SupplementaryMaterial
Eigen analysis of tree-ring records
2. Posing the eigen problem
BaoYang,1 Dmitry M. Sonechkin,2Nina M. Datsenko,3
Nadezda N. Ivashchenko,3 Chun Qin,1 and Jingjing Liu1
1. Duality of the eigen problem solutions for covariations of sampled data
Let us consider a matrix having N rows and K columns:
. (S1)
Rows of this matrix consist of individual tree-ring records of the same life-span.The number of these records is equal to N. Columns of this matrix correspond to 1,2, …,K years of the tree biological ages. If the tree-ring records of interest span a very time (on the order of ), then the number of the matrix rows will usually be less than the number of its columns: N < K. In such a case, any estimation of the intra-record covariation matrix,
(S2)
is very affected by the numerous sources of noise that inevitably exist in tree-ring records. Note, for the goal of simplicity the pre-factor N-1 will be omitted below.As a result, any solution of the eigen problem
(S3)
where and are called the eigen values and eigen vectors is very biased and aggravated by noises (Datsenko et al., 1983; North et al., 1982). However, it might be possible to obtain a reasonably reliable solution of this eigen problemby using simple tools of linear algebra because of a duality of (S3) to another eigen problem for the inter-record covariation matrix of the same tree sample.
Indeed, instead of the quadratic matrix of the order K, one canconsider the inter-record covariation matrix
. (S4)
This matrix is also quadratic. It has N rows and N columns, that is, its order is equal to N. If N < K,computing this matrix is easier and solving the eigen problem for this matrix (the pre-factor K-1 is also omitted)
. (S5)
can be more reliable than such an eigen problem solving for the matrix .
Multiplying the left-hand side and the right-hand side of Equation (S5) by yields:
(S6)
After simple manipulations with both sides of Equation (S6), it is easy to obtain:
(S7)
One can conclude from Equation (S7) that the eigen problem (S6) is dual to the eigen problem (S3) in principle, i.e.if k = 1,2, …, N, but if kN in Equation (S3),and there is a one-to-one correspondence betweenthe eigenvectors and .It is interesting to note that is nothing but the principal components (without normalization only) of the intra-record covariation matrix. On the other hand, are nothing but the principal components (also without normalization) of the inter-record covariation matrix.
Thus, in the case of a poor sampling (N < K),only the first N eigenvectors of the intra-record covariation matrix are reliable in principle, which correspond to the first eigenvalues . Moreover, these eigenvectors can be computed well if and only if both differences and are rather large. Therefore, in the case of the hyperbolic shape of the eigenvalue spectrum, and under the obligatory use of the inter-record eigen problem solution, a few of the largest eigenvalues (essentially less than K) canbe reliable estimated in fact.All of other eigen vectors must be disregarded. But, even after this is done, a care must be taken when a chronology is created by using only a few PCs because the above best estimations of the largest eigenvalues will surely be overestimated, even if the sample is very large (i.e., the number Nwill be commensurate with or greater than the number K) (Datsenko et al., 1983; North et al., 1982).
References
Datsenko, N. M., V. I. Perfilov, and D. M. Sonechkin (1983), A method for calculating the natural components of meteorological fields, Izvestyja, Atmospheric and Oceanic Physics, 19(4), 348–356, in Russian.
North, G. E., T. L. Bell, R. F. Graham, and F. J. Moeng (1982), Sampling errors in the estimation of empirical orthogonal functions, Monthly Weather Rev., 110, 699–703.
2. Results of solving the eigen problem for the intra-record covariation matrix of the Dulan tree-ring width extra sample of 56 long-lived trees
Figure S1. Chronologies of the Dulan ring width extra sample reconstructed usingonly one principal component from PC(1) throughPC(5).
Figure S2. Chronologies of the Dulan ring width extra sample reconstructed using only one principal component from PC(6) throughPC(10).
Figure S3. Chronologies of the Dulan ring width extra sample reconstructed using only one principal component from PC(11) throughPC(15).
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