Detection of trend changes in time series using Bayesian inference

Change points in time series are perceived as isolated singularities where two regular trends of a given signal do not match. The detection of such transitions is of fundamental interest for the understanding of the system’s internal dynamics or external forcings. In practice observational noise makes it difficult to detect such change points in time series. In this work we elaborate on a Bayesian algorithm to estimate the location of the singularities and to quantify their credibility. We validate the performance and sensitivity of our inference method by estimating change points of synthetic data sets. As an application we use our algorithm to analyze the annual flow volume of the Nile River at Aswan from 1871 to 1970, where we confirm a well-established significant transition point within the time series.

Figure 1

Realization of a time series of $n_{\text{obs}}=100$ data points generated by Eq. (6), whereas the mean is parametrized by $F_{\theta }\mathit{\beta }=5+0.22{\mathit{\zeta }}_{-}^{\theta }+0.08{\mathit{\zeta }}_{+}^{\theta }$ and the variance is modeled as ${\sigma }^2{\Omega }_{\theta ,\mathit{s}}={\left[1.6(1+0.2{\mathit{\zeta }}_{-}^{\theta }+0.1{\mathit{\zeta }}_{+}^{\theta })\right]}^2$.

Figure 2

Marginal posterior density $p\left(\theta \right|\mathit{y})$ for the time series in Fig. 1, represented in the lower panel. The maximum indicates the estimate of the change point $\widehat{\theta }=40.5$, whereas the dashed line marks the true parameter value of the underlying model.

Figure 3

Marginal posterior density $p(s_1,s_2|\mathit{y})$ for the time series in Fig. 1. Alongside are presented the projected posterior densities $p\left(s_1\right|\mathit{y})$ and $p\left(s_2\right|\mathit{y})$. The maxima indicate the estimates of the slope parameters $({\widehat{s}}_1,{\widehat{s}}_2)=(0.170,0.080),$ whereas the dashed lines mark the true values of the underlying model.

Figure 4

Marginal posterior density $p({\beta }_1,{\beta }_2|\mathit{y})$ for the time series in Fig. 1. Alongside are presented the projected posterior densities $p\left({\beta }_1\right|\mathit{y})$ and $p\left({\beta }_2\right|\mathit{y})$. The maxima indicate the estimates of the slopes $({\widehat{\beta }}_1,{\widehat{\beta }}_2)=(0.198,0.088)$, whereas the dashed lines mark the true values of the underlying model.

Figure 5

Marginal posterior density $p\left(\theta \right|\mathit{y})$ for a time series where only the mean undergoes a transition, represented in the lower panel. The maximum indicates the estimate of the change point $\widehat{\theta }=39.5$, whereas the dashed line marks the true parameter value of the underlying model.

Figure 6

Marginal posterior density $p\left(\theta \right|\mathit{y})$ for a time series where only the deviation undergoes a transition, represented in the lower panel. The maximum indicates the estimate of the change point $\widehat{\theta }=40.5$, whereas the dashed line marks the true parameter value of the underlying model.

Figure 7

The global maxima of the averaged posterior densities $\langle p_{n_{\text{obs}}}\left(\theta \right|\mathit{y})\rangle$ converge for increasing number of data points $n_{\text{obs}}$ toward a delta distribution located at the true change point value $\theta =135$ (dashed line).

Figure 8

Histograms of the global change point estimators ${\widehat{\theta }}_{n_{\text{obs}}}^i$ for $i=1,\cdots ,60$ realizations and with respect to $n_{\text{obs}}$ data points from the setting of Fig. 7. Even for $n_{\text{obs}}=100$ all global estimates ${\widehat{\theta }}_{100}^i$ lie in the relatively narrow interval $[121,148]$ around the true singularity $\theta =135$ (dashed lines).

Figure 9

Sum of local posterior densities weighted by the local Likelihood maxima (dash-dotted) and with respect to the Bayes factor (solid), computed for windows of $n_{\text{sub}}=50$ data points and $n_{\text{cp}}=30$ change points on uniform sampling grids. The dashed line marks the true singularity $\theta =80$.

Figure 10

Local Bayes factor (squares) scaled in deciban and obtained for the time series in Fig. 9. The shaded area encloses values whose support for none of the models is substantial (based on [36]). Values underneath this area strongly support a change point against a linear model, and vice versa.

Figure 11

Sum of local posterior densities weighted by the Bayes factor, computed for sub time series of $n_{\text{sub}}=50$ data points and $n_{\text{cp}}=30$ change points on uniform sampling grids. The parametrization of the mean is defined as $F\mathit{\beta }=14+0.2{\mathit{\zeta }}_{-}^{40}+0.1{\mathit{\zeta }}_{+}^{40}-0.25{\mathit{\zeta }}_{+}^{100}+0.3{\mathit{\zeta }}_{+}^{160}$ and the variance as ${\sigma }^2\Omega ={\left[1.6(1+0.2{\mathit{\zeta }}_{-}^{40}+0.03{\mathit{\zeta }}_{+}^{40}-0.05{\mathit{\zeta }}_{+}^{100}+0.1{\mathit{\zeta }}_{+}^{160})\right]}^2$.

Figure 12

Annual Nile flow containing a known change point at $\theta =1899$. The sum of localized posterior densities weighted with respect to the Bayes factor indicates a change point at $\widehat{\theta }=1898$ within its confidence interval $[1895,1901]$ of about $90\%$. The estimated underlying model reveals the most dominant transition in the behavior of the mean.