Path algorithms for fused lasso signal approximator with application to COVID‐19 spread in Korea

Summary

The fused lasso signal approximator (FLSA) is a smoothing procedure for noisy observations that uses fused lasso penalty on unobserved mean levels to find sparse signal blocks. Several path algorithms have been developed to obtain the whole solution path of the FLSA. However, it is known that the FLSA has model selection inconsistency when the underlying signals have a stair‐case block, where three consecutive signal blocks are either strictly increasing or decreasing. Modified path algorithms for the FLSA have been proposed to guarantee model selection consistency regardless of the stair‐case block. In this paper, we provide a comprehensive review of the path algorithms for the FLSA and prove the properties of the recently modified path algorithms' hitting times. Specifically, we reinterpret the modified path algorithm as the path algorithm for local FLSA problems and reveal the condition that the hitting time for the fusion of the modified path algorithm is not monotone in a tuning parameter. To recover the monotonicity of the solution path, we propose a pathwise adaptive FLSA having monotonicity with similar performance as the modified solution path algorithm. Finally, we apply the proposed method to the number of daily‐confirmed cases of COVID‐19 in Korea to identify the change points of its spread.

1. INTRODUCTION

Consider observations ${\{y_i\}}_{i=1}^n$ from a model

$$y_i={\mu }_i+{ϵ}_i,$$ (1)

where ${ϵ}_i\mathrm{s}$ are independently and identically from a distribution with mean 0 and variance ${\sigma }_n^2$. Here, we assume that true underlying signals have block structures with ${\mu }_1=\dots ={\mu }_{j_1}\ne {\mu }_{j_1+1}=\dots ={\mu }_{j_K}\ne {\mu }_{j_K+1}=\dots ={\mu }_n$. Under this assumption, it is essential to identify the unknown block structures to estimate the signal means. This problem is known as multiple‐change‐point detection. The literature contains various methods on multiple‐change‐point detection, including best subset selection (Yao, 1988; Yao & Au, 1989), circular binary segmentation (CBS) (Olshen et al., 2004), and wild binary segmentation (WBS) (Fryzlewicz, 2014).

This study focuses on the fused lasso signal approximator (FLSA) that finds sparse signal blocks using the fused lasso penalty (Tibshirani et al., 2005) on underlying mean levels. The FLSA obtains the signal estimate by minimising

$$f_{SFL}(\mu ;\mathbf{\text{y}},{\lambda }_1,{\lambda }_2)=\frac{1}{2}\Vert \mathbf{\text{y}}-\mathit{\mu }{\Vert }_2^2+{\lambda }_1\Vert \mathit{\mu }{\Vert }_1+{\lambda }_2\Vert \mathit{\mu }{\Vert }_{\text{TV}},$$ (2)

where $\mathbf{\text{y}}={(y_1,\dots ,y_n)}^T$, $\mathit{\mu }={({\mu }_1,\dots ,{\mu }_n)}^T$, $\Vert \mathit{\mu }{\Vert }_1={\sum }_{j=1}^n|{\mu }_j|$ is the ℓ ₁‐norm of $\mathit{\mu }$, and $\Vert \mathit{\mu }{\Vert }_{\text{TV}}={\sum }_{j=2}^n|{\mu }_j-{\mu }_{j-1}|$ is the total‐variation norm of $\mathit{\mu }$. Given ${\lambda }_1$ and ${\lambda }_2$, the FLSA obtains the sparse block solution for $\mathit{\mu }$, providing the estimates of both signal levels and block structures. As shown in Lemma A.1 in Friedman et al. (2007), the sparse estimate can be obtained directly from the minimiser ${\widehat{\mu }}_{(0,{\lambda }_2)}$ of $f(\mu ;\mathbf{\text{y}},0,{\lambda }_2)$ by the soft‐thresholding operator ${\widehat{\mu }}_{({\lambda }_1,{\lambda }_2)}={\text{Soft}}_{{\lambda }_1}({\widehat{\mu }}_{(0,{\lambda }_2)})$, where ${\text{Soft}}_a(\mathbf{\text{x}})={\left(\text{sign}(x_i)\max (|x_i|-a,0)\right)}_{i=1}^n$ for $\mathbf{\text{x}}\in {ℝ}^n$.

Furthermore, the FLSA has the model selection consistency (i.e. exact identification of the true block structure) when the underlying true block structure has no stair‐case blocks, where a stair‐case block denotes that three consecutive blocks that are either strictly increasing or decreasing (Rinaldo, 2009, 2014; Qian & Jia, 2016). The FLSA estimator is inconsistent in identifying the true signal blocks when the true block structure contains any stair‐case blocks. The preconditioned FLSA via Puffer transformation (Qian & Jia, 2016) and the modified path algorithm for FLSA (Son & Lim, 2019) are proposed to resolve this inconsistency. Recently, the error analysis for the FLSA has been done in literature (Lin et al., 2016).

Besides the theoretical properties of the FLSA, it is critical to obtain its solution efficiently. The entire solution path of the FLSA can be obtained using either the path algorithm for the FLSA proposed by Hoefling (2010) or the path algorithm for the generalised lasso by Tibshirani & Taylor (2011). These two solution path algorithms solve the problem (2) but still show the model selection inconsistency for identifying true blocks with stair‐case blocks. Contrarily, two modified path algorithms have been developed recently to resolve the model selection inconsistency of the FLSA. The preconditioned FLSA with the Puffer transformation (Qian & Jia, 2016) converts the original problem into a Lasso problem with reparameterisation and applies the Puffer transformation. Essentially, the solution path algorithm for the preconditioned FLSA is based on the path algorithm for Lasso (Bradely Efron et al., 2004). The modified path algorithm for the FLSA (mPath‐FLSA) by Son & Lim (2019) reinterprets the FLSA path algorithm using the distances between two adjacent blocks and proposes a new distance criterion that guarantees the model selection consistency regardless of a stair‐case block. Note that the original study does not provide the overall objective function of mPath‐FLSA.

Our main contributions through this study are as follows. First, we provide a comprehensive review of the FLSA path algorithms and prove the properties of the hitting times of the recently modified path algorithm. Specifically, we provide an explicit form of the objective function of mPath‐FLSA. Furthermore, we apply the new representation to show that the hitting times of mPath‐FLSA can fail to increase monotonically with respect to ${\lambda }_2$. For example, we suppose that there are four observations, $y_1=-0.032$, $y_2=0.787$, $y_3=-0.122$, and $y_4=0.207$. The hitting times for the FLSA solution path algorithm are $({\lambda }_2^{(0)},\dots ,{\lambda }_2^{(3)})=(\mathrm{0,0}.\mathrm{1097,0}.\mathrm{2731,0}.3352)$, which satisfy the monotone property (i.e. ${\lambda }_2^{(0)}\le \dots \le {\lambda }_2^{(3)}$). However, the hitting times of mPath‐FLSA are calculated as $({\lambda }_2^{(0)},\dots ,{\lambda }_2^{(3)})=(\mathrm{0,0}.\mathrm{0822,0}.\mathrm{2049,0}.1676)$, which are not monotone in ${\lambda }_2$. Section 3 provides additional details. Second, we propose a pathwise adaptive FLSA and its solution path algorithm to resolve the violation of the monotonicity of the solution path along ${\lambda }_2$. The proposed method adopts the weighted fusion penalty terms ${\sum }_{j=2}^n{w}_j^{(l)}|{\mu }_j-{\mu }_{j-1}|$, which are adaptively defined by the solutions at the hitting times ${({\lambda }_2^{(l)})}_{l=0}^{n-1}$ on the solution path along ${\lambda }_2$. Finally, we provide a comprehensive numerical comparison of the existing path algorithms with the proposed algorithm. We compare four methods based on the entire solution path along ${\lambda }_2$, the exact pattern recovery probability, and the estimation performance of the signal levels and block structures concerning the selection of tuning parameters.

The remainder of this paper is organised as follows. Section 2 contains a brief review of the existing solution path algorithms related to the FLSA. Section 3 presents a new interpretation of mPath‐FLSA besides the conditions for the non‐monotonicity of the solution path along ${\lambda }_2$. The pathwise adaptive FLSA and its solution path algorithm that satisfies the monotone property in the solution path along ${\lambda }_2$ are proposed in Section 4. In Section 5, a numerical study for measuring the probability of the exact recovery was conducted using the proposed algorithm and the other existing algorithms. Moreover, we compared the estimation performance with the optimal tuning parameter chosen by the Bayesian information criterion (BIC) and the extended BIC (EBIC). In Section 6, we apply the proposed method and existing FLSA path algorithms to the number of daily‐confirmed cases of COVID‐19 in Korea to identify events that affect the COVID‐19 spread. Finally, we conclude the paper with some remarks.

2. EXISTING SOLUTION PATH ALGORITHMS FOR FLSA

In this section, we review three existing solution path algorithms related to the FLSA; the path algorithm for the FLSA (Path‐FLSA) by Hoefling (2010), the path algorithm for the preconditioned FLSA (Path‐PFLSA) by Qian & Jia (2016), the modified path algorithm for the FLSA (mPath‐FLSA) by Son & Lim (2019). Henceforth, to focus on the identification of true mean blocks, we consider the FLSA with ${\lambda }_1=0$ as follows:

$$f_{FL}(\mu ;\mathbf{\text{y}},{\lambda }_2)=\frac{1}{2}\Vert \mathbf{\text{y}}-\mathit{\mu }{\Vert }_2^2+{\lambda }_2{\displaystyle \sum _{j=2}^n}|{\mu }_j-{\mu }_{j-1}|.$$ (3)

Note that we omit the review of the solution path algorithm for the generalised Lasso by Tibshirani & Taylor (2011) because its solution path for the FLSA is the same as Path‐FLSA.

2.1. Path Algorithm for the FLSA (Path‐FLSA)

The FLSA path algorithm proposed by Hoefling (2010) provides a whole solution path of the FLSA along ${\lambda }_2$ based on the fact that the solution path for ${\lambda }_2$ is a piece‐wise linear in ${\lambda }_2$. Originally, Hoefling (2010) proposes a path algorithm for the generalised FLSA problem

$$\underset{\mathit{\mu }}{\min }{f}_{FL}(\mathit{\mu };\mathbf{\text{y}},{\lambda }_2)=\frac{1}{2}\Vert \mathbf{\text{y}}-\mathit{\mu }{\Vert }_2^2+{\lambda }_2{\displaystyle \sum _{(i,j)\in E}}|{\mu }_i-{\mu }_j|,$$ (4)

where $E$ denotes a set of index pairs that correspond to fusion penalty terms. Here, we focus on the path algorithm for the one‐dimensional FLSA as in (3) (i.e. $E=\{(\mathrm{1,2}),(\mathrm{2,3}),\dots ,(p-1,p)\}$).

Specifically, let ${\{{\widehat{B}}_k\}}_{k=1}^{\widehat{J}({\lambda }_2)}$ be a partition of $\{1,\dots ,n\}$ defined from the solution ${\widehat{\mu }}^{FL}({\lambda }_2)$ of the FLSA at ${\lambda }_2$, where $\widehat{J}({\lambda }_2)-1$ is the number of change points in ${\widehat{\mu }}^{FL}({\lambda }_2)$. Further, we let ${\{{\lambda }_2^{(l)}\}}_{l=0}^{n-1}$ be hitting times on the solution path, with two adjacent sets in the partition defined by the previous hitting time fused on each of them. For example, there is an index $k\in \{1,\dots ,n-l+1\}$ on ${\lambda }_2={\lambda }_2^{(l)}$, such that ${\widehat{B}}_{k-1}^{(l-1)}$ and ${\widehat{B}}_k^{(l-1)}$ are fused as ${\widehat{B}}_{k-1}^{(l)}={\widehat{B}}_{k-1}^{(l-1)}\cup {\widehat{B}}_k^{(l-1)}$. Thus, the solution ${\widehat{\mu }}^{FL}({\lambda }_2)$ for ${\lambda }_2={\lambda }_2^{(l)}$ can be expressed explicitly as:

$${\widehat{\mu }}_i^{FL}({\lambda }_2)={\displaystyle \sum _{k=1}^{n-l}}{\widehat{\nu }}_k^{(l)}({\lambda }_2)I(i\in {\widehat{B}}_k^{(l)}),$$ (5)

where ${\widehat{b}}_k^{(l)}=|{\widehat{B}}_k^{(l)}|$, ${\widehat{\nu }}_k^{(l)}({\lambda }_2)=\frac{1}{{\widehat{b}}_k^{(l)}}{\sum }_{i\in {\widehat{B}}_k^{(l)}}{y}_i+{\widehat{c}}_k({\lambda }_2)={\overline{y}}_k+{\widehat{c}}_k({\lambda }_2)$,

$${\widehat{c}}_k({\lambda }_2)=\left\{\begin{array}{ccc}\frac{2{\lambda }_2}{{\widehat{b}}_k^{(l)}}& \text{if}& {\widehat{\nu }}_k({\lambda }_2)<{\widehat{\nu }}_{k-1}({\lambda }_2),{\widehat{\nu }}_{k+1}({\lambda }_2)<{\widehat{\nu }}_k({\lambda }_2)\\ -\frac{2{\lambda }_2}{{\widehat{b}}_k^{(l)}}& \text{if}& {\widehat{\nu }}_k({\lambda }_2)>{\widehat{\nu }}_{k-1}({\lambda }_2),{\widehat{\nu }}_{k+1}({\lambda }_2)>{\widehat{\nu }}_k({\lambda }_2)\\ 0& \text{if}& ({\widehat{\nu }}_k({\lambda }_2)-{\widehat{\nu }}_{k-1}({\lambda }_2))({\widehat{\nu }}_{k+1}({\lambda }_2)-{\widehat{\nu }}_k({\lambda }_2))>0\end{array}\right..$$

Now, suppose that we know a partition ${\{{\widehat{B}}_k^{(l)}\}}_{k=1}^{n-l}$ by ${\widehat{\mu }}^{FL}({\lambda }_2^{(l)})$, such that $\bigcup {\widehat{B}}_k^{(l)}=\{1,\dots ,n\}$, and consider ${\lambda }_2\in ({\lambda }_2^{(l)},{\lambda }_2^{(l+1)})$, where there is no fusion in the FLSA estimate. The FLSA solution satisfies the following equation:

$$\frac{\partial f}{\partial {\nu }_k}={\widehat{b}}_k^{(l)}{\nu }_k-\sum _{i\in {\widehat{B}}_k^{(l)}}{y}_i+{\lambda }_2(\partial |{\nu }_j-{\nu }_{j-1}|+\partial |{\nu }_j-{\nu }_{j+1}|)=0,$$ (6)

where $\partial |x|$ is a subdifferential of $|x|$, which is defined as $\partial |x|=1$ if $x>0$, −1 if $x<0$, and $z\in [-\mathrm{1,1}]$ if $x=0$. We can easily obtain the derivative of ${\widehat{\nu }}_k$ with respect to ${\lambda }_2$ from (6) as

$$\frac{\partial {\widehat{\nu }}_k}{\partial {\lambda }_2}=-\frac{\partial |{\widehat{\nu }}_k-{\widehat{\nu }}_{k-1}|+\partial |{\widehat{\nu }}_k-{\widehat{\nu }}_{k+1}|}{{\widehat{b}}_k^{(l)}}.$$ (7)

Thus, ${\widehat{\nu }}_k$ is linear in ${\lambda }_2\in ({\lambda }_2^{(l)},{\lambda }_2^{(l+1)})$ because the numerator and denominator of (7) are unchanged for the given interval $({\lambda }_2^{(l)},{\lambda }_2^{(l+1)})$. This property shows that the FLSA solution path is piecewise linear in ${\lambda }_2$.

Given the solution ${\widehat{\nu }}_k$ on the hitting time ${\lambda }_2^{(l)}$ and the derivative $\partial {\widehat{\nu }}_k/\partial {\lambda }_2$, the solution ${\widehat{\nu }}_k({\lambda }_2)$ for ${\lambda }_2\in ({\lambda }_2^{(l)},{\lambda }_2^{(l+1)})$ can be easily obtained as

$${\widehat{\nu }}_k({\lambda }_2)={\widehat{\nu }}_k({\lambda }_2^{(l)})+\frac{\partial {\widehat{\nu }}_k}{\partial {\lambda }_2}\delta ,$$ (8)

where $\delta ={\lambda }_2-{\lambda }_2^{(l)}$. Based on (8), the next hitting time is defined as

$${\lambda }_2^{(l+1)}=\underset{1\le k\le n-l-1}{\min }{h}_{k,k+1}({\lambda }_2^{(l)})=\underset{1\le k\le n-l-1}{\min }\frac{{\widehat{\nu }}_{k+1}({\lambda }_2^{(l)})-{\widehat{\nu }}_k({\lambda }_2^{(l)})}{\frac{\partial {\widehat{\nu }}_k}{\partial {\lambda }_2}-\frac{\partial {\widehat{\nu }}_{k+1}}{\partial {\lambda }_2}}+{\lambda }_2^{(l)}$$ (9)

Given the Eqs. 8 and (9), the Path‐FLSA starts with ${\lambda }_2^{(0)}=0$ and sequentially calculates the hitting times. The Path‐FLSA stores all solutions at hitting times $({\lambda }_2^{(0)},\dots ,{\lambda }_2^{(n-1)})$ and their derivatives to obtain the solution for any ${\lambda }_2$ with (8). Note that the FLSA solution path for generalised fusion penalty ${\lambda }_2{\sum }_{(i,j)\in E}|{\mu }_i-{\mu }_j|$ could be split for some edge set $E$. For example, two adjacent solutions ${\widehat{\mu }}_i^{(l-1)}$ and ${\widehat{\mu }}_j^{(l-1)}$ for $(i,j)\in E$ are fused at ${\lambda }_2^{(l)}$ (i.e. ${\widehat{\mu }}_i^{(l-1)}\ne {\widehat{\mu }}_j^{(l-1)}$ and ${\widehat{\mu }}_i^{(l)}={\widehat{\mu }}_j^{(l)}$) and split at ${\lambda }_2^{(t)}$ for some $t>l$ (i.e. ${\widehat{\mu }}_i^{(t-1)}={\widehat{\mu }}_j^{(t-1)}$ and ${\widehat{\mu }}_i^{(t)}\ne {\widehat{\mu }}_j^{(t)}$) with the generalised fusion penalty. However, the one‐dimensional FLSA (i.e. $E$ is an edge set $\{(\mathrm{1,2}),(\mathrm{2,3}),\dots ,(n-1,n)\}$ of a chain graph) only has hitting events on the solution path, as shown by Friedman et al. (2007) in Proposition 2.

2.2. Preconditioned FLSA with Puffer Transformation (PCD‐FLSA)

As explained in the Introduction, the FLSA faces the model selection inconsistency due to stair‐case blocks in the underlying signals. Qian & Jia (2016) connect the FLSA model selection inconsistency to the irrepresentable condition of the Lasso problem (Zhao & Yu, 2006) by the reparameterisation of the FLSA. To obtain the model selection consistency, Qian & Jia (2016) applied the Puffer transformation introduced by Jia & Rohe (2015) to the reparameterised FLSA, which is called the preconditioned FLSA with Puffer transformation (PCD‐FLSA).

Specifically, consider the reparameterisation of $\mu$ with $\theta$ as follows:

$$\left({\mu }_1={\theta }_1,{\mu }_2={\theta }_1+{\theta }_2,\dots ,{\mu }_n={\displaystyle \sum _{j=1}^n}{\theta }_j\right)\equiv \mathit{\mu }=\mathbf{\text{A}}\mathit{\theta },$$ (10)

where $\mathbf{\text{A}}$ is a lower triangular matrix with nonzero elements equal to one. This reparameterisation can represent the objective function of the FLSA as

$$f_L(\theta ;{\lambda }_2)=\frac{1}{2}\Vert \mathbf{\text{y}}-\mathbf{\text{A}}\mathit{\theta }{\Vert }_2^2+{\lambda }_2\Vert {\mathit{\theta }}_{[2:n]}{\Vert }_1,$$ (11)

where ${\mathit{\theta }}_{[2:n]}={({\theta }_2,\dots ,{\theta }_n)}^T\in R^{n-1}$. The solution of (11) can be obtained by

$${\widehat{\mathit{\theta }}}_{[2:n]}({\lambda }_2)=\underset{{\mathit{\theta }}_{[2:n]}}{\text{argmin}}\frac{1}{2}\Vert \widetilde{\mathbf{\text{y}}}-\widetilde{\mathbf{\text{X}}}{\mathit{\theta }}_{[2:n]}{\Vert }_2^2+{\lambda }_2\Vert {\mathit{\theta }}_{[2:n]}{\Vert }_1,\text{and}{\widehat{\theta }}_1({\lambda }_2)=\overline{y}-{\overline{\mathbf{\text{x}}}}^T{\widehat{\mathit{\theta }}}_{[2:n]}({\lambda }_2),$$ (12)

where $\overline{y}=(1/n){\sum }_{i=1}^n{y}_i$, $\widetilde{\mathbf{\text{y}}}=\mathbf{\text{y}}-\overline{y}{\mathbf{\text{1}}}_n$, ${\mathbf{\text{1}}}_n$ is an $n$‐dimensional vector of ones, $\widetilde{\mathbf{\text{X}}}=[{\mathbf{\text{x}}}_1-{\overline{x}}_1{\mathbf{\text{1}}}_n,\dots ,{\mathbf{\text{x}}}_{n-1}-{\overline{x}}_{n-1}{\mathbf{\text{1}}}_n]=\mathbf{\text{X}}-{\mathbf{\text{1}}}_n{\overline{\mathbf{\text{x}}}}^T$, $\overline{\mathbf{\text{x}}}={({\overline{x}}_1,\dots ,{\overline{x}}_{n-1})}^T$, ${\overline{x}}_j={\mathbf{\text{1}}}_n^T{\mathbf{\text{x}}}_j/n$, and $\mathbf{\text{X}}={\mathbf{\text{A}}}_{[1:n,2:n]}=[{\mathbf{\text{x}}}_1,\dots ,{\mathbf{\text{x}}}_{n-1}]$, which denotes that $\mathbf{\text{X}}$ is an $n\times (n-1)$ matrix defined by removing the first column of $\mathbf{\text{A}}$. Henceforth, we refer to the reparameterised FLSA problem (12) as the transformed Lasso.

Based on the reparameterisation, the pattern recovery for $\mathit{\mu }$ using the FLSA is equivalent to the sign consistency of the Lasso estimator for ${\mathit{\theta }}_{[2:n]}$. For a linear model $\widetilde{\mathbf{\text{y}}}=\widetilde{\mathbf{\text{X}}}{\mathit{\theta }}_{[2:n]}+\mathit{ϵ}$ with $E(\mathit{ϵ})=\mathbf{\text{0}}$, the conditions for the sign consistency of the Lasso estimator is well understood. One of the necessary conditions of the sign consistency of the Lasso estimator is the irrepresentable condition (Zhao & Yu, 2006) defined as follows:

$$\Vert {\widetilde{\mathbf{\text{X}}}}_{S^c}^T{\widetilde{\mathbf{\text{X}}}}_S{({\widetilde{\mathbf{\text{X}}}}_S^T{\widetilde{\mathbf{\text{X}}}}_S)}^{-1}\text{sign}({({\mathit{\theta }}_{[2:n]})}_S){\Vert }_{\infty }\le 1-\eta \text{for some}\eta \in (\mathrm{0,1}],$$ (13)

where $S=\{j:{({\theta }_{[2:n]})}_j\ne 0\}$. Theorem 2 of Qian & Jia (2016) shows that the irrepresentable condition (13) holds if and only if one of the following two conditions holds.

• (1)For a set of change points $\{j_1,\dots ,j_K\}$, $K=1$ or ${\max }_{1\le t<K}(j_{t+1}-j_t)=1$.
• (2)The underlying signal has no stair‐case blocks. That is, $({\mu }_{j_t}-{\mu }_{j_{t-1}})({\mu }_{j_{t+1}}-{\mu }_{j_t})<0$.

The abovementioned condition (2) shows the inconsistency of the FLSA in identifying true change points when underlying signals have stair‐case blocks.

Qian & Jia (2016) proposed a preconditioned FLSA with a Puffer transformation to resolve the inconsistency of the FLSA. The Puffer transformation for Lasso is proposed by Jia & Rohe (2015) to make the Lasso estimator sign consistent when the irreresentable condition does not hold for the design matrix $\mathbf{\text{X}}\in {ℝ}^{n\times p}$ with $n\ge p$. Let the singular value decomposition of $\widetilde{\mathbf{\text{X}}}$ be $\widetilde{\mathbf{\text{X}}}=\mathbf{\text{U}}\mathbf{\text{D}}{\mathbf{\text{V}}}^T$. The Puffer transformation is defined as ${\mathbf{\text{F}}}_{n\times n}=\mathbf{\text{U}}{\mathbf{\text{D}}}^{-1}{\mathbf{\text{U}}}^T$. Because $\widetilde{\mathbf{\text{X}}}\in {ℝ}^{n\times (n-1)}$ in (12), the results of Jia & Rohe (2015) are valid for the transformed Lasso problem in (12). Let $\mathbf{\text{Z}}=\mathbf{\text{F}}\widetilde{\mathbf{\text{X}}}$ and $\mathbf{\text{a}}=\mathbf{\text{F}}\widetilde{\mathbf{\text{y}}}$ be the preconditioned design matrix and the response vector, respectively. The preconditioned FLSA with the Puffer transformation solves

$$\underset{\mathbf{\text{b}}}{\min }\frac{1}{2}\Vert \mathbf{\text{a}}-\mathbf{\text{Z}}\mathbf{\text{b}}{\Vert }_2^2+{\lambda }_2\Vert \mathbf{\text{b}}{\Vert }_1.$$ (14)

Because ${\mathbf{\text{Z}}}^T\mathbf{\text{Z}}={\mathbf{\text{I}}}_n$, the solution of the PCD‐FLSA has the explicit form $\widehat{\mathbf{\text{b}}}({\lambda }_2)={\text{Soft}}_{{\lambda }_2}({\mathbf{\text{Z}}}^T\mathbf{\text{a}})$, where ${\text{Soft}}_a(\mathbf{\text{x}})={(\text{sign}(x_i)\max (|x_i|-a,0))}_{i=1}^n$. Thus, the solution path of $\widehat{\mathbf{\text{b}}}({\lambda }_2)$ is piecewise linear in ${\lambda }_2$, and the hitting times (or slope‐change points in the solution path) are ${\lambda }_{2,PCD-FLSA}^{(i)}=|{\mathbf{\text{Z}}}^T\mathbf{\text{a}}{|}_{(i)}$ for $i=1,\dots ,n-1$, where ${\mathbf{\text{x}}}_{(i)}$ is the $i\text{th}$ smallest value in a vector $\mathbf{\text{x}}$.

2.3. Modified Path Algorithm for FLSA (mPath‐FLSA)

Recently, Son & Lim (2019) provided a new interpretation of the path algorithm for the one‐dimensional FLSA (1D‐FLSA), which uses a new distance measure for two adjacent blocks to determine if they are fused. Furthermore, they used this interpretation to show the inconsistency of the FLSA in the presence of the stair‐case blocks and proposed a path algorithm with a modified distance measure that resolves this inconsistency.

Specifically, let ${\lambda }_2^{(0)}=0<{\lambda }_2^{(1)}<\dots ,<{\lambda }_2^{(n-1)}$ be the hitting times of the 1D‐FLSA. Further, let ${\{B_k^{(l)}\}}_{k=1}^{n-l}$ be a partition of $\{1,\dots ,n\}$ corresponding to the estimated blocks of $\widehat{\mathit{\mu }}({\lambda }_2)$ for ${\lambda }_2\in [{\lambda }_2^{(l)},{\lambda }_2^{(l+1)})$. Son & Lim (2019) developed a new distance measure $d({\widehat{B}}_{k-1}^{(l)},{\widehat{B}}_k^{(l)})$ for two adjacent blocks ${\widehat{B}}_{k-1}^{(l)}$ and ${\widehat{B}}_k^{(l)}$ defined as

$$d({\widehat{B}}_{k-1}^{(l)},{\widehat{B}}_k^{(l)})=\frac{|{\overline{y}}_{k-1}^{(l)}-{\overline{y}}_k^{(l)}|}{\frac{1}{{\widehat{b}}_{k-1}^{(l)}}s\left({\overline{y}}_{k-1}^{(l)}\right)+\frac{1}{{\widehat{b}}_k^{(l)}}s\left({\overline{y}}_k^{\left(l\right)}\right)},$$ (15)

where ${\widehat{b}}_k^{(l)}=|{\widehat{B}}_k^{(l)}|$, ${\overline{y}}_k^{(l)}=(1/{\widehat{b}}_k^{(l)}){\sum }_{i\in {\widehat{B}}_k^{(l)}}{y}_i$, and

$$s\left({\overline{y}}_k^{(l)}\right)=\left\{\begin{array}{ll}|\left(\partial |{\overline{y}}_k^{(l)}-{\overline{y}}_{k-1}^{(l)}|+\partial |{\overline{y}}_k^{(l)}-{\overline{y}}_{k+1}^{(l)}|\right)|/2& \text{for}k=2,\dots ,n-l-1\\ |(\partial |{\overline{y}}_k^{(l)}-{\overline{y}}_{k-1}^{(l)})|)|/2& \text{for}k=n-l\\ |\left(\partial |{\overline{y}}_k^{(l)}-{\overline{y}}_{k+1}^{(l)}|\right)|/2& \text{for}k=1\end{array}\right..$$ (16)

Son & Lim (2019) used the distance measure in (15) to show that the hitting time ${\lambda }_2^{(l+1)}$ in (9) is equivalent to $\frac{1}{2}{\min }_{k}d({\widehat{B}}_{k-1}^{(l)},{\widehat{B}}_k^{(l)})$. This interpretation shows that the 1D‐FLSA cannot fuse two consecutive stair‐case blocks with a finite ${\lambda }_2$ because the denominator of the distance measure is defined as 0.

The abovementioned phenomenon for two consecutive stair‐case blocks promoted Son & Lim (2019) to propose the modified distance measure that guarantees the consistency of the 1D‐FLSA estimator in identifying true block structures regardless of the stair‐case blocks. The modified distance measure $\delta ({\widehat{B}}_{k-1}^{(l)},{\widehat{B}}_k^{(l)})$ for two consecutive blocks ${\widehat{B}}_{k-1}^{(l)}$ and ${\widehat{B}}_k^{(l)}$ is defined as

$$\delta ({\widehat{B}}_{k-1}^{(l)},{\widehat{B}}_k^{(l)})=\frac{|{\overline{y}}_{k-1}^{(l)}-{\overline{y}}_k^{(l)}|}{1/{\widehat{b}}_{k-1}^{(l)}+1/{\widehat{b}}_k^{(l)}}.$$ (17)

A modified path algorithm is proposed using the modified distance measure with the hitting times defined as:

$${\lambda }_{2,mPath-FLSA}^{(l+1)}=\frac{1}{2}\underset{k}{\min }\delta ({\widehat{B}}_{k-1}^{(l)},{\widehat{B}}_k^{(l)})\text{for}l=0,\dots ,n-2.$$ (18)

Theorem 2 of Son & Lim (2019) shows that the 1D‐FLSA with a modified path algorithm consistently identifies the change points and signs of difference between two adjacent blocks. Note that the modified path algorithm guarantees a consistent estimator for identifying true block structures by adopting the modified distance measure $\delta (·,·)$, but this modification causes the modified path algorithm to solve a different problem that is not equivalent to the 1D‐FLSA, and the monotone increasing property of the hitting times could be violated (i.e. for some $l$, ${\lambda }_2^{(l)}>{\lambda }_2^{(l+1)}$). This phenomenon is explained in the next section.

3. NEW INTERPRETATION OF MODIFIED PATH ALGORITHM

In this section, we provide a novel interpretation of the mPath‐FLSA, including the exact target objective function of the mPath‐FLSA. It clarifies the differences between Path‐FLSA and mPath‐FLSA. Moreover, we illustrate a condition that violates the monotone increasing property of the hitting times in mPath‐FLSA.

Suppose we know that a partition ${\{{\widehat{B}}_k^{(l)}\}}_{k=1}^{n-l}$ of $\{1,\dots ,n\}$ for a given ${\lambda }_{2,mPath-FLSA}^{(l)}=\frac{1}{2}{\min }_k\delta ({\widehat{B}}_{k-1}^{(l-1)},{\widehat{B}}_k^{(l-)})$, where ${\lambda }_{2,mPath-FLSA}^{(l)}$ is the $l\text{th}$ hitting time of $\{y_1,y_2,\dots ,y_n\}$ based on the mPath‐FLSA. In this section, we denote ${\lambda }_{2,mPath-FLSA}^{(l)}$ as ${\lambda }_2^{(l)}$ for notational simplicity. Let ${\nu }_k^{(l)}$ be a parameter of the $k\text{th}$ block mean at ${\lambda }_2\in [{\lambda }_2^{(l)},{\lambda }_2^{(l+1)})$. A localised FLSA problem for a pair $(k-1,k)$ of the partition indices of ${\{{\widehat{B}}_j^{(l)}\}}_{j=1}^{n-l}$ is define as

$$f_l({\nu }_{k-1}^{(l)},{\nu }_k^{(l)};{\eta }_k)=\frac{1}{2}\sum _{j\in \{k-1,k\}}\sum _{i\in {\widehat{B}}_j^{(l)}}{(y_i-{\nu }_j^{(l)})}^2+{\eta }_k|{\nu }_k^{(l)}-{\nu }_{k-1}^{(l)}|\text{for}k=2,\dots ,n-l,$$ (19)

where ${\eta }_k\ge 0$ is a tuning parameter of the localised FLSA problem for $({\widehat{B}}_{k-1}^{(l)},{\widehat{B}}_k^{(l)})$. In Theorem 1, we show that the hitting times of the mPath‐FLSA are equivalent to the minimum hitting times of $(n-l-1)$ localised FLSA problems.

Theorem 1

Let ${\lambda }_2^{(l)}$ be the $l\mathit{\text{th}}$ hitting time of the modified path algorithm with $\mathbf{\text{y}}=(y_1,\dots ,y_n)$ and ${\lambda }_2^{(0)}=0$. Let ${\eta }_j^{(l+1)}$ be the hitting time of (19) for a partition pair $({\widehat{B}}_{j-1}^{(l)},{\widehat{B}}_j^{(l)})$. We define ${\tau }^{(l+1)}={\min }_{2\le j\le n-l}{\eta }_j^{(l+1)}$, which is the minimum hitting time of $f_l({\nu }_{j-1},{\nu }_j)$ for $j=2,\dots ,n-l$. Then, ${\eta }_j^{(l+1)}=\delta ({\widehat{B}}_{j-1}^{(l)},{\widehat{B}}_j^{(l)})$ and ${\lambda }_2^{(l+1)}={\tau }^{(l+1)}/2$.

The necessary and sufficient conditions of the solution using the subgradient approach of $f_l({\nu }_{j-1},{\nu }_j;{\eta }_j)$ are as follows:

$$\begin{array}{r}\frac{\partial f_l}{\partial {\nu }_{j-1}^{(l)}}={\sum }_{i\in {\widehat{B}}_{j-1}^{(l)}}{y}_i-{\widehat{b}}_{j-1}^{(l)}{\nu }_{j-1}^{(l)}-{\eta }_j\partial |{\nu }_j^{(l)}-{\nu }_{j-1}^{(l)}|=0\hfill \\ \frac{\partial f_l}{\partial {\nu }_j^{(l)}}={\sum }_{i\in {\widehat{B}}_j^{(l)}}{y}_i-{\widehat{b}}_j^{(l)}{\nu }_j^{(l)}+{\eta }_j\partial |{\nu }_j^{(l)}-{\nu }_{j-1}^{(l)}|=0,\hfill \end{array}$$ (20)

where ${\widehat{b}}_j^{(l)}=|{\widehat{B}}_j^{(l)}|$, and $\partial |x|$ is the subdifferential of $|x|$. Let ${\widehat{\nu }}_{j-1}^{(l)}$ and ${\widehat{\nu }}_j^{(l)}$ be the solutions of the Eq. (20). For ${\widehat{\nu }}_{j-1}^{(l)}>{\widehat{\nu }}_j^{(l)}$, the solution can be represented as ${\widehat{\nu }}_{j-1}^{(l)}={\overline{y}}_{j-1}^{(l)}-({\eta }_k/{\widehat{b}}_{j-1}^{(l)})$ and ${\widehat{\nu }}_j^{(l)}={\overline{y}}_j^{(l)}+({\eta }_k/{\widehat{b}}_j^{(l)})$. From ${\widehat{\nu }}_{j-1}^{(l)}>{\widehat{\nu }}_j^{(l)}$, the hitting time is ${\eta }_k^{(l+1)}=\frac{{\overline{y}}_{j-1}^{(l)}-{\overline{y}}_j^{(l)}}{1/{\widehat{b}}_{j-1}^{(l)}+1/{\widehat{b}}_j^{(l)}}=\frac{|{\overline{y}}_j^{(l)}-{\overline{y}}_{j-1}^{(l)}|}{1/{\widehat{b}}_{j-1}^{(l)}+1/{\widehat{b}}_j^{(l)}}$. Similarly, for ${\widehat{\nu }}_{j-1}^{(l)}<{\widehat{\nu }}_j^{(l)}$, the hitting time is ${\eta }_j^{(l+1)}=\frac{{\overline{y}}_j^{(l)}-{\overline{y}}_{j-1}^{(l)}}{1/{\widehat{b}}_{j-1}^{(l)}+1/{\widehat{b}}_j^{(l)}}=\frac{|{\overline{y}}_j^{(l)}-{\overline{y}}_{j-1}^{(l)}|}{1/{\widehat{b}}_{j-1}^{(l)}+1/{\widehat{b}}_j^{(l)}}$. Hence, ${\eta }_j^{(l+1)}=\delta ({\widehat{B}}_{j-1}^{(l)},{\widehat{B}}_j^{(l)})$. Based on the definition of the minimum hitting time ${\tau }^{(l+1)}$, ${\tau }^{(l+1)}$ is equivalent to ${\min }_{2\le j\le n-l}\delta ({\widehat{B}}_{j-1}^{(l)},{\widehat{B}}_j^{(l)})$ in the modified path algorithm. It proves ${\lambda }_2^{(l+1)}={\tau }^{(l+1)}/2$.

Theorem 1 offers a novel interpretation of mPath‐FLSA. The mPath‐FLSA lacks an overall objective function, and it finds the next fusion block sequentially by solving independent localised FLSA problems for two adjacent blocks, called local fused lasso signal approximator (LFLSA). Henceforth, we refer to the mPath‐FLSA by Son and Lim (2019) as Path‐LFLSA. Moreover, the hitting times for each sub‐problems can be considered as the distance between the two blocks. Consequently, Path‐LFLSA can be considered as a hierarchical clustering algorithm for $\{\mathrm{1,2},\dots ,n-l\}$ with distance metric $\rho (i,j)=\delta ({\widehat{B}}_i^{(l)},{\widehat{B}}_j^{(l)})$ if $|i-j|=1$ and $\rho (i,j)=\infty$ if $|i-j|>1$.

Furthermore, this interpretation can help identify conditions that violate the monotonicity in the hitting times of Path‐LFLSA. Theorem 2 shows the conditions for the occurrence of the non‐monotonic sequence terms.

Theorem 2

Let ${\lambda }_2^{(l)}={\min }_{2\le j\le n-l+1}{\eta }_j^{(l-1)}/2$ be the $l\mathit{\text{th}}$ hitting time of the Path‐LFLSA with $\mathbf{\text{y}}=(y_1,\dots ,y_n)$ and ${\lambda }_2^{(0)}=0$. Suppose that $k={\text{argmin}}_{2\le j\le n-l}{\eta }_j^{(l-1)}$, and the partition set ${\mathcal{P}}^{(l)}={\{{\widehat{B}}_j^{(l)}\}}_{j=1}^{n-l}$, where ${\widehat{B}}_i^{(l)}={\widehat{B}}_i^{(l-1)}$ for $1\le i\le k-2$, ${\widehat{B}}_{k-1}^{(l)}={\widehat{B}}_{k-1}^{(l-1)}\cup {\widehat{B}}_k^{(l-1)}$, and ${\widehat{B}}_i^{(l)}={\widehat{B}}_{i+1}^{(l-1)}$ for $k\le i\le n-l$. Then, the next hitting time ${\lambda }_2^{(l+1)}$ is less than ${\lambda }_2^{(l)}$ if ${\eta }_{k-1}^{(l+1)}/2<{\lambda }_2^{(l)}$ or ${\eta }_k^{(l+1)}/2<{\lambda }_2^{(l)}$. Moreover, the violation ${\lambda }_2^{(l)}>{\lambda }_2^{(l+1)}$ can be checked with the solution at ${\lambda }_2^{(l-1)}$. Let ${\omega }_k^{(l-1)}={\widehat{b}}_k^{(l-1)}/({\widehat{b}}_{k-1}^{(l-1)}+{\widehat{b}}_k^{(l-1)})$. The violation occurs if and only if the following inequality hold for $m=k-2$ $(2<k\le n-l)$ or $k+1$ $(2\le k<n-l)$:

$$|{\overline{y}}_k^{(l-1)}-{\overline{y}}_{k-1}^{(l-1)}|>\frac{{\widehat{b}}_m^{(l-1)}{({\widehat{b}}_{k-1}^{(l-1)}+{\widehat{b}}_k^{(l-1)})}^2}{{\widehat{b}}_{k-1}^{(l-1)}{\widehat{b}}_k^{(l-1)}{\sum }_{j\in \{m,k-1,k\}}{\widehat{b}}_j^{(l-1)}}\left|(1-{\omega }_k^{(l-1)}){\overline{y}}_{k-1}^{(l-1)}+{\omega }_k^{(l-1)}{\overline{y}}_k^{(l-1)}-{\overline{y}}_m^{(l-1)}\right|.$$ (21)

Due to the definition of ${\lambda }_2^{(l)}$, ${\lambda }_2^{(l)}\le {\eta }_j^{(l-1)}/2$ for $2\le j\le n-l+1$. After $l\mathit{\text{th}}$ hitting time, the set ${\widehat{B}}_{k-1}^{(l)}$ is defined by the union of ${\widehat{B}}_{k-1}^{(l-1)}$ and ${\widehat{B}}_k^{(l-1)}$, and the other blocks remain the same as in the previous partition ${\mathcal{P}}^{(l-1)}$. Thus, the pairs $({\widehat{B}}_{j-1}^{(l)},{\widehat{B}}_j^{(l)})$ for $j=2,\dots ,k-2,k+1,\dots ,n-l$ also remain the same as in the previous partition set ${\mathcal{P}}^{(l-1)}$, which implies ${\eta }_j^{(l)}={\eta }_j^{(l-1)}\ge {\lambda }_2^{(l)}$ for $j\ne k-1,k$. Therefore, the next hitting time ${\lambda }_2^{(l+1)}$ decreases only when ${\eta }_{k-1}^{(l+1)}/2<{\lambda }_2^{(l)}$ or ${\eta }_k^{(l+1)}/2<{\lambda }_2^{(l)}$. Moreover, based on this assumption, we can express the $l\mathit{\text{th}}$ hitting time as ${\lambda }_2^{(l)}={\eta }_k^{(l-1)}/2=|{\overline{y}}_k^{(l-1)}-{\overline{y}}_{k-1}^{(l-1)}|/(2/{\widehat{b}}_k^{(l-1)}+2/{\widehat{b}}_{k-1}^{(l-1)})$. Because ${\overline{y}}_{k-2}^{(l)}={\overline{y}}_{k-2}^{(l-1)}$, ${\overline{y}}_{k-1}^{(l)}=(1-{\omega }_k^{(l-1)}){\overline{y}}_{k-1}^{(l-1)}+{\omega }_k^{(l-1)}{\overline{y}}_k^{(l-1)}$, and ${\overline{y}}_k^{(l)}={\overline{y}}_{k+1}^{(l-1)}$, we can express ${\eta }_{k-1}^{(l+1)}$ and ${\eta }_k^{(l+1)}$ as ${\eta }_{k-1}^{(l+1)}=|(1-{\omega }_k^{(l-1)}){\overline{y}}_{k-1}^{(l-1)}+{\omega }_k^{(l-1)}{\overline{y}}_k^{(l-1)}-{\overline{y}}_{k-2}^{(l-1)}|/(1/({\widehat{b}}_k^{(l-1)}+{\widehat{b}}_{k-1}^{(l-1)})+1/{\widehat{b}}_{k-2}^{(l-1)})$ and ${\eta }_k^{(l+1)}=|(1-{\omega }_k^{(l-1)}){\overline{y}}_{k-1}^{(l-1)}+{\omega }_k^{(l-1)}{\overline{y}}_k^{(l-1)}-{\overline{y}}_{k+1}^{(l-1)}|/(1/({\widehat{b}}_k^{(l-1)}+{\widehat{b}}_{k-1}^{(l-1)})+1/{\widehat{b}}_{k+1}^{(l-1)}),$ respectively. Applying these observations to $2{\lambda }_2^{(l)}/{\eta }_{k-1}^{(l+1)}>1$ and $2{\lambda }_2^{(l)}/{\eta }_k^{(l+1)}>1$, we obtain the condition (21).

To describe the violation ${\lambda }_2^{(l)}>{\lambda }_2^{(l+1)}$, we consider three independent random variables $(Y_1,Y_2,Y_3)$ from the standard normal distribution $N(\mathrm{0,1})$. Corollary 1 shows that the probability of the violation ${\lambda }_2^{(1)}>{\lambda }_2^{(2)}$ is $4P((4Y_1+Y_2)/5<Y_3<Y_1)=4E\left(\mathrm{\Phi }\left(-3\sqrt{3}Z\right)|Z>0\right)\approx 0.121$ when we apply the Path‐LFLSA to $(Y_1,Y_2,Y_3)$, where $Z\sim N(\mathrm{0,1})$ and $\mathrm{\Phi }(z)$ is the cumulative distribution function of the standard normal distribution.

Corollary 1

Suppose $Y_1,Y_2,Y_3$ are random samples from $N(\mathrm{0,1})$. Let $Z$ be a standard normal random variable and $\mathrm{\Phi }(z)$ be the cumulative distribution function of $Z$. Let ${\lambda }_2^{(1)}={\min }_{j=\mathrm{2,3}}{\eta }_j^{(0)}/2$ and ${\lambda }_2^{(2)}={\eta }_2^{(1)}/2$ be the first and the second hitting times of the Path‐LFLSA with ${\lambda }_2^{(0)}=0$, respectively. Then, the probability $P({\lambda }_2^{(1)}>{\lambda }_2^{(2)})$ is equal to $4P((4Y_1+Y_2)/5<Y_3<Y_1)$. In addition, $P({\lambda }_2^{(1)}>{\lambda }_2^{(2)})$ can be calculated from the univariate conditional expectation $4E\left(\mathrm{\Phi }\left(-3\sqrt{3}Z\right)|Z>0\right)$.

A proof of Corollary 1 can be found in Appendix A of the supplementary material. The findings of this study reveal that the mPath‐FLSA, also called the Path‐LFLSA, is the clustering of the observed point based on the distance defined by the minimum of the next hitting times of the localised FLSA problems. We suggest using the indices $l=\mathrm{0,1},\dots ,n-1$ of hitting times $\{{\lambda }_2^{(l)}\}$ to represent the solution path of the Path‐LFLSA instead of directly drawing the solution path along ${\lambda }_2$, which avoids falsely split points in the solution path. This phenomenon is illustrated in Section 5.

4. PATHWISE ADAPTIVE FLSA

The Path‐LFLSA motivated us to propose the pathwise adaptive FLSA, which is a weighted FLSA with pathwise adaptive weights and guarantees the monotonicity of the hitting times. Specifically, let ${\lambda }_2^{(l)}$ be the $l\text{th}$ hitting time and ${\mathcal{P}}^{(l)}={\{{\widehat{B}}_j^{(l)}\}}_{j=1}^{n-l}$ be a partition of $\{\mathrm{1,2},\dots ,n\}$ at ${\lambda }_2^{(l)}$ such that ${\widehat{B}}_i^{(l)}\bigcap {\widehat{B}}_j^{(l)}=\varnothing$ for $i\ne j$ and ${\bigcup }_{j=1}^{n-l}{\widehat{B}}_j^{(l)}=\{1,\dots ,n\}$. For ${\lambda }_2\in ({\lambda }_2^{(l)},{\lambda }_2^{(l+1)}]$, the pathwise adaptive FLSA (PA‐FLSA) minimises

$$f_{PA}(\mathit{\mu };{\lambda }_2)=\frac{1}{2}{\displaystyle \sum _{i=1}^n}{(y_i-{\mu }_i)}^2+{\lambda }_2{\displaystyle \sum _{j=2}^n}{w}_{\mu ,j}^{(l)}|{\mu }_j-{\mu }_{j-1}|,$$ (22)

where

$$w_{\mu ,j}^{(l)}=\left\{\begin{array}{cc}\frac{1}{|{\overline{y}}_t^{(l)}-{\overline{y}}_{t-1}^{(l)}|}& \text{if}j\in {\widehat{B}}_t^{(l)},j-1\in {\widehat{B}}_{t-1}^{(l)},|{\overline{y}}_t^{(l)}-{\overline{y}}_{t-1}^{(l)}|>\frac{1}{M}\\ \hfill M\hfill & \text{otherwise}\hfill \end{array}\right.$$

and $M>0$ is a sufficiently large weight that fuses the corresponding two parameters. Equivalently, we can express the PA‐FLSA problem in the following reduced form:

$$f_l({\nu }_1,\dots ,{\nu }_{n-l};{\lambda }_2)=\frac{1}{2}{\displaystyle \sum _{j=1}^{n-l}}\sum _{i\in {\widehat{B}}_j^{(l)}}{(y_i-{\nu }_j)}^2+{\lambda }_2{\displaystyle \sum _{j=2}^{n-l}}{w}_j^{(l)}|{\nu }_j-{\nu }_{j-1}|,$$ (23)

where $w_j^{(l)}={|{\overline{y}}_j^{(l)}-{\overline{y}}_{j-1}^{(l)}|}^{-1}$ if $|{\overline{y}}_j^{(l)}-{\overline{y}}_{j-1}^{(l)}|>M^{-1}$ and $M$ otherwise. Note that $M>0$ is also used in the reduced form to avoid the division by zero besides the numerical instability. For example, we set $M=100$ in our numerical study.

We now investigate the properties of PA‐FLSA. First, we express the solution of the PA‐FLSA in Lemma 1.

Lemma 1

Suppose that there are $n$ intervals $[{\lambda }_2^{(0)},{\lambda }_2^{(1)})$, $[{\lambda }_2^{(1)},{\lambda }_2^{(2)})$, $\dots$, $[{\lambda }_2^{(n)},\infty )$ with ${\lambda }_2^{(0)}=0$, where the fused set is unchanged in each interval. For ${\lambda }_2\in [{\lambda }_2^{(l)},{\lambda }_2^{(l+1)})$, the solution of the pathwise adaptive FLSA estimator of (23) is given by

$${\widehat{\mu }}_i^{PA}({\lambda }_2)={\displaystyle \sum _{k=1}^{n-l}}{\widehat{\nu }}_k^{(l)}({\lambda }_2)I(i\in {\widehat{B}}_k^{(l)}),$$

where ${\widehat{\nu }}_k^{(l)}={\overline{y}}_k^{(l)}+{\widehat{c}}_k^{(l)}({\lambda }_2)$, ${\widehat{b}}_k^{(l)}=|{\widehat{B}}_k^{(l)}|$, $\partial |x|$ is the subdifferential of $|x|$, and

$${\widehat{c}}_k^{(l)}({\lambda }_2)=\left\{\begin{array}{cc}\frac{{\lambda }_2}{{\widehat{b}}_k^{(l)}}\left(w_{k+1}^{(l)}\partial |{\widehat{\nu }}_{k+1}^{(l)}-{\widehat{\nu }}_k^{(l)}|\right)& \text{if}k=1\\ -\frac{{\lambda }_2}{{\widehat{b}}_k^{(l)}}\left(w_k^{(l)}\partial |{\widehat{\nu }}_k^{(l)}-{\widehat{\nu }}_{k-1}^{(l)}|-w_{k+1}^{(l)}\partial |{\widehat{\nu }}_{k+1}^{(l)}-{\widehat{\nu }}_k^{(l)}|\right)& \text{if}2\le k\le n-l-1\\ -\frac{{\lambda }_2}{{\widehat{b}}_k^{(l)}}\left(w_k^{(l)}\partial |{\widehat{\nu }}_k^{(l)}-{\widehat{\nu }}_{k-1}^{(l)}|\right)& \text{if}k=n-l\end{array}\right.$$

The subdifferential of $f_l(·)$ gives the proof of Lemma 1 directly (Bertsekas, 1999).

Based on Lemma 1, we can anticipate the PA‐FLSA to avoid the inconsistency of the FLSA in the presence of the stair‐case blocks. Specifically, for the FLSA, the bias term ${\widehat{c}}_k$ is zero for the stair‐case blocks (i.e. $({\widehat{\nu }}_k-{\widehat{\nu }}_{k-1})({\widehat{\nu }}_{k+1}-{\widehat{\nu }}_k)>0$), implying that the FLSA fails to move the estimates of the stair‐case blocks. However, for the PA‐FLSA, the ${\widehat{c}}_k^{(l)}$ for $2\le k\le n-l-1$ is defined as:

$${\widehat{c}}_k^{(l)}({\lambda }_2)=\left\{\begin{array}{rl}\frac{{\lambda }_2}{{\widehat{b}}_k^{(l)}}\left(w_k^{(l)}+w_{k+1}^{(l)}\right)& \text{if}{\widehat{\nu }}_{k-1}^{(l)}>{\widehat{\nu }}_k^{(l)}\text{and}{\widehat{\nu }}_k^{(l)}<{\widehat{\nu }}_{k+1}^{(l)}\text{(local min.)}\\ -\frac{{\lambda }_2}{{\widehat{b}}_k^{(l)}}\left(w_k^{(l)}+w_{k+1}^{(l)}\right)& \text{if}{\widehat{\nu }}_{k-1}^{(l)}<{\widehat{\nu }}_k^{(l)}\text{and}{\widehat{\nu }}_k^{(l)}>{\widehat{\nu }}_{k+1}^{(l)}\text{(local max.)}\\ -\frac{{\lambda }_2}{{\widehat{b}}_k^{(l)}}\left(w_k^{(l)}-w_{k+1}^{(l)}\right)& \text{if}{\widehat{\nu }}_{k-1}^{(l)}<{\widehat{\nu }}_k^{(l)}\text{and}{\widehat{\nu }}_k^{(l)}<{\widehat{\nu }}_{k+1}^{(l)}\text{(increasing stair‐block)}\\ \frac{{\lambda }_2}{{\widehat{b}}_k^{(l)}}\left(w_k^{(l)}-w_{k+1}^{(l)}\right)& \text{if}{\widehat{\nu }}_{k-1}^{(l)}>{\widehat{\nu }}_k^{(l)}\text{and}{\widehat{\nu }}_k^{(l)}>{\widehat{\nu }}_{k+1}^{(l)}\text{(decreasing stair‐block)}\end{array}\right..$$

Thus, the PA‐FLSA has a nonzero bias term for the stair‐case blocks unless $|{\overline{y}}_k^{(l)}-{\overline{y}}_{k-1}^{(l)}|=|{\overline{y}}_{k+1}^{(l)}-{\overline{y}}_k^{(l)}|$.

Second, we investigate the monotone fusion property of the PA‐FLSA, which denotes that ${\widehat{\mu }}_j$ and ${\widehat{\mu }}_{j+1}$ remain fused at ${\lambda }_2^{\prime }>{\lambda }_2$ if ${\widehat{\mu }}_j$ and ${\widehat{\mu }}_{j+1}$ are fused at ${\lambda }_2$. The monotone fusion property for the FLSA was proved in Proposition 2 of Friedman et al. (2007). We show that the PA‐FLSA holds the monotone fusion property similar to the FLSA in Proposition 1. However, the PA‐FLSA adopts the weighted fusion penalty with the pathwise adaptive weights.

Proposition 1

In the PA‐FLSA, two parameters that are fused in the solution for ${\lambda }_2$ are fused for all ${\lambda }_2^{\prime }>{\lambda }_2$.

Appendix B of the supplementary material contains a proof of Proposition 1.

Finally, we show that the solution path of the PA‐FLSA is piecewise linear in the given interval $[{\lambda }_2^{(l)},{\lambda }_2^{(l+1)})$ and discontinuous at the hitting times, which changes the pathwise adaptive weights in the Theorem 3 mentioned below.

Theorem 3

The solution path of the PA‐FLSA is a piecewise linear function of ${\lambda }_2$ and discontinuous at the hitting times ${\lambda }_2^{(l)}$, which changes the pathwise adaptive weights. The next hitting time ${\lambda }_2^{(l+1)}$ can be obtained from the current solution at ${\lambda }_2^{(l)}$ by

$${\lambda }_2^{(l+1)}={\lambda }_2^{(l)}+\underset{\underset{{\displaystyle h_{k,k-1}^{(l)}}>0}{2\le k\le n-l}}{\min }{h}_{k,k-1}^{(l)},\text{where}{h}_{k,k-1}^{(l)}=\frac{{\widehat{\nu }}_k^{(l)}-{\widehat{\nu }}_{k-1}^{(l)}}{\frac{\partial {\widehat{\nu }}_{k-1}^{(l)}}{\partial {\lambda }_2}-\frac{\partial {\widehat{\nu }}_k^{(l)}}{\partial {\lambda }_2}}.$$ (24)

The partition ${\{{\widehat{B}}_j^{(l)}\}}_{j=1}^{n-l}$ is unchanged for ${\lambda }_2\in [{\lambda }_2^{(l)},{\lambda }_2^{(l+1)})$. Based on Lemma 1, we can represent the derivative of the solution with respect to ${\lambda }_2$ for a given interval $[{\lambda }_2^{(l)},{\lambda }_2^{(l+1)})$ as

$$\frac{\partial {\widehat{\nu }}_k}{\partial {\lambda }_2}=\left\{\begin{array}{ll}{({\widehat{b}}_k^{(l)})}^{-1}{w}_{k+1}^{(l)}\text{sign}({\widehat{\nu }}_{k+1}-{\widehat{\nu }}_k)& \text{for}k=1\\ -{({\widehat{b}}_k^{(l)})}^{-1}\left(w_k^{(l)}\text{sign}({\widehat{\nu }}_k-{\widehat{\nu }}_{k-1})-w_{k+1}^{(l)}\text{sign}({\widehat{\nu }}_{k+1}-{\widehat{\nu }}_k)\right)& \text{for}k=2,\dots ,n-l-1\\ -{({\widehat{b}}_k^{(l)})}^{-1}{w}_k^{(l)}\text{sign}({\widehat{\nu }}_k-{\widehat{\nu }}_{k-1})& \text{for}k=n-l\end{array}\right..$$

Thus, the solution paths of ${\widehat{\nu }}_k$ for $k=1,\dots ,n-l$ along ${\lambda }_2$ are linear in the given interval $[{\lambda }_2^{(l)},{\lambda }_2^{(l+1)})$ because ${\{w_k^{(l)}\}}_{k=1}^{n-l}$ and ${\{\text{sign}({\widehat{\nu }}_{k+1}-{\widehat{\nu }}_k)\}}_{k=1}^{n-l}$ are unchanged. Based on this linearity, for ${\lambda }_2\in [{\lambda }_2^{(l)},{\lambda }_2^{(l+1)}$), we can obtain the solution at ${\lambda }_2$ using the equation:

$${\widehat{\nu }}_k({\lambda }_2)={\widehat{\nu }}_k^{(l)}+\alpha \frac{\partial {\nu }_k}{\partial {\lambda }_2},$$

where ${\widehat{\nu }}_k^{(l)}$ is the minimiser of (23) at ${\lambda }_2={\lambda }_2^{(l)}$ and $\alpha ={\lambda }_2-{\lambda }_2^{(l)}\in [0,{\lambda }_2^{(l+1)}-{\lambda }_2^{(l)})$. We can find the next hitting time $h_{k,k-1}^{(l)}$ for a pair $({\widehat{\nu }}_k({\lambda }_2),{\widehat{\nu }}_{k-1}({\lambda }_2))$ of two solution paths from the equation ${\widehat{\nu }}_k({\lambda }_2)={\widehat{\nu }}_{k-1}({\lambda }_2)$, for $k=2,\dots ,n-l$, as

$$h_{k,k-1}^{(l)}=\frac{{\widehat{\nu }}_k^{(l)}-{\widehat{\nu }}_{k-1}^{(l)}}{\frac{\partial {\widehat{\nu }}_{k-1}}{\partial {\lambda }_2}-\frac{\partial {\widehat{\nu }}_k}{\partial {\lambda }_2}}\in (0,{\lambda }_2^{(l+1)}-{\lambda }_2^{(l)}].$$

Because these hitting times are only valid for the unchanged partition ${\{{\widehat{B}}_j^{(l)}\}}_{j=1}^{n-l}$ (i.e. there is no fusion for ${\lambda }_2\in ({\lambda }_2^{(l)},{\lambda }_2^{(l+1)})$), the next hitting time ${\lambda }_2^{(l+1)}$ should satisfy the following equation:

$${\lambda }_2^{(l+1)}-{\lambda }_2^{(l)}=\underset{\underset{{\displaystyle h_{k,k-1}^{(l)}}>0}{2\le k\le n-l}}{\min }{h}_{k,k-1}^{(l)}.$$

Consider two PA‐FLSA problems with ${(w_j^{(l)})}_{j=1}^{n-1}$ and ${(w_j^{(l+1)})}_{j=1}^{n-l-1}$ at ${\lambda }_2={\lambda }_2^{(l+1)}$ to show the discontinuity of the solution path of the PA‐FLSA. We denote the first problem as the PA‐FLSA( $l$) and the second problem as the PA‐FLSA( $l+1$) to distinguish the two problems. Suppose that $3<q<n-l$ and ${\widehat{\nu }}_q^{(l)}({\lambda }_2)$ and ${\widehat{\nu }}_{q-1}^{(l)}({\lambda }_2)$ of the PA‐FLSA( $l$) are fused at ${\lambda }_2^{(l)}$. Let $s({\widehat{\nu }}_k^{(l)},{\widehat{\nu }}_{k-1}^{(l)})=\partial |{\widehat{\nu }}_k^{(l)}-{\widehat{\nu }}_{k-1}^{(l)}|$, ${\widehat{b}}_{[q-1:q]}^{(l)}={\widehat{b}}_{q-1}^{(l)}+{\widehat{b}}_q^{(l)}$, and ${\overline{y}}_{[q-1:q]}^{(l)}=({\widehat{b}}_{q-1}^{(l)}{\overline{y}}_{q-1}^{(l)}+{\widehat{b}}_q^{(l)}{\overline{y}}_q^{(l)})/{\widehat{b}}_{[q-1:q]}$. Based on Lemma 1, we can express the solution ${\widehat{\nu }}^{(l)}({\lambda }_2^{(l+1)})$ of the PA‐FLSA( $l$) as

$${\widehat{\nu }}_k^{(l)}({\lambda }_2^{(l+1)})=\left\{\begin{array}{ll}{\overline{y}}_1^{(l)}+\frac{{\lambda }_2^{(l+1)}}{{\widehat{b}}_1^{(l)}}\left(w_2^{(l)}s({\widehat{\nu }}_2^{(l)},{\widehat{\nu }}_1^{(l)})\right)& \text{if}k=1\\ {\overline{y}}_k^{(l)}-\frac{{\lambda }_2^{(l+1)}}{{\widehat{b}}_k^{(l)}}\left(w_k^{(l)}s({\widehat{\nu }}_k^{(l)},{\widehat{\nu }}_{k-1}^{(l)})-w_{k+1}^{(l)}s({\widehat{\nu }}_{k+1}^{(l)},{\widehat{\nu }}_k^{(l)})\right)& \text{if}k\ne 1,q-1,q,n-l\\ {\overline{y}}_{[q-1:q]}^{(l)}-\frac{{\lambda }_2^{(l+1)}}{{\widehat{b}}_{[q-1:q]}^{(l)}}\left(w_{q-1}^{(l)}s({\widehat{\nu }}_{q-1}^{(l)},{\widehat{\nu }}_{q-2}^{(l)})-w_{q+1}^{(l)}s({\widehat{\nu }}_{q+1}^{(l)},{\widehat{\nu }}_q^{(l)})\right)& \text{if}k=q-1,q\\ {\overline{y}}_{n-l}^{(l)}-\frac{{\lambda }_2^{(l+1)}}{{\widehat{b}}_{n-l}^{(l)}}\left(w_{n-l}^{(l)}s({\widehat{\nu }}_{n-l}^{(l)},{\widehat{\nu }}_{n-l-1}^{(l)})\right)& \text{if}k=n-l\end{array}\right.$$

Based on this assumption, we can represent the partition ${\widehat{B}}_k^{(l+1)}$ into ${\widehat{B}}_k^{(l+1)}={\widehat{B}}_k^{(l)}$ for $1\le k\le q-2$, ${\widehat{B}}_{q-1}^{(l+1)}={\widehat{B}}_{q-1}^{(l)}\cup {\widehat{B}}_q^{(l)}$, and ${\widehat{B}}_k^{(l+1)}={\widehat{B}}_{k+1}^{(l)}$ for $q\le k\le n-l-1$. For notational simplicity, in the proof, we use the definition of the pathwise adaptive weight as $w_k^{(l)}=|{\overline{y}}_k^{(l)}-{\overline{y}}_{k-1}^{(l)}{|}^{-1}$. Thus, the following equations hold:

$${\overline{y}}_k^{(l+1)}=\left\{\begin{array}{cl}{\overline{y}}_k^{(l)}& \text{for}1\le k\le q-2\\ {\overline{y}}_{[q-1:q]}^{(l)}& \text{for}k=q-1\\ {\overline{y}}_{k+1}^{(l)}& \text{for}q\le k\le n-l-1\end{array}\right.,w_k^{(l+1)}=\left\{\begin{array}{cl}{w}_k^{(l)}& \text{for}2\le k\le q-2\\ {\left(|{\overline{y}}_{[q-1:q]}^{(l)}-{\overline{y}}_{q-2}^{(l)}|\right)}^{-1}& \text{for}k=q-1\\ {\left(|{\overline{y}}_{q+1}^{(l)}-{\overline{y}}_{[q-1:q]}^{(l)}|\right)}^{-1}& \text{for}k=q\\ w_{k+1}^{(l)}& \text{for}q+1\le k\le n-l-1\end{array}\right..$$

Then, by Lemma 1, the solution ${\widehat{\nu }}^{(l+1)}({\lambda }_2^{(l+1)})$ of the PA‐FLSA( $l+1$) can be represented as

$${\widehat{\nu }}_k^{(l+1)}({\lambda }_2^{(l+1)})=\left\{\begin{array}{ll}{\overline{y}}_k^{(l)}+\frac{{\lambda }_2^{(l+1)}}{{\widehat{b}}_k^{(l)}}\left(w_2^{(l)}s({\widehat{\nu }}_2^{(l+1)},{\widehat{\nu }}_1^{(l+1)})\right)& \text{if}k=1\\ {\overline{y}}_k^{(l)}-\frac{{\lambda }_2^{(l+1)}}{{\widehat{b}}_k^{(l)}}\left(w_k^{(l)}s({\widehat{\nu }}_k^{(l+1)},{\widehat{\nu }}_{k-1}^{(l+1)})-w_{k+1}^{(l)}s({\widehat{\nu }}_{k+1}^{(l+1)},{\widehat{\nu }}_k^{(l+1)})\right)& \text{if}2\le k\le q-3\\ {\overline{y}}_k^{(l)}-\frac{{\lambda }_2^{(l+1)}}{{\widehat{b}}_k^{(l)}}\left(w_k^{(l)}s({\widehat{\nu }}_k^{(l+1)},{\widehat{\nu }}_{k-1}^{(l+1)})-w_{k+1}^{(l+1)}s({\widehat{\nu }}_{k+1}^{(l+1)},{\widehat{\nu }}_k^{(l+1)})\right)& \text{if}k=q-2\\ {\overline{y}}_{[q-1:q]}^{(l)}-\frac{{\lambda }_2^{(l+1)}}{{\widehat{b}}_{[q-1:q]}^{(l)}}\left(w_{q-1}^{(l+1)}s({\widehat{\nu }}_{q-1}^{(l+1)},{\widehat{\nu }}_{q-2}^{(l+1)})-w_q^{(l+1)}s({\widehat{\nu }}_q^{(l+1)},{\widehat{\nu }}_{q-1}^{(l+1)})\right)& \text{if}k=q-1\\ {\overline{y}}_{q+1}^{(l)}-\frac{{\lambda }_2^{(l+1)}}{{\widehat{b}}_{q+1}^{(l)}}\left(w_q^{(l+1)}s({\widehat{\nu }}_q^{(l+1)},{\widehat{\nu }}_{q-1}^{(l+1)})-w_{q+2}^{(l)}s({\widehat{\nu }}_{q+1}^{(l+1)},{\widehat{\nu }}_q^{(l+1)})\right)& \text{if}k=q\\ {\overline{y}}_{k+1}^{(l)}-\frac{{\lambda }_2^{(l+1)}}{{\widehat{b}}_{k+1}^{(l)}}\left(w_{k+1}^{(l)}s({\widehat{\nu }}_k^{(l+1)},{\widehat{\nu }}_{k-1}^{(l+1)})-w_{k+2}^{(l)}s({\widehat{\nu }}_{k+1}^{(l+1)},{\widehat{\nu }}_k^{(l+1)})\right)& \text{if}q<k<n-l-1\\ {\overline{y}}_{n-l}^{(l)}-\frac{{\lambda }_2^{(l+1)}}{{\widehat{b}}_{n-l}^{(l)}}\left(w_{n-l}^{(l)}s({\widehat{\nu }}_{n-l-1}^{(l+1)},{\widehat{\nu }}_{n-l-2}^{(l+1)})\right)& \text{if}k=n-l-1\end{array}\right.$$

Let ${\widehat{\mu }}_i^{(l)}({\lambda }_2^{(l+1)})={\sum }_{j=1}^{n-l}{\widehat{\nu }}_j^{(l)}({\lambda }_2^{(l+1)})I(i\in {\widehat{B}}_j^{(l)})$ and ${\widehat{\mu }}^{(l+1)}({\lambda }_2^{(l+1)})={\sum }_{j=1}^{n-l-1}{\widehat{\nu }}_j^{(l+1)}({\lambda }_2^{(l+1)})I(i\in {\widehat{B}}_j^{(l+1)})$. A comparison of the two solutions ${\widehat{\mu }}^{(l)}({\lambda }_2^{(l+1)})$ and ${\widehat{\mu }}^{(l+1)}({\lambda }_2^{(l+1)})$ can easily spot that the absolute differences between $|{\mu }_i^{(l)}({\lambda }_2^{(l+1)})-{\mu }_i^{(l+1)}({\lambda }_2^{(l+1)})|$ for $i\in {\cup }_{j=q-2}^q{\widehat{B}}_j^{(l+1)}$ are nonzero if $w_{q-1}^{(l)}\ne w_{q-1}^{(l+1)}$ and $w_{q+1}^{(l)}\ne w_q^{(l+1)}$. This completes the proof of the discontinuity of the solution path of the PA‐FLSA.

The results in Theorem 3 show that the hitting times ${\lambda }_2^{(l)}$ of the PA‐FLSA increase monotonically. Unlike the original FLSA, the solution path of the PA‐FLSA is discontinuous at hitting times that change the pathwise adaptive weights. However, this discontinuity does not affect PA‐FLSA's performance in estimating true signal levels and identifying the change points. We will examine these phenomena in the numerical study.

5. NUMERICAL STUDY

This section presents a comprehensive numerical study. First, we numerically investigate the properties we have explored in the previous sections, including the violation of the monotone increasing property for the hitting times of the Path‐LFLSA and the discontinuity of the PA‐FLSA's solution path. Second, we numerically compare the probabilities of the exact pattern recovery of the four methods (Path‐FLSA, PCD‐FLSA, Path‐LFLSA, and PA‐FLSA) under four scenarios. The first three scenarios are based on Qian & Jia (2016), and the fourth is novel for this study. We also conduct a comparison of the performances of the four methods in estimating the signal levels and identifying the block structures with the optimal tuning parameters chosen by the Bayesian information criterion (BIC) (Schwarz, 1978) and the extended BIC (EBIC) (Chen & Chen, 2008) in Appendix C of the supplementary material.

5.1. Comparison of Whole Solution Paths

In this subsection, we compare the whole solution paths of the Path‐FLSA, PCD‐FLSA, Path‐LFLSA, and PA‐FLSA. We consider $y_i={\mu }_i^{\ast }+{ϵ}_i$, ${ϵ}_i\sim N(0,{\sigma }^2)$, $\mu ={({\mu }_1,\dots ,{\mu }_8)}^T$, and $\sigma =0.25$ to generate the underlying signal and noisy observations. The true mean value is set to ${\mu }_i^{\ast }={(0.\mathrm{5,0}.\mathrm{5,0},0,-0.5,-0.5)}^T$ and the noisy observation is generated as $\mathbf{\text{y}}={(0.\mathrm{4314,0}.4000,-0.\mathrm{2140,0}.5188,-0.2379,-0.4435)}^T$. We applied the four methods to obtain the whole solution paths along ${\lambda }_2$. The solution paths of the Path‐FLSA, PCD‐FLSA, Path‐LFLSA, and PA‐FLSA are depicted in Figure 1. The solution path of the Path‐LFLSA seems to have a split event around ${\lambda }_2=0.2$, while the Path‐LFLSA actually has fusion events only. We also report the hitting times of the solution paths from Path‐FLSA, PCD‐FLSA, Path‐LFLSA, and PA‐FLSA in Table 1 for details.

FIGURE 1.

Solution paths of Path‐FLSA, PCD‐FLSA, Path‐LFLSA, and PA‐FLSA for $\mu ={({\mu }_1,\dots ,{\mu }_6)}^T$

TABLE 1.

Hitting times of the whole solution paths from the four methods with the noisy observation $\mathbf{\text{y}}$ with ${\lambda }_2^{(0)}=0$

Method	${\lambda }_2^{(1)}$	${\lambda }_2^{(2)}$	${\lambda }_2^{(3)}$	${\lambda }_2^{(4)}$	${\lambda }_2^{(5)}$
Path‐FLSA	0.0314	0.1832	0.2056	0.5266	0.8330
PCD‐FLSA	0.0314	0.2056	0.6140	0.7328	0.7567
Path‐LFLSA	0.0078	0.0514	0.1832	0.1317	0.4165
PA‐FLSA	0.0314	0.0521	0.1322	0.1838	0.2682

Table 1 shows that the fourth hitting time ${\lambda }_2^{(4)}=0.1317$ is less than the third hitting time ${\lambda }_2^{(3)}=0.1832$ for Path‐LFLSA, violating the monotone increasing property of the hitting times. Contrarily, the other solution path algorithms satisfy the monotone increasing property of the hitting times. As demonstrated in Theorem 2, this violation ${\lambda }_2^{(3)}>{\lambda }_2^{(4)}$ can be checked with the solution at the second hitting time ${\lambda }_2={\lambda }_2^{(2)}$. At ${\lambda }_2={\lambda }_2^{(2)}$, ${\overline{y}}^{(2)}={(0.4157,-0.\mathrm{2140,0}.5188,-0.3407)}^T$, ${\widehat{b}}^{(2)}={(\mathrm{2,1},\mathrm{1,2})}^T$, ${\eta }^{(2)}=({\eta }_2^{(2)},{\eta }_3^{(2)},{\eta }_4^{(2)})={(0.\mathrm{2099,0}.\mathrm{1832,0}.2865)}^T$. Thus, $k={\text{argmin}}_{j=\mathrm{2,3},4}{\eta }_j^{(2)}=3$, $w_3^{(2)}={\widehat{b}}_3^{(2)}/({\widehat{b}}_2^{(2)}+{\widehat{b}}_3^{(2)})=0.5$, and $m=k-2=1$ or $m=k+1=4$ in the condition (21). Thus, the following condition (21) holds for $m=1$:

$$0.7328=|{\overline{y}}_3^{(2)}-{\overline{y}}_2^{(2)}|>\frac{{\widehat{b}}_1^{(2)}{({\widehat{b}}_2^{(2)}+{\widehat{b}}_3^{(2)})}^2}{{\widehat{b}}_2^{(2)}{\widehat{b}}_3^{(2)}{\displaystyle {\sum }_{j=1}^3}{\widehat{b}}_j^{(2)}}\left|({\overline{y}}_2^{(2)}+{\overline{y}}_3^{(2)})/2-{\overline{y}}_1^{(2)}\right|=0.5266.$$

As mentioned in Section 3, we suggest drawing the solution path of the Path‐LFLSA along indices of its hitting times, which is equivalent to the order of fusion events. We depict the solution paths along ${\lambda }_2$ and the indices of the hitting times in Figure 2. The solution path along the indices of the hitting times (Figure 2 (b)) is more readable than the solution path along ${\lambda }_2$ (Figure 2 (a)).

FIGURE 2.

Solution paths of Path‐LFLSA along ${\lambda }_2$ and indices of hitting times for $\mu ={({\mu }_1,\dots ,{\mu }_6)}^T$

Figure 1 also shows that the solution path of the PA‐FLSA has discontinuous points at the hitting times. For example, the values of ${\widehat{\nu }}^{(2)}({\lambda }_2^{(3)})$ and ${\widehat{\nu }}^{(3)}({\lambda }_2^{(3)})$ in the proof of Theorem 3 are different, where ${\widehat{\nu }}^{(2)}({\lambda }_2^{(3)})={(0.\mathrm{3107,0}.1764,-0.2597)}^T$ and ${\widehat{\nu }}^{(3)}({\lambda }_2^{(3)})={(0.\mathrm{3107,0}.1672,-0.2505)}^T$. Table 1 shows that the hitting times of the PA‐FLSA are monotonically increasing, supporting Theorem 3.

Finally, we compared the fused sets at the hitting times. From Figure 1, the partitions from the Path‐FLSA are obtained as ${\widehat{B}}^{(1)}=\{\{\mathrm{1,2}\},\{3\},\{4\},\{5\},\{6\}\}$, ${\widehat{B}}^{(2)}=\{\{\mathrm{1,2}\},\{\mathrm{3,4}\},\{5\},\{6\}\}$, ${\widehat{B}}^{(3)}=\{\{\mathrm{1,2}\},\{\mathrm{3,4}\},\{\mathrm{5,6}\}\}$, ${\widehat{B}}^{(4)}=\{\{\mathrm{1,2},\mathrm{3,4}\},\{\mathrm{5,6}\}\}$, and ${\widehat{B}}^{(5)}=\{\{\mathrm{1,2},\mathrm{3,4},\mathrm{5,6}\}\}$ for ${\lambda }_2^{(1)},\dots ,{\lambda }_2^{(5)}$, respectively. Path‐LFLSA and PA‐FLSA have the same partitions ${\widehat{B}}^{(1)}=\{\{\mathrm{1,2}\},\{3\},\{4\},\{5\},\{6\}\}$, ${\widehat{B}}^{(2)}=\{\{\mathrm{1,2}\},\{3\},\{4\},\{\mathrm{5,6}\}\}$, ${\widehat{B}}^{(3)}=\{\{\mathrm{1,2}\},\{\mathrm{3,4}\},\{\mathrm{5,6}\}\}$, ${\widehat{B}}^{(4)}=\{\{\mathrm{1,2},\mathrm{3,4}\},\{\mathrm{5,6}\}\}$, and ${\widehat{B}}^{(5)}=\{\{\mathrm{1,2},\mathrm{3,4},\mathrm{5,6}\}\}$ for ${\lambda }_2^{(1)},\dots ,{\lambda }_2^{(5)}$, respectively. Thus, we observe that Path‐FLSA, Path‐LFLSA, and PA‐FLSA contain the true partition $\{\{\mathrm{1,2}\},\{\mathrm{3,4}\},\{\mathrm{5,6}\}\}$. Contrarily, the partitions from the PCD‐FLSA are obtained as ${\widehat{B}}^{(1)}=\{\{\mathrm{1,2}\},\{3\},\{4\},\{5\},\{6\}\}$, ${\widehat{B}}^{(2)}=\{\{\mathrm{1,2}\},\{3\},\{4\},\{\mathrm{5,6}\}\}$, ${\widehat{B}}^{(3)}=\{\{\mathrm{1,2},3\},\{4\},\{\mathrm{5,6}\}\}$, ${\widehat{B}}^{(4)}=\{\{\mathrm{1,2},\mathrm{3,4}\},\{\mathrm{5,6}\}\}$, and ${\widehat{B}}^{(5)}=\{\{\mathrm{1,2},\mathrm{3,4},\mathrm{5,6}\}\}$ for ${\lambda }_2^{(1)},\dots ,{\lambda }_2^{(5)}$, respectively. The PCD‐FLSA fails to contain the true partition, and this difference is because of the Puffer transformation in the PCD‐FLSA. Note that the last observation describes the estimates of the methods from the one data set $\mathbf{\text{y}}$ only, not implying that the PCD‐FLSA generally fails to contain the true partition. In the following subsection, we evaluate the exact pattern recovery probabilities estimated by 1000 data sets to compare the general performance of containing the true partition.

5.2. Comparison of Exact Pattern Recovery Probabilities

In this section, we compare the probabilities of the exact pattern recovery of the four methods similar to Qian & Jia (2016), where the exact pattern recovery indicates that the solution path of the given method contains the solution of the true block structures. We considered four scenarios for the underlying block structures. Three scenarios are taken directly from Qian & Jia (2016) to reproduce their results and conduct a fair comparison, in which two scenarios have no stair‐case blocks, and one scenario has stair‐case blocks. Besides, the last scenario for our numerical study was a true block structure with multiple stair‐case blocks. These four scenarios can be divided into two categories based on the existence of the stair‐case blocks, which is equivalent to the failure of the irrepresentable condition of the transformed Lasso in (13). Again, for a fair comparison, we set $n=430$ and consider $\sigma =0.\mathrm{05,0},10,\dots ,0.5$ like Qian & Jia (2016). The noisy observations are from $y_i={\mu }_i+{ϵ}_i$, where ${ϵ}_i\sim N(0,{\sigma }^2)$ and ${\mu }_i$ are specified case by case. We generated 1000 data sets and checked the inclusion of the true block structures with the given solution path to measure the probability of the exact pattern recovery. To obtain the confidence intervals of the exact pattern recovery probabilities, we repeat the procedure for estimating the exact pattern recovery 50 times as well.

5.2.1. Two scenarios without stair‐case blocks

First, we considered the two true signal structures ${\mu }^{(1)}$ and ${\mu }^{(2)}$ used in Qian & Jia (2016) for the scenarios $\mathbf{\text{S1}}$ and $\mathbf{\text{S2}}$, respectively. The two true signal vectors for $\mathbf{\text{S1}}$ and $\mathbf{\text{S2}}$ are respectively defined as

$${\mu }_i^{(1)}=\left\{\begin{array}{rcl}0& & 1\le i\le 40\\ 2& & 41\le i\le 80\\ -1& & 81\le i\le 120\\ 3& & 121\le i\le 160\\ 0& & 161\le i\le 200\\ 2& & 201\le i\le 240\\ 0& & 241\le i\le 280\\ 2& & 281\le i\le 320\\ -1& & 321\le i\le 360\\ 3& & 361\le i\le 400\\ 0& & 401\le i\le 430\end{array}\right.\text{and}{\mu }_i^{(2)}=\left\{\begin{array}{rcl}0& & 1\le i\le 15\\ 2& & 16\le i\le 30\\ 0& & 31\le i\le 60\\ 2& & 61\le i\le 120\\ 0& & 121\le i\le 210\\ 2& & 211\le i\le 240\\ 0& & 241\le i\le 255\\ 2& & 256\le i\le 370\\ 0& & 371\le i\le 385\\ 2& & 386\le i\le 400\\ 0& & 401\le i\le 430\end{array}.\right.$$

With these two true signal vectors, the left hand sides (LHS) of the irrepresentable condition of the transformed Lasso are 0.975 and 0.9826 for ${\mu }^{(1)}$ and ${\mu }^{(2)}$, respectively. These values denote that the irrepresentation conditions hold for two signal vectors because there exists a constant $\eta$ such that $0<\eta \le 1-\text{LHS}\le 1$. Figure 3 depicts the true signals and noisy observations for two scenarios with $\sigma =0.25$. Scenarios $\mathbf{\text{S1}}$ and $\mathbf{\text{S2}}$ have ten change points in the true signal. The differences between $\mathbf{\text{S1}}$ and $\mathbf{\text{S2}}$ are the lengths of the blocks and signal levels.

FIGURE 3.

True signals and noisy observations for two scenarios $\mathbf{\text{S1}}$ and $\mathbf{\text{S2}}$

Figure 4 depicts the estimated probabilities and their 95% confidence intervals of the exact pattern recovery of Path‐FLSA, PCD‐FLSA, Path‐LFLSA, and PA‐FLSA for $\mathbf{\text{S1}}$ and $\mathbf{\text{S2}}$. We confirmed that the probabilities of the exact pattern recovery of Path‐FLSA and PCD‐FLSA have the same shapes as in the simulation results in Qian & Jia (2016). Moreover, Figure 4 shows an interesting finding that the estimated probabilities of the exact pattern recovery of Path‐LFLSA and PA‐FLSA are almost similar and greater than those of Path‐FLSA and PCD‐FLSA for $\sigma \ge 0.25$. Specifically, for $\sigma =0.5$ and case $\mathbf{\text{S2}}$, the probabilities of the exact pattern recovery for Path‐LFLSA and PA‐FLSA are estimated as 0.422 and 0.458, respectively, while those of Path‐FLSA and PCD‐FLSA are zero. For $\mathbf{\text{S1}}$ and $\mathbf{\text{S2}}$, we further observe that PA‐FLSA has a larger pattern recovery probability than Path‐LFLSA for relatively large $\sigma$ levels $(\sigma \ge 0.35)$.

FIGURE 4.

Plots of the estimated probabilities of the exact pattern recovery under $\sigma =0.\mathrm{05,0}.10,\dots ,0.5$ for $\mathbf{\text{S1}}$ and $\mathbf{\text{S2}}$. Dashed lines denote the 95% confidence intervals of the exact pattern recovery probabilities

5.2.2. Two scenarios with stair‐case blocks

Second, we consider two true signal structures ${\mu }^{(3)}$ and ${\mu }^{(4)}$ for the scenarios $\mathbf{\text{S3}}$ and $\mathbf{\text{S4}}$, respectively, where ${\mu }^{(3)}$ is from Qian & Jia (2016) and ${\mu }^{(4)}$ is a novel case with multiple stair‐case blocks. The true signal vectors ${\mu }^{(3)}$ and ${\mu }^{(4)}$ for $\mathbf{\text{S3}}$ and $\mathbf{\text{S4}}$ are defined as follows:

$${\mu }_i^{(3)}=\left\{\begin{array}{rcl}0& & 1\le i\le 100\\ -2& & 101\le i\le 110\\ -0.1& & 111\le i\le 210\\ 2& & 211\le i\le 220\\ 0.1& & 221\le i\le 320\\ -2& & 321\le i\le 330\\ 0& & 331\le i\le 430\end{array}\right.\text{and}{\mu }_i^{(4)}=\left\{\begin{array}{rcl}-2.4& & 1\le i\le 50\\ -1.8& & 51\le i\le 100\\ -1.2& & 101\le i\le 150\\ -0.6& & 151\le i\le 200\\ 0& & 201\le i\le 250\\ 0.6& & 251\le i\le 300\\ 1.2& & 301\le i\le 350\\ 1.8& & 351\le i\le 400\\ 2.4& & 401\le i\le 430\end{array}.\right.$$

As in the cases of $\mathbf{\text{S1}}$ and $\mathbf{\text{S2}}$, we calculated the LHS of the irrepresentable condition of the transformed Lasso for $\mathbf{\text{S3}}$ and $\mathbf{\text{S4}}$. Both the LHS values for ${\mu }^{(3)}$ and ${\mu }^{(4)}$ were calculated as 1.0. These LHS values indicate that the irrepresentation conditions failed for two signal vectors because a constant $\eta$ such that $0<\eta \le 1-\text{LHS}=0$ does not exist. Figure 5 depicts the true signals and noisy observations for two scenarios $\mathbf{\text{S3}}$ and $\mathbf{\text{S4}}$ with $\sigma =0.25$. Scenarios $\mathbf{\text{S3}}$ and $\mathbf{\text{S4}}$ have six and ten change points, respectively.

FIGURE 5.

True signals and noisy observations for two scenarios S3 and S4

Figure 6 depicts the estimated probabilities and their 95% confidence intervals of the exact pattern recovery of Path‐FLSA, PCD‐FLSA, Path‐LFLSA, and PA‐FLSA for $\mathbf{\text{S3}}$ and $\mathbf{\text{S4}}$. We can compare the characteristics of the four methods when the true signal has the stair‐case blocks. First, Path‐FLSA fails totally in finding the exact pattern (i.e. true block structure) for $\mathbf{\text{S3}}$ and $\mathbf{\text{S4}}$ because the estimated probabilities of the exact pattern recovery of Path‐FLSA for $\mathbf{\text{S3}}$ and $\mathbf{\text{S4}}$ are zero for all $\sigma =0.05,\dots ,0.5$. This observation supports the inconsistency of the FLSA under the existence of the stair‐case blocks. Second, Path‐LFLSA and PA‐FLSA are more robust to the noise level (i.e. error variance) than PCD‐FLSA. The estimated exact pattern recovery probabilities of Path‐LFLSA and PA‐FLSA decreased slower that of PCD‐FLSA. For example, the exact pattern recovery probability of PCD‐FLSA decreased from 1.000 at $\sigma =0.05$ to 0.001 at $\sigma =0.15$ while that of Path‐LFLSA and PA‐FLSA decreased from 1.000 and 1.000 at $\sigma =0.05$ to 0.549 and 0.595 at $\sigma =0.15$ for $\mathbf{\text{S4}}$, respectively.

FIGURE 6.

Plots of the estimated probabilities of the exact pattern recovery under $\sigma =0.\mathrm{05,0}.10,\dots ,0.5$ for $\mathbf{\text{S3}}$ and $\mathbf{\text{S4}}$. Dashed lines denote the 95% confidence intervals of the exact pattern recovery probabilities

Overall, Path‐LFLSA and PA‐FLSA outperformed Path‐FLSA and PCD‐FLSA concerning the exact pattern recovery probability for all cases from $\mathbf{\text{S1}}$ to $\mathbf{\text{S4}}$. For Path‐LFLSA and PA‐FLSA, PA‐FLSA (Path‐LFLSA) was better than Path‐LFLSA (PA‐FLSA) for S1, S2, and S4 (S3), but the differences were small.

6. APPLICATION TO COVID‐19 INFECTION IN KOREA

In this section, we apply the four methods Path‐FLSA, PCD‐FLSA, Path‐LFLSA, and PA‐FLSA to the daily‐confirmed cases of COVID‐19 infection in Korea. We downloaded a dataset of COVID‐19 infection status from the Public Data Portal (http://data.go.kr) by using OpenAPI through https://openapi.data.go.kr/openapi/service/rest/Covid19/getCovid19InfStateJson with the authorised key. The downloaded dataset contained 9 variables, including date of state (stateDt), a time of state (stateTime), and the number of cumulative confirmed cases of COVID‐19 (decideCnt). We set a target period as a period from 03‐01‐2020 to 03/31/2022 and used the cases whose times of state were 0:00. To obtain the daily‐confirmed cases of COVID‐19, we applied the first differencing to the number of cumulative confirmed cases, where the difference between the cumulative confirmed cases of COVID‐19 stated at 0:00 on 03/01/2020 and 03/02/2020 as the daily‐confirmed cases occurred on 03/01/2020.

We considered a logarithmic transformation to stabilise the variance of the observation before applying the four methods because the FLSA‐based procedures are sensitive to the noise level, as shown in Section 5. In addition, the number of daily confirmed cases is dramatically increasing from January 2022 in Korea. For example, the number of daily confirmed cases was 621,204 on March 16, 2022, while the number of daily confirmed patients was 3831 on January 1, 2022. Figure 7 depicts the log‐transformed daily‐confirmed cases of the target period with a gray line.

FIGURE 7.

Plots of the logarithmic transformed daily confirmed cases of COVID‐19 (), the estimates of Path‐FLSA (), PCD‐FLSA (), Path‐LFLSA (), and PA‐FLSA () with the EBIC

To select the optimal tuning parameter, we used the following EBIC proposed by Chen & Chen (2008), which showed better results for the FLSA as described in Appendix C of the supplementary material,

$$\text{EBIC}({\lambda }_2)=n\log (\text{RSS}({\lambda }_2))+\widehat{J}({\lambda }_2)\log (n)+\log \left(\genfrac{}{}{0ex}{}{n}{\widehat{J}({\lambda }_2)}\right),$$

where $\text{RSS}({\lambda }_2)={\sum }_{i=1}^n{(y_i-{\widehat{\mu }}_i^{DB}({\lambda }_2))}^2$, ${\widehat{\mu }}_i^{DB}({\lambda }_2)={\overline{y}}_{{\widehat{B}}_j}$ for $i\in {\widehat{B}}_j({\lambda }_2)$, ${\overline{y}}_{{\widehat{B}}_j}=|{\widehat{B}}_j({\lambda }_2){|}^{-1}{\sum }_{i\in {\widehat{B}}_j({\lambda }_2)}{y}_i$, and $\widehat{J}({\lambda }_2)$ is the number of the estimated blocks at ${\lambda }_2$.

Figure 7 depicts the estimates by Path‐FLSA, PCD‐FLSA, Path‐LFLSA, and PA‐FLSA. The number of identified change points for Path‐FLSA, PCD‐FLSA, Path‐LFLSA, and PA‐FLSA were 82, 172, 55, and 39, respectively. As shown in Figure 7 (a), the Path‐FLSA identified many change points in the periods that seems to have trends or stair‐case blocks and also missed several local changes. For example, the Path‐FLSA missed the change points for a local peak from 05/04/2020 to 05/06/2020, while both the Path‐LFLSA and PA‐FLSA identified the change points on 05/03/2020 and 05/06/2020. Figure 7 (b) shows that the PCD‐FLSA estimates seem to have many false‐positive change points and also identify much more change points compared with the others. As shown in the case of $\mathbf{\text{S4}}$ having the stair‐case blocks, the pattern recovery probabilities of the Path‐FLSA and PCD‐FLSA decrease very quickly when the noise level increases. With the residual of the fitted models, the estimates of the error standard deviation by Path‐FLSA, PCD‐FLSA, Path‐LFLSA, and PA‐FLSA are 0.24, 0.14, 0.18, and 0.20, respectively. Thus, this observation is also consistent with the comparison result of the pattern recovery probability for $\mathbf{\text{S4}}$. From the above observations, we focused on the comparison of the identified change points by the Path‐LFLSA and PA‐FLSA. We also depict the histograms and Q‐Q plots of the residuals of the four methods in Appendix D of the supplementary material. From the figures in Appendix D of the supplementary material, the residuals of the four methods are not exactly following the normal distribution but seem to follow the normal distribution within the interval $[-1.\mathrm{5,1}.5]$. It is also worth noting that the FLSA tends to find many change points when the underlying signal has a trend because the FLSA model is adequate for the piecewise constant mean model. For identifying the trend‐change points, we refer to the ${\ell }_1$ trend filtering method proposed by Kim et al. (2009), which is more suitable to identify sparse trend‐change points.

Table 2 reports the identified change points by Path‐LFLSA and PA‐FLSA with the corresponding indices, dates, and debiased estimates. The gray rows in Table 2 denote the identified change points by either the Path‐LFLSA or the PA‐FLSA only. There are 32 change points commonly identified by the Path‐LFLSA and PA‐FLSA, and there are also 23 (7) change points identified by the Path‐LFLSA (PA‐FLSA) only. Among the seven change points identified by the PA‐FLSA only, the six change points at 03/07/2020, 08/12/2020, 09/11/2020, 10/24/2020, 01/01/2021 and 11/01/2021 are closed to the six change points identified by the Path‐LFLSA only, where most of the differences are one or two days. The change point by the PA‐FLSA on 01/31/2022 seems to be an intermediate point within the period having a trend. For the change points by the Path‐LFLSA only except for the six change points corresponding to the change points by the PA‐FLSA only, most of the identified change points catch the local peaks.

TABLE 2.

The estimated change points of Path‐LFLSA and PA‐FLSA

Path‐LFLSA			PA‐LFLSA			Path‐LFLSA			PA‐LFLSA
Index	Date	${\widehat{\mu }}^{DB}$	Index	Date	${\widehat{\mu }}^{DB}$	Index	Date	${\widehat{\mu }}^{DB}$	Index	Date	${\widehat{\mu }}^{DB}$
6	03/06/2020	535.34				306	12/31/2020	992.99
			7	03/07/2020	507.43				307	01/01/2021	984.45
10	03/10/2020	232.84	10	03/10/2020	199.89	315	01/09/2021	757.46	315	01/09/2021	749.87
35	04/04/2020	97.92	35	04/04/2020	97.92	322	01/16/2021	526.01	322	01/16/2021	526.01
39	04/08/2020	47.23	39	04/08/2020	47.23	351	02/14/2021	393.59
48	04/17/2020	26.23	48	04/17/2020	26.23	355	02/18/2021	561.75
64	05/03/2020	9.89	64	05/03/2020	9.89	372	03/07/2021	392.68
67	05/06/2020	3.91	67	05/06/2020	3.91	401	04/05/2021	460.51	401	04/05/2021	424.31
69	05/08/2020	15.72				420	04/24/2021	663.27
75	05/14/2020	30.47				468	06/11/2021	589.20
86	05/25/2020	19.05	86	05/25/2020	21.65	478	06/21/2021	444.14
112	06/20/2020	45.73				492	07/05/2021	683.50	492	07/05/2021	599.00
117	06/25/2020	34.80				520	08/02/2021	1437.73	520	08/02/2021	1437.73
138	07/16/2020	51.02				571	09/22/2021	1760.66
141	07/19/2020	33.56				579	09/30/2021	2617.28
147	07/25/2020	60.68	147	07/25/2020	47.02	587	10/08/2021	2001.65
163	08/10/2020	31.85				604	10/25/2021	1422.00
			165	08/12/2020	33.91				611	11/01/2021	1784.54
167	08/14/2020	85.90	167	08/14/2020	131.79	625	11/15/2021	2134.56	625	11/15/2021	2242.17
185	09/01/2020	296.16	185	09/01/2020	296.16	639	11/29/2021	3372.36	639	11/29/2021	3372.36
194	09/10/2020	162.48				646	12/06/2021	5002.52	646	12/06/2021	5002.52
			195	09/11/2020	159.73	665	12/25/2021	6532.64	665	12/25/2021	6532.64
202	09/18/2020	121.88	202	09/18/2020	119.86	688	01/17/2022	978.03	688	01/17/2022	3978.03
210	09/26/2020	87.69				695	01/24/2022	7080.12	695	01/24/2022	7080.12
212	09/28/2020	44.60							702	01/31/2022	16194.46
234	10/20/2020	77.29				705	02/03/2022	18074.79	705	02/03/2022	23355.52
			238	10/24/2020	79.16	709	02/07/2022	36739.06	709	02/07/2022	36739.06
256	11/11/2020	112.47	256	11/11/2020	115.98	716	02/14/2022	54343.44	716	02/14/2022	54343.44
261	11/16/2020	211.76	261	11/16/2020	211.76	723	02/21/2022	99131.02	723	02/21/2022	99131.02
269	11/24/2020	333.21	269	11/24/2020	333.21	730	02/28/2022	158836.04	730	02/28/2022	158836.04
285	12/10/2020	564.42	285	12/10/2020	564.42	737	03/07/2022	226706.55	737	03/07/2022	226706.55

Note: The rows in gray denote the change points found by only one method.

On the other hand, interestingly, most of the commonly identified change points correspond to the period of the social distancing announced by the Ministry of Health and Welfare of Korea. For example, April 4, 2020, corresponds to the end of the intensive social distancing period from March 22, 2020, to April 5, 2020. In addition, the time periods between change points are similar to the periods of the social distancing policy by the Ministry of Health and Welfare of Korea, where the policy periods are usually two or three weeks. Finally, we should address that the sequence of the daily confirmed cases is observed through time, and it seems to have time‐dependent trends. Thus, the main assumptions of the FLSA for the model selection consistency is hardly satisfied, and then the FLSA‐based methods do not guarantee the model selection consistency. For example, among the commonly identified change points, the change points within four periods (03/10/2020 ∼ 05/06/2020, 11/16/2020 ∼ 12/10/2020, 11/15/2021∼ 12/25/2025, 02/03/2022 ∼ 03/07/2022) seem to be intermediate points within the periods having either an increasing or a decreasing trend.

However, this COVID‐19 spread example addresses that the Path‐LFLSA and PA‐FLSA are still applicable to find the main change points caused by an external event in a stable period (i.e. a period with a low trend effect). For example, both Path‐LFLSA and PA‐FLSA succeeded in finding the beginning day of the second wave of the COVID‐19 pandemic in Korea on August 15, 2020, when a mass rally was held near Gwanghwamun Square in Seoul, where the identified change point on August 14, 2020, in Table 2 denotes that the underlying mean value was changed on August 15, 2020. In addition, the identified change point on November 11, 2020, is close to the beginning day (11/4/2020) of the third wave reported in Seong et al. (2021) and the identified change point at 07/05/2021 is also close to the beginning day (7/7/2021) of the fourth wave 1. Recently, the number of daily confirmed cases has been rapidly increasing, and the fifth wave had was begun on 01/26/2022 2. Both Path‐LFLSA and PA‐FLSA found the change points on 01/24/2022, which is close to the beginning day of the fifth wave. The identified change points related to the second to the fifth waves are highlighted in red in Table 2.

7. CONCLUSION

In this study, we provide a new interpretation of the modified path algorithm for the FLSA by discovering the exact optimisation problems corresponding to the modified path algorithm, called Path‐LFLSA. Our discovery demonstrates that the modified path algorithm's hitting times are not monotonically increasing, and the violation of the monotone increasing property for the next hitting time can be verified by comparing the solution from the previous hitting time. We propose a pathwise adaptive FLSA with a weighted fusion penalty to recover the monotonicity of the hitting times. The comprehensive numerical study illustrates the whole solution paths of the four methods, including three existing ones and the proposed PA‐FLSA, and it also shows that the Path‐LFLSA and PA‐FLSA are less sensitive to noise levels for pattern recovery than the Path‐FLSA and PCD‐FLSA. Furthermore, our numerical study in Appendix C of the supplementary material provides a practical guideline for choosing the optimal tuning parameters of the Path‐LFLSA and PA‐FLSA that outperform Path‐FLSA and PCD‐FLSA to identify the true block structures and estimate the true signal levels. The application of Path‐LFLSA and PA‐FLSA with the optimal tuning parameter selection by EBIC to the number of daily‐confirmed cases of COVID‐19 infection found the change points related to the beginning days of the COVID‐19 pandemic waves from the second to the fifth in Korea.

Supporting information

insr12521‐sup‐0001‐Supp_PATH_FLSA_rev_v1.pdf

Son, W. , Lim, J. , and Yu, D. (2022) Path algorithms for fused lasso signal approximator with application to COVID‐19 spread in Korea. International Statistical Review, 10.1111/insr.12521.