Because a strictly monotone map fixing both endpoints is increasing, the change of variables formula

t = h (s)

gives

∥ Y ∥^{2} = \int_{0}^{1} X (h^{- 1} (t))^{2} d t = \int_{0}^{1} X (s)^{2} h^{'} (s) d s .

Assuming

E ∥ X ∥^{2} > 0

A = \frac{E \int _{0}^{1} X ( s ) ^{2} h ^{'} ( s ) d s}{E \int _{0}^{1} X ( s ) ^{2} d s} .

The Hilbert-space variance identity then gives

E ∥ Y - μ ∥^{2} = E ∥ Y ∥^{2} - ∥ μ ∥^{2} = A E ∥ X ∥^{2} - ∥ μ ∥^{2} = A E ∥ X - ν ∥^{2} + A ∥ ν ∥^{2} - ∥ μ ∥^{2} .

Under the regularity needed to interchange expectation and differentiation,

E h^{'} (s) = 1

because

E h (s) = s

. Hence

A = 1 + \frac{\int _{0}^{1} Cov ( X ( s ) ^{2} , h ^{'} ( s )) d s}{E ∥ X ∥ ^{2}} .

Thus

A = 1

exactly when the integrated covariance in the numerator vanishes. In particular, this holds when the amplitude

X

and the time warp

h

ii

Solution

For these increasing warps, the square-root velocity function is

Q (h) (t) = h^{'} (t)

. The chain rule gives

Q (h_{i} \circ γ) (t) = Q (h_{i}) (γ (t)) γ^{'} (t) .

Therefore

∥ Q (h_{i} \circ γ) - Q (h_{j} \circ γ) ∥_{2}^{2} = \int_{0}^{1} ∣ Q (h_{i}) (γ (t)) - Q (h_{j}) (γ (t)) ∣^{2} γ^{'} (t) d t = \int_{0}^{1} ∣ Q (h_{i}) (u) - Q (h_{j}) (u) ∣^{2} d u,

where the last equality again uses the change of variables formula. Taking square roots proves invariance under common right composition by

γ

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