**6. Examples**

**Figure 1** below, illustrates the difference between analysis, commencing with the MSBV, of global mean temperature and the area averaged Northern midlatitude (NML) temperature. While the step change after 1996 is very obvious in the zonal data, the change is sometimes disputed in the global signal. **Table 5**, adds strong support to the contention that the so-called hiatus was a significant event, but not on the basis of trends. The 1988 event in the NML corresponds to an atmospheric reorganisation and extensive biophysical changes regionally [10]. All of the change-points occur in data which is otherwise stationary.

**Figure 2** below, illustrates the contribution to reasoning about the nature of decadal climate regimes which follows from a reasoned classification scheme. If the global temperatures are averaged over smaller areas, and then step-change points are calculated it becomes more likely that the data will present as stationary. This shows that the zonal data are not homogeneous with respect to regime shifts; that regime shifts are more regional. Note also the difference between land (almost always stationary) and ocean (less so), supporting the ocean as being more complex. There is also a tendency for land shifts to be a year or two delayed.

**Figure 1.**

*Adapted from R2019, Figure Ch3.11. Step-like shifts in the Northern extra-tropics compared to those detected in global data.*

**MSBV**

**Zone**

**229**

 **First**

**Internal**

**Internal**

**Data**

**Residuals**

 **Data**

**Residuals**

 **Data**

**Residuals**

 **Data**

**Residuals**

 **First**

**ANOVA-Internal**

**ANOVA-Trend**

**ANCOVA-Changepoint(Pr)**

**Changed**

**Year**

**Shift(Pr)**

**Change**

**(Pr)**

**segment**

**segment**

**segment**

**Changed**

**Shift**

**Trend**

**segment**

**Year**

30 N–60 N 30 N–60 N

30 N–60 N

Global

Global

Global

**Table 5.** *Adapted from R2019 Table A4.1.30: For the UR tests red text denotes results of tests where the data length may affect precision. NS =* 

*\*\* 0.001 > p < =0.01, \* 0.01 < p > =0.05.*

 1997

 0.16

0.005

 NS

 1979

 0.12

 0.009

 NS

 S

 NS SSS

S

 S

S

 S

 1997 *non-stationary,*

 *S =*

*stationary.*

 *\*\*\* p < = 0.001,*

 \*\*\*

—

 S

 NS

 S

 S

 S

 1946

 \*\*

 \*

 \*\*\* \*\*

 1930

 0.25

 0.003

 NS

 S

 NS

 S

 NS

 S

 S

 S

 1914

 \*\*\*

 \*

 \*\*\*

 1997

 0.43

 0.019

NS

 1988

 0.37

0.014

NS

 S

 NS SSS

NS

 S

NS

S

1997

 \*\*\*

—

\*\*

 S

 NS

 S

 S

 S

 1964

 \*\*

—

\*\*\*

 1921

 0.34

 0.001

 NS

 S

 NS

 S

 NS

 S

 S

 S

 1921

 \*\*\*

—

\*\*\*

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

**(°C)**

**Change**

**(0C/Yr)**

**KPSS-L**

**KPSS-T**

**ADF**

**Zivot Andrews**

**ANOVA/ANCOVA**

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

## **7. Conclusions**

The principal contribution of this paper is to expand on the use of misspecification testing to strengthen reasoning about abrupt shifts in time-series. We focus on climate data records and statistical model specification with respect to the data. Probing the misspecification of statistical models helps ensure that tests better represent probative criteria, and better distinguish between them.


*\*\* 0.001 > p < =0.01, \* 0.01 < p > =0.05.*

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

**229**

Koninkliijk Nederlands Meteorologisch Instituut (KNMI) make available the

Institute for Space Studies and United States National Climatic Data Center.

Other data has been sourced from, Met Office Hadley Centre, NASA, Goddard

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.

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

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.

Eight shifts random shift level (mean 1.5 σ) are added at defined times (Shifts of

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

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

**Year 1954 1982 1998 2029 2035 2054 2070 2096 Total**

*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.*

**Dataset Break model Defined Linear model Quad model** A. 0.6802 0.6959 0.0241 0.0001 B. 0.0034 0.0270 0.0000 0.9108 C. 0.0000 0.0039 0.0000 0.5205 D. 0.2870 0.0024 0.0000 0.0457

*Studentized Breusch-Pagan Test results. Green denotes 0.01 < p < 0.05, red 0.01 > p, black p > 0.05. A null*

1.00 (2.27)

0.61 (1.39)

0.94 (2.14)

**0.31 (0.70)**

5.34 (12.14)

0.85 (1.93)

KNMI Climate Explorer and this was a valuable resource.

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

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

shifts as detected by MSBV and as originally defined.

1.5σ are less than MSBV reliability threshold) (**Table 6**).

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

0.72 (1.64)

The degree of autocorrelation is consistent with the findings of [67].

produce set B, set C is the defined steps plus red-noise and D is red-noise only.

**A. Appendix**

Shifts in K (σ)

**Table 6.**

**Table 7.**

**231**

0.57 (1.30)

**0.34 (0.77)**

*hypothesis of homoscedasticity is rejected for low p-values.*

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

#### **Figure 2.**

The TM/SI framework has been suggested as a variation on previously discussed inductive frameworks. While it assists adequate testing of physical hypotheses; none the less, climate is a complex system [66]. Thus the ongoing assessment of testing procedures, and with it model specification.

Misspecification testing supports severe testing. Severe testing is strengthened by improved one to one mapping between physical features and statistical test outcomes. The tests outlined here assist a probative analysis, firstly by adding nuance to the findings, and secondly by providing the basis of a change-point classification, they assist strong reasoning. They have been selected because they are individually automatable and complementary, and the utility of this has been indicated in the case study. The chain of reasoning involved in the use of multiple tests is complex but the final classification scheme is compact and as seen, informative.

A basis has also been established for potentially detecting signatures of a data composition misspecification whereby features emerge or submerge in composited data due to averaging of signals (especially ones moving in time and space). The signature is a reduction in non-stationarity when signals are decomposed or segmented using the MSBV as seen in **Figure 2**. The same issue also affects both autocorrelation and trend analysis simply because step-like dislocations in data are generally deceptive for the regressions embedded in many general methods.

### **Acknowledgements**

R.N. Jones is a Professorial Fellow of Victoria Institute for Strategic Economic Studies in Melbourne. J. H. Ricketts was the holder of a Victoria University postgraduate research scholarship. The anonymous reviewers of a joint paper and a joint conference paper, and two thesis reviewers, all contributed substantial improvements and assisted development of the ideas expressed here. We would like to acknowledge with both gratitude and sorrow the lasting influence of our colleague and friend, Dr. Penny Whetton, who passed away too soon.

### **Notes**

A number of tables and figure are adapted from the PhD thesis of JH Ricketts [13], mostly chapter 4. A peer reviewed joint paper [21] and a joint conference paper [21] are also sourced.

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

Koninkliijk Nederlands Meteorologisch Instituut (KNMI) make available the KNMI Climate Explorer and this was a valuable resource.

Other data has been sourced from, Met Office Hadley Centre, NASA, Goddard Institute for Space Studies and United States National Climatic Data Center.
