In the Pacific Ocean and 19 sites in the Indian Ocean, covering

In the Pacific Ocean and 19 sites in the Indian Ocean, covering a large portion of the two oceans (Fig. 1). Data from piston core samples2 and details of the chemical analysis methods have been reported elsewhere2,3.MethodsSamples and data.Fundamentals of Independent Component Analysis. ICA was first applied to the field of geochemistry to decode mantle isotopic signals by Iwamori and Albar e11, who clearly demonstrated that ICA can successfully extract independent geochemical signals from the data structure. The principles of ICA have been concisely NecrosulfonamideMedChemExpress Necrosulfonamide reviewed11,12, and the theoretical background and AZD3759 site numerical methods have been described in detail9. The crucial point of ICA is to extract the non-Gaussianity inherent in the data structure. According to the central limit theorem, the sum of independent random variables becomes closer to a Gaussian variable than any of the originalScientific RepoRts | 6:29603 | DOI: 10.1038/srepwww.nature.com/scientificreports/random variables. Conversely, when a sum of variables deviates maximally from a Gaussian distribution, it is equivalent to one of the independent source signals or ICs. To apply ICA to the compositional data of deep-sea sediments, we assumed that the data included a mixture of geochemical components characterised by mutually independent signatures. Our ICA problem is the basic linear mixing model represented as follows, which is the same expression as that used in other traditional multivariate analytical methods such as PCA:X = SA, (1)where X is the observed data matrix of which the rows and columns correspond to the individual samples and observed variables (i.e. elemental contents), respectively, S is an independent source matrix whose columns correspond to ICs, and A is a linear mixing matrix. In the ICA literature9, the model is usually explained by using the random variable vector, as discussed in the Supplementary Information. However, for consistency with our practical calculation52, we denote our ICA model by equation (1) using matrices, which is mathematically equivalent to that in the literature9. Both S and A are unknown and need to be estimated. First, the data matrix X is centred according to the mean of each variable. Next, the whitened data matrix Z is calculated byZ = XK, (2)where K is the whitening matrix generated from the eigenvalues and eigenvectors of the covariance matrix of X, both of which can be obtained by using a basic PCA algorithm9. At that time, each column in Z has zero mean and unit variance, and any pair of new variables is mutually uncorrelated but not necessarily independent. Then, to make the variables statistically independent, the whitened axes are rotated until the projection of the whitened data on the axes gives a histogram, regarded as a probability density function along each axis, with maximum non-Gaussianity. Here, the estimated independent source matrix S is obtained byS = ZW = XKW, (3)where W, known as the un-mixing matrix, is an orthogonal matrix for rotating the whitened data, Z and is numerically determined when an evaluation function of non-Gaussianity converges. As the evaluation function, we used negentropy J(y) approximated byJ (y ) = c [EG (y ) – EG ( )]2 , (4)where y is a random variable of zero mean and unit variance (i.e. whitened and rotated data), c is an arbitrary constant, and is a Gaussian variable of zero mean and unit variance. The function G is defined asy2 G (y ) = – exp ,(5)following refs 9,11. We employed t.In the Pacific Ocean and 19 sites in the Indian Ocean, covering a large portion of the two oceans (Fig. 1). Data from piston core samples2 and details of the chemical analysis methods have been reported elsewhere2,3.MethodsSamples and data.Fundamentals of Independent Component Analysis. ICA was first applied to the field of geochemistry to decode mantle isotopic signals by Iwamori and Albar e11, who clearly demonstrated that ICA can successfully extract independent geochemical signals from the data structure. The principles of ICA have been concisely reviewed11,12, and the theoretical background and numerical methods have been described in detail9. The crucial point of ICA is to extract the non-Gaussianity inherent in the data structure. According to the central limit theorem, the sum of independent random variables becomes closer to a Gaussian variable than any of the originalScientific RepoRts | 6:29603 | DOI: 10.1038/srepwww.nature.com/scientificreports/random variables. Conversely, when a sum of variables deviates maximally from a Gaussian distribution, it is equivalent to one of the independent source signals or ICs. To apply ICA to the compositional data of deep-sea sediments, we assumed that the data included a mixture of geochemical components characterised by mutually independent signatures. Our ICA problem is the basic linear mixing model represented as follows, which is the same expression as that used in other traditional multivariate analytical methods such as PCA:X = SA, (1)where X is the observed data matrix of which the rows and columns correspond to the individual samples and observed variables (i.e. elemental contents), respectively, S is an independent source matrix whose columns correspond to ICs, and A is a linear mixing matrix. In the ICA literature9, the model is usually explained by using the random variable vector, as discussed in the Supplementary Information. However, for consistency with our practical calculation52, we denote our ICA model by equation (1) using matrices, which is mathematically equivalent to that in the literature9. Both S and A are unknown and need to be estimated. First, the data matrix X is centred according to the mean of each variable. Next, the whitened data matrix Z is calculated byZ = XK, (2)where K is the whitening matrix generated from the eigenvalues and eigenvectors of the covariance matrix of X, both of which can be obtained by using a basic PCA algorithm9. At that time, each column in Z has zero mean and unit variance, and any pair of new variables is mutually uncorrelated but not necessarily independent. Then, to make the variables statistically independent, the whitened axes are rotated until the projection of the whitened data on the axes gives a histogram, regarded as a probability density function along each axis, with maximum non-Gaussianity. Here, the estimated independent source matrix S is obtained byS = ZW = XKW, (3)where W, known as the un-mixing matrix, is an orthogonal matrix for rotating the whitened data, Z and is numerically determined when an evaluation function of non-Gaussianity converges. As the evaluation function, we used negentropy J(y) approximated byJ (y ) = c [EG (y ) – EG ( )]2 , (4)where y is a random variable of zero mean and unit variance (i.e. whitened and rotated data), c is an arbitrary constant, and is a Gaussian variable of zero mean and unit variance. The function G is defined asy2 G (y ) = – exp ,(5)following refs 9,11. We employed t.