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To compute the log-signature over any interval $I$, we granularise at a fixed stream resolution $n$ to obtain the interval $\tilde I = [k_1/2^n, k_2/2^n)$, and then compute -:::{math} -\mathrm{LogSig}(\tilde I) = \log\biggl(\prod\_{k=k_1}^{k_2-1} +```{math} +\mathrm{LogSig}(\tilde{I}) = \log\biggl(\prod_{k=k_1}^{k_2-1} \exp(\mathrm{LogSig}(D_k^n))\biggr). -::: +``` The $\mathrm{LogSig}(D_k^n)$ terms on the right-hand-side are either retrieved from the cache, or computed from the underlying source. 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