$\ell_1$ Trend Filtering

Detecting trends in twitter time series

“…From a probabilistic perspective, this is equivalent to assuming that n t has a Gaussian prior distribution. ), we obtain the popular L 1 filter [1]. Although no probabilistic interpretation was offered by Kim et al [1], it is easy to check that it amounts to maximum likelihood estimation when the residuals are modelled as Gaussian.…”

“…), we obtain the popular L 1 filter [1]. Although no probabilistic interpretation was offered by Kim et al [1], it is easy to check that it amounts to maximum likelihood estimation when the residuals are modelled as Gaussian. Importantly, they also mention in passing a possible extension to include the detection of peaks.…”

“…We follow this approach, and model the peak values z t as a Pareto distribution, p z (z t ) = 2 z ( 2 +1) t , with 2 the shape parameter. 1 Pareto distributions are heavy-tailed distributions. 2 This means that a large multiplicative effect of a peak on the observed number of tweets can still be attributed a large probability by our model.…”

“…2.4 we will prove that this particular prior also induces sparsity: many peak values ⇣ t will be equal to 0 in the maximum likelihood estimate for and ⇣. More specifically, 1 The scale parameter is equal to 1. 2 Given a random variable A ⇠ Exponential( ), a random variable B ⇠ Pareto(↵), and an arbitrarily large factor r 2 R, there always exists a value s 2 R such that P (B > s) > r · P (A > s).…”

“…The L 1 filter [1] is arguably the state-of-the-art method in trend detection. This and other related works are discussed in Section 3.…”