Which of the following is a linear data transformation method?

Question

BlackTom AI · Accepted Answer

PCA is a linear transformation because each principal component is a linear combination of the original variables: y = W^T x, where W is a matrix whose columns are orthonormal eigenvectors of the data covariance matrix.

1) Relevant concepts
- Linear transformation: a map T(x) satisfying T(ax + b) = aT(x) + T(b); in vector/matrix form T(x) = A x for some matrix A (affine maps include an added translation).
- PCA: after centering the data, principal components are computed as Y = Xc W where Xc = X − 1μ^T and W contains eigenvectors of the covariance; each column of Y is a linear combination of the original centered variables.
- Other transforms: logarithm and Box–Cox are nonlinear (power/log functions); standardization (x' = (x − μ)/σ) is affine (translation plus scaling), not a pure linear map unless the mean is zero.

2) Step-by-step reasoning and equations
- PCA mapping (on centered data): let x be an original variable vector and W = [w1 ... wk]. Principal components are y = W^T x. This is matrix multiplication (linear). If X is n×p, Xc = X − 1μ^T, then Y = Xc W → each y_j = w_j^T x (linear combination).
- Standardization: x' = (x − μ)/σ = (1/σ) x − μ/σ. This equals A x + b with A = 1/σ and b = −μ/σ, so it is affine, not strictly linear (because of the translation term −μ/σ).
- Log transform: x' = log(x) is nonlinear; it does not satisfy linearity (log(a x1 + b x2) ≠ a log x1 + b log x2).
- Box–Cox transform: x'(λ) = (x^λ − 1)/λ (for λ ≠ 0) is a nonlinear power transform.

3) Why PCA is correct
PCA produces new features that are linear combinations of the original features (weighted sums). That is the defining property of a linear transformation in multivariate data processing. Although PCA commonly includes a centering step (translation), the mapping from centered original variables to principal components is linear.

4) Why the other options are incorrect
- a) Logarithmic: nonlinear (log is non-linear).
- b) Standardization: affine (includes subtraction of mean), not a pure linear map unless mean = 0.
- c) Box–Cox: nonlinear power transformation.

Conclusion: PCA (option d) is the linear data transformation method. This matches the provided answer: d.

Which of the following is a linear data transformation method?单项选择题

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