# Estimate the signal This function estimates the signal in a given data sequence `x` with change-points at `cpt`. The type of the signal depends on whether the change-points represent changes in a piecewise-constant or continuous, piecewise-linear signal. For more information see Details below. ```r est_signal(x, cpt, type = c("mean", "slope")) ``` ## Arguments - `x`: A numeric vector containing the given data. - `cpt`: A positive integer vector with the locations of the change-points. If missing, the `ID_pcm` or the `ID_cplm` function (depending on the type of the signal) is called internally to extract the change-points in `x`. - `type`: A character string, which defines the type of the detected change-points. If ```type = ``mean''```, then the change-points represent the locations of changes in the mean of a piecewise-constant signal. If ```type = ``slope''```, then the change-points represent the locations of changes in the slope of a continuous, piecewise-linear signal. ## Returns A numeric vector with the estimated signal. ## Details The data points provided in `x` are assumed to follow $$ X_t = f_t + \sigma\epsilon_t; t = 1,2,...,T, $$ where $T$ is the total length of the data sequence, $X_t$ are the observed data, $f_t$ is a one-dimensional, deterministic signal with abrupt structural changes at certain points, and $\epsilon_t$ is white noise. We denote by $r_1, r_2, ..., r_N$ the elements in `cpt` and by $r_0 = 0$ and $r_{N+1} = T$. Depending on the value that has been passed to `type`, the returned value is calculated as follows. * For ```type = ``mean''```, in each segment $(r_j + 1, r_{j+1})$, $f_t$ for $t \in (r_j + 1, r_{j+1})$ is approximated by the mean of $X_t$ calculated over $t \in (r_j + 1, r_{j+1})$. * For ```type = ``slope''```, $f_t$ is approximated by the linear spline fit with knots at $r_1, r_2, ..., r_N$ minimising the $l_2$ distance between the fit and the data. ## Examples ```r single.cpt.pcm <- c(rep(4,1000),rep(0,1000)) single.cpt.pcm.noise <- single.cpt.pcm + rnorm(2000) cpt.single.pcm <- ID_pcm(single.cpt.pcm.noise) fit.cpt.single.pcm <- est_signal(single.cpt.pcm.noise, cpt.single.pcm$cpt, type = "mean") three.cpt.pcm <- c(rep(4,500),rep(0,500),rep(-4,500),rep(1,500)) three.cpt.pcm.noise <- three.cpt.pcm + rnorm(2000) cpt.three.pcm <- ID_pcm(three.cpt.pcm.noise) fit.cpt.three.pcm <- est_signal(three.cpt.pcm.noise, cpt.three.pcm$pcm, type = "mean") single.cpt.plm <- c(seq(0,999,1),seq(998.5,499,-0.5)) single.cpt.plm.noise <- single.cpt.plm + rnorm(2000) cpt.single.plm <- ID_cplm(single.cpt.plm.noise) fit.cpt.single.plm <- est_signal(single.cpt.plm.noise, cpt.single.plm$cpt, type = "slope") ``` ## Author(s) Andreas Anastasiou, a.anastasiou@lse.ac.uk

est_signal function