*5.5.1 Level stationarity*

The KPSS-L test is used here with an expectation that segments of climate data in which a change-point occurs contain a step-like shift but may also contain a change of trend. Hence it is used as a cross check. Further, once the deterministic internal shift and trend components are removed the residual should be both level and trend stationary. Level non-stationarity in the segment and level stationarity in the residuals supports the existence of a change-point.

### *5.5.2 Trend stationarity*

Data with a provisional change of trend is expected to be non-trend-stationary. Data with constant trend and a step-like change may show as trend stationary depending on the assumptions of the specific test. The KPSS-T test and the ADF test as formulated here may return different results in the presence of a step-like shift and no trend change, with the ADF test showing trend stationarity and the KPSS showing non-stationarity.

#### *5.5.3 Unit root/non-stationarity in the absence of any deterministic change*

The presence of a unit root may cause the data to mimic either a step-like change or a change of trend. In either case the MSBV can return a step-like change. All four unit-root tests are expected to detect this, with the ADF being less powerful, partly due to a potential to overfit autocorrelation lags. Since the detection method has provisionally detected a change-point, tests on the residuals would likely all show non-stationarity, and similarly testing of the segment itself. The ZA test would likely be the most powerful.

**Properties of Test**

Ruling assumptions

Null hypothesis, H0

Contrast hypothesis, H1

If exogenous change and UR present

If exogenous change but UR not present

Unit root but no exogenous change

Plus Unit Root

No Unit Root

No Unit Root

**225**

**ADF (trend and drift)**

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

No exogenous change.

I(1) Unit Root after allowing

autocorrelation and trend.

Presumption of trend stationarity

Accept H0, i.e.

May accept H0, i.e. UR

Accept H0, i.e.

When multiple exogenous changes are present

Accept H0, i.e.

May accept H0, i.e. UR

After removal of all exogenous change

Prefer H1, trend stationarity

If Unit Root Accept H0, i.e. UR

If Unit Root Accept H0, i.e. UR

UR

UR

When at most a single exogenous change is present

for

UR

**KPSS (level stationarity)**

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

**KPSS (trend stationarity)**

change.

stationarity

I(0) Unit Root I(1) Unit Root Deterministic with

UR

If step-change only then will accept H0, trend stationarity. A strong trend change will prefer H1, i.e. UR

No exogenous change. No exogenous

Prefer H1, i.e. UR Prefer H1, i.e.

Prefer H1, i.e. UR Prefer H1, i.e.

Prefer H1, i.e. UR Prefer H1, i.e.

Prefer H1, i.e. UR May prefer H1,

Prefer H1, i.e. UR Prefer H1, i.e.

Prefer H1, i.e. UR Prefer H1, i.e.

Accept H0, stationarity (unless residual trend

remains)

After removal of main exogenous change but with exogenous change still present

UR

UR

UR

UR

Accept H0, trend stationarity (unless residual trend remains)

i.e. UR if exogenous trend changes present

If constant trend and step-change then will accept H0, stationarity. Otherwise prefer H1, i.e.

UR

Stationarity Trend

**ZA**

change).

Not combined exogenous change and unit root. At most one exogenous change (shift or trend

I(1) Unit Root with drift and no exogenous change

possible exogenous change at a date

May prefer H1, i.e. exogenous change

Accept H0, i.e. UR

Accept H0, i.e. UR

Accept H0, i.e. UR

Prefer H1, exogenous change (even if there is

May prefer H1 if exactly one exogenous change remains. H0 of more than

none)

one.

May accept H0, i.e. UR

change

Prefer H1, i.e. exogenous

#### *5.5.4 Unit root/non-stationarity in the presence of deterministic change*

This is a complex issue. The combination of UR and deterministic trend is potentially explosive [58]. On the other hand the climate system is physically bounded and so at worst the combination may appear as step-like. If tests support unit root in both the segment data and its residuals, either a genuine unit root is present or multiple deterministic changes are. Data with apparent UR that disappears in the residuals is consistent with a single deterministic change. However data with multiple change points is misspecified for all tests.

#### *5.5.5 Misspecification due to use of averaging*

As discussed, the TM/SI identifies the sampling model as a point of consideration, and data conditioning may itself be a misspecification to a given investigation. Climate data is not homogenous. Averaging is often assumed to increase the signal to noise ratio (S/N) but more localised features may fall below detectability thresholds. For step-like changes occurring at different times in different components, the steps are diluted but potentially, autocorrelation is induced. Further, if the changes differ slightly in time over a number of components then the deterministic shift-like changes may be confused with either stochastic or deterministic trend. Similarly trend changes: Only if the step-like or trend change happens simultaneously across all processes will the S/N increase. Data conditioning methods that imposes or presume smoothing may turn steps into trends. If autocorrelation is present as part of the signal in the component's data together with trend, the situation is still more complex.

**Table 1** summarises the conditions that can be diagnosed with these UR tests. **Tables 2** and **3** provide interpretations.

#### **5.6 The averaging of multiple datasets with autocorrelation**

The "order" of an AR process is the number of lags, and also the polynomial order required to fit the error terms. The Dickey-Fuller equation (Eq. (6)) describes an autoregressive single lag, i.e. an AR(1) process. The sum of two AR(1) processes is most compactly represented as an autoregressive-moving average (ARMA) process of greater order, ARMA(2,1) [65].

If *p* and *q* are the lag order of processes, then two AR processes combine into an ARMA process, where the first parameter of the ARMA is the order of the AR part, and the second is the order of the moving average (MA) part.

$$AR(p) + AR(q) = ARMA(p + q, \max(p, q))\tag{11}$$


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


#### **Table 1.**

*Unit root tests used and their main assumptions (reproduced from R2019, Table Ch4.1.2). Possibilities not formally considered may deceive these tests by supporting either the null or contrast hypotheses.*


In Chapter 4 of R2019 the KPSS-T, ADF and ZA tests are combined to provide a classification scheme for change-points (**Table 4**). Using this classification scheme it became straight-forward to determine that regime changes over land and ocean

> We accept the single exogenous change, but the residuals are not stationary, leaving open the possibility of undetected features. The ZA has reverted from Exogenous/ stationary to Endogenous/non-stationary in the residuals, consistent with a single exogenous change plus a presumptive unit root. The presumptive unit root in the residuals is not reliably separable from multiple change-points below detectability.

> We accept the step-change detected by the MSBV as the single exogenous change with no stochastic trend. The residuals are stationary supporting the single change-

> cast doubt on the MSBV. The ZA test does not revert from endogenous/nonstationary and neither do the other tests. Hence the removal of a single change-point has had no apparent effect. Multiple change-points on top of a non-stationary

> We may be dealing with a pair of exogenous changes. The ZA reverted from nonstationary to stationary with other tests consistent with this. Potentially a single additional undetected change-point, since two exogenous changes may be classified

point. The ZA test does not change from exogenous/stationary Single, N/A We accept the step-change detected by the MSBV, without a valid ZA result, noting

Non-stationary We have evidence that the data segment contains sufficient non-stationarity as to

background is too complex a situation to detect with these tests.

that there is insufficient data to probe further.

as an endogenous change in the ZA. Stationary Possible false positive or weak change in stationary data

*Extended from R2019 Table Ch4.1.5: classifications of data segments.*

N/A Not classifiable/indeterminate

*Reproduced from R2019 Table Ch4.1.4: Expected outcomes of the KPSS-T and ADF tests, given data with a*

**Initial data with a presumptive step**

*KPSS-T H0* not rejected accept as Stationary. *ADF H0* rejected accept as Stationary.

*KPSS-T H0* rejected accept as Non stationary. *ADF H0* not rejected accept as Non stationary.

**Table 3.**

Single, nonstationary

Single, Stationary

Multiple, Stationary

**Table 4.**

**227**

**Residual with internal step and trends removed**

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

*KPSS-T H0* not rejected accept as Stationary. *ADF H0* rejected accept as

*KPSS-T H0* rejected accept as Non stationary. *ADF H0* not rejected accept as Non stationary.

*KPSS-T H0* not rejected accept as Stationary. *ADF H0* rejected accept as

*KPSS-T H0* rejected accept as Non stationary. *ADF H0* not rejected accept as Non stationary.

Stationary.

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

Stationary.

*presumptive step-like change plus a variety of different conditions.*

**Classification Reasoning and interpretation**

**Interpretations**

a trend change

did not have a trend change

Residual is stationary, the single change-point

Location of a single change-point misidentified so that the trend is also miscalculated

Residual is stationary and change-point included

The data segment is non-stationary and the provisional change-point may be a false positive. Residual is non-stationary. The initial segment contained a step and/or trend change.

**change**

#### **Table 2.**

*Reproduced from R2019 Table Ch4.1.3: Expected outcomes of the Zivot Andrews test, given data with a presumptive step-like change plus a variety of additional conditions. The first and second columns define results of the tests on the initial data segment and the residual with internal step and trend removed. The last column lists interpretations of the pairs of results.*

Treating the result of *AR*ð Þþ 1 *AR*ð Þ¼ 1 *ARMA*ð Þ 2, 1 as an AR(1) process may be deceptive. And yet in many analyses, the issue of the composition of the data is at best brushed off, and autocorrelation is in general approximated as AR(1). In R2019 apparent unit root-like behaviour in some zonal ocean temperature data sets resolves to deterministic shifts at different times in sub-sectors of those zones, and this affects the determination of change-points.

#### **5.7 Reasoning about change-points**

It is possible to examine the data segment and its residual and to have greatly increased confidence that the change-point methods are adequate to the task, and to broadly classify change-points detected as potentially affected by (a) misspecification of the detection tests with data out of its applicability range, (b) random-walks, (c) presence of undetected change-points, (d) some forms of model family misspecification.

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


#### **Table 3.**

*Reproduced from R2019 Table Ch4.1.4: Expected outcomes of the KPSS-T and ADF tests, given data with a presumptive step-like change plus a variety of different conditions.*

In Chapter 4 of R2019 the KPSS-T, ADF and ZA tests are combined to provide a classification scheme for change-points (**Table 4**). Using this classification scheme it became straight-forward to determine that regime changes over land and ocean


#### **Table 4.**

*Extended from R2019 Table Ch4.1.5: classifications of data segments.*

differ in complexity. Sharpening the testing, it also further supported the principal findings of R2019, that abrupt shifts relate directly to warming; in their extent, frequency and intensity; and more so at finer scale. For this paper the last two additional classes apply when ANCOVA does not support a change-point. An example is provided in the appendix.
