Transfer entropy could not tell which way climate and tree rings are linked

Transfer entropy is widely used to read which of two time series drives the other. Tree rings give a case where the answer is known: summer warmth shapes the ring laid down that year, and the ring has no meaningful effect on the following summer. We ran a nearest-neighbour transfer entropy estimator on 135 ring-width chronologies from the International Tree-Ring Data Bank, about 78 years each, against June to August temperature, under a plan hash-locked before any estimate on real data. On real data the estimator detected climate to ring at 15.6 percent of sites (8.8 to 26.2) and ring to climate at 11.9 percent (4.9 to 22.5), above the 10 percent reverse limit fixed in advance.c1 Read backwards in time, the same series gave 19.3 and 22.2 percent, so the estimator was reading no arrow of time.c3 With a climate-to-ring coupling planted in synthetic rings, the false direction ranked at or above the true one at 52.6 percent of sites at the real coupling strength and at 40.0 percent with strong coupling, where direction accuracy reached 0.600 against the 0.8 required.c2 The estimator detected coupling and misread its direction. It gets no direction-reading status for this kind of data.

Schreiber [1] introduced transfer entropyDefinition: transfer entropy information theory (Schreiber 2000) The information, in bits, that the past of one series adds about the next value of a second series beyond what the second series' own past already gives. A larger value one way than the other is commonly read as the direction of influence. Excludes plain correlation or mutual information, which are the same in both directions, and any proof of a physical cause. full definition as a measure that separates information exchanged between two systems from information they share through common history, and showed that it can tell a driving element from a responding one. It is now used in climate, ecology, neuroscience and finance to assign a direction to a link between two series. For continuous data it is usually estimated with the nearest-neighbour method of Kraskov, Stoegbauer and Grassberger [2], in the conditional form of Frenzel and Pompe [3].

The method has known failure modes. Smirnov [4] examined spurious causalities with transfer entropy; later papers that cite that work report that the method can fail to recover preset one-way couplings and can give the wrong direction. We read Smirnov's paper only through those citing papers and its title. Runge and colleagues [6] reviewed causal inference from time series in the Earth sciences and set out the conditions under which such methods can be trusted. Sugihara and colleagues [5] proposed convergent cross mappingDefinition: convergent cross mapping nonlinear dynamics and ecology (Sugihara and colleagues 2012) A test that rebuilds the state of one series from its own lagged values and asks how well that rebuilt state predicts the other series. The driven series is expected to predict the driver. Excludes regression of one series on the lagged values of the other. full definition as a direction test for weakly coupled ecological systems, and we ran it beside transfer entropy as a second probe.

What a user of these methods needs is a known-answer testDefinition: known-answer test measurement and method validation A test of a method on a case where the true answer is fixed by physics or by construction, so that any wrong reading is a fault of the method. Excludes agreement with another method, which can share the same fault. full definition on data like their own. Annual tree rings and summer temperature give one. Summer warmth shapes ring width at cold sites, and a ring cannot set the weather. Our searches, run on 9 November 2026, found general studies of transfer entropy direction failure and no planted-direction test on ring-width data. That is a search result, not proof that none exists.

The plan was written as a preregistrationDefinition: preregistration research method (metascience) A dated written plan for a study, fixed and hash-locked before any result is seen: the question, the data, the statistic, the controls, the forecasts and the kill conditions. Excludes a plan revised after looking at the outcome, and a protocol written only for the record after the run. full definition and hash-locked on 31 October 2026 at 05:19 UTC, before any transfer entropy, cross mapping or lagged statistic was computed on the real data. Before the lock the estimator was tested on synthetic autoregressive series only, and from the real data only site counts and series lengths were read.

The decision the study informs was written into the same file: may a direction instrument of this kind be used to read the direction of links between other ecological series? The rule set in advance was yes only if the as-written kill conditionDefinition: kill condition research method (our usage, after Popper's falsification) A rule written in the preregistration that, when met, decides against the claim with no room for judgement after the run. Excludes reasons found after the run for setting a result aside. full definition and the planted-truth anchor both pass, or, for a labelled variant, if the variant's kill condition and its anchor both pass. Otherwise the instrument gets no direction-reading status.

The plan forecast that lag 0 would break the test. A summer and the ring of the same year are simultaneous at annual resolution, so a same-year term carries no time order and can show as a detection in both directions. The design therefore named one variant in advance, the same test with lag 0 removed, and stated that no other variant would be tried.

Rings: ring-width chronologies from the International Tree-Ring Data Bank [7], held by NOAA's National Centers for Environmental Information as public data with a request to cite the contributors. A site entered if its series was gap-free from 1901 with at least 50 years. This gave 135 sites in 57 two-degree blocks: 51 in the Alps, 24 in Fennoscandia and 60 elsewhere, with series of 64 to 91 years and a median of 78.

Climate: June to August mean temperature from the nearest cell of CRU TS 4.10 [8] [9], released under the Open Database License 1.0 with attribution to the Climatic Research Unit, University of East Anglia. June to August precipitation was run as a secondary, descriptive target.

Both series were passed through a high-pass filterDefinition: high-pass filter signal processing A filter that keeps fast variation and removes slow trends. Here each series minus a Gaussian smooth of width 10 years. Excludes a fixed polynomial detrend. full definition, each minus a Gaussian smooth with a width of 10 years, and scaled to unit variance. The mean absolute same-year correlation between ring and temperature was 0.218.

For a source series $X$ and a target $Y$, the transfer entropy at lag $l$ is the conditional information

estimated in bits with the KSG estimator in conditional formDefinition: KSG estimator, conditional form statistics and information theory The k-nearest-neighbour estimate of information from samples given by Kraskov, Stoegbauer and Grassberger and extended to conditional information by Frenzel and Pompe, here with 4 neighbours and the maximum norm. Excludes estimates that first bin the data into histograms. full definition with $k=4$. It was computed at lags 0, 1, 2 and 3 on one common set of years. The site statistic is the largest of the four:

Forward means climate to ring and reverse means ring to climate.

Each site statistic was compared with 199 iAAFT surrogatesDefinition: iAAFT surrogate time-series statistics, surrogate data A copy of a series built by repeated rounds of Fourier phase randomisation and rank reordering, so that it keeps both the original spectrum, and with it the autocorrelation, and the original set of values, while its timing against any other series is lost. Excludes a shuffle of single years, which destroys the autocorrelation. full definition of the source series, each run through the same estimator and maximised over the same lags. The p value is

with $k$ surrogates at or above the real statistic and $n=199$, so the smallest possible value is 0.005. A site counts as a detection at p 0.05 or below. The detection rateDefinition: detection rate statistics, our usage The share of sites at which the score in one direction beats its own surrogates at p 0.05 or below. Excludes a rate corrected for testing 135 sites, which is reported separately. full definition of an arm is the share of its 135 sites with a detection, and its interval is a block bootstrapDefinition: block bootstrap statistics An interval made by resampling whole 2 degree blocks of sites with replacement, so that neighbouring sites that share climate move together and the interval is not too narrow. Excludes a bootstrap over single sites, and a binomial interval that treats sites as independent. full definition over the 57 two-degree blocks.

For each arm, direction accuracyDefinition: direction accuracy method validation, our usage The share of sites at which the climate-to-ring score is larger than the ring-to-climate score. 0.5 is what a coin would give. Excludes whether either score is significant. full definition is

where $m$ is the number of sites whose forward statistic exceeds the reverse one and $N=135$. A coin gives 0.5.

Convergent cross mapping used an embedding dimension of 3, a delay of 1 year, the full library, leave-one-out prediction and 99 iAAFT surrogates of the predicted series. It was scored with the same detection and accuracy rules.

The preregistered variant is the same test with lag 0 removed from both directions, so eq. 2 takes the largest of lags 1 to 3 for the real data and for the surrogates. It was computed in the same run and is labelled as a variant throughout.

Every controlDefinition: control statistics and experimental design A version of the data with the structure under test removed (for example shuffled or pooled), scored the same way as the real data, so the real score has something to beat. Excludes a comparison with a different dataset or with a published number. full definition and synthetic arm ran in the same run as the real data.

The planted checksDefinition: planted check research method; called a positive control in experimental science Data built so that the right answer is known, run through the same analysis before the real data, to show that the analysis can see the effect when it is there. Excludes any check made on the real data, which cannot show what the analysis misses. full definition build synthetic rings with the true direction known. Each starts from an iAAFT copy $S_t$ of the site's ring series and adds the site's own climate $C_t$:

At real strength, $a$ is the site's real same-year correlation. In the strong arm the two coefficients are 0.6 and 0.3. Each synthetic arm was drawn once per site.

The planted anchor passes only if, in the strong arm, the forward rate exceeds 0.8 and direction accuracy exceeds 0.8, and if the no-coupling and shared-trend arms keep both rates at or below 0.10.

Table 1 gives the detection rates and direction accuracy in every arm, and Figure 1 draws the rates with their intervals.

On real data the estimator detected climate to ring at 15.6 percent of sites (8.8 to 26.2) and ring to climate at 11.9 percent (4.9 to 22.5), above the 10 percent reverse limit fixed in advance.c1 The forward rate beat the far-site pairing, whose mean was 0.056 and 95th percentile 0.082 (p 0.048, the smallest value 20 pairings allow). The reverse rate failed the limit by a narrow margin, and its interval spans 0.10. The kill condition fired. Direction accuracy was 0.519 against a coin's 0.5.

Read backwards in time, the same series gave 19.3 and 22.2 percent, so the estimator was reading no arrow of time.c3 The mean transfer entropy at lag 0 was 0.044 bits in both directions, which fits the forecast that the same-year term carries no order.

The planted anchor failed. With a climate-to-ring coupling planted in synthetic rings, the false direction ranked at or above the true one at 52.6 percent of sites at the real coupling strength and at 40.0 percent with strong coupling, where direction accuracy reached 0.600 against the 0.8 required.c2 With strong coupling, detection rose to 0.978 forward and 0.919 reverse: the estimator saw the coupling and could not place its direction. The no-coupling arm stayed inside its limit (0.074 and 0.037). The shared-trend world produced 12.6 and 17.0 percent, above the 0.10 limit even after the high-pass.c5 The common-driver arm stayed under 0.10 (0.059 and 0.096). At a coupling near 0.2 this says more about low power than about any ability to separate a shared driver from a direct link.

Figure 2 sets direction accuracy in each arm against the coin and the required bar.

The variant without lag 0 passed on real data, 8.9 percent forward and 3.7 percent reverse, and missed the planted effect: with strong coupling it detected the forward link at 2.2 percent of sites, so its pass cannot be credited.c4 Its real forward rate beat its pairing 95th percentile of 0.060 and its accuracy was 0.533. In the planted strong arm its accuracy was 0.630, and its shared-trend rates were 0.104 both ways, just over the limit.

Convergent cross mapping pointed the wrong way on real data: 18.5 percent forward against 28.9 percent reverse, accuracy 0.319. With strong planted coupling its accuracy was 0.378.

Correcting across sites with the Benjamini-Hochberg ruleDefinition: Benjamini-Hochberg correction statistics, multiple testing A rule that adjusts a set of p values so that, among the tests called significant, the expected share of false calls stays at or below a chosen rate, here 0.05. The adjusted value is called q. Excludes the stricter Bonferroni rule, which bounds the chance of any false call. full definition left no detections on real data in any direction. With 199 surrogates the smallest p is 0.005, so the correction over 135 sites can reject only when a large share of sites sit near that floor. It did so in the strong planted arm (0.978 forward) and nowhere in the real data. Over the six preregistered family tests, the reverse rate against its nominal level (q 0.002) and both cross-mapping rates (q below 1e-7) were rejected; the forward and variant-forward rates against the pairing gave q 0.057, limited by the 20-pairing floor.

By region, Fennoscandia, where the mean same-year correlation was 0.318, gave 0.375 forward and 0.292 reverse; the Alps 0.137 both ways; other sites 0.083 and 0.033.

Table 2 sets the forecasts written before the run against the outcomes.

The test has a known answer, and the estimator did not find it. On real data the two directions were detected at similar rates, reversing time made no difference, and when we planted the right direction the estimator ranked the wrong one at or above it at about half the sites. This is an honest nullDefinition: honest null research method (our usage) A working test that found no effect beyond a stated bound, a forecast that failed, or an error of our own found and fixed, reported with the same weight as a positive result. Excludes a void run, and a result left unpublished because it was negative. full definition: the method's failure on this kind of data, measured against a truth we built in.

Two mechanisms fit the pattern. The same-year term is symmetric, as forecast, and it dominates the estimate at these lengths. The lagged terms are small, and conditioning on the ring's own previous year can absorb a planted lag-one effect, because the previous ring already holds the previous summer. That second reading is an inference from a look after the run: real data showed mean transfer entropy at lag 1 of 0.035 bits forward and 0.002 reverse, an asymmetry in the known direction, while the planted lag-one arms showed about 0.01 or less. We did not test it.

The result agrees in kind with the general failures reported for transfer entropy [4] and with the caution of Runge and colleagues [6]. Its contribution is narrower: a measured failure on public ring-width data, at the length and coupling strength that ring studies have.

Two later rounds asked the same question with a different instrument: lagged Granger testsDefinition: Granger test econometrics and time-series statistics A regression test of whether the past values of one series improve the prediction of another beyond that series' own past. For Gaussian data it is equivalent to transfer entropy. Excludes a test of same-year effects, and proof of a physical cause. full definition read across the whole panel of sites at once. On 161 held-out high-latitude sites (finding F675narrowed in part), the panel index pointed from summer to ring at +1.206 (95% interval +0.871 to +1.591) against a far-pairing 95th percentile of +0.213, and flipped under time reversal. Two of its locked first-difference checks missed by small margins (0.584 against 0.6, and 0.117 against 0.10), so its own kill condition fired. On a third panel of 145 held-out sites (finding F676), with the planted prior-summer term fitted to the measured sign, the panel index was +1.204 against a pairing 95th percentile of +0.461, every binding check but one passed, and the kill condition did not fire. That panel method earned a limited direction status at that coupling strength. Neither round changes the result for transfer entropy read site by site.