Neliösummalla

Neliösummalla is the instrumental/adessive form of the Finnish noun neliösumma, which means the sum of squares. In mathematical Finnish, neliösummalla is used to indicate that a calculation or description is based on or performed with the sum of squares. For example, phrases like neliösummalla laskettu etäisyys or neliösummalla mitattu varianssi appear in technical texts, where the focus is on the quantity Σ xi^2 rather than its plain form.

Neliösumma refers to the sum of the squares of a set of numbers or vector components. For numbers x1, x2, ..., xn, the neliösumma is Σ xi^2 = x1^2 + x2^2 + ... + xn^2. This quantity is closely related to the Euclidean norm: the Euclidean length of a vector x = (x1, ..., xn) is ||x||2 = sqrt(Σ xi^2), and the square of the norm equals the neliösumma. The concept is nonnegative and is zero only when all components are zero.

In statistics and data analysis, neliösumma appears in various contexts, such as the residual sum of squares in regression and the total sum of squares in variance analysis. In machine learning, the term is encountered in discussions of L2 regularization, where the penalty is proportional to the sum of squares of model parameters. The neliösumma thus serves as a fundamental quantitative measure across disciplines that deal with magnitudes of numbers or parameters.