bootstrap method should be adopted (e.g., Davison and Hinkley 1997), since each sample is not independent in this case. In this method, samples are not drawn one at a time, but blocks of fixed length ðkÞ are drawn randomly. Considering the ratio of the sample number to the effective sample number, which can be estimated as ð1 þ rÞ/ð1 rÞ, we set k to 3, though this estimation is slightly conservative. For instance, for the 3-month data, Dec., Jan., and Feb. data in the same winter form one block such that they cannot be divided. Thus, the independent sample number is 53. From these blocks, we draw 53 samples that permit overlapping, which comprises one bootstrap sample. Then, we calculate angles with the kurtosis minima from this bootstrap sample. By repeating it B times, we make the frequency distribution of the angles with kurtosis minima. Significance can be determined from frequencies at the angles of the AO and NCM patterns in this figure; when the frequencies are zero or negligible, the angles with kurtosis minima significantly differ from those of the AO or NCM patterns. The results of the second test, as a matter of course, do not show significance.

8.

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Independent components from the numerical experiment data

Significance can never be obtained from the observed data because the data period is short. We then use numerical experiment data, seeking independent components and making significance tests. The data used is the MRI data. First, it is necessary to grasp the characteristics of this data. It is important to recognize differences from the real data. The lower parts in Tables 1 and 2 show the variance of the SLP for each month and average kurtoses for several periods, respectively. Compared with the NCEP-NCAR data, the variances in March and April are large, while those in January and February are small; the variance in winter is considerably homogeneous. On the basis of this, the average kurtoses are close to zero for all the winter periods. Next, EOF analysis is performed on the SLP. Here we will show the results of the 4-month data for later convenience, but similar results are obtained from the 3-month to 6-month data. EOFs 1 and 2 are shown in Figs. 9a and b, respectively. Since the overall features are similar to the observations, they can be identified as the AO and NCM patterns. However, the positions of the center of action in the Pacific, for both EOFs 1 and 2, are shifted south-

Fig. 9. Patterns of (a) EOF1 and (b) EOF2 in the MRI SLP 4-month data. Their percent variances are 27.9% and 12.8%. Amplitudes are obtained by the regression on PC1 and PC2. The contour interval is 1.2 hPa.

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Fig. 10. (a) Regression of the SLP northward of 20 N on PC1 in the Atlantic sector in the MRI 4-month data. (b) As in (a) except in the Pacific sector. Others are the same as in Fig. 1.

eastward, in comparison with the observations. Note that this feature will have various influences on the following analyses. One more major difference from the real data is that there are coherent variations between the Atlantic and Pacific sectors to a certain extent. This can be clarified by calculating the correlation coefficients between the two regions (not shown). Consequently, by making EOF analysis in the Atlantic (Pacific) sector, and regressing the Northern Hemisphere on PC1 of this analysis, we see patterns extending into the Pacific (Atlantic) sector. This is shown in Fig. 10. The reason for the occurrence of coherent variations between the Atlantic and Pacific sectors may be that the resolution of this model is as relatively coarse as T42 (triangular truncation with maximum wavenumber 42). Although the pattern in Fig. 10a is different from the observed NAO pattern, it will be called the NAO pattern due to the fact that localized patterns do not appear in this experiment. Note that relative magnitudes of the Atlantic and Pacific regions are largely different between Figs. 10a and 9a. It should again be pointed out that the center of action in the Pacific in Fig. 10b is shifted southeastward. Hence, the regressed pattern of Z500 on PC1 is not the PNA pattern (not shown).2 This is one defect in this experiment. Thus, the pattern of Fig. 10b cannot be called the PNA pattern, but we may name it as the Pacific-Arctic Oscillation

because this pattern exhibits a seesaw between the Pacific and Arctic regions. However, from the consistency of the description in this paper, it will be referred to as the ‘‘PNA’’ pattern. Independent components are searched on the plane spanned by hemispherical EOFs 1 and 2. The distributions of kurtosis for various periods are shown in Fig. 11. All the distributions have minima at similar angles. By comparing Fig. 11 with observations (Fig. 5), we can find two major differences. The four lines except those for the 9-month and allmonth data, are close to each other, indicating that the kurtosis is almost negative. This is consistent with the fact that for these four periods, the variances are similar to each other, and the average kurtoses are almost zero. The other difference is that dips associated with the minima of the ‘‘PNA’’ pattern decreases rapidly with increase in data periods. The reason for this is not clear, but it may be because the structure of the ‘‘PNA’’ pattern in this experiment is different from observed one, as stated above. 2 Although a glimpse of the PNA pattern is discernible, a seesaw with the East Siberian Sea is considerably stronger. On the other hand, the regression of the Z500 on PC1 in the Pacific sector shows the PNA pattern. This means that EOF1 of the SLP and that of the Z500 do not necessarily correspond to each other.

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Fig. 11. As in Fig. 5 except for the MRI SLP data.

The patterns at the kurtosis minima for the 4-month data are shown in Fig. 12. They are similar to those in Fig. 10. The spatial correlation is 0.95 between Figs. 12a and 10b, and 0.98 between Figs. 12b and 10a. It can be said that they are the NAO and ‘‘PNA’’ patterns. Next, significance is tested, by using the autocorrelation of PC1 (0.343 for the 6-month data).

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First, the test for kurtosis minima is conducted. Data having 99% significance are the 3- to 6-month data for the NAO, and the 3- to 5-month data for the ‘‘PNA.’’ Note that the 9month and all-month data never show significance, therefore the increase in the sample number does not necessarily lead to an increase in significance for such heterogeneous data. The result for the 4-month data, which is the most significant, is shown in Fig. 13. The probability of accidentally selecting each minimum value is 0.00% for the NAO and 0.01% for the ‘‘PNA.’’ Therefore, the minimum values of kurtosis for both the components are 99% significant. The second test, i.e., the test for angles with kurtosis minima is conducted. Since the 4month data is tested, the moving block bootstrap method of k ¼ 4 (one winter data form one block) is used, although k ¼ 4 is considerably conservative. The results are shown in Fig. 14. The probability that the NAO takes minima is unity, and its overlap with the AO (angle of 0 ) is almost negligible. The NAO is therefore a significant independent component, being clearly distinguished from the AO. The probability that ‘‘PNA’’ takes minima is 87%. For the remaining 13%, it does not take minima. However, it never overlaps with the NCM

Fig. 12. Independent patterns on the two-dimensional plane spanned by EOF1 and EOF2 in the MRI SLP 4-month data. They are the patterns at (a) 51 and (b) 156 corresponding to the minima in the thick solid line in Fig. 11. Kurtoses (percent variances) are 0.44 (21.7%) for (a) and 0.59 (26.7%) for (b).

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Fig. 13. Relative frequency of kurtosis minima constructed as follows: First, we make 350 4 (months) twodimensional samples corresponding to PC1 and PC2 by the Markov process, and calculate kurtosis minima. Then, by repeating this 10 000 times, we obtain the relative frequency. The bin width is 0.05. The amplitudes (square roots of the variances in Table 1) for December to March for the MRI SLP data are multiplied for each group of 350 out of 1440 samples. The vertical lines indicate the two minima of kurtosis, and the numerals express values of the distribution function (i.e., integrated values from y to each minimum value).

(angle of 90 ), so that it is impossible for the NCM to be independent. For the 4-month data having the highest significance, we seek independent components, extending the space dimension. Other period data (the 3-, 5-, and 6-month data) yield similar results. We can stably extend the space dimension. The number of independent components gradually increases; two for 3 through 5 dimensions, and three for 6 dimensions. However, in the case of 6 dimensions, the influence of noise is observed, so that we may terminate the inspection in 5 dimensions. Anyway, since we do not require many components for the present purpose, we refer to only two main components. They are almost the same as those in Fig. 12 up to 5 dimensions. Thus, we observe that independent components are the NAO and ‘‘PNA,’’ even when the space dimension is expanded.

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Fig. 14. Relative frequency of angles with kurtosis minima. This figure is constructed by using the moving block bootstrap method with k ¼ 4, and repeating the calculation of the angles with kurtosis minima 10 000 times. The bin width is 5 , and numerals above the two lines show accumulated relative frequencies integrated over the ‘‘PNA’’ and NAO regions.

Finally, independent components are explored in the Z500 for the 4-month data. The number of independent components gradually increases with the space dimension; one for 2 and 3 dimensions, two for 4 dimensions, and three for 5 and 6 dimensions. Although there are four components for 7 dimensions, the orthogonality among the patterns of independent components is not preserved, therefore 6 dimensions should be the largest dimension for the inspection. Three patterns for 6 dimensions are illustrated in Fig. 15. Figure 15a can be identified as the NAO pattern. Although the pattern is shifted westward, compared with the SLP pattern, it is a well-known characteristic of the NAO pattern. Indeed, when the Z500 is regressed on PC1 of the SLP Atlantic sector, this pattern appears. Figure 15c corresponds to the ‘‘PNA’’ pattern, exhibiting a seesaw between the Pacific and the East Siberian Sea. The regression of Z500 on PC1 of the SLP Pacific sector coincides with this pattern. A wavetrain pattern over Eurasia is seen in Fig. 15b. As compared with the Eurasian pattern (Wallace and Gutzler 1981), however, it is out of phase by about 90 .

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Fig. 15. Independent patterns in the 6 dimensional phase space spanned by EOF1-EOF6 in the MRI Z500 4-month data. Angles between two patterns after the whitening are 81.7 between a and b, 77.4 between a and c, and 83.1 between b and c. Kurtoses (percent variances) are 0.75 (19.9%), 0.46 (11.0%), and 0.43 (17.8%) from the left.

All the above results indicate that the NAO is an independent component. The kurtoses associated with this pattern are very small. Hence, we believe that the observed NAO is also an independent component. On the other hand, the PNA has several obscure points in this experiment, so that we cannot give final conclusions. However, it is likely that a mode with the center of action associated with the Aleutian low is independent. It can be concluded again that the AO and NCM are not independent components but mixed ones. 9.

Discussion

It is fair to note drawbacks in ICA used in this study. The following three should be pointed out: (1) Even though a signal is independent, it cannot be extracted if its PDF is near a normal distribution. (2) When three or more independent components with similar amplitudes are located on the two-dimensional plane, they cannot be extracted properly (imagine a sum of three trigonometric functions with similar amplitudes and different phases). (3) Even when two independent components are located on the two-dimensional plane, there is a possibility that one component is hidden, if kurtosis of the other component is very large as an absolute value (imagine a sum of two trigonometric functions with quite different amplitudes). However, it is unlikely that our results are contaminated by the drawbacks (1) and (2). Concerning (2), the probability that three or

more independent components are placed onto the 2-dimensional plane in very high dimensional phase space is infinitesimally small. The case that the AO and NCM are also independent (but not extracted as independent components) is furthermore extreme; in addition to that the four independent modes are located on the same plane, the AO and NCM patterns are perpendicular, and the AO just coincides with the sum of the NAO and PNA. We can say that such chance is impossible (see also Itoh 2007). As to (1), it is unlikely that although the AO is independent, it could not be extracted as an independent mode due to a nearGaussian distribution; since the AO has features similar to the NAO in many aspects, it never happens that only their kurtoses are largely different. Drawback (3) may contaminate the present result. For instance, it may reflect this drawback that the number of independent components are different between the SLP and Z500, even for the same dimension of phase space. As a method to reveal hidden components, the deflation algorithm is known, in which further independent components are sought after the dimension of phase space is diminished by withdrawing the contribution of extracted independent components. However, this method was not adopted in this study, because we were worried about the appearance of false components due to noise. Advantages and disadvantages of the deflation algorithm will be a future study. There is one more important problem relat-

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Table 3. Patterns with minimum kurtoses when the correlation between the ‘‘NAO’’ and ‘‘PNA’’ signals is 0.1, 0.2, and 0.3 for the 3-point seesaw system (model 1 with mutual correlation). Relative amplitudes at the three points (N, A, and P) are shown. Calculations are made by using 10 000 samples. ‘‘NAO’’ pattern Correlation

‘‘PNA’’ pattern

d

N

A

P

N

A

P

0.1

0.333

1.000

0.989

0.011

1.000

0.004

1.004

0.2

0.500

1.000

1.011

0.011

1.000

0.025

0.975

0.3

0.655

1.000

0.920

0.080

1.000

0.135

0.866

ing to the notion of independence. ICA gives statistically independent components, resulting in the NAO and PNA in this case. However, in the real world, it is unlikely that the NAO and PNA are completely independent. I02 mentions that the correlation between the NAO and PNA signals is 0.17. In such cases, does ICA used in this study still hold? In other words, do the results obtained so far have validity? This issue will be considered in the remainder of this section. Methods to extract source signals with mutual correlation are not called ICA but separation of dependent sources and so on. Various methods have been proposed, depending on the characteristics of correlation. Among them, a method that measures non-Gaussianity like ICA is one of promising methods (e.g., Caiafa and Proto 2006). Then, we revise the 3-point seesaw model in which ‘‘NAO’’ and ‘‘PNA’’ signals have certain correlations, examining whether the method using kurtosis still has validity. Expected results will be that the ‘‘NAO’’ pattern slightly changes to have a weak amplitudes at P from the original ð1; 1; 0Þ pattern. The ‘‘PNA’’ pattern will also have a similar result. At least two models may be designed, depending on the nature of correlation. 1. The ‘‘NAO’’ and ‘‘PNA’’ have weak correlation by the mediation of a third mode. 2. Although the ‘‘NAO’’ and ‘‘PNA’’ usually vary independently, they are synchronized with each other for short periods. Thus, they have weak correlation. In the following, we will consider these models, for which we assume all source signals with zero-mean and unit variance. Also, yN , yA , yP ,

r1 , and r2 are the same as in (4), and r6 , r7 , r8 appearing later are defined similarly. The first model may be constructed as follows: yN ðtÞ ¼ r1 ðtÞ þ r2 ðtÞ þ ðd1 þ d2 Þr6 ðtÞ;

ð9aÞ

yA ðtÞ ¼ ðr1 ðtÞ þ d1 r6 ðtÞÞ;

ð9bÞ

yP ðtÞ ¼ ðr2 ðtÞ þ d2 r6 ðtÞÞ:

ð9cÞ

When d1 ¼ d2 ðd1 ¼ d2 Þ, a third mode similar to the AO (NCM) is mediated. Suppose a correlation coefficient between the ‘‘NAO’’ and ‘‘PNA’’ signals is c, then d1 d2 c ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð1 þ d12 Þð1 þ d22 Þ

ð10Þ

In this case, since there are three independent components located on the 2-dimensional plane, it is unclear what results can be obtained. Then, for c ¼ 0:1; 0:2, and 0.3, we perform simulations, giving random numbers (sum of two uniform random numbers) for ‘‘NAO’’ and ‘‘PNA’’ signals, and obtain patterns with kurtosis minima. The result is shown in Table 3, where d1 ¼ d2 ð1 dÞ is assumed. The source signals are well identified for all the correlations. The distribution of kurtoses at each angle (not shown) shows that as the correlation increases, the difference between kurtosis maxima and minima decreases, therefore the identification of the source signals becomes difficult. However, when correlations are about 0.2 or less, the difference is still large, and there is no difficulty in the identification. The second model may be written as follows: yN ðtÞ ¼ r7 ðtÞ þ r8 ðtÞ;

ð11aÞ

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Table 4. Same as Table 3 except for model 2. ‘‘NAO’’ pattern Correlation

N

A

P

N

A

P

0.1

1.000

0.924

0.077

1.000

0.093

0.908

0.2

1.000

0.832

0.167

1.000

0.178

0.822

0.3

1.000

0.769

0.230

1.000

0.223

0.777

yA ðtÞ ¼ r7 ðtÞ;

ð11bÞ

yP ðtÞ ¼ r8 ðtÞ:

ð11cÞ

Although this equation system is formally the same as (4), the difference is that r7 and r8 have correlation. We make the two time series with a correlation of c such that Nc samples out of N total samples have the same values, while the other samples have independent values. Note that this model has the following defect; Nc points are located on the EOF1 axis in phase space, therefore the frequency at zero amplitude of EOF2 is very large. This means that EOF2 has a large, positive kurtosis, which is far from the reality. Calculations are performed in a similar manner to model 1, and the result is shown in Table 4. Compared with Table 3, this result has the characteristics that with increasing c, amplitudes at A for the ‘‘NAO’’ pattern or P for the ‘‘PNA’’ pattern considerably decrease, while amplitudes at the other point increase. However, overall features are those expected. Looking at the distribution of kurtosis at each angle (not shown), positive kurtoses at the ‘‘NCM’’ are conspicuous for c ¼ 0:2 and 0.3, as stated above. Except this point, we can easily identify independent components. Thus, as long as the 3-point seesaw system is used, there is no problem even though ‘‘NAO’’ and ‘‘PNA’’ signals have weak correlation. However, the above conclusion is based on results from the simple models so that it lacks generality. A more general approach will be a future work. 10.

‘‘PNA’’ pattern

Conclusions and remarks

The Arctic Oscillation (AO, or annular mode) is examined to determine whether it is true or not by using independent component analysis (ICA). ICA can distinguish between true and

apparent oscillations, when it is assumed that true oscillations are mutually independent. Equivalently, apparent oscillations are dependent, and are derived from linear combinations of independent components. The basic idea of the discrimination is that independent and dependent oscillations have different probability density functions (PDFs); PDFs of independent (dependent) oscillations are far from (close to) Gaussian distributions under the assumption that PDFs of independent oscillations are nonGaussian. For this purpose, the NCEP-NCAR reanalysis data (NCEP-NCAR data) is used. In addition, the present climate experiment data of the Meteorological Research Institute (MRI data) is also used, because a longer period data is needed to obtain statistical significance. It is necessary to properly select data periods for performing ICA. The right selection is to use the periods of data whose variances almost coincide with each other. Under this data selection, the amplitudes of well-mixed components show normal distribution. Next, we mention two major issues in ICA applied to meteorology: the treatment of noise, and the relationship between the dimension of phase space in the analysis and the number of independent components. We make several considerations regarding these issues. One finding is that the patterns associated with independent components after whitening are not perpendicular to each other under the existence of noise. Finally, the method to obtain independent components by kurtosis is the most relevant to meteorology. This method is then used throughout the present paper. On the basis of the above considerations and proposals, ICA is performed on the NCEPNCAR data. Independent components are found for the sea level pressure (SLP) and

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H. ITOH et al.

500 hPa height (Z500) fields. They are the North Atlantic Oscillation (NAO) and the Pacific-North American Oscillation (PNA). Another component is also independent. Thus, the AO is an apparent mode derived from them. In addition to the AO (the first EOF mode in the Northern Hemisphere SLP), the second EOF mode referred to as the negative correlation mode between the Atlantic and Pacific (NCM) can also be understood as an apparent mode. However, since the period of the data is too short, statistical significance cannot be obtained. Then, ICA is performed on the MRI data. This data have the characteristic that the center of action in the Pacific shifts southeastward, compared with observations, and that variations over the Atlantic and Pacific regions are somewhat coherent. As a result, the discrimination between the NAO and AO is not as clear as the observed data. Even though, it can be completely determined to the fullest extent that the NAO is independent. Moreover, a component with a seesaw pattern between the Pacific and polar region is also independent in both the SLP and Z500. It can be concluded once again that the AO and NCM are not independent but mixed components. The inescapable conclusion to be drawn from the above results is that the NAO is an independent component. Further, it can be said that the PNA is independent to a considerable extent. However, in the case of the PNA, further studies are needed, especially to determine the positions of the center of action. More importantly in this paper is that there is no evidence from both of the NCEP-NCAR and MRI data that the AO and NCM are independent components, i.e., true modes. In our analysis, there is one inconsistent procedure: We performed the EOF analysis over the Atlantic and Pacific sectors, identifying the NAO and PNA. However, we have not examined whether or not these oscillations are really independent. In the MRI data, the ‘‘PNA’’ exhibits complicated features. If ICA is applied to the Pacific sector, more convincing results may be obtained. A study addressing sectorial independent components and their relation to hemispherical components will appear in a subsequent paper. Since periods of observational data are not

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sufficiently long to obtain significance, output data from long-term integration experiments simulating the present climate are very important to judge significantly whether the AO is true or apparent. Now, many long-term integration data have been accumulated for the IPCC 4th Assessment Report. By performing ICA on these data, we can obtain more conclusive results. If all the data indicate the same result, the issue of the true or apparent AO will be settled. We believe that this direction is the optimum one to settle the controversy. This study is now underway. It is general in meteorology to obtain leading modes by using EOF analysis. Certainly, there is no problem in using it as the medium of data reduction. On the other hand, there have been many cases in which leading modes are treated as true modes having physical entities. However, as stated in this paper, when there are two oscillations that have a common center of action, EOF analysis necessarily produces an apparent mode as a leading mode. Therefore, some of leading EOF modes that have been interpreted as physical modes may be apparent modes, and should be reexamined. In doing so, ICA plays an important role in distinguishing between true and apparent modes. We hope that ICA will be widely used in the meteorological community. Acknowledgments We would like to express our hearty appreciation to two anonymous reviewers for giving many appropriate and constructive comments. We think the paper was greatly improved by these comments. NCEP-NCAR reanalysis data were provided by the NOAA-CIRES Climate Diagnostics Center, USA, from their website at http://www.cdc.noaa.gov. This research is supported by a Grant-in-Aid for Scientific Research of the Japanese Ministry of Education, Science, Sports and Culture. The GFD-DENNOU Library of the Japanese meteorological community and GrADS were used to draft the figures. References Ambaum, M.H.P., B.J. Hoskins, and D.B. Stephenson, 2001: Arctic Oscillation or North Atlantic Oscillation? J. Climate, 14, 3495–3507. Caiafa, C.F. and A.N. Proto, 2006: Separation of sta-

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tistically dependent sources using an L 2 distance non-Gaussianity measure. Signal Processing, 86, 3404–3420. Christiansen, B., 2002: Comments on ‘‘True versus apparent Arctic Oscillation.’’ Geophys. Res. Lett., 29, 10.1029/2002GL016051. Davison, A.C. and D.V. Hinkley, 1997: Bootstrap methods and their application, Cambridge University Press, Cambridge, 582 pp. Deser, C., 2000: On the teleconnectivity of the ‘‘Arctic Oscillation.’’ Geophys. Res. Lett., 27, 779–782. Hyvarinen, A., J. Karhunen, and E. Oja, 2001: Independent component analysis. John Wiley & Sons, 504 pp. Itoh, H., 2002: True versus apparent Arctic Oscillation. Geophys. Res. Lett., 29, 10.1029/ 2001GL013978. Itoh, H., 2007: Reconsideration of the true versus apparent Arctic Oscillation. J. Climate, in press. Mori, A., N. Kawasaki, K. Yamazaki, M. Honda, and

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