L0TFinv (version 0.1.0)

L0TFinv.opt: The inverse L0 trend filtering with optimal change points

Description

Fit the input data points to a piecewise constant or piecewise linear trend with optimal change points.

Usage

L0TFinv.opt(y = y, kmax = kmax, q = q, first = 0, last = 1, penalty = "bic")

Value

An S3 object of type "L0TFinvopt". A list containing the fitted trend results:

sic

Information criterion value with a penalty term of \(2\log(\log(n)) \times \log(n)\)

bic

Information criterion value with a penalty term of \(2 \times \log(n)\)

mse

The mean square error between the fitted trend and the input data

y

The input data points

betaopt

The fitted \(\hat{\boldsymbol{\beta}}\) coefficients with optimal change points

yopt

The fitted trend with optimal change points

Aopt

The set of position indicators of the fitted change points with optimal change points

kopt

Optimal number of change points

beta.all

A data frame with dimensions \(n \times k_{\text{max}}\), where each column represents the fitted \(\hat{\boldsymbol{\beta}}\) coefficients corresponding to a given number of change points

y.all

A data frame with dimensions \(n \times k_{\text{max}}\), where each column represents the fitted estimated trend corresponding to a given number of change points

A.all

A list of length \(k_{\text{max}}\), where each element corresponds to the set of position indicators of change points under a given number

Arguments

y

The input data points

kmax

The maximum number of change points

q

0 or 1. Correspond to a piecewise constant or piecewise linear trend

first

The value ranges from 0 to 1. Represent the minimum percentile point where a change point may occur. If 'first' = 0.01, it means that change points cannot appear in the first 1% of the data points. If 'first' = 0, there is no constraint on the position of the change point.

last

The value ranges from 0 to 1. Represent the maximum percentile point where a change point may occur. If 'last' = 0.99, it means that change points cannot appear in the last 1% of the data points. If 'last' = 1, there is no constraint on the position of the change point.

penalty

'sic' or 'bic' penalty

Details

Let the fitted trend be denoted as \(\hat{\boldsymbol{y}}\), then $$\text{sic} = n \times \log(\frac{1}{n}\|\boldsymbol{y}-\hat{\boldsymbol{y}}\|_2^2)+2\log(\log(n)) \times \log(n) \times \text{df}(\hat{\boldsymbol{y}})$$ and $$\text{bic} = n \times \log(\frac{1}{n}\|\boldsymbol{y}-\hat{\boldsymbol{y}}\|_2^2)+2\times \log(n) \times \text{df}(\hat{\boldsymbol{y}}).$$ The term \(\text{df}(\hat{\boldsymbol{y}})\) represents the degrees of freedom for the estimated trend, where \(\text{df}(\hat{\boldsymbol{y}})=k+q+1\). Here, \(k\) refers to the number of change points in the estimated trend.

Examples

tau = c(0.1, 0.3, 0.4, 0.7, 0.85)
h = c(-1, 5, 3, 0, -1, 2)
BlocksData <- SimuBlocksInv(n = 500, sigma = 0.2, seed = 50, tau = tau ,h = h)
res <- L0TFinv.opt(y=BlocksData$y, kmax=20, q=0, first=0.01, last=1, penalty="bic")
print(res$Aopt)
print(BlocksData$setA)
plot(BlocksData$x, BlocksData$y, xlab="", ylab="")
lines(BlocksData$x, BlocksData$y0, col = "red")
lines(BlocksData$x, res$yopt, col = "lightgreen")

tau1 = c(0.4, 0.6, 0.7)
h1 = c(-3, 5, -4, 6)
a0 = -10
WaveData <- SimuWaveInv(n = 500, sigma = 0.1, seed = 50, tau = tau1, h = h1, a0 = a0)
res1 <- L0TFinv.opt(y=WaveData$y, kmax=10, q=1, first=0, last=0.99, penalty="sic")
print(res1$Aopt)
print(WaveData$setA)
plot(WaveData$x, WaveData$y, xlab="", ylab="")
lines(WaveData$x, WaveData$y0, col = "red")
lines(WaveData$x, res1$yopt, col = "lightgreen")