## Abstract

Phase analyses of the annual cycle of monthly temperature time series that date back to the eighteenth century show trending behavior that has been difficult to interpret. Negative trends in the estimated phase have been identified with precession of Earth’s axis of rotation, but the implied later onset of seasons is at odds with recent satellite measurements and with the phenological record. Positive trends in the phase and the occurrence of trends of both signs in temperature time series from geographically nearby locations have remained mysterious. This paper shows that there is a mathematical equivalence between trends in phases and seasonally differing warming trends, in particular more intense warming in winters than in summers. Using temperature time series from 16 Northern Hemispheric locations reaching back to the eighteenth century and a statistical model that can estimate the seasonal warming trends, the authors reject the hypothesis that the timing of the seasons in these locations is jointly driven by precession.

## 1. Introduction

A salient feature of long instrumental temperature records is trends in the estimates of the phases of the annual cycle. These trends have widely differing slopes of either sign depending on the location, and many geographically nearby locations have opposite signs. Negative trends in the phase correspond to phase delays (later onset of seasons) and positive trends correspond to phase advances (earlier onset of seasons). The implications of differing trend slopes for the timing of the seasons have been difficult to interpret (Thomson 1995; Mann and Park 1996; Stine et al. 2009; Serreze et al. 2009; Screen and Simmonds 2010). Thomson (1995) explained negative trends with precession of Earth’s axis of rotation, most notably the one in the Central England Temperature (CET) series, since the trend slope is numerically in the vicinity of the precession constant in this particular case. Thomson therefore concluded that “… in much of the world the dominant frequency of the seasons is one cycle per anomalistic year […], not one cycle per tropical year …” (Thomson 1995, p. 59).

The proposition of precession as an explanation of the observed trends in phase has been controversially discussed in, for example, White et al. (1996), who point out that the finding of a phase trend slope close to the precession constant of −50.3 arcsec yr^{−1} (expressed as −14.2 days per millennium in this paper) is not sufficient to conclude that one cycle per anomalistic year is the dominant frequency. Karl et al. (1996) demonstrate that daily temperature data from the United States cannot confirm the finding in Thomson (1995). Using satellite data, Stine et al. (2009) report a phase advance, not delay, over land. Many phenological studies report an earlier onset of spring; see, for example Schwartz and Reiter (2000), Parmesan (2007), and Thompson and Clark (2008).

Other explanations for the phase trends have been proposed, for example loss of sea ice, which increases the heat capacity of the ocean and dampens the surface temperature response (Manabe and Stouffer 1980; Manabe et al. 1992; Mann and Park 1996; Stroeve et al. 2007; Dwyer et al. 2012); decreases in soil moisture, reducing the thermal mass of land (Seager et al. 2007; Sheffield and Wood 2008; Stine et al. 2009; Dai 2011); changes in atmospheric shortwave absorptivity due to aerosols (Stine et al. 2009); and changes in the relative influence of land and ocean at different locations, in particular the Pacific–North American pattern and the northern annular mode (Stine and Huybers 2012).

In this paper we show that trends in the phase of the annual cycle are mathematically equivalent to seasonally differential warming (i.e., trends in temperature that undergo slope changes from month to month). Using instrumental monthly temperature time series from 16 Northern Hemispheric locations reaching back to the eighteenth century, we show that the widely differing slopes of the trends in phase can equivalently be understood as differences in seasonal temperature trends. Therefore, instead of interpreting the observed trends in phase as indicative of a change in the astronomical timing of the seasons with respect to the Gregorian calendar, one can alternatively understand them as the result of seasonally differential warming that varies across locations.

The phenomenon of seasonally varying temperature trends is well documented in the literature (see, e.g., IPCC 2007, 2013; Balling et al. 1998; Harvey and Mills 2003; Vogelsang and Franses 2005; Cohen et al. 2012; Proietti and Hillebrand 2016). In particular, a stronger warming trend in Northern Hemispheric winter months than in summer months has been observed. Spatial differences in warming trends are described, for example, in Hansen et al. (2010), in particular in their Figs. 2 and 4.

Seasonally varying trends in temperatures can be appreciated in Fig. 1, which shows the 12-month series of temperature measurements for the CET time series 1753–2015, first described in Manley (1974) and Parker et al. (1992) and maintained by the UK Met Office Hadley Centre, and the De Bilt temperature time series 1706–2015, maintained by the Royal Netherlands Meteorological Institute (KNMI; van Engelen and Nellestijn 1995). In this paper all data are monthly, and seasonality refers to the annual cycle described by the 12 months making up the year. The black lines in Fig. 1 are least squares fits of linear-trend-plus-intercept models for each month. The estimated slopes are positive with a single exception for each series, June in the case of CET and September in the case of De Bilt. For both series, winter months show steeper trend slopes than summer months. The imposition of a single, stable linear trend on each month series ignores the acceleration in warming during the more recent part of the record and is just shown for illustration here.

To account for this acceleration and to provide a more reliable estimate of the seasonally varying temperature trends, we estimate the monthly intercepts and trends in temperature for 16 Northern Hemispheric locations using a statistical model proposed in Proietti and Hillebrand (2016). This model contains a random walk component that captures nonlinear increases in temperatures. We establish a mathematical equivalence between seasonal trends in temperature and trends in phase, and we translate the estimates of seasonal trends in temperature obtained from the model into estimates of the equivalent trends in the phase. The stochastic structure of the model allows for the construction of confidence intervals around the phase trend estimate. We test, and reject, the null hypothesis that the trends in phase in all 16 locations are jointly driven by precession.

In the next section we discuss the precession hypothesis formulated in Thomson (1995) and how it implies seasonally varying trends in month plots such as in Fig. 1. In section 3 we show that, conversely, seasonally varying trends in month plots also imply trends in phase, and we apply the model proposed in Proietti and Hillebrand (2016) to estimate these trends. In section 4, we show that for the temperature time series considered here, precession is a statistically unlikely explanation for the observed trends in the phase of the annual cycle in temperatures. The last section concludes.

## 2. The precession hypothesis

To build up some intuition for our result, we revisit Thomson’s (1995) findings, which employ complex demodulation to extract the phase series from the temperature series. Given a temperature series , complex demodulation calculates the series

where *L* is a bandwidth parameter of a linear filter , *i* is the imaginary unit, and is the fundamental frequency of interest, in the case of monthly data . To fix terms, we refer in the following to as the unfiltered complex demodulate, and to as the filtered complex demodulate. Since is a complex number, it can be written

with amplitude and phase , where Re and Im denote the real and imaginary parts of a complex number, respectively. This paper focuses on the phase series .

Figure 2 shows the estimated monthly phase series from complex demodulation of the CET and De Bilt time series with a zero-order Slepian filter with cutoff frequency . This is the same method as employed in Thomson (1995), with a similar bandwidth. For the CET series we only consider observations after the inception of the Gregorian calendar in 1752. The two phase series show qualitatively very different behavior over time, despite the geographical proximity of central England and the Netherlands. While the CET phase series displays a pronounced downward trend, the De Bilt phase series is slightly upward trending. The oscillations of the two phase series have a degree of semblance: for example, the downward movement after 1950 that was emphasized in Thomson (1995) is visible and has since reverted in both series.

Least squares fitting of a linear trend to the CET phase series yields an estimate of the slope of −18 days per millennium, a bit larger in absolute value than the precession constant of −14.2 days per millennium. The De Bilt phase series, on the other hand, shows a positive trend with a slope of 4.6 days per millennium. Table 1 (see columns 3–8) shows the estimated phase shifts from least squares fits of linear trends to the phase series for all temperature series we consider in this paper across different filter lengths for the filtered complex demodulate. Only about half of the series consistently show negative trends in phase, and the estimates differ widely.

If we assume that the CET series does contain the precession signal, what do the implied month plots look like? A simple model for such a monthly temperature process can be written as

This is a periodic process with a fundamental frequency that captures the annual cycle of the seasons, with a low frequency that captures the precession periodicity of about 26 000 years, and with a constant amplitude *A*.

Figure 3 shows the month plots implied by model (3) for a period of 263 years, corresponding to the length of the CET series after the inception of the Gregorian calendar. The constant amplitude is set to °C, corresponding to the mean amplitude of the CET series. The month plot curves shown in red are changes from year to year *k* for each month . From model (3), they are given by

where with being an index that counts years and an index that counts months, where is the Gauss bracket. Since the precession period is very long compared to the fundamental period of one year, the sinusoidal curves in (4) appear as straight lines over 263 years. As can be seen in Fig. 3, the seasonal periodicity moves from the solid sinusoid to the dashed sinusoid, the negative term in the phase translates to a later and later occurrence of the seasons in the Gregorian calendar year. The model in Eq. (3) implies that half of the slopes of the month plots are negative and half of them are positive. Intuitively, a later onset of spring and summer in the first half of the year on the Northern Hemisphere implies that temperatures decrease, whereas the later onset of fall and winter in the second half of the year implies that temperatures increase. From the intuition of Fig. 3 it is not difficult, however, to imagine an additional positive trend in temperature that would move the dashed sinusoid upward. The result would be month plots similar to those of the CET and De Bilt series in Fig. 1. A time-varying amplitude also has an influence. A decrease in amplitude warms winters and cools summers, and an increase in amplitude does the opposite.

## 3. Seasonally varying temperature trends and the phase

Figure 3 and Eq. (4) show that trending phases, for example due to precession, lead to seasonally varying trends in temperature. In this section, we show that the converse is also true: seasonally varying trends in temperature imply trending behavior in the phase. This opens up the possibility to interpret the observed trends in phase as evidence of seasonally varying sensitivities of different locations to a global warming trend, rather than concluding that temperatures in central England follow precession, while temperatures in De Bilt, in the Netherlands, do not.

Consider a temperature process of monthly observations with seasonally varying intercepts and linear trend slopes

where is the indicator function that takes value 1 for month *m*, , and zero otherwise. As in Eq. (2), let , where is an index that counts years. Then, our main result is that the phase process of annual observations for the filtered complex demodulate of the process in Eq. (5) is given by the function

where the constants and are given by

Equation (6) is derived in appendix B both for complex demodulation and for the alternative method of trigonometric regression. Trigonometric regression is used, for example, in Stine et al. (2009). Since the arc tangent function is continuous and monotonic, the slope of the phase is determined by the slope of the argument function , and the sign of the slope of the phase is given by . Whether the phase is upward or downward sloping, and to which degree, thus depends on the configuration of intercepts and trend coefficients , which differ from place to place as a consequence of local conditions.

A simple least squares estimate of monthly intercepts and trend slopes over the entire sample period as in Fig. 1 is too restrictive since it ignores the acceleration in warming in the later part of the record. In Proietti and Hillebrand (2016), we propose a model for temperature time series that addresses this issue by specifying a permanent and a transitory component. The permanent component consists of a deterministic trend and a stochastic trend component. The deterministic trend component has the form studied above, and the stochastic trend component is a random walk on which the different months load differentially. The stochastic trend captures the acceleration of warming in the later part of the record, since it is a stochastic process that is integrated of order one. It can accommodate runs of upward movements, and it is not restricted to revert to a mean. The transitory component, on the other hand, is a periodic autoregressive (AR) model and captures fluctuations of the temperature series around the trend components.

Using the symbols for the permanent and for the transitory component, the model has the following formulation:

All the disturbances and are serially and mutually uncorrelated. The selection vector serves the role of the indicator function, taking value 1 in the position corresponding to the current month and zero elsewhere, so that if the *t*th observation refers to month *m*, , , , and . The vector is characterized by the periodic property .

The permanent component consists of the seasonal intercept temperatures and the seasonal slope coefficients as they were considered in Eq. (5). In addition, allows for stochastic and nonlinear trending behavior through the random walk , which is of one order of stochastic integration, as specified by the third equation in (7). This component can therefore absorb the acceleration of warming in the later part of the record. The seasons can load on this component differentially, through the coefficients , .

The source of randomness in the stochastic trend component, , cannot account for fluctuations around the permanent component; it only describes the trending behavior within the permanent component. Therefore, we introduce a second source of randomness, , which is a white noise process that generates a periodic autoregressive series through the fourth equation in (7). This periodic autoregressive component captures the dispersion of the temperature observations around the permanent component . A motivation for the seasonal specification of this autoregressive component can be appreciated in Fig. 1: the dispersion of the temperature observations around the trend lines varies from month to month. For example, for both the CET and the De Bilt series, the variances in December, January, and February are visibly larger than those of June, July, and August. The model captures these seasonally differing variances in the parameters and .

The model has five parameter vectors: and , for a total of 60 parameters. We employ a periodic spline function to reduce each vector to five elements, for a total of 25 parameters. For the details of the periodic spline method and the maximum likelihood estimation of model (7) in general, we refer to Proietti and Hillebrand (2016).

## 4. Results

Figure 4 shows the estimated deterministic trend lines in the month plots of CET and De Bilt series in the left panels, and the estimated permanent components in the right panels. The deterministic trend lines in the month plots are the result of a maximum-likelihood fit of model (7) and not least squares fits. All monthly trends in both CET and De Bilt series are positive. Note that the pattern that winter months show steeper slopes than summer months is confirmed in our dataset. The stochastic trend components in the right panels display the dramatic increase in warming in the twentieth and twenty-first centuries, with a marked upward jump after 1980.

Table 2 reports the maximum likelihood estimates of the seasonal intercepts and the seasonal slopes for the CET and De Bilt series. Standard errors are implied by the stochastic structure of Eq. (7). With the exceptions of the summer months (May, June, and July for CET, June and July for De Bilt), the estimated slope coefficients are significantly different from zero. The results for the other temperature series are reported in appendix C.

We are now in a position to check how well Eq. (6) captures the phase series obtained by complex demodulation of the temperature series. To this end, we plug the maximum-likelihood estimates of intercepts and trend slopes from the model in Eq. (7) into Eq. (6). Figure 5 displays the resulting curves (dotted lines) together with the phase series from complex demodulation (solid lines) for CET and De Bilt. The curves implied by Eq. (6) capture the trends in the phase series from complex demodulation very well: they are qualitatively similar to the least squares lines shown in Fig. 2. The 95% confidence bands are generated by 1000 simulations and subsequent maximum-likelihood estimations of Eq. (7) using the estimated parameter vector from the data series as data-generating parameters in the simulations. The confidence intervals thus reflect the model’s explicit uncertainty in the error terms and as well as the estimation uncertainty from the finite sample.

Table 1 reports the slopes implied by Eq. (6) in days per millennium in the second column, together with the standard errors calculated from the simulations. The numerical values are reasonably close to the ones obtained from least squares fits of linear trends to the phase series obtained from complex demodulation; certainly the signs match. Five out of the 16 time series display a slope coefficient of the same order of magnitude as the precession constant of −14.2 days per millennium, and four time series contain the precession constant in a 95% confidence interval around the slope estimate.

To test the null hypothesis that the slope of the phase of the annual cycle equals the precession constant in all 16 temperature time series considered here, we conduct 16 *t* tests with a Bonferroni correction for the significance level (Holm 1979; Efron 2012). The significance level in each individual two-sided *t* test is set to 0.05/16 = 0.003125. Table 3 reports the corresponding confidence intervals.

Since nine of the confidence intervals do not contain the precession constant, we can reject the null hypothesis that the slope of the phase of the annual cycle equals the precession constant in all 16 temperature time series. It is therefore statistically highly unlikely that the slope coefficients in Table 1 are all noisy estimates of the precession constant.

The Bonferroni rule guarantees that the family-wise error rate, defined as the probability of rejecting any true null making up a multiple test, does not exceed the level *α*. A rejection according to the Bonferroni rule leads to the same decision as one following more powerful procedures such as Holm (1979) and Benjamini and Hochberg (1995). See Efron (2012) for a review.

We report the month plots, the estimated permanent components, and the phase plots for the series other than CET and De Bilt in appendix C. In appendixes D to F, we report a number of robustness checks for our result. One possibility is that the finding is influenced by the different starting and ending dates. In appendix D, we repeat the procedure used in this paper on the eight temperature time series that span the common sample period 1779–2015. Appendix D (see Table D1) reports the confidence intervals for a two-sided *t* test with Bonferroni correction . For seven of the eight locations, with the single exception of the CET series, the confidence interval does not include the precession constant, and we thus reject the null hypothesis that the phase of the annual cycle of the eight series follows precession. Also in appendix D (Figs. D1 and D2) are shown the phase plots with filtered complex demodulate, phase trend according to Eq. (6), and confidence interval obtained from 1000 simulations and estimations of the model in Eq. (7) using the parameters estimated on the 1779–2015 sample period.

Another possibility is that our result is influenced by the early instrumental warm bias reported in Böhm et al. (2010). Before the period 1850–70, thermometers in the greater Alpine region were not sufficiently shielded from sunlight by screens, and the Histalp network (http://www.zamg.ac.at/histalp/dataset/station/csv.php) provides temperature time series that are corrected for this effect. We have selected 11 locations available on Histalp for comparison in this paper: For comparison with Munich, we use the Histalp series Muenchen Stadt, for Milan, we use the Histalp series Milano-Brera, for Prague, we use the Histalp series Brno-Turany, for Vienna, both GHCN and Histalp use data from Hohe Warte, and for Budapest, we use the Histalp series Budapest-Loerinc Airport. We further consider the temperature time series from Karlsruhe, Stuttgart-Schnarrenberg, Regensburg, Hohenpeissenberg, Kremsmuenster, and Innsbruck-Universitaet, which all cover a period from the eighteenth century to 2015. Appendix E (see Table E1) reports the confidence intervals for a two-sided *t*-test with Bonferroni correction . Only 3 of the 11 confidence intervals contain the precession constant. Also, appendix figures (Figs. E1 and E2) show the phase plots with filtered complex demodulate, phase trend according to Eq. (6), and confidence interval obtained from 1000 simulations and estimations of the model in Eq. (7) using the parameters estimated on the Histalp data.

In appendix F, we study the consequences of a sample split in 1950; that is, we estimate Eq. (7) on the period before 1950 (for all series) and then separately on the period after 1950 (for the series that continue until 2015). Thomson (1995) and Stine et al. (2009) noted changes in phase trends around the middle of the twentieth century, perhaps due to anthropogenic forcing. Appendix F (see Table F1) shows the Bonferroni-corrected confidence intervals for both subperiods. For the pre-1950 period, precession as a joint driver of the phases is clearly rejected just as on the full sample. For the post-1950 period, the confidence intervals are very wide due to the small number of observations. We can still formally reject the null hypothesis that all series on the post-1950 sample are jointly driven by precession, but the rejection hinges on a single series, Vienna. On the other hand, the appendix figures (Figs. F1–F3) show that while the confidence intervals have become wider, the qualitative behavior of the phases after 1950 has undergone a shift toward more positive phase trends (i.e., phase advances), which contradict precession. Several series that exhibit negative trends on the full sample display positive trends on the post-1950 sample, such as Stockholm, Berlin, and Vienna, consistent with the evidence presented in Stine et al. (2009). We conclude from these robustness checks that our finding that precession does not jointly determine the phase trends in the 16 time series considered in this paper is confirmed.

## 5. Conclusions

We have shown that there is a mathematical equivalence between seasonally differential warming in monthly temperature data, a well-documented phenomenon, and trending behavior in the estimated phases of the annual cycle in these time series. The slope of the phase trend depends on the specific configurations of monthly mean temperatures and monthly temperature trends, which vary across locations. We have shown that the trends in the estimated phase series for long instrumental temperature records from 16 Northern Hemispheric locations are well described by the mathematical expression for the phase derived in this paper, which is a function of the intercepts and trend slopes of monthly temperature. Employing a statistical model to estimate the seasonally varying trends and intercepts, we can test and reject the null hypothesis that precession jointly drives the phase changes.

## Acknowledgments

The authors acknowledge support from CREATES, funded by the Danish National Research Foundation under Grant DNRF78. We are grateful for comments from two anonymous referees as well as from Neil Ericsson, Peter Grabarczyk, Peter Hansen, David Hendry, Soren Johansen, Steffen Lauritzen, Felix Pretis, Timo Terasvirta, and Martin Wagner, and seminar and conference participants at CREATES, the University of Copenhagen, the Technical University Dortmund, the OxMetrics User Conference 2016 at George Washington University, the International Symposium on Forecasting 2016 in Santander, the conference on Econometric Models of Climate Change 2016 in Aarhus, and the annual meeting of the Arctic Research Centre 2016 in Sandbjerg. We thank Daniel D. Morgan for proofreading the manuscript. All remaining errors are ours.

### APPENDIX A

#### Data Description

The selection of monthly temperature time series in this paper is motivated by the list in Table 13.1 of Bradley and Jones (1992). We employ only those time series that date back to the eighteenth century. Many of the series have station numbers in the Global Historical Climatology Network (GHCN); others (CET, De Bilt, and Stockholm) are homogenized series maintained by national meteorological institutes. All series were obtained through the KNMI Climate Explorer of the Royal Netherlands Meteorological Institute (climexp.knmi.nl), with the exception of the CET series (www.metoffice.gov.uk/hadobs/hadcet/data/download.html) and the Copenhagen series (research.dmi.dk/data/). The GHCN series are all adjusted monthly data.

Table A1 lists the series, the GHCN station number, the sample period, and the number of missing observations. Imputation of missing data is only needed for complex demodulation, in which case we replace the missing observations with their expectation conditional on the available sample and the maximum likelihood parameter estimates obtained by our reference model (7). The Kalman filter and smoothing algorithm that are relevant to model (7), once it is cast in state-space form, enables the evaluation of the conditional expectation of the missing given the observed sample . All the inference concerning our model, including the phase estimates discussed in section 4, do not require any imputation, as the missing values are automatically handled by the estimation filters. See Durbin and Koopman (2012) for details.

### APPENDIX B

#### Derivation of Eq. (6)

##### a. Complex demodulation

The unfiltered complex demodulate of the process as defined in Eq. (5) is given by

We let , with and . Then, for a given year *k* and month *m*, since and for all , the unfiltered complex demodulate is

Consider now the filtered complex demodulate

Let , where we sometimes write and when it seems appropriate to emphasize the dependence. Then, because ,

We write this sum as

with functions , , and .

Note that , , and are all periodic in *t* with period 12. In the sum above, they are demodulated with , leaving constants

for each month . A low-pass filter operating on these sequences of constants therefore returns their average value.

In the last summand, , the counter is a step function in *j* with center *k*. The low-pass filter returns the mean value *k*.

We have that the average of is

the average of is

and the average of is

Renaming and splitting up into real and imaginary parts, together with

yields Eq. (6).

##### b. Trigonometric regression

The regression equation for the temperature *x* is given by

where *M* is the window length of the trigonometric regression in years, thus for a total of regressions, is the fundamental frequency, and is a stationary error term. The phase estimate is

We have that

and for ,

Note that the filtered complex demodulate is given by

Let and for all *j*, then

are asymptotically equivalent, and the derivation of Eq. (6) shown above also holds for trigonometric regression.

### APPENDIX C

#### Results for Other Series

### APPENDIX D

#### Robustness Check: Common Period 1779–2015

The procedure used in this paper on the eight temperature time series that span the common sample period 1779–2015 is presented in Table D1. The phase plots with filtered complex demodulate, phase trend according to Eq. (6), and confidence interval obtained from 1000 simulations and estimations of the model in Eq. (7) using the parameters estimated on the 1779–2015 sample period are shown in Figs. D1 and D2.

### APPENDIX E

#### Robustness Check: Early Instrumental Warm Bias

Table E1 reports the confidence intervals for a two-sided *t*-test with Bonferroni correction . Only 3 of the 11 confidence intervals contain the precession constant. Figures E1 and E2 show the phase plots with filtered complex demodulate, phase trend according to Eq. (6), and confidence interval obtained from 1000 simulations and estimations of the model in Eq. (7) using the parameters estimated on the Histalp data.

### APPENDIX F

#### Robustness Check: 1950 Sample Split

Appendix F (Table F1) shows the Bonferroni-corrected confidence intervals for the pre-1950 period and the post-1950 period. The appendix figures (Figs. F1–F3) show sample splits of the phase estimates from complex demodulation together with estimate of phase trend.

## REFERENCES

*Climate since 1500 AD*. Routledge, 679 pp.

*Time Series Analysis by State Space Methods*. 2nd ed. Oxford University Press, 346 pp.

*Large-Scale Inference: Empirical Bayes Methods for Estimation, Testing, and Prediction*. Cambridge University Press, 263 pp.

*Climate Change 2007: The Physical Science Basis*. S. Solomon et al., Eds., Cambridge University Press, 996 pp.

*Climate Change 2013: The Physical Science Basis*. T. F. Stocker et al., Eds., Cambridge University Press, 1535 pp.

_{2}concentration in the atmosphere

_{2}. Part II: Seasonal response

## Footnotes

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