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Regularization techniques in joinpoint regression


Joinpoint regression models are popular in various situations (modeling different trends in economics, mortality and incidence series or epidemiology studies and clinical trials). The literature on joinpoint regression mostly focuses on either the frequentist point of view, or discusses Bayesian approaches instead. A model selection step in all these scenarios considers only some limited set of alternatives, from which the final model is chosen. We present a different model estimation approach: the final model is selected out of all possible alternatives admitted by the data. We apply the \(L_{1}\)-regularization idea and via the sparsity principle we identify significant joinpoint locations to construct the final model. Some theoretical results and practical examples are given as well.

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The authors would like to express sincere thanks to both referees and the editors for proposing valuable suggestions that led to the improvement of the paper.

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Correspondence to Matúš Maciak.



Proof of Lemma 1

Let us start with the minimization formulation given in (4). Note that the vector of unknown parameters \(\varvec{\beta } = (a_{1}, b_{1}, b_{2} - b_{1} , \dots , b_{n - 1} - b_{n - 2})^\top \in \mathbb {R}^{n}\) can be decomposed into two parts: the first two parameters \(a_{1}, b_{1} \in \mathbb {R}\) represent the overal intercept and slope for a classical linear regression and are not penalized in the \(L_{1}\) penalty. The second part consists of differences \((b_{2} - b_{1}) , \dots , (b_{n - 1} - b_{n - 2})\) which we denoted as \(\varvec{\beta }_{(-2)} = (\beta _{2}, \dots , \beta _{n - 1})^\top \in \mathbb {R}^{n - 2}\). These parameters are all included in the LASSO penalty in (4).

Thus, the minimization problem (4) can be equivalently expressed as

$$\begin{aligned} \begin{array}{c} ~\\ Minimize\\ {\varvec{\beta }_{0} \in \mathbb {R}^2, \varvec{\beta }_{(-2)} \in \mathbb {R}^{n}} \end{array} \frac{1}{n}\left\| \varvec{Y} - \left( \mathbb {X}_{0}, \mathbb {X}_{(-2)}\right) \left( \begin{array}{c} \varvec{\beta }_{0} \\ \varvec{\beta }_{(-2)} \end{array} \right) \right\| _{2}^{2} + \lambda _{n} \sum _{j = 2}^{n - 1} |\beta _{j} |, \end{aligned}$$

where \(\varvec{\beta }_{0} = (a_{1}, b_{1})^{\top }\), \(\varvec{\beta }_{(-2)} = (b_{2} - b_{1}, \dots , b_{n - 1} - b_{n - 2})^{\top }\) and \(\mathbb {X}_{0}\) is a matrix consisting of the first two columns of the design matrix in (5) and \(\mathbb {X}_{(-2)}\) is a matrix consisting of all remaining columns.

We can define a hat matrix

$$\begin{aligned} \mathbb {H} = \mathbb {X}_{0} \Big (\mathbb {X}_{0}^{\top } \mathbb {X}_{0}\Big )^{-1} \mathbb {X}_{0} \end{aligned}$$

for a projection into a linear span of the first two columns of the design matrix \(\mathbb {X}\) and just by using simple matrix algebra computation we can verify that the projection of \(\varvec{Y}\) (into a linear span of columns of \(\mathbb {X}\)) defined as \(\mathbb {X}_{0}\widehat{\varvec{\beta }}_{0} + \mathbb {X}_{(-2)}\widehat{\varvec{\beta }}_{(-2)}\), for \(\widehat{\varvec{\beta }} = (\widehat{\varvec{\beta }}_{0}^\top , \widehat{\varvec{\beta }}_{(-2)}^\top )^\top \) being the solution of (7), can be equivalently expressed as \(\mathbb {H}\varvec{Y} + (\mathbb {I} - \mathbb {H})\mathbb {X}_{(-2)}\widehat{\varvec{\beta }}_{(-2)}\) where \(\widehat{\varvec{\beta }}_{(-2)}\) now solves the minimization problem

$$\begin{aligned} \begin{array}{c} ~\\ Minimize\\ \varvec{\beta }_{(-2)} \in \mathbb {R}^{n - 2} \end{array} \frac{1}{n} \Vert (\mathbb {I} - \mathbb {H})\varvec{Y} - (\mathbb {I} - \mathbb {H})\mathbb {X}_{(-2)}\varvec{\beta }_{(-2)}\Vert _{2}^{2} + \lambda _{n} \Vert \varvec{\beta }_{(-2)}\Vert _{1}. \end{aligned}$$

We only need to put \(\widetilde{\varvec{Y}} = (\mathbb {I} - \mathbb {H})\varvec{Y}\) and \(\widetilde{\mathbb {X}} = (\mathbb {I} - \mathbb {H})\mathbb {X}_{(-2)}\), which completes the proof of Lemma 1. \(\square \)

Proof of Theorem 1

We start from the assertion of Lemma 1. The design points are all drawn from the interval (0, 1) and without any loss of generality we will assume that they are centered (\(\sum _{i = 1}^{n} X_{i} = 0\)). Using the fact that \(\widehat{\varvec{\beta }}_{(-2)} = (\widehat{\beta }_{2}, \dots , \widehat{\beta }_{n - 1})^\top \in \mathbb {R}^{n - 2}\) is the minimizer of (9) we obtain that

$$\begin{aligned} \frac{1}{n}\Big \Vert \widehat{\varvec{Y}} - \widetilde{\mathbb {X}}\widehat{\varvec{\beta }}_{(-2)} \Big \Vert _{2}^{2} + \lambda _{n} \Big \Vert \widehat{\varvec{\beta }}_{(-2)} \Big \Vert _{1} \le \frac{1}{n}\Big \Vert \widetilde{\varvec{Y}} - \widetilde{\mathbb {X}}\varvec{\beta }_{(-2)} \Big \Vert _{2}^{2} + \lambda _{n} \Big \Vert \varvec{\beta }_{(-2)} \Big \Vert _{1}, \end{aligned}$$

for \(\varvec{\beta }_{(-2)} = (\beta _{2}, \dots , \beta _{n - 1}) \in \mathbb {R}^{n - 2}\) being the true but unknown vector of parameters. Using the model formula we have

$$\begin{aligned}&\frac{1}{n}\Big \Vert \widetilde{\mathbb {X}} \Big (\varvec{\beta }_{(-2)} - \widehat{\varvec{\beta }}_{(-2)} \Big ) \Big \Vert _{2}^{2} \le \frac{2}{n} \Big (\varvec{\beta }_{(-2)} - \widehat{\varvec{\beta }}_{(-2)} \Big )^\top \widetilde{\mathbb {X}}^{\top }\varvec{\varepsilon } + \lambda _{n}\Big ( \Vert \varvec{\beta }_{(-2)} \Vert _{1} - \Vert \widehat{\varvec{\beta }}_{(-2)} \Vert _{1} \Big )\\&\quad \le \frac{2}{n} \Big (\varvec{\beta }_{(-2)} - \widehat{\varvec{\beta }}_{(-2)} \Big )^\top \widetilde{\mathbb {X}}^{\top }\varvec{\varepsilon } + \lambda _{n} \left( \sum _{j; \beta _{j} \ne 0} |\beta _{j}| - |\widehat{\beta }_{j}| \right) - \lambda _{n} \sum _{j; \beta _{j} = 0} |\widehat{\beta }_{j}|, \end{aligned}$$

where we distinguished for two disjoint sets of indexes for elements of \(\varvec{\beta }_{(-2)} = (\beta _{2}, \dots , \beta _{n - 1})^\top \) and \(\widehat{\varvec{\beta }}_{(-2)} = (\widehat{\beta }_{2}, \dots , \widehat{\beta }_{n - 1})^\top \) depending on whether the true value of each parameter is zero or not. Recall, that we assume that \(\varvec{\beta }_{(-2)}\) is a sparse vector with most elements being zeros.

Using now the definition of matrices \(\mathbb {H}\) in (8), \(\widetilde{\mathbb {X}}\) in the proof Lemma 1 and the design matrix \(\mathbb {X}\) in (5) we obtain that

$$\begin{aligned}&\Big (\varvec{\beta }_{(-2)} - \widehat{\varvec{\beta }}_{(-2)} \Big )^\top \widetilde{\mathbb {X}}^{\top }\varvec{\varepsilon } \nonumber \\&\quad = \sum _{k = 2}^{n - 1} \big ( \widehat{\beta }_{k} - \beta _{k} \big ) \big ( X_{k} - X_{k - 1} \big ) \left[ \sum _{i = k}^{n} \varepsilon _{i} - \sum _{i = 1}^{n} \sum _{j = k}^{n} \Big (\frac{1}{n} + \frac{X_{i}X_{j}}{\sum _{l = 1}^{n} X_{l}^{2}}\Big ) \varepsilon _{i} \right] \\&\quad \le \sum _{k = 2}^{n - 1}\big ( \widehat{\beta }_{k} - \beta _{k} \big ) \left[ \sum _{i = k}^{n} \underbrace{\Big ( 1 - \sum _{j = k}^n \Big ( \frac{1}{n} + \frac{X_{i}X_{j}}{\sum _{l = 1}^{n} X_{l}^{2}} \Big )}_{m_{i}(\varvec{X}, n, k) } \Big )\varepsilon _{i}\right. \ \nonumber \\&\qquad \left. - \sum _{i = 1}^{k - 1} \Big (\underbrace{\sum _{j = k}^n \Big ( \frac{1}{n} + \frac{X_{i}X_{j}}{\sum _{l = 1}^{n} X_{l}^{2}} \Big )}_{h_{i}(\varvec{X}, n, k) }\Big ) \varepsilon _{i} \right] \nonumber \end{aligned}$$

Note, that expressions \(h_{i}(\varvec{X}, n, k)\) and \(m_{i}(\varvec{X}, n, k)\) are only defined as some combinations of elements of the projection matrices \(\mathbb {H}\) and \((\mathbb {I} - \mathbb {H})\). Therefore, conditionally on \(\varvec{X} = (X_{1}, \dots , X_{n})^\top \) we have that

$$\begin{aligned} \sum _{i = 1}^{n} \widetilde{\varepsilon }_{i} = \sum _{i = k}^{n} m_{i}(\varvec{X}, n, k)\varepsilon _{i} + \sum _{i = 1}^{k - 1} - h_{i}(\varvec{X}, n, k) \varepsilon _{i}, \end{aligned}$$

for \(\widetilde{\varepsilon }_{i} = - h_{i}(\varvec{X}, n, k) \varepsilon _{i}\), if \(i \le k - 1\), and \(\widetilde{\varepsilon }_{i} = m_{i}(\varvec{X}, n, k)\varepsilon _{i}\), otherwise, is a sum of independent Gaussian random variables with zero means and variances at most \((\kappa _{0} + 1)^{2} \sigma ^2\) as one can easily verify that uniformly for any \(n \in \mathbb {N}\), \(i = 1, \dots , n\) and \(k = 2, \dots , n - 1\) it holds that

$$\begin{aligned} Var ( h_{i}(\varvec{X}, n, k) \varepsilon _{i})&\le \sigma ^2 \left[ \sum _{j = k}^n \Big ( \frac{1}{n} + \frac{1}{\sum _{l = 1}^{n} X_{l}^{2}} \Big )\right] ^2 \le \sigma ^2 \left[ \sum _{j = k}^n \Big (\frac{\kappa _{0} + 1}{n}\Big )\right] ^2\\&= \sigma ^2 \left[ \Big (\frac{n - k + 1}{n}\Big )(\kappa _{0} + 1)\right] ^2 \le (\kappa _{0} + 1)^2 \sigma ^2, \end{aligned}$$

and analogously also

$$\begin{aligned} Var ( m_{i}(\varvec{X}, n, k) \varepsilon _{i})&= \sigma ^2 \left[ 1 - \sum _{j = k}^n \Big ( \frac{1}{n} + \frac{X_{i}X_{j}}{\sum _{l = 1}^{n} X_{l}^{2}} \Big )\right] ^2 \\&\le \sigma ^2 \left( \frac{k - 1}{n} - \frac{X_{i} \sum _{j = k}^{n}X_{j}}{\sum _{l = 1}^{n} X_{l}^{2}}\right) ^2\\&\le \sigma ^2 \left( 1 + \frac{2 n}{\sum _{l = 1}^{n}X_{l}^{2}} + \frac{(n - k + 1)^2}{(\sum _{l = 1}^{n}X_{l}^{2})^2}\right) \le (\kappa _{0} + 1)^2 \sigma ^2, \end{aligned}$$

where in both cases we used the eigenvalue restriction assumption given as \(\sum _{l = 1}^{n}X_{i}^2 \ge n/\kappa _0 > 0\).

Using now Theorem 3.8 from Massart (1996) we obtain that

$$\begin{aligned} P\left( \frac{1}{n}\sum _{i = 1}^{n} \widetilde{\varepsilon }_{i} > \frac{\lambda _{n}}{2}\right) \le 2 \exp {\Big (- \frac{\lambda _{n}^2 n }{8 (\kappa _{0} + 1)^2 \sigma ^2}\Big )}, \end{aligned}$$

and thus, for \(\lambda _{n} = \sqrt{\frac{\log n}{n}} \sigma K\) we have that with probability at least \(1 - 2e^{-n}\) it holds that

$$\begin{aligned} \frac{1}{n} \Big \Vert \widetilde{\mathbb {X}} \Big (\varvec{\beta }_{(-2)} - \widehat{\varvec{\beta }}_{(-2)} \Big ) \Big \Vert _{2}^{2} \le&\ \lambda _{n} \sum _{k = 2}^{n - 1} | \widehat{\beta }_{k} - \beta _{k} | + \lambda _{n} \Big ( \sum _{j; \beta _{j} \ne 0} |\beta _{j}| - |\widehat{\beta }_{j}| \Big ) \\&- \lambda _{n} \sum _{j; \beta _{j} = 0} |\widehat{\beta }_{j}|, \end{aligned}$$

and given the fact that the first sum can be decomposed with respect to true zero and nonzero parameters as \(\sum _{k = 2}^{n - 1} | \widehat{\beta }_{k} - \beta _{k} | = \sum _{k; \beta _{k} \ne 0}|\widehat{\beta }_{k} - \beta _{k}| + \sum _{k; \beta _{k} = 0}|\widehat{\beta }_{k}|\) we finally obtain that with probability at least \(1 - 2e^{-n}\) it holds that

$$\begin{aligned}&\frac{1}{n} \Big \Vert \widetilde{\mathbb {X}} \Big (\varvec{\beta }_{(-2)} - \widehat{\varvec{\beta }}_{(-2)} \Big ) \Big \Vert _{2}^{2} \le 2 \lambda _{n} \sum _{k; \beta _{k} \ne 0} |\beta _{k}| \le 2 K \sigma \sqrt{\frac{\log n}{n}} \overline{\beta } M, \end{aligned}$$

where \(M \in \mathbb {N}\) is the maximum number of changepoints in the mode and \(\overline{\beta }\) is some maximum allowed magnitude change. Using now the unitary property of matrix \((\mathbb {I} - \mathbb {H})\) we can directly apply the assertion of Lemma 1 to get the result of Theorem 1. \(\square \)

Let us conclude that for some specific scenarios we can provide stronger asymptotic results. For instance, if the design points \(X_{1}, \dots , X_{n}\) are fixed and moreover, equidistant, we have \((X_{i} - X_{i - 1})\) of the order 1 / n for all \(i = 2, \dots , n\) which can be used to improve the inequality in (10). Similarly, if the design points are random but with some given distribution the distribution can be used to bound terms \((X_{i} - X_{i - 1})\) in probability which also improves the overall rate at the end.

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Maciak, M., Mizera, I. Regularization techniques in joinpoint regression. Stat Papers 57, 939–955 (2016).

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  • Joinpoint regression
  • Segmented regression
  • Piecewise linear
  • Changepoints
  • Regularization
  • Model selection

Mathematics Subject Classification

  • 62J07
  • 62F99
  • 62P25