**A. Appendix**

During sensitivity testing of the detection and characterization tests in R2019 simulations were run, including assessments of (a) the effects of shifts single and multiple shifts below detection thresholds, (b) multiple shifts close in time, (c) high levels of autocorrelation, (d) state switching between deterministic and stochastic data, and (e) curvilinear trends. This illustrative example is an extension of one part of that work.

## **A.1 Synthetic climate-like data**

Following R2019, a suite of four artificial multi-step time series ('A' to 'D') was constructed and analyzed by MSBV then validation tests were run against both the shifts as detected by MSBV and as originally defined.

A is an artificial 200 year annual temperature consisting of random data (and a standard deviation, σ, of 0.44) with lag 1 autocorrelation of 25%, lag 7 autocorrelation of 10%, centered about zero, plus a quadratic trend curve rising 2.1 degrees. The degree of autocorrelation is consistent with the findings of [67].

Eight shifts random shift level (mean 1.5 σ) are added at defined times (Shifts of 1.5σ are less than MSBV reliability threshold) (**Table 6**).

To assess the suite presence of UR with deterministic trends plus shifts, shifts without trends, and UR alone, red-noise (summed white noise, μ = 0, σ = 0.44) was added to A to produce set B, set C is the defined steps plus red-noise and D is red-noise only.

#### *A.1.1 Studentized Breusch-Pagan test for heteroskedasticity*

The studentized BP test was run for the disjoint regression of breaks detected (Break model), and also for the breaks as defined (**Table 7**). A linear model and a


**Table 6.**

*Adapted from R2019, Table Ch4.1.6 Synthetic Data Timing and extent of Shifts. Total Rise is shown both as anomaly and as standard deviations. Shifts of <0.5 are not guaranteed to be found by MSBV and are bolded.*


**Table 7.**

*Studentized Breusch-Pagan Test results. Green denotes 0.01 < p < 0.05, red 0.01 > p, black p > 0.05. A null hypothesis of homoscedasticity is rejected for low p-values.*

quadratic model were also run for comparison. Data sets A and D appear to have homoskedastic residuals for their breaks given the detected shifts, and yet A is deterministic and D is non-deterministic. Datasets B and C, on the other hand appear homoskedastic given a quadratic model. Note that the SBP operates under an i.i.d assumption which is violated by sets B, C and D.

The breaks returned by the MSBV, and breaks defined, for A both form an adequate model. The breaks returned by the MSBV for D form an adequate model whereas the defined set – not present in D – does not. B contains a curvilinear trend, plus along with C, shifts which also induce an apparent curvilinearity. As can be seen from **Figure 3** the MSBV does quite well at locating the change-points.

Note: The data tested, residuals of detected change-points, will almost always appear to be deterministic when tested by the UR tests. This is because the residual

of deterministic signal is expected to be deterministic, whereas the residual of a purely non-deterministic signal from which a deterministic components has been subtracted *acquires* a deterministic component and appears mean-reverting, i.e. I (0). There is no current method for dealing with multiple deterministic changes in a UR time series, and blended series such as B and C will not meet the criteria of a UR series. In fact looking at D only through the lens of the SBP and UR tests of the residuals does not distinguish it from a deterministic time series like A. The difference only becomes apparent when the individual change-points are tested (see

**Defined A B C D**

2029 2029(+S) 2029(++S) 2029(+NS) 2030(NS)

2070 2070(+S) 2073(S) 2069(+NS)

2096 2091() 2095(+) 2095(+)

*Changes defined and years detected in each dataset. Annotations denote segment classification, + is single change-point, ++ possible multiple changes, S is stationary, NS is non-stationary. Red denotes ANCOVA*

2088(S) 2082(++S) 2082(+)

1954 1954(+S) 1926(NS)

*Severe Testing and Characterization of Change Points in Climate Time Series*

1998 1998(+S) 1998(+S) 1998(NS)

2054 2049(S) 2049(S) 2055(+NS)

1912(++S) 1912(NS) 1908(+NS)

1971(++S) 1970(+NS) 1970(NS)

1944(+NS) 1950(+NS)

The full analysis results are available on-line at https://cdn.intechopen.com/

Set A. One pair of defined change-points violated an assumption of the MSBV that rejects shifts within a seven year refractory period (defined as 2029, 2035), selecting only 2029 which registers as a strong shift embedded in stationary data with an internal trend (notably the ZA test of the residuals locates 2035). When the data is broken up according to the defined shifts, 2035 registers as a strong shift in non-stationary data, and evidence for the internal trend weakens. The defined small shift in 2054 following 2035 was attributed to 2049 after 2029 but not supported by ANCOVA and the segment was classified as non-stationary. The ZA suggests a change in 2034 but non-stationarity in the residuals. All other change-points were detected as defined and classified as single change-points in trend-stationary data. Datasets B though D represent increasingly UR dominated data. For B (combining deterministic trends and red noise), the only detected shift that is classified as a single shift in stationary data is 1998, all prior being classified as having possible multiple sub-detection shifts, and all following being rejected by ANCOVA

although the segments are classed as stationary. Sets C (UR with shifts) and D (UR

only), show that the MSBV by itself is vulnerable to non-determinism.

**Tables 8** and **9**).

**233**

*A.1.2 Analysis of individual change-points*

public/docs/230558\_files.zip.

1982 1979(+S)

*DOI: http://dx.doi.org/10.5772/intechopen.98364*

2035

**Table 8.**

*p < =0.05.*

#### **Figure 3.**

*Top: Blue is data set D, red is break-segments determined by MSBV. Magenta is auto-correlated noise plus quadratic trend. Orange is the defined shifts. Middle: Dataset C is shown as orange (black breaks), dataset B is blue (red breaks). Bottom: dataset A in blue, red is break-segments determined by MSBV. Vertical grey reference lines indicate defined shifts.*


*Severe Testing and Characterization of Change Points in Climate Time Series DOI: http://dx.doi.org/10.5772/intechopen.98364*

#### **Table 8.**

*Changes defined and years detected in each dataset. Annotations denote segment classification, + is single change-point, ++ possible multiple changes, S is stationary, NS is non-stationary. Red denotes ANCOVA p < =0.05.*

of deterministic signal is expected to be deterministic, whereas the residual of a purely non-deterministic signal from which a deterministic components has been subtracted *acquires* a deterministic component and appears mean-reverting, i.e. I (0). There is no current method for dealing with multiple deterministic changes in a UR time series, and blended series such as B and C will not meet the criteria of a UR series. In fact looking at D only through the lens of the SBP and UR tests of the residuals does not distinguish it from a deterministic time series like A. The difference only becomes apparent when the individual change-points are tested (see **Tables 8** and **9**).

#### *A.1.2 Analysis of individual change-points*

The full analysis results are available on-line at https://cdn.intechopen.com/ public/docs/230558\_files.zip.

Set A. One pair of defined change-points violated an assumption of the MSBV that rejects shifts within a seven year refractory period (defined as 2029, 2035), selecting only 2029 which registers as a strong shift embedded in stationary data with an internal trend (notably the ZA test of the residuals locates 2035). When the data is broken up according to the defined shifts, 2035 registers as a strong shift in non-stationary data, and evidence for the internal trend weakens. The defined small shift in 2054 following 2035 was attributed to 2049 after 2029 but not supported by ANCOVA and the segment was classified as non-stationary. The ZA suggests a change in 2034 but non-stationarity in the residuals. All other change-points were detected as defined and classified as single change-points in trend-stationary data.

Datasets B though D represent increasingly UR dominated data. For B (combining deterministic trends and red noise), the only detected shift that is classified as a single shift in stationary data is 1998, all prior being classified as having possible multiple sub-detection shifts, and all following being rejected by ANCOVA although the segments are classed as stationary. Sets C (UR with shifts) and D (UR only), show that the MSBV by itself is vulnerable to non-determinism.


*Numbers of change-points assigned to each class. Note that C and D differ from A and B by having non-stationary residuals, where as B differs from A by displaying evidence of undetected multiple change-points.*

The principal indication that a change-point dominated time-series has an underlying difference stationarity (i.e. red, or brown noise) is given by examination

*Severe Testing and Characterization of Change Points in Climate Time Series*

of the segmentation and not the residuals.

*DOI: http://dx.doi.org/10.5772/intechopen.98364*

**Author details**

**235**

James Ricketts\* and Roger Jones

Victoria University, Melbourne, Australia

provided the original work is properly cited.

\*Address all correspondence to: james.ricketts@live.vu.edu.au

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

*Severe Testing and Characterization of Change Points in Climate Time Series DOI: http://dx.doi.org/10.5772/intechopen.98364*

The principal indication that a change-point dominated time-series has an underlying difference stationarity (i.e. red, or brown noise) is given by examination of the segmentation and not the residuals.
