# The crux with reducing emissions in the long-term: The underestimated “now” versus the overestimated “then”

• Matthias Jonas
• Piotr Żebrowski
Open Access
Original Article

## Abstract

The focus of this perspective piece is on memory, persistence, and explainable outreach of forced systems, with greenhouse gas (GHG) emissions into the atmosphere serving as our case in point. In the light of the continued increase in emissions globally vis-à-vis the reductions required without further delay until 2050 and beyond, we conjecture that, being ignorant of memory and persistence, we may underestimate the “inertia” with which global GHG emissions will continue on their increasing path beyond today, thus, also leading to the amount of reduction that can be achieved in the future being overestimated. This issue is at the heart of mitigation and adaptation. For a practitioner, this translates to the problem of how persistently an emissions system behaves when subjected to a specified mitigation measure and which emissions level to adapt to for precautionary reasons in the presence of uncertainty. Memory allows us to reference how strongly the past can influence the “near-term future” of the system or (what we define as) its explainable outreach. We consider memory to be an intrinsic property of a system, retrospective in nature; and persistence to be a consequential (i.e., observable) feature of memory, prospective in nature and reflecting the tendency of a system to preserve a current state (including trend). Persistence depends on the system’s memory which, in turn, reflects how many historical states directly influence the current one. The nature of this influence can range from purely deterministic to purely stochastic. Different approaches exist to capture memory. We capture memory generically with the help of three characteristics: its temporal extent and both its weight and quality over time. The extent of memory quantifies how many historical data directly influence the current data point. The weight of memory describes the strength of this influence (fading of memory), while the quality of memory steers how well we know the latter (blurring of memory). Capturing fading and blurring of memory in combination is novel. In a numerical experiment with the focus on systemic insight, we cast a glance far ahead by illustrating one way to capture memory, and to understand how persistence plays out and how an explainable outreach of the system can be derived even under unfavorable conditions. We look into the following two questions: (1) Do we learn properly from the past? That is, do we have the right science in place to understand and treat memory appropriately? And (2) being aware that memory links a system’s past with its near-term future, do we quantify this outreach in a way that is useful for prognostic modelers and decision-makers? The latter question implies another question, namely, whether we can differentiate between and specify the various characteristics of memory (i.e., those mentioned above) by way of diagnostic data-processing alone? Or, in other words, how much system understanding do we need to have and to inject into the data-analysis process to enable such differentiation? Although the prime intention of our perspective piece is to study memory, persistence, and explainable outreach of forced systems and, thus, to expand on the usefulness of GHG emission inventories, our insights indicate the high chance of our conjecture proving true: being ignorant of memory and persistence, we underestimate, probably considerably, the “inertia” with which global GHG emissions will continue on their historical path beyond today and thus overestimate the amount of reductions that we might achieve in the future.

## Keywords

Earth system science Greenhouse gas Emissions inventory Data analysis Memory Persistence Explainable outreach Uncertainty

## 1 Introduction

This article is a perspective piece on research still to be conducted and elaborated in scientific papers. Its focus is on forced systems exhibiting memory, with greenhouse gas [GHG] emissions into the atmosphere serving as an example. Here, we ask two simple questions: (1) Do we learn properly from the past? And (2) being aware that memory allows us to reference how strongly a system’s past can influence its near-term future, do we quantify this “outreach” in a way that is useful for prognostic modelers and decision-makers?

The answer to both questions is no, as we argue below, the reason being that we do not have the right science in place to address these questions adequately. Embarking on such research will lead us to new grounds needing to be mapped and explored—which is why we opt for a perspective piece before anything else. We are lacking tools to fully understand the nature of memory in systems and its effect. As a first consequence, the questions lead us to the need to fuse deterministic and statistical data analysis to complement process modeling (question 1) and to the need to establish (what we call) the explainable outreach [EO] of data series with memory (question 2).

As a further consequence, the two questions together point to an important, widely observed problem: we typically underestimate the tendency of systems with memory to continue on their historical path into the future. In the case of GHG emissions, we face the need to reduce emissions instantaneously. Being ignorant of memory and persistence, a consequential feature of memory, we underestimate—most likely, even considerably—the tendency with which GHG emissions, currently still increasing globally (Carbon Brief 2017), will continue on their historical path beyond today; and thus we overestimate the amount of reductions that we can achieve in the near future. This issue is at the heart of mitigation and adaptation. For a practitioner, it translates to the problem of how persistent an emissions system's behavior is when it is subjected to a specified mitigation measure and which emissions level to adapt to for precautionary reasons in the presence of uncertainty.

Although GHG emissions, notably CO2, into the atmosphere serve as our prime real-world example, we gain a more fundamental understanding of the problem by initially working under lab conditions, i.e., with synthetic data (as done here). Nonetheless, any other series of data (observations) of systems with memory, e.g., atmospheric CO2 concentrations or average surface air temperature changes, can also be used, preferably in combination.

## 2 Motivating a broader perspective of memory

It is in the context of analyzing the structural dependencies in the stochastic component of time series where the terms “memory” and “persistence” commonly appear and are widely discussed.

However, we start from an accurate and precise world where we observe memory (precisely). In this world, time series analysis, being statistically grounded, is inappropriate for deal with memory (noise does not exist). This must be done differently. Only then do we make the step, common in physics, to an accurate and imprecise world, the realm of time series analysis.

Our intention in defining memory independently (i.e., outside) of time series analysis is not to exclude time series analysis, but to explore (1) how memory defined in an accurate and precise world “spills over” and impacts time series analysis in an accurate and imprecise world and (2) the advantage of treating memory (more specifically, its characteristics) consistently for both the deterministic and the stochastic part of a time series.

In the further course of the perspective piece, we will encounter additional advantages that will justify this two-pronged approach. These include not being restricted to working with small numbers which are, not unusually, also in the noise range (a consequence of making time series stationary); and being able to introduce gradual blurring of memory (increase in uncertainty back in time) as the unavoidable twin effect accompanying the fading of memory (decrease in memory back in time). To our knowledge, these two memory characteristics have not been addressed in combination to date.

Above and beyond, there is another fundamental difference to time series analysis: We do not attempt to predict. Instead, we make use of the memory characteristics of a time series to determine its explainable outreach [EO] (as visualized in Section 4 and demonstrated in Section 6).

### 2.1 Motivating by way of example

Our example is a brief, detail-avoiding, but insightful detour to reinforce why understanding memory in an accurate and precise world initially is important (while we motivate only in Section 2.2 why understanding memory at all is fundamental). The example starts from a univariate time series, here given by emissions of GHGs, notably CO2, into the atmosphere. We know that emissions contain memory—furnaces (a placeholder for any anthropogenic source of emissions) installed years ago still contribute to emissions accounted today. Figure 1 illustrates this situation graphically where conditions are assumed to be constant over a given time; this means that, during that time, furnaces phase in instantaneously (here at a rate increasing linearly over time) and phase out linearly (here over 3 years). This does not restrict generality; any other phase-in/phase-out combination is conceivable as well. It is further assumed that furnaces can be distinguished by age. Finally, Fig. 1 does not display furnaces by number but by their emissions (which add up linearly).

It can be argued that emissions from newly installed furnaces as well as total emissions are better known than emissions from older furnaces. On the one hand, countries that make estimates of CO2 emissions under the UNFCCC (United Nations Framework Convention on Climate Change) provide annual updates in which they add another year of data to the time series and revise the estimates for earlier years (bottom-up approach) (Marland et al. 2009; Hamal 2010; UNFCCC 2018). On the other hand, high-resolution carbon-observing satellites are able to provide emission accounts (top-down approach) (e.g., Oda et al. 2018), such as the Japanese Greenhouse Gases Observing Satellite [GOSAT] (Matsunaga and Maksyutov 2018) and NASA’s Orbiting Carbon Observatory-2 [OCO-2] (Hakkarainen et al. 2016).

This suggests that our starting point, Fig. 1, should be modified, as follows: At each point in time, we assume that we are able to “measure” precisely both CO2 emissions from newly installed furnaces and total CO2 emissions (which corresponds to knowing the red and the red-blue-orange bars in Fig. 1).

Two additional modifications help us to illustrate the impact of memory. We anticipate nonlinearity and periodicity increasing in importance. For this reason, emissions from newly installed furnaces will still phase in instantaneously, but at an increasing rate greater than linear, here following a second-order polynomial, superimposed by a sine function (see red curve yQS ≔ yQuad + ySin in Fig. 2). (To recall, we keep mathematical details to a minimum. These will be mentioned later, to the extent relevant, in Section 5.)
Total emissions shall come as a simple moving average, here over the last 7 years (including today), with weights decreasing (not linearly as in Fig. 1 but) exponentially back in time. The weighting will stay constant (equivalent to requesting that all furnaces phase out alike) and is determined such that its value 6 years back in time (excluding today) is only 0.05 (which reflects our cut-off level; see Section 5). Constructing a simple moving average of a time series can also be considered as convolution or applying a linear filter: yQS _ wM ≔ yQuad _ wM + ySin _ wM = yQS × M = (yQuad + ySin) × M, where M denotes the applied memory model or filter, and wM (and no distinct index, respectively) the worlds we perceive, that is, with memory (see black curve in Fig. 2) and without memory. Determining M—what we are ultimately interested in—with the knowledge of both input (yQS) and output (yQS _ wM) is called deconvolution. Table 1 provides a summary of what we know/do not know and facilitates the reading of Fig. 2.
Table 1

The world we perceive

Trend (not observable individually)

Periodicity (not observable individually)

Noise (n.a.; will be dealt with in Sections 5 and 6)

Observable as a whole (here constructed)

w/o M

y Sin

y QS

with M

y Sin _  wM

y QS _  wM

The example shows that determining M can/should happen outside of time series analysis (e.g., by way of deconvolution). Knowing M is also key in determining the true trend of yQS _ wM (i.e., yQuad _ wM). In further consequence, yQS _ wM cannot/not easily be detrended correctly (see Reg _ O2(yQS _ wM) in Fig. 2) if M is unknown. But correct detrending is tacitly presupposed in making time series stationary.

### 2.2 Motivating more generally

As mentioned, our focus is on systems with memory, typical in Earth system sciences. In addition, we restrict ourselves to forced systems. In many cases, we do know that a system possesses memory, e.g., because it does not respond instantaneously to the forcing it experiences—which is what a system with no memory would do. (This becomes visible in Fig. 2: yQuad _ wM lags behind yQuad.)

Such systems are typically dealt with by using one of two approaches: (1) process-based modeling (e.g., Solomon et al. 2010) and (2) time series analysis (e.g., Belbute et al. 2017). With reference to our example, the first approach would be equivalent to modeling the stock, i.e., the phase-in and phase-out, of furnaces over time; the second approach would be equivalent to subtracting trend and (ideally) periodicity from a time series to determine the dependency structures in the random residuals.

Each approach comes with strengths and weaknesses. The first approach is useful in gaining systemic insight and understanding of, how a system responds to an applied forcing and close to or beyond systemically important thresholds. However, this approach typically goes hand-in-hand with an increased need for data and may fall short of describing reality correctly (thus, also projections into the future). By way of comparison, the second approach is purely data-driven. Its strength lies in interpreting a data series statistically, i.e., without seeking to understand (at least, not primarily) the processes impacting and governing the system. However, this approach requires correct detrending; dealing, as a result of this, with potentially small numbers; and addressing the challenge of reflecting the statistical properties of (what time series analysts call) the data-generating process underlying the series’ residuals.

Of course, this data-generating process also leaves its mark on the deterministic part of the series. But this is simply not considered or overlooked. Take memory as an example. Testing the residual of a time series for stationarity does not imply that we capture the influence of M on the deterministic part of the time series. This can easily be shown with the help of our example (cf. Figures 2 and 1): Change the trend of yQS from nonlinear back to linear, omit periodicity but add white noise instead; and then apply a simple moving average (our memory model M)—which will not change the linear trend per se. That is, in making the time series stationary, a linear trend is subtracted from the time series without awareness being created that the trend had been impacted by memory. We recall that the situation is more challenging in the case where yQS _ wM exhibits a nonlinear trend. Correct detrending is then not/not easily possible any longer unless M is known. In general, it can be said that the two-pronged characterization of systems in terms of trend and fluctuations around that trend is widely recognized (e.g., Kantelhardt 2004), but memory is not understood as an important element connecting the two.

Hence, the question (referred to as question 1 above) arises as to how to treat memory (more specifically, its characteristics) consistently across both the deterministic and the stochastic part of a time series? This question needs to be addressed by applying a deterministic-stochastic approach, initially under controlled (i.e., lab) conditions. This approach would contribute to filling the aforementioned gap in science and can be considered as falling in between process-based modeling and time series analysis.

#### 2.2.1 From an accurate and precise world

When we speak of memory in a deterministic context, we refer to the momentum of a system over “some” historical time range. In the physical world of Newtonian systems, momentum indicates a system’s inertia. If the momentum is not constant, this indicates how much the system deviates from an inert state due to an external force. Likewise, systems with memory also exhibit inertia, called persistence below. (For inertia in the climate system, refer to, e.g., IPCC 2001: SPM: Q5.)

#### 2.2.2 …to an accurate and imprecise world

There are different approaches to capturing memory. We capture memory generically with the help of three characteristics: its temporal extent, and both its weight and quality over time (see Sections 5 and 6). The extent of memory quantifies how many historical data directly influence the current data point. The weight of memory describes the strength of this influence (fading of memory), while the quality of memory steers how well we know the latter (blurring of memory). Capturing fading and blurring of memory in combination is novel.

Memory allows us to refer to how strongly a system’s past can influence its near-term future. This is by virtue of persistence, the counterpart to memory which, just like momentum, is not to be confused with prediction. The question of interest to us (referred to as question 2 above) is how well we need to know the various characteristics of memory (e.g., the ones mentioned above) in order to delineate a system’s near-term future, which we do by means of (what we call) the system’s explainable outreach [EO] Or, put differently, how much system understanding do we need to have and to inject into the data analysis process to enable the discrimination of memory characteristics and the quantification of the EO? We have reasons to be optimistic that the system’s EO can be derived under both incomplete knowledge of memory and imperfect understanding of how the system is forced. This we illustrate in Section 6.

It appears striking that it is reality—here we refer to the need to reduce GHG emissions instantaneously—which underscores the need to research these two questions: Being ignorant of memory and persistence, we may underestimate, even considerably, the “inertia” with which global GHG emissions will continue on their ever-increasing path beyond today and thus overestimate the amount of reductions that we can achieve in the near future.

## 3 Literature review

Prior to approaching memory and persistence, it is useful to start by decomposing a time series X(t) = XTrend(t) + S(t)XNoise into a deterministic component XTrend(t), accounting for all systematic or deterministic processes, and a stochastic component S(t)XNoise, consisting of stationary noise XNoise(t) and a scaling function S(t) to describe (e.g., climate) variability (Mudelsee 2014: Section 1). In proceeding, we simplify further—it is here where we deviate from time series analysis—by assuming that memory acts like a linear filter allowing the impact of memory on the two components to be evaluated separately; likewise, persistence as a consequential feature of memory, resulting from the system’s dynamics (e.g., when its current state is directly influenced by previous states) and/or from the structural stochastic dependencies in the observed process.

However, in capturing the deterministic component, memory effects are typically not the main concern. This component is usually described without identifying memory, e.g., by means of suitable regression methods (e.g., linear/polynomial or non-parametric), or through an equation which is supposed to reflect the system’s dynamics and whose parameters are estimated from the data (so-called data assimilation techniques, e.g., Abarbanel 2013). It is usually in models where memory is captured explicitly, e.g., by introducing time delays in response to an applied forcing. Common to all approaches is that they consider the stochastic component as white noise, thereby ignoring its structural dependencies.

Nonetheless, mathematical techniques do exist to deal with memory without being associated with this term explicitly, depending on how memory is realized. For instance, in our example, memory is realized by means of a simple moving average (as a natural first choice), which can also be understood as convolution of a time series containing no memory. Another example is the empirical mode decomposition [EMD] method (Wu et al. 2007), which allows a skillful detrending by perceiving any trend in the data as an intrinsic property of the data but which, to our knowledge, has not yet been applied with the focus on memory.

And yet, it is probably in the context of analyzing structural dependencies in the stochastic component where the terms “memory” and “persistence” most often appear and are widely discussed. However, these terms are not strictly/formally defined as they are regarded as statistical properties resulting from the structural dependencies of time series. As a consequence, this leaves room for interpretation, and scientific communities may understand these terms differently; they may also apply different methods to analyze them. Table 2 gives an overview of the terminology used by various scientific communities when they refer to memory and persistence; and how they interpret the two.
Table 2

How memory and persistence are understood and interpreted by various scientific communities

Field

Terminology

Interpretation

Literature

Climate analysis

Memory, dependence (distinguishing between short-term/short-range and long-term/long-range)

Rate of decay of the autocorrelation function (considered geometrically bounded; but also with exponential, power-rate, or hyperbolic decay)

Caballero et al. (2002), Palma (2007), Franzke (2010), Mudelsee (2014), Lüdecke et al. (2013), Barros et al. (2016), Belbute and Pereira (2017)

Also persistence

Long-range memory (also checked by spectral or fluctuation analysis)

Economy and finance

Serial dependence, serial correlation, memory, dependence

Statistical dependence in terms of the correlation structure with lags (mostly long memory, i.e., with long lags)

Lo (1991), Chow et al. (1995), Barkoulas and Baum (1996), Dajcman (2012), Hansen and Lunde (2014)

Also persistence

Positive autocorrelation

Geophysics and physics

Persistence, dependence, also memory (mostly long-term)

Correlation structure in terms of Hurst exponent or power spectral density; but also system dynamics expressed by regularities and repeated patterns

Majunmdar and Dhar (2001), Kantelhardt et al. (2006), Lennartz and Bunde (2009, 2011)

The performance of methods to capture memory in the stochastic component (e.g., by means of ARIMA and ARFIMA models) relies on how nonlinear nonstationary data are (pre-) processed, i.e., on how the deterministic component (trend, seasonality, cycles, etc.) is estimated and removed (if need be) in order to obtain, ideally, a stationary series of residuals. In this approach, memory is associated with the structural dependencies in the series’ residuals and not with the deterministic component, which serves only as a means to obtain a (simplified) statistical model to describe the residuals.

However, capturing the deterministic component unskillfully/incorrectly results in artificial structural dependencies in the residuals—which brings us back to our example and the quest for a coordinated approach to treat memory consistently for both the deterministic and the stochastic component. To our knowledge, such a method is not available.

In pursuing a coordinated approach, a number of additional and well-known problems must be expected. For instance, the deterministic component may be overly complex, so that one may be inclined to prefer describing it stochastically. Or the time series may not be long enough to treat long memory unambiguously, causing an apparent trend and thus a dichotomy between the deterministic and stochastic component. This dichotomy is known to several scientific disciplines, e.g., in econometrics, hydrology, and climatology (Mudelsee 2014: Section 4.4). It is (also) for these reasons that we strongly advocate for an approach under controlled lab conditions initially, which would allows us to explore from when on these and other problems begin to become relevant, or whether these problems can be overcome by simple but effective means, e.g., by swapping ignorance for (greater) uncertainty. (This we show in Sections 5 and 6.)

## 4 Summary of our current understanding of memory, persistence, and explainable outreach

Before proceeding with a numerical experiment in Sections 5 and 6, we summarize our current understanding of memory, persistence, and explainable outreach below:
1. 1.

We consider memory as an intrinsic property of a system, retrospective in nature; and persistence as a consequential (and observable) feature of memory, prospective in nature. Persistence is understood as reflecting the tendency of a system to preserve a current value or state and depends on the system’s memory which, in turn, reflects how many historical values or states directly influence the current one. The nature of this influence can range from purely deterministic to purely stochastic.

2. 2.

Deriving the EO of a data series should not be confused with prediction (nor with perfect forecasting).

3. 3.

Figure 3 illustrates the idea of using EOs as a reference for prognostic modelers and decision-makers. We visualize the EO as an uncertainty wedge, which is derived from the historical data of a time series only (applying a mode of learning and testing) and then shifted to its end (= today). Shifting the EO to the end of the series’ historical data has to happen untested and would therefore not be permitted. However, the only reason this forward shift is still done is to provide a bridge into the immediate future, thus a reference for prognostic modelers and decision-makers. Prognostic scenarios falling outside (above or below) the EO, as well as scenarios falling within, but eventually extending beyond the EO, are no longer in accordance with the series’ past—allowing a decision-maker to inquire about the assumptions made in constructing a forward-looking scenario and to interpret these in terms of how effective planned measures (e.g., emission reductions) need to be and/or how long the effectiveness of these measures remains uncertain. Shifting the EO to today requires a conservative systems view which ensures that the system is not exposed to surprises it had not experienced before.

4. 4.

Deriving the EO of a data series must not be confused with signal detection. Signal detection requires the time to be determined at which changes in the data series outstrip uncertainty—which is not done here. Take, e.g., the scenario falling above the EO in the figure. The time at which its signal steps out of the uncertainty wedge does not coincide with the temporal extent of the EO.

5. 5.

Assuming persistence to be an observable of memory, Fig. 3 would allow persistence to be quantified (not defined). Given its directional positioning, the red-shaded EO in the figure may be described with the help of two parameters, its extent L and its aperture A at the end. We would then say that a data series with a long and narrow EO (the ratio L/A would be great) exhibits a greater persistence than a data series with a short and wide EO (the ratio L/A would be small).

## 5 Methodology

Our numerical experiment simplifies and extends our example presented in Section 2.1 (trend: preserved; periodicity: omitted; noise: introduced). Mathematical details are mentioned to the extent necessary to keep systemic insight in the focus and to cast a glance far ahead by illustrating one way to capture memory, and also to understand how persistence plays out and how an EO can be derived. We discuss the numerical experiment intensively in Section 7 with respect to how consequential it is, where it underperforms, and the questions it provokes. However, the experiment does not exhibit fundamental shortfalls. It does not restrict generalization, while allowing the important research issues which we will be facing in deriving the EO of a data series to be spotted.

The experiment is geared toward making the EO concept applicable. Figure 4 visualizes the different “worlds of knowledge” with which we are confronted in the experiment. Some of its features are excessively exaggerated to better understand how memory and persistence perform even under unfavorable conditions, as, for instance, under a forcing which is weak and a memory whose extent is short in relation to the noise which is superimposed. It is such conditions which may make it necessary to address ignorance by simple but effective means, e.g., greater uncertainty.

The numerical experiment is composed of two steps. In the first step, we obliterate the knowledge of our control, a second-order polynomial, by applying a high level of noise. In the second step, we offset this obliteration back in time by introducing memory in terms of extent, weight, and quality. This allows the reconstruction, at least partially, of what had been obliterated before. The qualitative and quantitative characteristics of this incomplete reconstruction remain to be investigated against a reference. The intention behind this step-wise procedure is to develop an understanding of how strongly memory plays out and leads to persistence. (Section 6 visualizes this process graphically and also offers a way of deriving an EO.)

We work with four functions dependent on t (with t = 1, ..., 35; sufficiently long for illustration purposes) which can be interpreted as time series with time t measured in years. The functions can be understood as reflecting four observers [O] who perceive an accurate world differently: precisely (O1, O2) or imprecisely (O3, O4); and with perfect knowledge (O1); and limited (O2, O4) or no memory (O3) (see also upper half of Fig. 4). For the sake of simplicity, we drop physical units and proceed dimensionlessly.

All four observers have complete knowledge of their worlds (i.e., t extending from 1 to 35). We introduce two additional observers (O5, O6) later when we split the time series into past (t from 1 to 7) and future (t from 8 to 35). Their knowledge is incomplete because they see the historical part of the time series only (see also lower half of Fig. 4).

### 5.1 The world of observer O1 is described by

yQuad: O1’s observations are accurate and precise and can be perfectly described by a second-order polynomial, serving as both forcing and control. Its coefficients are chosen such that its initial part exhibits a quasi-linear behavior:
$${y}_{Quad}(t)={a}_0+{a}_1t+{a}_2{t}^2$$
(5.1)
here with a0 = 1, a1 =  − 0.025, and a2 = 0.0025.

### 5.2 The world of observer O2 is described by

yQuad _ wM: yQuad with memory [M]. M is chosen arbitrarily (7 years here; justified below) but making sure that it is shorter than the quasi-linear range of yQuad (thus adding to the unfavorableness of conditions). Each value of yQuad _ wM is constructed as sum over the seven last values of yQuad (including today), the weights of which decrease exponentially back in time:
$${y}_{Quad\_ wM}\left({t}_k\right)=\sum \limits_{j=0}^6{e}^{-{c}_j}{y}_{Quad}\left({t}_{k-j}\right)$$
(5.2)
for tk = k (k = 1, ..., 35) and yQuad = 0 for tk − j =  − 5, ..., 0 (k − j =  − 5, ..., 0); and with $${e}^{-{c}_j}$$ steering the weight of memory. The exponential weighting is determined such that its value 6 years back in time (excluding today) is only 0.05, which reflects our cutoff level (extent of memory). That is, only 5% of a 6-year-old yQuad value contributes to constructing the yQuad _ wM value of today (c = ln 0.05/(−6) = 0.50). The weighting stays constant during the construction of yQuad _ wM and is not yet normalized (which we leave for later).

The notion of memory in connection with yQuad _ wM may not appear straightforward because, ideally, yQuad requires the values of only three points (years) to be entirely determined for all time, all the way from the beginning to the end. On the other hand, we use a memory extent of 7 years when we construct yQuad _ wM with the help of yQuad. Thus, it may be argued that a finite memory becomes meaningless because each individual point of yQuad _ wM carries “full memory.” However, the situation changes if yQuad _ wM is perceived as the extreme outcome of a thought experiment in which the noise surrounding each point of yQuad _ wM decreases to zero eventually.

### 5.3 The world of observer O3 is described by

YQwN: yQuad with noise. YQwN is derived by practically “obliterating” the second-order polynomial character of yQuad by means of great noise (allowing the extent of memory to be chosen freely for the time being), here expressed in relative terms:
$${Y}_{QwN}(t)=\left({a}_0+{a}_1t+{a}_2{t}^2\right)\;\left(1+N\;u\right)={y}_{Quad}(t)\;\left(1+N\;u\right)$$
(5.3)
where N is a scaling factor and the values uk are taken randomly from the u (standard normal) distribution. The equation describes a parabola with a noise component of N × 100% on top of yQuad.

In general, we deal with noise in the order of N ≈ 0.10 (that is, N× 100 %  ≈ 10%). Here, we increase N by one order, namely to N = 3.0 (that is, N × 100 %  = 300%), which may result in YQwN as a whole being perceived as random noise with some directional drift, if at all, rather than a signal that is clearly visible, albeit superimposed by noise.

### 5.4 The world of observer O4 is described by

YQwN _ wM: YQwN with M (7 years). YQwN _ wM is given by
$${Y}_{QwN\_ wM}\left({t}_k\right)=\sum \limits_{j=0}^6{e}^{-{c}_j}{y}_{Quad}\left({t}_{k-j}\right)\;\left[1+\left(1-D{e}^{-{d}_j}\right)N\;{u}_{k-j}\right]$$
(5.4)
with $$1-D{e}^{-{d}_j}$$ steering the quality of memory. This term is determined such that it allows only 0.05 parts (5%) of random noise for today, meaning that our memory is fairly precise; while it allows 0.95 parts (95%) of random noise when our memory becomes as old as 6 years (excluding today), meaning that our memory is highly imprecise (D = 0.95 and d = ln(0.05/0.95)/(−6) = 0.49). Or, if interpreted systemically in a furnace-GHG emissions context, the contribution of old furnaces to today’s emissions is not only smaller than that of newer furnaces, but their contribution is also less well known. The quality weighting stays constant during the construction of YQwN _ wM and can easily be refined.

To summarize, in introducing memory, we make use of three characteristics: its temporal extent (here dealt with by way of “insightful decision”), and both its weight and quality over time. We show in Section 6 that memory can, but need not, allow partial reconstruction of what had been previously obliterated.

## 6 Results

Equations (5.1)–(5.4) allow multiple experiments. A new experiment is launched with a new set of uk taken randomly from the standard normal distribution, while all other parameters are kept constant. Each experiment consists of two parts: (I) construction and graphical visualization of yQuad, yQuad _ wM, YQwN, and YQwN _ wM; and (II) linear regression of the first seven points of YQwN _ wM. The deeper understanding of part II is (1) that we now split the world with respect to time into two parts, past (t = 1, ..., 7) and future (t = 8, ..., 35); making, in particular, the step from observer O4 who has complete knowledge of his/her world—the world which we ultimately experience and have to deal with—to observers (O5 and O6; see also lower half of Fig. 4) who have incomplete knowledge of that world, namely, of its historical part only (7 years; in accordance with the extent of memory); and (2) that these observers can perceive the historical part of the “O4 world” only by way of linear regression, at best.

### 6.1 Part I: construction and graphical visualization of yQuad, yQuad _ wM, YQwN, and YQwN _ wM

Figure 5a and b show the graphical visualization of an experiment. Fig. 5a shows yQuad (orange), yQuad _ wM (black), YQwN (blue), and YQwN _ wM (red), while Fig. 5b shows only YQwN (blue) and YQwN _ wM (red). Dashed lines indicate second-order regressions and their coefficients of determination (R2) which were determined using Excel. The purpose of showing the second-order regressions of yQuad, yQuad _ wM, and YQwN in Fig. 5a and of YQwN _ wM in Fig. 5b, along with their R2 values, is to facilitate understanding. Knowing that our control is a second-order polynomial, these regressions and their R2 values allow the obliteration of yQuad to be followed by its incomplete reconstruction thereafter.
The experiment is very insightful because it is not (yet) as successful as we wish it to be. As expected, the application of great noise obliterates yQuad. The blue points (YQwN) do not seem to follow a clear trend. Still, if one wanted to assign a second-order regression to these points just for the sake of it, the regression would exhibit (here) a concave curvature—which would be opposite to the convex curvature of yQuad—and a low R2 value of 0.005 (cf. also Table 3), confirming the complete obliteration of yQuad.
Table 3

Supplementing Figs. 5 and 6: Compilation of regression parameters and coefficients of correlation, the latter between: (1) yQuad and yQuad _ wM (invariant); (2) yQuadand YQwN (variable); (3) YQwN _ wM and YQwN _ wM − 7yr (variable) with YQwN _ wM − 7yr being identical to YQwN _ wM but shifted backward in time (year 8 becomes year 1, year 9 year 2, and so on, while dropping the first 7 years of YQwN _ wM); and (4) yQuad and YQwN _ wM (variable). The first correlation coefficient indicates that limiting only the extent of memory back in time is not sufficient to overcome the “full memory” of yQuad. Correlation coefficients 2 and 3 seem to confirm that applying a high level of noise completely obliterates the second-order polynomial character of yQuad and that memory does not extend beyond 7 years. Finally, correlation coefficient 4 seems to confirm that memory (that is, YQwN _ wM) nullifies much of the obliteration brought about by YQwN

 Polynomial/regression for a 2 a 1 a 0 R 2 y Quad 0.0025 − 0.0250 1.0000 1.000 y Quad _  wM 0.0044 0.0016 1.8079 0.9857 Y QwN − 0.0037 0.1564 − 0.7961 0.005 Y QwN _  wM 0.0023 0.0496 0.9113 0.5138 Y Lin, 7 yr – − 0.2434 2.1639 0.5142 Y Lin _   exp , 7 yr – − 0.3966 2.9095 0.9049 Coefficient of correlation between 1) yQuad & yQuad _ wM Influence of memory (w/o noise) 0.99 2) yQuad & YQwN Influence of noise (obliteration) 0.02 3) YQwN _ wM & YQwN _ wM − 7yr Influence of memory after 7 years (w noise) 0.06 4) yQuad & YQwN _ wM Influence of memory in the presence of noise (reconstruction) 0.71

YQwN _ wM overcomes much of that obliteration, bringing the curvature back to convex and increasing the R2 value substantially, here to greater than 0.5 (cf. also Table 3).

### 6.2 Part II: linear regression of the first seven points of yQwN _ wM

Figure 6 expands Fig. 5 (see also lower half of Fig. 4). Figure 6a shows a linear regression called R1_Y_QwN_wM_hist_uw (in the figure) and YLin, 7yr (in Table 3) for the first seven points of YQwN _ wM where we assume that it is only these seven points of YQwN _ wM that an observer (O5 hereafter) knows. It is this assumption—knowing the extent of memory—that requires discussion. In deriving the linear regression, the seven points are weighted equally (unit weighting [uw] back in time), resulting in a low R2 value of about 0.51 but, more importantly, in the wrong direction (downward). Note that the overall direction of YQwN _ wM is upward (cf. also Table 3).

By way of contrast, in deriving the linear regression in Fig. 6b the first seven points are weighted exponentially [ew] over time. Here, we assume that an observer (O6 hereafter) like observer O5, knows not only the first seven points (i.e., the extent of memory) of YQwN _ wM but also the weight of memory over time—an assumption that also requires discussion. The exponential weighting (the same as that underlying YQwN _ wM) results in a more confident linear regression called R1_Y_QwN_wM_hist_ew (in the figure) and YLin _  exp , 7yr (in Table 3) with an R2 value of about 0.90 and an even greater downward trend (− 0.40 versus − 0.24; cf. Table 3). Figure 6b also shows the confidence bands belonging to YLin _  exp , 7yr for the first 7 years [inConf] and beyond, the latter by means of the out-of-sample [outConf] continuation of the seven-year confidence band. As can be seen, YQwN _ wM crosses the 7-year confidence band from below to above and falls above the out-of-sample confidence band.

The reason for selecting this (unsuccessful), rather than another (successful) experimental realization is to prepare for the question as to whether we can make use of repeated regression analyses to capture the immediate future of YQwN _ wM? This will cause the experimental outlook to change from unsuccessful to promising.

We now repeat the above experiment multiple times (see also lower half of Fig. 4). Table 4 summarizes the results of 100 consecutive experiments where YQwN _ wM falls within the (in-sample and out-of-sample) confidence band of YLin _  exp , 7yr for a time that corresponds to two times the extent of memory (= 14.5 years in the numerical setup). These experimental realizations are denoted by “1: YQwN _ wM in.” All other experiments without exception by “0: YQwN _ wM out.” This repetition indicates how often shifting an EO with an extent of 7 years to today (here: year 7) is justified, using “one times the extent of memory” as reference for both the shift and the extent of the EO. Table 4 indicates that this is the case in 42% of all experiments.
Table 4

Summary of results of 100 consecutive experiments where YQwN _ wM falls within the (in-sample and out-of-sample) confidence bands of YLin _  exp , 7yr for a time that corresponds to two times the extent of memory (= 14.5 years in the experiment). The individual experimental realizations are denoted by “1: YQwN _ wM in”; all others by “0: YQwN _ wM out”; indicating how often it is justified to shift the EO to today (here: year 7)

Grouping of experiments

Coefficient of determination for

Coefficient of correlation for

No. of exp.

Y Lin _   exp , 7 yr

Y Lin, 7 yr

Y QwN _  wM

YQwN _ wM & YQwN _ wM − 7yr

No grouping

0.58 ± 0.32

0.50 ± 0.31

0.53 ± 0.22

0.10 ± 0.23

0.19 ± 0.35

0.62 ± 0.26

100

$$0:{Y}_{QwN\_ wM}\ out$$

0.72 ± 0.26

0.60 ± 0.30

0.55 ± 0.22

0.15 ± 0.22

0.20 ± 0.37

0.65 ± 0.27

58

$$1:{Y}_{QwN\_ wM}\ in$$

0.38 ± 0.30

0.35 ± 0.27

0.50 ± 0.21

0.03 ± 0.22

0.17 ± 0.32

0.59 ± 0.25

42

0 : YQwN _ wM out and R2 of YLin _  exp , 7yr > 0.30

0.77 ± 0.19

0.64 ± 0.28

0.56 ± 0.21

0.15 ± 0.21

0.19 ± 0.37

0.67 ± 0.24

53

1 : YQwN _ wM in and R2 of YLin _ exp < 0.70

0.27 ± 0.22

0.25 ± 0.19

0.50 ± 0.23

0.04 ± 0.23

0.19 ± 0.34

0.61 ± 0.25

34

0 : YQwN _ wM out and R2 of YLin _ exp , 7yr > 0.50

0.82 ± 0.13

0.68 ± 0.26

0.54 ± 0.21

0.13 ± 0.20

0.17 ± 0.36

0.66 ± 0.25

48

1 : YQwN _ wM in and R2 of YLin _ exp , 7yr < 0.50

0.18 ± 0.11

0.19 ± 0.15

0.50 ± 0.22

0.03 ± 0.23

0.18 ± 0.33

0.61 ± 0.26

27

But we can learn more than just success and failure from the statistics. Table 4 also suggests that the R2 value of YLin _  exp , 7yr, as well as that of YLin, 7yr, seems to be the right leverage point to differentiate “0-experiments” from “1-experiments.” In the numerical setup given here, a grouping of experiments depending on whether the R2 value of YLin _  exp , 7yr is greater or smaller than 0.50 seems to be a success. This is shown in the fact that the R2 values of YLin _  exp , 7yr, as well as those of YLin, 7yr, do not overlap:
$${\displaystyle \begin{array}{lll}{Y}_{Lin\_\exp, 7 yr}& {R}^2>0.50& 0.82\pm 0.13=\left[0.69,0.95\right]\\ {}& {R}^2<0.50& 0.18\pm 0.11=\left[0.07,0.29\right]\\ {}{Y}_{Lin,7 yr}& {R}^2& 0.68\pm 0.26=\left[0.42,0.94\right]\\ {}& {R}^2& 0.19\pm 0.15=\left[0.04,0.34\right]\end{array}}$$

In addition, Table 4 indicates that the obliteration of yQuad appears to be slightly greater on average for “1-experiments” than for “0-experiments” (cf. coefficients of correlation between yQuad and YQwN: 0.03 ± 0.23 versus 0.13 ± 0.20). However, it seems that “1-experiments” perform, on average, slightly better from a reconstruction perspective than “0-experiments.” In fact, they almost catch up (cf. coefficients of correlation between yQuad and YQwN _ wM: 0.61 ± 0.26 versus 0.66 ± 0.25).

In a nutshell, Table 4 confirms what common sense tells us: a world perceived too precisely is difficult to “project” even into the immediate future. Conversely, it is much easier to achieve if we are confronted with a highly imprecise world (forcing us to acknowledge our ignorance). This insight is not counter-intuitive but reflects prudent decision-making: In planning ahead, one is always well-advised to factor in some (not necessarily all) adverse effects that one can think of and that can easily occur. It is exactly this insight which tells us that (1) we should avoid following the footsteps of perfect forecasting to derive the EO of a data series (cf. Section 4) and (2) we can even derive a robust EO if we resist the temptation to describe the world we perceive too precisely.

## 7 Discussion

Our perspective piece in general and our numerical experiment in particular touch on a number of is sues the most pertinent of which we discuss below where we state what we know or believe we know and what we do not yet know.

Our approach of capturing memory is not restricted to identifying memory in time series of GHG emissions, in our case between emissions from newly installed furnaces (world without memory) and total emissions (world with memory). However, GHG emissions serve only as a case in point. Any two time series which are causally linked by memory (“memory-linked”) can be used instead, e.g., global GHG emissions and atmospheric concentrations or global mean surface temperature change.

We argue for and prefer using (at least, initially) synthetic data over real-world data. Synthetic data allow us to test the limits of deriving memory and the robustness of EOs by varying trend, periodicity, and stochastic remainder widely and in any combination.

In the numerical experiment in Sections 5 and 6, we assumed knowleged of the extent of memory in deriving a system’s EO. However, this piece of a priori information is not critical. According to our current understanding, the temporal extent of memory appears to be more important than both its weight (fading of memory) and quality (blurring of memory) back in time. From preliminary autocorrelation experiments, it appears that the extent of memory can be approximated even under unfavorable conditions. Our numerical experiment furthermore indicates that deriving an EO under unfavorable conditions is still possible too. However, the question of what constitutes favorable conditions needs to be explored thoroughly.

We assumed that we were able to repeat our numerical experiment multiple times, that is, that we would have available multiple realizations of a time series. But this does not disqualify us from using real-world data series which represent single realizations only. If applied in a piecemeal (learning and testing) mode, our method of deriving an EO is also applicable to analyzing real data series. However, the length of the data series relative to the extent of memory will affect the robustness of the EO. The main purpose of repeating experiments multiple times is to understand how long a data series needs to be to ensure robustness. Although we find that a system’s EO can be derived even under unfavorable conditions, we it is precisely for these reasons that we consider the time series used in the numerical experiment to be too short to determine the parameters of the underlying (here second-order) trend with high quality. That is, the insight that an EO can be derived under unfavorable conditions does not release us from an in-depth exploration of the issue of length of a data series vis-à-vis complexity of trend spotting.

But we see a trade-off evolving between the ignorance of understanding the past, e.g., by way of diagnostic modeling, and the robustness of the EO. Overestimating the precision of historical data may lead to an overly complex model for deriving an EO. This is problematic in two ways: First, it is known that over-parametrized diagnostic models are not necessarily optimal for use under prognostic conditions. Second, and of importance here, historical “surprises” (e.g., large outliers of random nature) are not treated as such; that is, the model tries to mimic them with the consequence that surprises are not expected to occur in the future. Adopting larger error margins for historical data while keeping model complexity to a minimum leads to a more stable trend and an EO sufficiently wide to allow for future surprises.

## 8 Summary and outlook

We opted for a perspective piece to discuss memory, persistence, and explainable outreach of forced systems, with GHG emissions into the atmosphere serving as our case in point. In the light of the continued increase in emissions globally vis-à-vis the reductions in them are required without further delay until 2050 and beyond, we conjecture that, being ignorant of memory and persistence, we may underestimate the “inertia” with which global GHG emissions will continue on their increasing path beyond today, which thus would also lead to overestimation of the amount of the reduction able to be achieved in the future. This issue is at the heart of mitigation and adaptation. For a practitioner, it translates to the problem of how persistently an emissions system behaves when subjected to a specified mitigation measure and what emissions level to adapt to for precautionary reasons in the presence of uncertainty.

We consider memory to be an intrinsic property of a system, retrospective in nature, and persistence to be a consequential (i.e., observable) feature of memory, prospective in nature and reflecting the tendency of a system to preserve a current state (including trend). Persistence depends on the system’s memory which, in turn, reflects how many historical states directly influence the current state. The nature of this influence can range from purely deterministic to purely stochastic.

Different approaches exist to capture memory. We capture memory generically with the help of three characteristics: its temporal extent and both its weight and quality over time. The extent of memory quantifies how many historical data directly influence the current data point. The weight of memory describes the strength of this influence (fading of memory), while the quality of memory steers how well we know the latter (blurring of memory). Capturing fading and blurring of memory in combination is novel.

In a numerical experiment with the focus on systemic insight, we cast a glance far ahead by illustrating one way to capture memory, and to understand how persistence plays out and how an EO can be derived even under unfavorable conditions. We discuss the experiment intensively with respect to how consequential it is, where it underperforms, and the research questions it provokes. However, the experiment does not exhibit fundamental shortfalls and does not restrict generalization.

We look into the following two questions:
1. 1.

Do we have the right science in place to understand and treat memory appropriately?

2. 2.

Given that memory links a system’s past with its near-term future and in the belief that knowing the system’s explainable outreach is of use for prognostic modelers and decision-makers, how well do we need to know the various characteristics of memory (e.g., those mentioned above), and can we differentiate and specify them by way of diagnostic data-processing alone, in order to delineate the system’s EO? Or, in other words, how much systems understanding do we need to have and to inject into the data-analysis process in order to enable such differentiation?

With regard to question 1, we demonstrate that treating memory appropriately requires its characteristics to be treated consistently for both the deterministic and the stochastic part of a time series, leading us onto new grounds to be mapped and explored, and multiple deterministic-stochastic approaches to be tested under controlled (i.e., lab) conditions initially and to be discussed widely—which is why we opt for a perspective piece before anything else. Any such approach falls in-between process-based modeling and time series analysis and contributes to closing the gap between the two.

With regard to question 2, we note that we study memory and persistence in a forward mode, and not yet in a backward mode; this means that we can contribute to this question only by way of initial insight. To this end, we can mention that we have reasons for optimism that the system’s EO can be derived under both incomplete knowledge of memory and imperfect understanding of how the system is forced. But we learn (1) that we should avoid following the footsteps of perfect forecasting to derive the EO of a data series; and (2) that we can derive a robust EO even if we resist the temptation to describe the world we perceive too precisely.

Determining the temporal extent of memory in the presence of (possibly great) noise is an important step, if not the most important. But we are interested in more, namely, how memory evolves back in time in terms of both weight and quality. That is, we are left with the challenge of acquiring a deeper systemic understanding to substantiate how memory plays out over time (exponentially, as in our numerical example, or otherwise). It remains to be seen whether meeting that challenge is feasible.

Although the prime intention of our perspective piece is to study memory, persistence, and explainable outreach of forced systems and, thus, to expand on the usefulness of GHG emission inventories, our insights indicate the high chance of our conjecture proving true: being ignorant of memory and persistence, we underestimate, probably considerably, the “inertia” with which global GHG emissions will continue on their historical path beyond today and thus we overestimate the amount of reductions that we might achieve in the future.

## Notes

### Acknowledgments

Open access funding provided by International Institute for Applied Systems Analysis (IIASA). The authors are grateful to IIASA and the Earth System Sciences Research Program of the Austrian Academy of Sciences to financing this research.

### Acronyms and nomenclature

Aaperture (of the EO)

ARIMAautoregressive integrated moving average

ARFIMAautoregressive fractionally-integrated moving average

EMDempirical mode decomposition

EOexplainable outreach

ewexponential weighting

GHGgreenhouse gas

GOSATGreenhouse Gases Observing Satellite

IIASAInternational Institute for Applied Systems Analysis

inConfin-sample confidence band

IPCCIntergovernmental Panel on Climate Change

Llength (extent of EO)

lolower

Mmemory

O observer

OCO-2Orbiting Carbon Observatory-2

outConfout-of-sample confidence band

Rcoefficient of determination

SPMsummary for policymakers

ttime

UNFCCCUnited Nations Framework Convention on Climate Change

upupper

uwunit weighting

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