Herman Wold

Herman Wold (1908-1992) was professor in Uppsala and Göteborg, Sweden. He developed algorithms on intuitive basis. His first major work was on Time Series Analysis. There he developed decompositions of the time series. His results have become fundamental in theoretical time series. The techniques he used have been applied in other fields, like spectral analysis, function algebras and other fields of applied mathematics. Later he worked with economics and large interdependent economic/social systems. When working with these large interdependent systems, he used arrow diagrams to depict the dependencies of the variables. An example of his arrow diagrams is the following one.

Here we have two systems, specified by the variables x=(x₁,x₂, ..., x_K) and y=(y₁, ..., y_M). The latent variable t is to represent the x-variables and the latent variable u the y-variables. The diagram shows one set of latent variables (t,u), but there can be many. Herman Wold was concerned with that the t-variables should both represent the x-variables well and also the u-variables. The same should hold for the u-variables concerning the y- and t-variables. He used successive projections onto the column and row spaces of the data, X and Y in order to to estimate the actual values of the latent variables. He named his class of algorithms for NIPALS, Nonlinear Iterative PArtial Least Squares. Prof. Svante Wold, Umeå, Sweden, his son, simplified the algorithms and gave a different geometric interpretations of the successive projections. PLS has been applied to many fields of applied sciences with great success. In chemometrics it is one of the favored methods of analysis. Statisticians had difficulties in understanding the use of arrow diagrams for analysis and the algorithmic formulation of PLS. This has caused statisticians to be critical towards the use of Partial Least Squares methods in analyzing data.

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