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In vrunge/slopeOP: Optimal partitioning algorithm for change-in-slope problem with a finite number of states

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slopeOP Vignette

Vincent Runge

LaMME, Evry University

June 13, 2019

Introduction

The slopeOP function

Options for constraining inference

Introduction

The package slopeOP is designed to segment univariate data $y_{1:n} = {y_1,...,y_n}$ by a continuous piecewise linear signal with restriction on starting/ending values for the segments. The changepoint vector $\overline{\tau} = (\tau_0 < \cdots < \tau_{k+1}) \in \mathbb{N}^{k+2}$ defines the $k+1$ segments ${\tau_i+1,...,\tau_{i+1}}$, $i = 0,...,k$ with fixed bounds $\tau_0 = 0$ and $\tau_{k+1} = n$. We use the set $S_n = {\hbox{changepoint vector } \overline{\tau} \in \mathbb{N}^{k+2}}$ to define the minimal global cost given by

$$Q_n = \min_{\overline{\tau} \in S_n}\left[ \sum_{i=0}^{k}\lbrace \mathcal{C}(y_{(\tau_i+1):\tau_{i+1}}) + \beta \rbrace \right]\,,$$

where $\beta > 0$ is a penalty parameter and $\mathcal{C}(y_{u:v})$ is the minimal cost over the segment ${u,...,v}$. The penalty $\beta$ is understood as an additional cost when introducing a new segment.

The cost for data $y_{\tau+1:t}$ with linear interpolation from value $a$ to value $b$ is given by

$$ \mathcal{C}(y_{\tau+1:t},a,b) = \sum_{i=\tau+1}^{t}\left(y_i - [a + (b-a)\frac{i-\tau}{t-\tau}]\right)^2\,, $$

which can be computed in constant time with the formula

$$\mathcal{C}(y_{\tau+1:t},a,b) = S_{t}^2 - S_{\tau}^2 +\frac{b^2-a^2}{2} + \frac{a^2 + ab +b^2}{3}(t-\tau) + \frac{(b-a)^2}{6(t-\tau)}$$

$$\quad\quad\quad\quad\quad\quad - \frac{2}{t-\tau} \left( (ta-b\tau)[S_{t}^1 - S_{\tau}^1] + (b-a)[S_{t}^+ - S_{\tau}^+]\right)\,,$$

where

$$ S^1_t = \sum_{i=1}^t y_i\quad , \quad S^2_t = \sum_{i=1}^t y_i^2\quad \hbox{and} \quad S^+t = \sum{i=1}^t iy_i\quad \hbox{for all} \,\, t \in {1,...,n}\,. $$

To address the continuity constraint, we introduce the function $v \mapsto Q_t(v)$ which is the optimal penalized cost up to position $t$ with a last infered value equal to $v$ (at position t). The idea is then to update a set

$$ \mathcal{Q}t = {Q_t(v), v= v{min},...,v_{max}}\,, $$

at any time step $t \in {1,...,n}$. $v_{min}$ and $v_{max}$ are the bounds of the interval of possible ending values for the considered data to segment. They can be determined in a preprocessing step.

The new update with continuity constraint takes the form

$$ Q_t(v) = \min_{0 \le \tau < t}\left( \min_{u}{Q_{\tau}(u) + \mathcal{C}(y_{\tau+1:t},u,v) + \beta}\right)\,, $$

where the presence of the same value $u$ in $Q_{\tau}$ and the cost realizes the continuity constraint. At initial step we simply have $Q_0(v) = -\beta$.

The slopeOP function

We install the package from Github:

#devtools::install_github("vrunge/slopeOP")
library(slopeOP)

We simulate data with the function slopeData given the indices for extremal values and in second vector these values to reach for corresponding indices. The last parameter is the standard deviation of a normal standard noise.

data <- slopeData(c(1,100,200,300,500), c(0,1,0,3,2), 1)

The changepoint detection is done by using the function slopeOP

slopeOP(data, c(0,1,2,3), 10)

## $changepoints
## [1]   1 101 201 307 500
## 
## $parameters
## [1] 0 1 0 3 2
## 
## $globalCost
## [1] 496.4254
## 
## attr(,"class")
## [1] "slopeOP"

In slopeOP function, the parameter type is channel by default. With type equal to channel we use the monotonicity property in optimal cost matrix to reduce time complexity. If it is equal to pruning we prune some positions using a theorem taking into account unseen data.

Options for constraining inference

Parameter constraint can be set to up which corresponds to a restriction to nondecreasing vector parameter.

data <- slopeData(c(1,150,200,350,500,750,1000), c(71,73,70,75,77,73,80), 1)
slopeOP(data, 71:80, 5, constraint = "up")

## $changepoints
## [1]    1   63  254  356  828 1000
## 
## $parameters
## [1] 71 72 72 75 75 80
## 
## $globalCost
## [1] 1768.407
## 
## attr(,"class")
## [1] "slopeOP"

With constraint equal to updown the infered vector is unimodal (with a maximum).

data <- slopeData(c(1,150,200,350,500,750,1000), c(71,73,70,75,78,73,75), 1)
slopeOP(data, 71:80, 5, constraint = "updown")

## $changepoints
## [1]    1  316  317  502  697 1000
## 
## $parameters
## [1] 71 73 74 78 74 74
## 
## $globalCost
## [1] 1407.936
## 
## attr(,"class")
## [1] "slopeOP"

We also can limit the angles between successive segments with constraint equal to smoothing and the parameter minAngle in degree.

data <- slopeData(c(1,30,40,70,100,150,200),c(70,80,70,80,70,80,70),0.5)
slopeOP(data,70:80,5, constraint = "smoothing", minAngle = 170)

## $changepoints
##  [1]   1   7  19  25  28  34  40  46  47  53  62  68  73  79  94 100 103
## [18] 149 154 160 200
## 
## $parameters
##  [1] 71 72 76 77 77 76 74 73 73 74 77 78 78 77 72 71 71 79 79 78 70
## 
## $globalCost
## [1] 319.9376
## 
## attr(,"class")
## [1] "slopeOP"

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vrunge/slopeOP documentation built on Dec. 23, 2021, 4:12 p.m.