“orthogonal polynomials”的概念、定义、翻译、参考文献-科学参考

orthogonal polynomials

Mathematics

A set of polynomials {p_k(x)}, k ≥ 0 are said to be orthogonal over an interval [a,b] if for some positive weighting function w(x),

$\int_{a}^{b} w (x) p_{i} (x) p_{j} (x) d x = 0$

when i ≠ j. See Chebyshev polynomials, Hermite polynomial, Legendre polynomials.

Statistics

Polynomials designed to simplify calculations when a response variable is believed to be related to a polynomial function of an explanatory variable. If x₁, x₂,…, x_n are n observations on the variable X, then the polynomials Φ₁(x) and Φ₂(x) are orthogonal if $orthogonal polynomials$ If we wish to fit the model $orthogonal polynomials$ then we will need (see multiple regression model) to invert the matrix product X′X, where $orthogonal polynomials$ This may not be easy, since X′X will often be nearly singular. One solution is to rewrite the model in the form $orthogonal polynomials$
where Φ₀(x)=1 and Φ_k(x) is a polynomial of degree k (k=1, 2,…, m). With n observations on x, denoted by x₁, x₂,…, x_n, the polynomials are chosen to be orthogonal, i.e. to satisfy the requirements that $orthogonal polynomials$ The revised X matrix has a typical element Φ_k(x_j) and is such that the product X′X is now a diagonal matrix and is therefore easily inverted.
Often the appropriate degree of dependence on x (in other words, the value of m) is unknown. The reformulation also makes it easy to select an appropriate model without the need for further iterations. In the ANOVA table the contributions for each polynomial can be listed separately.

单词	orthogonal polynomials
释义	orthogonal polynomials Mathematics A set of polynomials {p_k(x)}, k ≥ 0 are said to be orthogonal over an interval [a,b] if for some positive weighting function w(x), $\int_{a}^{b} w (x) p_{i} (x) p_{j} (x) d x = 0$ when i ≠ j. See Chebyshev polynomials, Hermite polynomial, Legendre polynomials. Statistics Polynomials designed to simplify calculations when a response variable is believed to be related to a polynomial function of an explanatory variable. If x₁, x₂,…, x_n are n observations on the variable X, then the polynomials Φ₁(x) and Φ₂(x) are orthogonal if $orthogonal polynomials$ If we wish to fit the model $orthogonal polynomials$ then we will need (see multiple regression model) to invert the matrix product X′X, where $orthogonal polynomials$ This may not be easy, since X′X will often be nearly singular. One solution is to rewrite the model in the form $orthogonal polynomials$ where Φ₀(x)=1 and Φ_k(x) is a polynomial of degree k (k=1, 2,…, m). With n observations on x, denoted by x₁, x₂,…, x_n, the polynomials are chosen to be orthogonal, i.e. to satisfy the requirements that $orthogonal polynomials$ The revised X matrix has a typical element Φ_k(x_j) and is such that the product X′X is now a diagonal matrix and is therefore easily inverted. Often the appropriate degree of dependence on x (in other words, the value of m) is unknown. The reformulation also makes it easy to select an appropriate model without the need for further iterations. In the ANOVA table the contributions for each polynomial can be listed separately.
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