Compressed Counting (CC)} was recently proposed for approximating the $alpha$th frequency moments of data streams, for $0<alpha leq 2$. Under the relaxed strict-Turnstile model, CC dramatically improves the standard algorithm based on symmetric stable random projections}, especially as $alphato 1$. A direct application of CC is to estimate the entropy, which is an important summary statistic in Web/network measurement and often serves a crucial feature for data mining. The Renyi entropy and the Tsallis entropy are functions of the $alpha$th frequency moments; and both approach the Shannon entropy as $alphato 1$. A recent theoretical work suggested using the $alpha$th frequency moment to approximate the Shannon entropy with $alpha=1+delta$ and very small $|delta|$ (e.g., $<10^{-4}$). In this study, we experiment using CC to estimate frequency moments, Renyi entropy, Tsallis entropy, and Shannon entropy, on real Web crawl data. We demonstrate the variance-bias trade-off in estimating Shannon entropy and provide practical recommendations. In particular, our experiments enable us to draw some important conclusions: (1) As $alphato 1$, CC dramatically improves {em symmetric stable random projections} in estimating frequency moments, Renyi entropy, Tsallis entropy, and Shannon entropy. The improvements appear to approach infinity. (2) Using {em symmetric stable random projections} and $alpha = 1+delta$ with very small $|delta|$ does not provide a practical algorithm because the required sample size is enormous.
تحميل البحث