A Journey Into a Traveler's Mental Map: Duration, Distance, and Speed Experienced in High‐Speed Trains

Abstract

Our study investigates how high‐speed transport affects humans’ perception of time, space, and speed. We tested 247 passengers traveling on the French Paris‐Lyon high‐speed train (TGV) who estimated the traveled distance, elapsed duration, and instantaneous and average speeds, while rating their confidence, felt speed of time passage, and emotional state. Environmental variables in this In Situ design included seating orientation (forward or backward) and train acceleration (positive, null [constant high‐speed], or negative [deceleration]). Our findings show that participants’ magnitude (duration, distance) estimations scaled with magnitudes. Estimations also exhibited typical features such as central tendency, which interestingly scaled to the midpoint of the journey (∼69 min) rather than the expected ∼15 min. Surprisingly, we found no influence of emotions or felt speed of time passage on magnitude and speed estimates. Instead, participants’ prior knowledge of the journey's total duration and distance better explained their magnitude estimations than their on‐the‐fly speed estimates. Seating orientation did not significantly affect estimations, whereas acceleration partly did. Participants demonstrated reliable metacognitive ratings of their duration and instantaneous speed estimates, but not of their distance and average speed estimates. Altogether, our results show that in passive high‐speed transport, passengers may rely on cognitive heuristics (mental maps) rather than sensorimotor integration (path integration) to estimate magnitudes. We discuss the implications of our work for understanding mental representations of magnitudes and speed in real‐life situations.

1. Introduction

The representation of time and space is essential for animals whose primary characteristic is to move around in the world. Numerous studies conducted in laboratories dissect the neurobiological and psychological representations of time and space in the brain, but much lab knowledge is born out of stringent, and sometimes ecologically invalid, experimental designs. Whether lab knowledge can generalize to ecological settings raises a central question (Holleman, Hooge, Kemner, & Hessels, 2020), which requires some adaptation of paradigmatic work to ensure that the collected data in real life conform to the scientific method and rigor of controlled lab conditions (Vallet & van Wassenhove, 2023). For instance, most laboratory experiments are conducted in stillness, when the world is fully dynamic. To an extreme, with technological advances, we can be transported at speeds that our bodies were not evolutionarily designed to achieve on their own, which puzzled our precursors in absurd and funny ways (“The Automobile Brain,” 1904). Still, an average walking pace typically produces a self‐displacement of around 5 km (∼3.1 miles) per hour when hundreds can be traveled in an hour with high‐speed trains or planes. Modes of transport can be argued as ecological in that they have been part of our daily life for more than a century, from bikes to planes. Our current modes of transportation have likely increased the relative appreciation of distances with respect to time, but how is this integrated by our brains? To say it differently, our modern society increases its time efficiency by shrinking space, but how this may affect our personal experiences of time and space is unknown.

In this study, we tested participants’ time, space, and speed perception in French high‐speed trains called “Train à Grande Vitesse” (TGV). Our adaptation of a lab design to the TGV factorized two ecological variables that were the acceleration and the seating orientation of passengers (in line with the proposed In Situ design; Vallet & van Wassenhove, 2023). We adapted the experience sampling method (ESM; Christensen, Barrett, Bliss‐Moreau, Lebo, & Christensen, 2003) to our context. ESM consists of asking individuals at random times during their day about their experience so as to capture an individual's experience in the context of everyday life. Herein, naïve train passengers were selected on the fly during their journey. Individuals were not a priori prepared to be tested or part of our study, they were travelers in the context of a real train travel, and they agreed to contribute to the survey on the spot. Hence, all measurements can be considered “retrospective estimations” as no attention was a priori given to speed, distance traveled, or duration elapsed since departure beyond that typically given by travelers in this situation.

One main goal of our study was to survey the perception of duration and distance (a.k.a. magnitude estimations; Walsh, 2003) as well as (average and instantaneous) speeds during a real‐world TGV ride. Magnitude estimations are characterized by distinct profiles: while magnitude estimates scale with magnitudes, an overestimation of short magnitudes and an underestimation of large magnitudes is often reported (Petzschner, Glasauer, & Stephan, 2015). This regression effect is accompanied by a centering of the estimations to the mean of the tested magnitude distribution. In prospective time estimations and within design, this effect is interpreted as resulting from the tested stimulus distribution, well explained by the Bayesian framework. In retrospective time estimations, a single value rather than a distribution is tested per participant; therefore, a regression effect in retrospective tasks would likely rely on long‐term priors such as cultural ones (Balcı et al., 2023; Chaumon et al., 2022). Previous studies tested retrospective duration estimation at the time scales of minutes, such as the ones tested here (Brown & Stubbs, 1998; Grondin and Plourde, 2007). These early findings suggested that shorter durations were overestimated, and longer durations were underestimated in agreement with a regression effect in retrospective time estimation. More recently, large‐scale studies reported a regression point around 15 min, suggesting a common reference duration for all individuals across cultures for tasks conducted in real‐life (Balcı et al., 2023; Chaumon et al., 2022). Herein, we thus sought to replicate regression effects in retrospective estimation of duration, and to extend the approach to distance and speed estimation, in the context of high‐speed transportation as explained in more detail below. We also inquired about participants’ felt speed of the passage of time (Wearden, 2015; Wearden, O'Donoghue, Ogden, & Montgomery, 2014), explored the effect of their emotional states on magnitude estimation (Droit‐Volet & Gil, 2009; Droit‐Volet & Meck, 2007), and their ability to rate their performances, namely, metacognition using confidence ratings and signed error monitoring (Akdoğan & Balcı, 2017; Kononowicz & van Wassenhove, 2019; Öztel, Eskenazi, & Balcı, 2021). Our aim was to validate typical lab findings (which we describe below) in the wild and explore the impact of high‐speed transport on the representation of time, space, and speed.

Our refined understanding of the neural bases for the representation of space and time during real‐world navigation is mainly based on rodents exploring a maze (Moser, Kropff, & Moser, 2008). In humans, navigation can be explored using mobile neuroimaging (Aghajan et al., 2017; Delaux et al., 2021; Maoz, Stangl, Topalovic, & Suthana, 2023) or virtual reality (Petzschner & Glasauer, 2011; Robinson & Wiener, 2021; Wiener, Michaelis, & Thompson, 2016). During self‐motion, an individual can continuously update its position by using multisensory information, a mechanism classically referred to as path integration (Gallistel, 1990; Mittelstaedt & Mittelstaedt, 1980). As the individual moves, duration and distance are also naturally entangled through the individual's speed of displacement (Issa, Tocker, Hasselmo, Heys, & Dombeck, 2020), which predicts a tight link between the representation of distance traveled and elapsed duration during self‐motion. In line with this, the estimation of duration and distance has long been proposed to rely on common neural mechanisms and metrics (Bueti & Walsh, 2009; Walsh, 2003), a theory of magnitude (ATOM) compatible with the representation of abstract quantities in the brain, including numerosity (Dehaene & Brannon, 2011; Gallistel, 1990; Petzschner et al., 2015). To date, however, a great majority of studies exploring magnitudes rely on studies conducted in the stillness of a lab, rarely using dynamic displays (Coull, Charras, Donadieu, Droit‐Volet, & Vidal, 2015; Lambrechts, Walsh, & Wassenhove, 2013; Martin, Wiener, & Van Wassenhove, 2017) or virtual reality (Glasauer, Schneider, Grasso, & Ivanenko, 2007; Robinson & Wiener, 2021; Thurley & Schild, 2018). Herein, we put this key question to the test of a real‐world situation, which comes with peculiar constraints on the path integration hypothesis.

First, in public transportations (e.g., trains, planes), participants passively experience self‐motion at speeds that largely exceed humans’ biological endowment. In self‐motion, motor efferent copies and proprioception can help provide estimates of the distance or the timing of movements. In a seated, passive, and fast mode of transportation, these inputs are not available to the path integration system (e.g., one cannot count and integrate one’ walking steps toward a magnitude estimation). Our first question was thus to what extent sensory signals such as optic flow and acceleration can impact magnitude estimation during passive linear motion, such as the one experienced in public transportation.

Second, passive motion in trains is mostly linear. Perceptually, humans are less sensitive to linear displacements (Kingma, 2005; MacNeilage, Turner, & Angelaki, 2010). While the vestibular system contributes to our sense of embodiment (Lopez, 2015) by establishing a model of where the body is located and heading to, it is most critical for angular displacements (Mittelstaedt & Mittelstaedt, 1980). Additionally, the two distinct vestibular functions that are the perception of self‐motion (vection) and the orientation in space (Seemungal, 2014) are calibrated by visual inputs: in full light, which are the testing conditions herein, visual cues dominate self‐motion perception (Glasauer, Amorim, Viaud‐Delmon, & Berthoz, 2002). The speed of visual motion has been shown to lengthen duration estimation either in passive viewing (Kanai et al., 2006) or during passive vection while virtually driving (van Rijn, 2014). Other studies have shown that motion and speed alter distance and numerosity but not necessarily duration estimation (Martin et al., 2017). Our second question, therefore, tested the effect of change of speed on magnitude estimation in real‐life.

Interestingly, Dallal, Yin, Nekovářová, Stuchlík, and Meck (2015) also showed that vestibular lesions do not impair timing in a navigation task, suggesting that time and space estimation are not the outcomes of a generalized magnitude system as proposed in ATOM, but rather the outcome of a self‐referential metric system. Consistent with this proposal, spatial distances have long been proposed to be represented through vector computations, which necessitate time estimations so that distances result from the product of time and speed (Gallistel, 1989, 1990). Additional observations suggest that passive motion appears to be coded later than active motion (Cullen, 2019), indicating the susceptibility of magnitude estimation to be affected by cognitive factors during passive motion in addition to path integration signals (Arthur, Philbeck, & Chichka, 2009). In this context, our third question asked whether a typical magnitude profile could be found for duration and distance estimation in TGVs. Magnitude profiles are typically tested from sensory displays in the lab and characterized by the scaling of magnitude and variance estimations with the objective magnitude as well as a central tendency effect (Petzschner et al., 2015). Both scalar and vector representations would predict a magnitude profile for duration and distance.

To further disentangle the scalar and vector code hypothesis, we explored the links between magnitudes (duration, distance) and average speed using three distinct models deriving information from participants’ estimates (presumably derived from path integration) or priors (declarative knowledge). We formulated two competing working hypotheses regarding the effect of acceleration on the accuracy of magnitude estimation. Following ATOM, a first working hypothesis was that higher speeds yield larger magnitude estimations due to the higher number of events per unit of time or space (Brown, 1995; Bueti & Walsh, 2009; Fortunato, Togoli, & Bueti, 2023; Fraisse, 1963; Guyau, 1923). Under this working hypothesis, duration and distance estimation should be longer for the fastest speeds (and null acceleration). Alternatively, the second working hypothesis was that magnitudes scale with the rate of change (Brown, 1995; Gallistel, 1990; Martin et al., 2017) so that the larger the acceleration or the deceleration, the larger the estimated magnitudes. As for speed estimation, we expected that the rate of change would improve accuracy so that speed estimates realized during speed changes (acceleration or deceleration of the train during the journey) would be more accurate than those realized at constant speed, thanks to sensory updating.

Additional predictions based on lab experiments were that participants’ duration and passage of time would be related so that shorter durations would be associated with faster passages of time (Droit‐Volet & Wearden, 2016; Lamprou‐Kokolaki, Nédélec, Lhuillier, & van Wassenhove, 2024; Nicolaï, Chaumon, & van Wassenhove, 2024). Travelers’ self‐reported emotional states were expected to be explanatory factors in their temporal estimates. For instance, the more bored travelers reported to be, the longer we expected their duration estimates to be (Droit‐Volet & Meck, 2007; Droit‐Volet & Gil, 2009). Last, we collected metacognitive ratings on duration, distance, and speed estimates to gather additional insights on such measurements and notably explore whether noisy conditions of the real‐world train journey prevented metacognitive inferences.

2. Materials and methods

2.1. Participants

Two hundred and seventy‐nine participants were recruited on the fly among the passengers of the Paris‐Lyon or Lyon‐Paris train journeys. Travelers were informed about the general goals of our study, and told about their rights and the use of their data would they agree to take part in the study. When they agreed to contribute to the study, we asked them to read and to sign a written informed consent form in accordance with the Declaration of Helsinki (2013). The study was validated by the Ethics Committee for Research (CER) of Paris‐Saclay University (CER‐Paris‐Saclay‐2018‐034‐A2).

Due to their lack of compliance with the instructions of the study, 16 out of 279 participants were discarded from the dataset prior to the analysis: two participants answered only one question; one participant's responses were not timed; nine participants got on or off board at an intermediary stop in the journey; four participants answered the questionnaire during an acceleration or deceleration episode of an intermediary stop. An additional 16 participants were removed because their knowledge about the total duration or distance of the journey was inaccurate according to the following criterion: their responses were below or above the first or the third quartile, respectively, yet participants answered “yes, I know exactly the duration or distance of the journey” (see Procedure relative to the beginning of the questionnaire). Hence, a total of 247 participants (104 females; 111 males; 32 undeclared; 35.2 ± 14.1 years old) were included for analysis in the study. Additional outliers’ removal was conducted as a function of the different analyses, as explained later in the text when needed.

2.2. Experimental factors: Orientation and acceleration

Although participants were selected on the fly during the journeys, the experimenters counterbalanced their seating orientation in the train: 133 participants were seated in the direction of the train motion (“forward”), and 112 participants were seated in the opposite direction (“backward”). The seating orientation manipulated the direction of the optic flow. This factor is referred to as Orientation throughout the text.

Participants could be grouped as a function of the acceleration episode during the journey, which was identified manually (see Section S1.2). This resulted in 34 participants tested during a positive acceleration (start of the journey), 166 participants tested during a null acceleration (constant speed), and 33 participants tested during negative acceleration, that is, deceleration (end of the journey). These numbers were proportional to the duration of each episode, so that the whole train journey was sampled equally with respect to time in the journey. This factor is referred to as Acceleration throughout the text.

2.3. Procedure

Questionnaires were delivered in paper format and started by inquiring participants about their basic knowledge of the journey. In the first series of yes/no questions, participants were asked to report whether they knew the total duration and the total distance of the journey. Irrespective of their answer, they were asked to provide their best numerical estimate for each (e.g., 2 h and 47 min or 257 km, respectively). They were also asked to provide their best estimate of the maximal speed the train could travel (in km/h). These initial estimates are participants’ declarative knowledge of the duration, distance, and speed.

In a second series of questions, participants were asked to estimate the elapsed duration, traveled distance, and average speed since departure, as well as the instantaneous speed of the train at the moment they answered the question. The order of the questions was pseudorandomized across participants. Each magnitude estimation (duration, distance) was tested on a separate page. The estimations of average and instantaneous speeds were tested on the same page.

Estimates had to be given both graphically (Fig. 1a−c) and numerically. In the graphical estimation, participants were given a clock, a map of France, and a speedometer for duration, distance, and speed estimations, respectively (Fig. 1a−c). They were instructed to indicate their subjective estimation of duration, distance, and instantaneous and average speed with a pencil mark as represented by the blue crosses in Fig. 1a−c. For their numerical estimation, participants reported the durations in hours and minutes, the traveled distances in kilometers, and the instantaneous and average speeds in kilometers per hour.

Fig. 1.

Questionnaire and experimental protocol. To perform their graphical estimations, participants were provided with (A) a schematic clock, (B) the map of France highlighting the departure and arrival cities of the journey, and (C) a speedometer. Red lines are the construction lines for digitizing the graphical estimates into a unitless ratio, subsequently multiplied by the participant's self‐reported knowledge of the total duration, distance, or speed of the journey. (D) The questionnaire was a 3 (Acceleration) × 2 (Orientation) between‐participants design. The journey was segmented into three episodes as a function of the acceleration of the train (Positive, Null, Negative), which defined our three‐level factor Acceleration. The two‐level factor Orientation was defined by participants’ orientation in the train, facing forward or backward with respect to the train direction (therefore, changing the optic flow as forward or backward (Vallet & van Wassenhove, 2023)). Black dots are actual measurements of the speed of the train at different times during the journey that were obtained from the train event recorder. Red dots are reconstructed speeds at missing times (see Procedure). Vertical black bars are acceleration boundaries.

Importantly, in our daily lives, individuals often have prior knowledge (like the total duration) about the activity they engage in. Hence, the first questions regarding knowledge of the travel and the graphical estimation might have encouraged them to rely on their prior knowledge, which could reflect this ecological aspect. However, in contrast to traditional laboratory‐based estimations, the graphical format necessarily limited the estimation, which could not be overestimated at the end of the journey. It is also possible that questions regarding knowledge of the travel may have limited the numerical estimations, in the sense that participants would not provide an estimate higher than their actual knowledge (see the discussion in Section 4.1).

After each estimation, participants rated their confidence level using a 4‐point Likert scale ranging from “not confident” to “very confident.” Additionally, following their estimation of duration, participants were also asked to provide a signed metacognitive rating of their estimate using a 4‐point Likert scale with the ranked options “underestimated,” “rather underestimated,” “rather overestimated,” and “overestimated” (Kononowicz & van Wassenhove, 2019; Chaumon et al., 2022).

The times at which participants provided their estimation were collected and written down by the experimenters—for example, “at 12h33, the participant estimated the distance; at 12h34, the participant estimated the instantaneous speed.” This time‐stamping procedure was accurate to the minute. It allowed extracting the objective duration, distance, or speed at the time of data collection and identifying the acceleration episode during which participants performed their estimation (Fig. 1d).

At the end of the questionnaire, participants rated their subjective feeling about the speed of the passage of time (passage of time judgment, PoTJ), ranging from “very slow” to “very fast” (Wyrick & Wyrick, 1977) from the departure to the moment of their rating—that is, the same period as their magnitude estimation. Participants were also asked to rate their emotional states over that time period, including their level of happiness, sadness, and boredom, ranging from “not at all” to “very” (Droit‐Volet & Wearden, 2016).

Last, participants were asked to freely describe the strategies they used to perform their estimations. We clustered strategies into the six most common categories: (1) no strategy or declaring to have responded by chance (19 participants); (2) perceptual strategy (such as using visual or auditory cues, 105 participants); (3) logical or mathematical strategies such as reporting “v = d/t” or “1/3 of the time means 1/3 of the distance” (13 participants); (4) measurement tools despite being explicitly instructed not to do so (e.g., watches, time on phones or computers; 45 participants, which were not considered outliers; Fig. S2); (5) previous knowledge: for instance, high familiarity with the journey, or the train driver giving speed or duration information at the beginning of the journey (39 participants); (6) no response to the question (12 participants).

Prior to any, and depending on the type of, analyses, the questionnaire data were digitized, normalized, and scored. This process is fully described in Supplementary Materials (Section S1.1). The objective duration, distance, and instantaneous speed data were retrieved (or reconstructed when missing) from the train journey track recordings, which were supplied by the French railway company (SNCF). This process is described in Supplementary Materials as well (Section S1.2).

2.4. Outliers’ removal

In all between‐participant analyses (i.e., all figures but Fig. 3), data points showing a relative error above the central 95th percentiles were removed. The relative errors were computed as: $\frac{(subjectiveestimate-objectivevalue)}{objectivevalue}$. This procedure removed ∼6% of data points per estimation (duration: 5.7%; distance: 6.5%; instantaneous speed: 6.5%; average speed: 6%). Using a similar criterion for the within‐participants analysis reported in Fig. 3, 40 participants out of 207 were discarded from this analysis (i.e., 16.2%).

Fig. 3.

Estimated speed is not predicted by estimated distance and duration. (A) Linear regression between estimated average speed and modeled average speed. Thick gray line corresponds to the regression curve. (B) Correlation between relative estimated distance and duration, within participants. Dashed black lines delimit X and Y data below/above 1. Any point below/above the horizontal dashed black bar corresponds to an underestimated/overestimated distance. Any point before/after the vertical dashed black bar corresponds to an underestimated/overestimated duration. Red numbers correspond to the number of observations below/before/above/after the dashed black lines. (C) Linear regressions between estimated distance and modeled distance. We tested three different ways of computing the modeled distance. The gray line corresponds to the modeled distance from the estimated average speed and duration estimation from each participant, labeled estimated speed. The green line corresponds to the modeled distance from knowledge of the maximum speed of participants and the estimated duration, labeled maximum speed. The pink line corresponds to the modeled distance from the estimated duration, the knowledge of the total duration, and the knowledge of the total distance reported by each participant, labeled knowledge. (D) Same as C, but for duration. In each plot, dots correspond to individual participant responses, shaded areas to 95% confidence interval of the regression parameters, and thin black line corresponds to the unity line.

2.5. Statistical analyses

Considering our theoretically driven working hypotheses, we built statistical models tailored to test each working hypothesis separately as opposed to using a single model including all the factors as a data‐driven and exploratory research would. Our dataset is publicly available (https://doi.org/10.5281/zenodo.14895063) so that exploratory analysis can be conducted by anyone interested in replicating or exploring additional effects.

All statistical analyses were performed in the R statistical environment (version 4.1.2 R Core Team, 2021). Prior to any analysis, we, first, verified that response modality (numerical vs. graphical) had no significant effects on their relative estimates (Section S2.1 and Fig. S1) and, second, showed that self‐reported strategy had no impact on participants’ estimates (Section S2.2 and Fig. S2). We also checked that there was no significant effect of the presence or absence of an intermediate stop on participants’ duration estimation nor on their relative duration estimation (Section S2.3 and Fig. S3). For each scientific question with dedicated working hypotheses, we computed ordinary least square (OLS) linear regression models between the variable and the factors of interest using the lm() function from the base R package “stats” (version 4.1.2 R Core Team, 2021). The independence, normality, and homogeneity of the residuals were verified for each linear regression using the package performance (Lüdecke, Ben‐Shachar, Patil, Waggoner, & Makowski, 2021). When necessary, the F‐values of the main effects and the interactions, standard errors (SEs), corresponding p‐values, and partial η square (η ² _p ) or η square (η ² _, if there is only one factor) of each factor from each model were computed with the anova() function from the lmerTest package (Kuznetsova, Brockhoff, & Christensen, 2017). For post‐hoc pairwise contrasts, we reported unstandardized regression coefficients (beta), |t| or |z| values, and corresponding p‐values computed with the pairs() function from the emmeans package (Lenth et al., 2021).

In perceptual studies, the log‐log function is typically used to analyze the link between the psychological reality of a magnitude and its objective value (Balcı et al., 2023). Hence, to investigate the estimation profile of each type of estimation (duration, distance, instantaneous, and average speed), we performed OLS linear regressions between the logarithm of the single estimation score and the logarithm of the objective values (continuous variables extracted from the reconstructed train data).

We used Pearson's correlation to investigate whether participants had a consistent bias in duration and distance estimation. For instance, we used this approach to verify whether a participant who overestimated a duration also overestimated distance.

To understand how participants computed the average speed, and which link existed between their duration and distance estimations, we used a modeling approach. We modeled the average speed estimation using each participant's distance and duration estimation score. We then looked at the linear relationship between the modeled average speed based on their magnitude estimation, and the estimated average speed effectively reported by participants.

To investigate the link between distance and duration estimation, we compared the variance explained (R ²) of OLS linear regression models constructed between the numerical estimated duration (or distance) and the modeled duration (or distance), hypothesizing three possible computations (Fig. 3c,d):

• 1.
  Estimated average speed: Participants used the combination of one magnitude (duration or distance) and their average speed estimate to derive the other magnitude (distance or duration, respectively).

  $$\mathrm{mod}.\mathrm{dist}\mathrm{ance}=\mathrm{est}.\mathrm{speed}*\mathrm{est}.\mathrm{dura}\mathrm{tion}$$

  $$\mathrm{mod}.\mathrm{dura}\mathrm{tion}=\mathrm{est}.\mathrm{speed}*\mathrm{est}.\mathrm{dist}\mathrm{ance}$$

With mod. for modeled and est. for estimated.

• 2.
  Reported maximum speed: Participants used their estimation of the maximum speed of the train reported at the beginning of the questionnaire to convert one magnitude into another.

  $$\mathrm{mod}.\mathrm{dist}\mathrm{ance}=\max .\mathrm{speed}*\mathrm{est}.\mathrm{dura}\mathrm{tion}$$

  $$\mathrm{mod}.\mathrm{dura}\mathrm{tion}=\max .\mathrm{speed}*\mathrm{est}.\mathrm{dist}\mathrm{ance}$$

With max. for the self‐reported maximum train speed.

• 3.
  Knowledge strategy: Participants used their prior knowledge of the journey to infer another magnitude. In this scenario, we modeled one magnitude (e.g., duration) as the ratio of the other magnitude (e.g., distance) divided by the reported total magnitude for the journey (e.g., total distance) multiplied by the other total magnitude of the journey (e.g., total duration):

  $$\mathrm{mod}.\mathrm{dist}\mathrm{ance}=\frac{\mathrm{est}.\mathrm{dura}\mathrm{tion}}{\mathrm{total}\mathrm{dura}\mathrm{tion}}*\mathrm{total}\mathrm{dist}\mathrm{ance}$$

  $$\mathrm{mod}.\mathrm{dura}\mathrm{tion}=\frac{\mathrm{est}.\mathrm{dist}\mathrm{ance}}{\mathrm{total}\mathrm{dist}\mathrm{ance}}*\mathrm{total}\mathrm{dura}\mathrm{tion}$$

Considering that the estimation scores were computed with self‐reported declarative knowledge, we used only the numerical estimates for this analysis (Fig. 3c,d). This was realized to prevent circularity in the data as our individual scoring method took into account declarative knowledge (see Section S1.1), which could inflate the variance explained by our model for this specific analysis (indeed, the same outcome is found when using estimation scores, but the variance explained is higher than when using numerical estimations only).

To investigate the effect of our two factors (Acceleration and Orientation), we computed the relative magnitude and speed estimates by taking the ratio between the subjective estimate and its objective value. We, then, used OLS linear models to investigate the effects of Acceleration (positive, null, negative) and Orientation (forward, backward) on participants’ relative estimates.

To investigate the effect of emotional states on duration estimation or PoTJs, we used Pearson partial correlation using the pcor.test function from the ppcor package to take into account the collinearity between the Likert scales of the emotional states. When partial correlation was significant, we computed the pairwise post‐hoc comparison with the emmeans package (Lenth et al., 2021). For completeness and replication, we also report a single model approach (with all the emotional states as factors) in Supplementary Materials (Sections S2.6 and S2.7). For the metacognitive ratings, very few participants used the extremes of the scale (“underestimate”: 9; “rather underestimate”: 110; “rather overestimate”: 107; “overestimate”: 3). Therefore, we converted the 4‐point Likert scale into a two‐level factor metacognitive rating (“underestimate”: 119; “overestimate”: 110). We computed an OLS linear model with the relative duration estimation as the predicted variable as a function of a two‐level factor for the metacognitive rating.

To investigate the effects of confidence, we looked at the relationship between the absolute relative error of each estimation, separately, and the self‐reported level of confidence with OLS linear models.

3. Results

3.1. Magnitude and speed estimation profiles

The first analysis compared participants’ subjective estimations of duration, distance, and instantaneous speed with objective measurements to characterize the general estimation profiles. For this, we performed linear regressions of the logarithm of participants’ subjective estimates (predicted variable) as a function of the logarithm of the objective values (predictor). For visualization purposes, we also plotted the data without the log (Fig. 2).

Fig. 2.

Duration, distance, instantaneous, and average speed profiles. Linear regression of subjective estimates as a function of the objective measurements for duration (A, blue), distance (B, red), instantaneous and average speed (C, D, green and yellow, respectively). The identity line is indicated in black. Regression lines are colored. Shaded areas indicate the 95% confidence interval of the regression parameters. The intersection between the regression and the identity lines is indicated in red. Small inserts in each plot correspond to the regression at the log‐log scale.

As predicted, we found that a significant part of the variance in duration estimation was explained by the objective duration (R ² =  .85, F(1, 245) = 1343.47, p <  .001, adj. R ² =  .85) with a significant and positive relationship between the two (beta = 0.94, 95% CI [0.89, 0.99], t(245) = 36.65, p < .001; Std. beta = 0.92, 95% CI [0.87, 0.97]). We also found that the greater part of the variance in distance estimation was significantly explained by the objective distance (R ² =  .85, F(1, 229) = 1279.51, p < .001, adj. R ² =  .85) with a significant and positive relationship (beta = 0.75, 95% CI [0.71, 0.79], t(229) = 35.77, p < .001; Std. beta = 0.92, 95% CI [0.87, 0.97]).

The variance explained by the objective instantaneous speed was lower in the instantaneous speed model (R ² = .65, F(1, 229) = 416.77, p < .001, adj. R ² = .64). Very little variance was significantly explained by the objective average speed in the average speed model (R ² = .17, F(1, 230) = 46.52, p < .001, adj. R ² = .16). Although the variance explained was lower for instantaneous and average speeds, there was still a significant and positive relationship between their subjective estimates and their objective measurements (instantaneous speed: beta = 0.76, 95% CI [0.68, 0.83], t(229) = 20.41, p < .001; Std. beta = 0.80, 95% CI [0.73, 0.88]; average speed: beta = 0.29, 95% CI [0.21, 0.38], t(230) = 6.82, p < .001; Std. beta = 0.41, 95% CI [0.29, 0.53]).

None of the magnitude or speed profiles showed increased variance with increased objective value, commonly expected following Weber's law for within‐individual designs (Fig. 2). However, we found that small magnitudes tended to be overestimated, while large magnitudes tended to be underestimated, which conformed to Vierordt's law despite the between‐individual design (see Chaumon et al., 2022; Balcı et al., 2023). The regression point designating the value at which participants were most accurate are indicated in Fig. 2a,b, in red dashed lines. Thus, to exemplify this effect in our data, we found that a duration of 5 min was overestimated to 8 min, but a duration of 120 min was underestimated to 117.4 min; a distance of 50 km was overestimated to 81 km, whereas a distance of 400 km was underestimated to 375 km. In magnitude estimations, regression points were surprisingly located very close to the middle of the journey (69 min for duration; 265 km for distance).

3.2. The role of declarative knowledge in distance and duration estimation

Despite participants’ accurate estimations of duration and distance, their average speed estimates were quite unsatisfactory. This was surprising given that the accurate estimation of two magnitudes should suffice to infer an accurate average speed using the simple ratio of scalars ($speed=\frac{distance}{duration}$). Herein, we thus explored further how participants may have inferred their estimation of duration, distance, and average speed. In other words, did participants draw their estimations from experience only, or did they use their declarative knowledge (initial questions about their knowledge of the journey) to estimate duration, distance, and average speed?

First, we asked whether participants inferred the average speed with the expected calculation, namely, $est.averagespeed=\frac{est.distance}{est.duration}$. For each participant, we computed the expected average speed estimate (modeled average speed) based on the ratio of the estimated distance and duration (Fig. 3a). We fitted an OLS linear model to the estimated average speed with the modeled average speed as a single predictor. We found no significant linear relationship between the estimated and the modeled average speeds (R ² = 7.94e‐03, F(1, 207) = 1.66, p = .199, adj. R ² = 3.15e‐03; Fig. 3a). This was partially expected considering the estimation profiles (Fig. 2a,b,d).

One possible reason for this result could also be that each individual's bias for duration and distance was not consistent within participants, meaning that even though participants were accurate, one may have slightly overestimated a duration while underestimating the distance, resulting in an inaccurate average speed estimation. To check for these potential individual biases across magnitude estimations, we computed Pearson's correlation between estimated duration and distance across all participants. We found a highly significant and moderate positive relationship between the two (r = .46, 95% CI [0.34, 0.56], t(207) = 7.36, p < .001; Fig. 3b). This moderate positive correlation appeared to be driven by a large number of participants overestimating both duration and distance. About the same number of participants showed an inverse bias across magnitudes, suggesting that there was no consistent bias across magnitudes (Fig. 3b, red numbers).

In light of this, we compared three possible strategies that participants could have used to link duration and distance estimations. The first model, derived from the reasoning laid out above, posits that participants used their estimation of one magnitude and of average speed to estimate the other magnitude (distance or duration) (Fig. 3c,d, gray line). This computation is similar to the one hypothesized in Fig. 3a. Based on our previous analysis (Fig. 3a), we expected this model to be the least explanatory of the data.

In the second model, participants assumed that the train was mostly traveling at its maximum speed during the journey. They may have used their knowledge of the train maximum speed reported at the beginning of the questionnaire (max speed; Fig. 3c,d, green line) and computed:

$$\mathrm{mod}.\mathrm{dist}\mathrm{ance}=\mathrm{est}.\mathrm{dura}\mathrm{tion}\ast \max .\mathrm{speed}$$

$$\mathrm{mod}.\mathrm{dura}\mathrm{tion}=\frac{\mathrm{est}.\mathrm{dist}\mathrm{ance}}{\max .\mathrm{speed}}$$

In the third model, participants used their prior knowledge of the distance or duration of the journey to infer the other magnitude and the average speed. Hence, we modeled one magnitude (e.g., modeled duration) based on the ratio of the other magnitude estimation (e.g., estimated distance) with participants’ reported knowledge of the total magnitude for the journey (e.g., total distance) multiplied by their knowledge of the other total magnitude for the journey (e.g., total duration) (Fig. 3c,d, pink line), namely:

$$\mathrm{mod}.\mathrm{dist}\mathrm{ance}=\frac{\mathrm{est}.\mathrm{dura}\mathrm{tion}}{\mathrm{total}\mathrm{dura}\mathrm{tion}}*\mathrm{total}\mathrm{dist}\mathrm{ance}$$

$$\mathrm{mod}.\mathrm{dura}\mathrm{tion}=\frac{\mathrm{est}.\mathrm{dist}\mathrm{ance}}{\mathrm{total}\mathrm{dist}\mathrm{ance}}*\mathrm{total}\mathrm{dura}\mathrm{tion}$$

Similar to our previous analysis, we fitted OLS linear models to the estimated distance (or duration) with a single predictor of modeled distance (or duration). The best model was chosen based on the largest variance explained (R ²). As stated in the statistical analyses section, our analyses used numerical estimations only (as opposed to estimation scores combining graphical and numerical estimations) to prevent circularity.

For distance (Fig. 3c), we found that the variance explained (R ²) was larger for the third model knowledge (R ² = .88, F(1, 205) = 1564.37, p < .001, adj. R ² = .88) than for the other two models, which showed comparable explained variance (maximum speed: R ² = .67, F(1, 203) = 410.47, p < .001, adj. R ² = .67; estimated average speed: (R ² = .70, F(1, 205) = 469.01, p < .001, adj. R ² = .69)). For duration (Fig. 3d), the pattern was the same although the R ² was slightly lower (third model based on knowledge: R ² = .77, F(1, 205) = 668.39, p < .001, adj. R ² = .76; second model based on reported maximum speed: R ² = .69, F(1, 203) = 442.47, p < .001, adj. R ² = .68; first model based on estimated average speed: R ² = .69, F(1, 205) = 449.24, p < .001, adj. R ² = .69).

Therefore, the computation that participants may have used to link duration and distance estimations was likely using the third model, which relied on the declarative knowledge of the journey's duration, distance, and the estimation of the other magnitude. Additionally, our analysis suggests that distance estimation is better explained by the duration than the duration is explained by the distance. This suggests that participants may have used a time heuristic to estimate traveled distances, and not the other way around. This is also supported by the fact that a majority of participants reported knowing the exact duration of the journey (78% of participants), while a majority reported not knowing the exact distance (82% of participants), as per the initial inquiry of our questionnaire.

3.3. Acceleration, but not orientation, impacts participants’ estimation of magnitude and speed estimation

Seeing that participants may have used a knowledge heuristic to infer duration, distance, and speed, we expected that Acceleration and Orientation may not massively affect magnitude and speed estimations. To verify this, we fitted a linear model with the two factors Acceleration (3: positive, null, negative) and Orientation (2: forward, backward) to each type of relative estimation (duration, distance, instantaneous, and average speed).

The linear models fits were significant for all but for duration estimation although the variance explained was systematically low: duration (R ² = .03, F(5, 225) = 1.19, p = .313, adj. R ² = 4.19e‐03), distance (R ² = .39, F(5, 223) = 28.56, p < .001, adj. R ² = .38), instantaneous speed (R ² = .11, F(5, 224) = 5.37, p < .001, adj. R ² = .09), and average speed (R ² = .43, F(5, 224) = 33.99, p < .001, adj. R ² = .42).

Nevertheless, significant effects of the Acceleration factor were found for the distance, instantaneous, and average speed estimates but not for duration estimates (Fig. 4a). For duration, Acceleration had no significant effect on the relative estimated duration (F(2, 225) = 2.17, p = .116; η ² _p = 0.02, 95% CI [0.00, 1.00]). However, Acceleration significantly altered distance estimates (F(2, 223) = 69.34, p < .001; η ² _p = 0.38, 95% CI [0.30, 1.00]) so that participants overestimated distances during positive acceleration compared to constant speed (beta = 1.33, SE = 0.12, t(223) = 11.34, p < .001) and deceleration (beta = 1.53, SE = 0.16, t(223) = 9.77, p < .001). No differences of Acceleration were found between constant speed and deceleration (beta = 0.20, SE = 0.12, t(223) = 1.62, p = .239).

Fig. 4.

Main effect of the Acceleration (positive, null, negative) on magnitudes and speed estimations. (A) Box plots showing the distribution of each relative estimation (blue: duration; red: distance; green: instantaneous speed; yellow: average speed) for each acceleration. The median line inside the box indicates the median of the distribution, while the upper and lower boundaries of the box represent the third and first quartiles, respectively. Whiskers extend to the minimum and maximum values, not exceeding 1.5 times the interquartile range from the box. Estimations beyond the whiskers are shown as individual points. Significant differences between the distributions are indicated with an asterisk. Nonsignificant differences between the distributions are indicated with n.s. (B) Scatter plot of the relative estimation (same color code as A) according to their objective values. The acceleration episodes are indicated by a change in transparency. Interestingly, we can see that the patterns captured by acceleration episodes match the patterns observed with the objective value.

Acceleration significantly altered estimates of instantaneous speed (F(2, 224) = 12.17, p < .001; η ² _p = 0.10, 95% CI [0.04, 1.00]) so that participants overestimated instantaneous speed during positive acceleration compared to constant speed (beta = 0.18, SE = 0.05, t(224) = 3.45, p = .002) but not compared to deceleration (beta = −0.02, SE = 0.07, t(224) = −0.38, p = .925). At constant speed (null acceleration), participants significantly underestimated instantaneous speed compared to deceleration (beta = −0.20, SE = 0.05, t(224) = −3.98, p < .001).

Acceleration altered the estimation of average speed (F(2, 224) = 84.70, p < .001; η ² _p = 0.43, 95% CI [0.35, 1.00]), which were overestimated during positive acceleration compared to null (beta = 1.05, SE = 0.09, t(224) = 11.07, p < .001) and negative (beta = 1.16, SE = 0.12, t(224) = 9.67, p < .001) accelerations. No differences were found between the null and the negative accelerations (beta = 0.10, SE = 0.09, t(224) = 1.19, p = .462). Interestingly, the overestimation of average speed follows closely the effect of objective magnitudes so that the faster the average speed is, the smaller the overestimation (Fig. 4b).

The factor Orientation did not significantly affect the estimations of magnitudes or speed (Section S2.4, Fig. S4): duration (F(1, 225) = 0.91, p = .342; η ² _p = 4.02e‐03, 95% CI [0.00, 1.00]), distance (F(1, 223) = 4.53e‐03, p = .946; η ² _p = 2.03e‐05, 95% CI [0.00, 1.00], instantaneous speed (F(1, 224) = 0.45, p = .504; η ² _p = 1.99e‐03, 95% CI [0.00, 1.00]), and average speed (F(1, 224) = 0.43, p = .515; η ² _p = 1.89e‐03, 95% CI [0.00, 1.00]).

We also found no significant interactions between Acceleration and Orientation for any type of estimation: duration (F(2, 225) = 0.36, p = .701; η ² _p = 3.15e‐03, 95% CI [0.00, 1.00]), distance (F(2, 223) = 2.06, p = .129; η ² _p = 0.02, 95% CI [0.00, 1.00]), instantaneous speed (F(2, 224) = 1.04, p = .354; η ² _p = 9.24e‐03, 95% CI [0.00, 1.00]), and average speed (F(2, 224) = 0.08, p = .926; η ² _p = 6.82e‐04, 95% CI [0.00, 1.00])).

For completeness, and data exploration, a similar analysis testing the effect of Acceleration and Orientation on PoTJs was conducted (Section S2.5). We found no significant effects of Acceleration or Orientation on the felt speed of time (Fig. S5).

3.4. Lack of evidence of a link between duration estimation and emotional state or passage of time in high‐speed trains

We then investigated the relation between the felt speed of the passage of time (asked with a PoTJ), an alternative measure of the subjective time estimation (e.g., Wearden, 2015), and self‐reported emotional ratings (e.g., Droit‐Volet & Meck, 2007), considering that both measurements have been previously linked to duration estimations.

First, we investigated the link between relative duration estimates and emotional states (happiness, sadness, and boredom) by computing three Pearson partial correlations for each emotional state. Partial correlation allows us to test the correlation between our measure of interest (here, duration estimation) and a factor (one emotional state) while taking into account the influence of other factors (the two other emotional states here). We found no significant correlations between the relative duration estimates and the self‐reported emotional states: happiness (r_p ​(219) = −.04, p = .556, 95% CI [−0.17, 0.09]; Fig. 5a), sadness (r_p ​(219) = −.04, p = .543, 95% CI [−0.17, 0.08]; Fig. 5b), and boredom (r_p ​(219) = .10, p = .141, 95% CI [−0.03, 0.23]; Fig. 5c). An additional analysis corroborating these results is reported in Supplementary Material, Section S2.6.

Fig. 5.

Emotional states alter the speed of passage of time, not duration. Boxplots of the relative duration estimation as a function of the level of (A) happiness, (B) sadness, (C) boredom, and (G) passage of time feeling self‐reported by the participants on a 4‐point Likert scale: (A) from “not happy at all” to “very happy”; (B) from “not sad at all” to “very sad”; (C) from “not at all” to “very much”; (G) from ”slow” to “fast”. The median line inside each box indicates the median of the distribution, while the upper and lower boundaries of the box represent the third and first quartiles, respectively. Whiskers extend to the minimum and maximum values, not exceeding 1.5 times the interquartile range from the box. Estimations beyond the whiskers are shown as individual points. (D) Scatter plot of the self‐reported speed of the passage of time (PoTJ) as a function of the self‐reported happiness level. (E) Scatter plot of the PoTJ as a function of the self‐reported sadness level. (F) Scatter plot of the PoTJ as a function of the self‐reported boredom level. (D−F) The size of the dot corresponds to the number of participants at each level for both axes. The actual number of participants at each level is noted in black. Horizontal black bars and stars indicate significantly different levels of emotional states and n.s. indicates nonsignificant contrast.

Second, to investigate the link between duration estimates and PoTJ, we computed an OLS linear regression model with the relative duration estimations as the dependent variable, and the PoTJ level as the explanatory variable. We found no significant relations between duration estimations and PoTJ (R ² = .01, F(3, 224) = 0.92, p = .433, adj. R ² = −1.07e‐03; Fig. 5g).

Finally, we investigated the link between PoTJ and emotional states (happiness, sadness, and boredom) by computing three Pearson partial correlations for each emotional state. We found a significant relationship between PoTJ and boredom only (r_p ​(219) = −.33, p < .001, 95% CI [−0.45, −0.21]; Fig. 5f), not for sadness (r_p ​(219) = −.02, p = .822, 95% CI [−0.146, 0.116]; Fig. 5e) nor for happiness (r_p ​(219) = −.06, p = .391, 95% CI [−0.07, 0.19]; Fig. 5d). Post‐hoc comparisons revealed that the boredom level 1 was different from the level 2 and 3, the boredom level 2 was different from the level 3 (Table 1 and Fig. 5f). An additional analysis corroborating these results is reported in Supplementary Material, Section S2.7.

Table 1.

Statistical results of PoTJ for each contrast of emotional level per type of emotional state

Emotional state	Contrast	Beta	s.e.m.	DoF	z‐value	p
Boredom	“1” – “2”	0.99	0.32	1	3.06	.012*
	“1” – “3”	2.27	0.39	1	5.87	< .001*
	“1” – “4”	−0.57	1.15	1	−0.49	.961
	“2” – “3”	1.28	0.41	1	3.16	.009*
	“2” – “4”	−1.56	1.18	1	−1.32	.547
	“3” – “4”	−2.84	1.20	1	2.37	.083

Note. Estimation type indicates the type of estimation for each row. Contrast corresponds to the contrast of the row by indicating which level of the 4‐point Likert scale is compared. The s.e.m. is the standard error to the mean. Dof is the degree of freedom. P‐values were adjusted with the Tukey method to take into account multiple comparisons. Stars and bold p‐values indicate significant p‐values lower than .05.

3.5. Confidence rating and temporal metacognition onboard

Last, we investigated participants’ confidence ratings on all estimations and their temporal metacognition. First, we tested whether the self‐reported confidence level on a 4‐point Likert scale would reflect participants’ accuracy, quantified as the absolute relative error of the participants’ estimations (Yallak & Balcı, 2022). For each estimation type (duration, distance, instantaneous speed, average speed), we computed, separately, an OLS regression model with the absolute relative error as the predicted variable and the confidence level (4‐point Likert scale) as the predictor.

For duration estimates, the confidence level significantly affected the absolute relative error (F(3, 229) = 12.02, p < .001; η ² = 0.14, 95% CI [0.07, 1.00], Fig. 6a). Post‐hoc comparisons between each confidence level revealed that only the contrast “rather confident”−“very confident” was not significant (Table 2).

Fig. 6.

Metacognitive judgments. (A−D) Confidence ratings in magnitude and speed estimates. Boxplot of the absolute relative error as a function of the confidence level reported by participants using a 4‐point Likert scale (from “not at all confident” to “very confident”) for duration (A, blue), distance (B, red), instantaneous speed (D, green), and average speed (D, yellow). (E) Temporal metacognitive judgments. Boxplot of the relative estimated duration as a function of the direction of the duration error. The median line inside the box indicates the median of the distribution, while the upper and lower boundaries of the box represent the third and first quartiles, respectively. Whiskers extend to the minimum and maximum values, not exceeding 1.5 times the interquartile range from the box. Individual points are individual estimates. Stars and horizontal black lines correspond to significant post‐hoc contrast.

Table 2.

Statistical results of absolute relative estimation for each contrast of confidence level per type of estimation

Estimation type	Contrast	Beta	Standard error	Degree of freedom	t‐value	p
Duration	“not at all confident” – “not confident”	0.26	0.09	229	2.97	.017*
	“not at all confident” – “rather confident”	0.37	0.08	229	4.53	< .001*
	“not at all confident” – “very confident”	0.45	0.09	229	5.10	< .001*
	“not confident” – “rather confident”	0.11	0.04	229	3.10	.012*
	“not confident” – “very confident”	0.19	0.05	229	3.92	< .001*
	“rather confident” – “very confident”	0.08	0.04	229	1.99	.193
Distance	“not at all confident” – “not confident”	0.11	0.16	227	0.65	.915
	“not at all confident” – “rather confident”	0.14	0.16	227	0.86	.825
	“not at all confident” – “very confident”	−0.43	−0.37	227	−1.15	.658
	“not confident” – “rather confident”	0.03	0.09	227	0.38	.981
	“not confident” – “very confident”	−0.53	0.34	227	−1.55	.410
	“rather confident” – “very confident”	−0.57	0.34	227	−1.65	.353
Instantaneous speed	“not at all confident” – “not confident”	0.04	0.04	224	0.94	.781
	“not at all confident” – “rather confident”	0.11	0.04	224	2.88	.023*
	“not at all confident” – “very confident”	0.16	0.06	224	2.49	.063
	“not confident” – “rather confident”	0.07	0.02	224	3.26	.007*
	“not confident” – “very confident”	0.12	0.06	224	2.20	.126
	“rather confident” – “very confident”	0.05	0.06	224	0.87	.818
Average speed	“not at all confident” – “not confident”	0.05	0.12	227	0.41	.977
	“not at all confident” – “rather confident”	−0.02	0.13	227	−0.20	.997
	“not at all confident” – “very confident”	0.05	0.26	227	0.20	.997
	“not confident” – “rather confident”	−0.08	0.07	227	−1.02	.738
	“not confident” – “very confident”	0.01	0.24	227	0.01	1.0000
	“rather confident” – “very confident”	0.08	0.24	227	0.33	.988

Note. Estimation type indicates the type of estimation for each row. P‐values were adjusted with the Tukey method to take into account multiple comparisons. Stars and bold p‐values indicate significant p‐values lower than .05.

For distance estimates, there was no significant effect of the confidence level on the absolute relative error (F(3, 227) = 1.09, p = .355; η ² = 0.01, 95% CI [0.00, 1.00], Fig. 6b). As expected, none of the post‐hoc contrasts were significant (Table 2).

For the instantaneous speed, the confidence level significantly affected the absolute relative error (F(3, 224) = 5.85, p < .001; η ² = 0.07, 95% CI [0.02, 1.00], Fig. 6c). Post‐hoc contrasts were significant for the “not at all confident”−“rather confident” and “not confident”−“rather confident” (Table 2).

For average speed, there was no significant effect of the confidence level on the absolute relative error (F(3, 227) = 0.36, p = .779; η ² = 4.78e‐03, 95% CI [0.00, 1.00], Fig. 6d). As expected, none of the post‐hoc contrasts were significant (Table 2).

Second, we explored participants’ temporal metacognition and tested whether they could report the direction of their duration error as in a trial‐by‐trial design (Kononowicz & van Wassenhove, 2019). We computed an OLS linear model to predict the relative estimated duration by the self‐reported direction error (2‐level factor: “underestimated”; “overestimated”). When participants reported “underestimated,” their relative duration estimates should be below 1; when participants reported “overestimated,” their relative duration estimates should be above 1. Although the variance explained was weak, the model was statistically significant (R ² = .04, F(1, 227) = 9.15, p = .003, adj. R ² = .03). The relative duration estimates were lower when participants reported “underestimated” compared to when they reported “overestimated” (beta = 0.10, 95% CI [0.04, 0.17], t(227) = 3.02, p = .003; Std. beta = 0.39, 95% CI [0.14, 0.65]; Fig. 6e).

Finally, we tested whether the Acceleration and Orientation factors influenced participants’ temporal metacognition. We constructed an ordinal linear model with the self‐reported direction error (4‐point Likert scale) as the predicted variable, and Acceleration and Orientation as the predictive factors. We found no main significant effects of Orientation (LR Chisq = 2.78, Df = 1, p = .095) nor Acceleration (LR Chisq = 2.27, Df = 2, p = .321) on temporal metacognitive performance. However, their interaction was significant (LR Chisq = 7.18, Df = 2, p = .028). In the interaction, the difference Backward – Forward was significant only during the positive acceleration episode (beta = 1.65, SE = 0.80, z(228) = 2.05, p = .040), meaning that during acceleration, participants reported overestimating more when facing backward than when facing forward.

4. Discussion

Our study aimed at characterizing the estimation of traveled distance, elapsed duration, and speed (both average and instantaneous) during a high‐speed train journey. We found that magnitude estimations presented features of classical magnitude profiles such as the central tendency effect, despite using a between‐subject design. Instantaneous and average speed profiles were less reliable. Average speed profiles did not show a scalar magnitude profile and did not result from the simple ratio of distance by duration estimates. Additional investigations revealed that the link between distance and duration estimation was better explained by using participants’ prior knowledge rather than their on‐the‐fly estimates. Congruent with this, the seating orientation had no significant effect on magnitude or speed estimates, although acceleration partially did so, with a positive acceleration resulting in an overestimation of magnitudes compared to a null (constant high‐speed) or a negative acceleration (deceleration). Consistent with the accessibility of magnitude and speed information to high‐level cognition, participants showed accurate confidence levels with task performance in both magnitudes and speed estimation. Additionally, participants showed the ability to perform well on temporal metacognitive judgments with a partial effect of acceleration. Surprisingly, the estimation of duration was not related to the felt speed of time or to participants’ self‐reported emotional states. Below, we discuss the implications of our findings for the representation of mental magnitudes and speed in real‐life situations.

4.1. Mental magnitudes regress to the midpoint of the journey

We found that passengers could accurately estimate distance and duration when passively transported at high‐speed that exceeds their biological endowment. Consistent with typical magnitude estimation profiles reported in the laboratories, we found that duration and distance estimation scaled with their objective magnitude and displayed a regression effect or central tendency toward the mean (Hollingworth, 1910; Stevens & Greenbaum, 1966). Regression effects and central tendency are typically accounted for by the range of stimuli tested in a given experimental context (Jazayeri & Shadlen, 2010; Petzschner & Glasauer, 2011; Petzschner et al., 2015; Glasauer & Shi, 2021). The central tendency typically lies around the midpoint of the distribution (Hollingworth, 1910; Stevens & Greenbaum, 1966) in a within‐subjects design (i.e., multiple samples per participant).

Here, however, we tested a between‐subjects design in which data points were independently sampled (i.e., one sample per participant). Previous studies using a similar retrospective approach (Chaumon et al., 2022; Balcı et al., 2023) reported a regression point around 15 min, suggesting a common reference duration for all individuals (and across cultures) for tasks conducted in the wild. In these and the present study, participants were tested once and did not know a priori that they would be asked to estimate the duration (and herein, the distance) that has just elapsed. Hence, no cognitive strategies to count time (and herein, space) could be set in place by participants before and going forward into the test. In our study, we could have expected similar outcomes for the regression point, namely, a central tendency around a quarter of an hour. However, this is not what we found. Our study instead showed regression points located at ∼69 min for duration and ∼265 km for distance (Fig. 2a,b). A major difference between the studies is contextual: in prior work carried out online (Chaumon et al., 2022; Balcı et al., 2023), participants were not informed of the duration of the tests and had no a priori knowledge of how long the tests were going to last in total. In the current design, travelers knew in advance the duration of their trip, and to some extent its distance, as they explicitly self‐reported prior to the tests. Their regression points (at the population level) fell nearly halfway through the trip for both magnitudes, suggesting a cognitive heuristic for the retrospective estimation of magnitudes. Additionally, and in contrast to typical prospective timing tasks, we found no evidence for scalar variability in retrospective estimates (Balcı et al., 2023). This may provide a quantitative distinction between prospective and retrospective duration estimation (Hicks, Miller, & Kinsbourne, 1976; Hicks, 1992) in real‐life situations, converging with recent work showing a neural dissociation between prospective and retrospective estimations (Azizi, Polti, & van Wassenhove, 2023). From our current findings, the observation can be extended to distance estimation, thereby highlighting a possible novel role of attentional orientation to time and space, and its effect on the scalar noise of magnitude estimation. Alternatively, a Bayesian framework in which the total duration of an event is known may also predict a central tendency scaled to the 50% point of the total duration, and the masking of scalar variability due to the bounded maximum of the known duration. This, however, was not tested in this study and could feed additional modeling work. Lab or virtual reality experiments could further explore this possibility in finer details. Additionally, in the absence of a generative model but possibly compatible with a Bayesian framework, we develop below one possible interpretation for this observation.

One possible limitation of the experience sampling approach is that participants may have kept track of time prior to filling out the questionnaires. If the majority of participants frequently looked at the time (e.g., their phone) during their journey, this could have explained the near midpoint regression point, but we tried to control for it with the reported strategy question at the end of our questionnaire. Instead, we suggest that prior declarative knowledge allows anchoring magnitude estimation by setting prior boundaries on the duration and distance of the trip, in a cognitive route kind of way. As participants were recruited on the fly, they had no incentive to keep precisely track of the distance (and arguably, of the duration) of their trip prior to being asked these questions. It is, therefore, likely that a first approximation would call about prior knowledge, which we collected before the questionnaire and which helped scale magnitude estimations. Assuming that participants built a cognitive map of their journey, the two main temporal landmarks demarcating the trip are the departure and arrival in both time and space. Prior declarative knowledge may thus anchor magnitude estimations, resulting in more precise and accurate estimations toward the end of the journey—which is contrary to the prediction of scalar variability. A trend for this pattern can be observed in Fig. 4b for duration and distance estimation. One possible alternative, in a Bayesian framework, would be that scalar variability remains but is masked by the reduced variability at the end of the travel because of the bounded priors. To validate further this working hypothesis, various contexts with different known total durations could be explored. Additionally, the regression point would be half the Euclidean distance between these two spatiotemporal landmarks which, bounded by an accurate knowledge of the trip, approximates the middle of the actual duration and distance of the journey. Similarly, in Bayesian terms, it is plausible that the 50% point be a result of the bounded magnitudes being tested. Note that in such a Bayesian framework, the use of bounded graphical estimations could implicitly encourage participants to rely on bounded priors, even for numerical estimations, due to the initial questions regarding prior knowledge. This could facilitate the observation of the 50% regression point in our data.

Our cognitive heuristic hypothesis is falsifiable. One prediction is that pending participants have an accurate prior belief of the duration and distance of a trip (or any activity), a similar regression to the midpoint should be found, scaled to the actual duration or distance of the activity. A second prediction is that blinding participants as to which journey they are embarking on (thereby preventing access to prior knowledge) should revert to a sampling heuristic approximating the 15 min rate observed across cultures (Chaumon et al., 2022; Balcı et al., 2023). The cognitive map heuristics adopted from declarative knowledge may explain why scalar variability was not observed in our experiment, although it was seen in previous reports using the same duration estimation task in the absence of declarative knowledge about the full task duration (Chaumon et al., 2022; Balcı et al., 2023). Complementary to this, a Bayesian framework including bounded prior knowledge may help decipher the origin of our empirical observations, and this would be interesting to explore in a more general context of magnitude estimations, making use of symbols and cognitive maps (i.e., the fourth level in the recently proposed “navigation toolbox”; Jeffery et al., 2024).

4.2. Heuristics in the estimation of magnitudes and speed

Participants displayed accurate estimations of magnitudes, suggesting that they should have minimally been able to compute the average speed. Our modeling approach suggested otherwise in that the estimated magnitudes were surprisingly not predictive of participants’ average speed accuracy (Figs. 2d and 3a). Speed estimates could also not be accounted for by systematic biases between estimated durations and distances (Fig. 3b). The fact that participants were so imprecise in their speed estimates, despite accurate magnitude estimations, raises the question of what links these internal representations.

As previously alluded to, duration (time) and speed (rate) have been proposed to underlie the representation of distance during movement (Gallistel, 1989, 1990). In self‐motion, the reproduction of a distance traveled is as accurate as its duration (Berthoz, Israël, Georges‐François, Grasso, & Tsuzuku, 1995), yielding the proposal that traveled distance is computed as the integral of the velocity of motion derived by the duration of motion (Berthoz et al., 1995; Seemungal, Glasauer, Gresty, & Bronstein, 2007). The derivation of spatial estimates from time estimates has also been deemed compatible with internal models of self‐motion (Green, Shaikh, & Angelaki, 2005). When vestibular information is insufficient—as in the passive linear vection in high‐speed trains tested here—time estimation can be used to approximate traveled distance (Israël et al., 2004). Compatible with these views, a recent model building on the spatial navigation system of the brain provides an account of timing estimation resulting from the integral of the rate of changes experienced during self‐motion (Issa et al., 2020).

Here, during passive motion and as predicted in the introduction from a vector computation view, we found that the model relying on prior declarative knowledge of the journey was most successful in predicting performance outcomes (Fig. 3c,d). The R ² was also larger for the distance explained by the duration, and the participant's a priori knowledge of the full distance than for the duration explained by the distance, and the participant's a priori knowledge of the full duration (Fig. 3c,d). This result appears consistent with the notion that time (duration) was used by participants to estimate distance, not the contrary. Another piece of evidence favors this temporal heuristic when looking at participants’ metacognitive ratings of self‐confidence: participants provided confidence levels that were well aligned with their accuracy for duration estimation, but much less so for distance and speed estimates.

As previously noted before, we cannot exclude that some participants accessed temporal information before the questionnaire, favoring a temporal strategy for distance estimation, and possibly explaining why duration estimations were so accurate compared to previous reports in this range of durations (Tobin, Bisson, & Grondin, 2010). Nevertheless, the current results are consistent with the setting up of an elaborate internal map of the train journey, in which time, rather than space, may be tracked.

4.3. The absence of strong acceleration and seating orientation (optic flow) effects

Another aim of the study was to investigate how available sensory information impacts magnitude estimation. Two types of sensory information were factorized in the In Situ design (Vallet & van Wassenhove, 2023): the seating orientation (modulating the optical flow as forward or backward) and the acceleration, which could be positive (acceleration at the onset of the journey), null (constant high‐speed), or negative (deceleration at the end of the journey). Within the path integration view, our working hypothesis was that acceleration and optical flow would both impact participants’ estimation of duration.

In laboratory experiments, it is well known that participants overestimate the duration of a moving stimulus (Brown, 1995), whose speed positively correlates with subjective time (Kanai & Watanabe, 2006). This effect has been reported when the environment moves around the stimulus (Mate, Pires, Campoy, & Estaún, 2009), suggesting that the optic flow could yield a similar effect. In a realistic task using a car simulator, increasing the speed of the optic flow yielded participants to overestimate duration (van Rijn, 2014). However, a recent study showed that the velocity and the density of events in virtual reality did not affect the estimation of duration and rather impacted the speed of the passage of time (Jording, Vogel, Viswanathan, & Vogeley, 2022), consistent with prior reports highlighting the resilience of duration estimates for visual motion and speed changes (Martin, Wiener, van Wassenhove, 2019). Herein, we effectively found no significant effects of seating orientation or of acceleration on duration estimation or felt passage of time.

To the contrary, a small but significant effect of acceleration was found on distance and speed estimations, yielding an overestimation when the acceleration was positive. However, this effect may well be confounded with the regression to the mean effect with a local overestimation of small magnitudes located at the beginning of the journey, when acceleration occurred (Fig. 2). In the current experimental design, small magnitude estimation and positive acceleration were collinear, preventing us from making a strong inference on the possible effect of acceleration on these estimations although the fact remains that duration was not affected by it. Future work could overcome this issue and test refined psychometric tasks on separate individuals.

4.4. Dissociation of duration and passage of time

The relation between the felt speed of the passage of time and duration estimation is a current area of investigation, as PoTJs have been argued to better capture participants’ subjective phenomenology in real life than duration (Wearden et al., 2014; Wearden, 2015; Droit‐Volet & Wearden, 2016). Converging with this idea, clinical observations suggested that depressive patients experience time as passing slowly yet display no changes in their estimation of duration compared to a healthy group (Thönes & Oberfeld, 2015). However, the two measures are not entirely disconnected: for instance, longer durations are perceived to pass slower than shorter durations (Droit‐Volet, Trahanias, & Maniadakis, 2017; Nicolaï et al., 2024), time errors increase with the slower speed of time (Martinelli & Droit‐Volet, 2022), and the density of events yields an overestimation of duration but speeds up subjective time‐in‐passing (Lamprou‐Kokolaki et al., 2024).

Herein, we found a lack of evidence of a link between changes in subjective duration and participants self‐report of time passing faster or slower (Fig. 5). One possibility is that the implicit scale at which participants formulated their judgments differed: durations were asked since the departure of the train, whereas speeds of the passage of time estimations may have been answered in a more instantaneous fashion despite our instructions. Given the relation between duration scales and the perceived speed of the passage of time (Droit‐Volet et al., 2017; Nicolaï et al., 2024), this may have affected the outcomes. One possible limitation of our protocol is that we used a 4‐point Likert scale, while previous studies typically used a 5‐ or 7‐point Likert scale. Most of our participants used the middle option of the Likert scale, reducing our measure to a 2‐point Likert scale. However, this pattern has been reported in other experiments showing the distribution of the data as well (Chaumon et al., 2022; Nicolaï et al., 2024).

Additionally, we found no relation between duration estimations and emotional states of happiness, sadness, and boredom self‐reported by participants. There was a significant relationship between the felt speed of time and the self‐reported boredom: when participants reported not being bored (boredom level 1), they tended to rate the speed of time as passing faster than when they reported being bored (boredom level 4). All in all, our results suggest that mechanisms for duration estimation and passage of time may be partially distinct and may rely on distinct information than what has been typically reported.

4.5. Metacognition in the wild

The metacognitive judgments in this study were exploratory and designed to test whether metacognitive assessments can be realized in such a complex environment with naïve participants. In line with a magnitude system, participants tested in the lab can monitor their metric error for duration, distance, and numerical judgments (Yallak & Balcı, 2021, 2022). While this is typically assessed using confidence ratings (absolute errors of the estimation), it can also be evaluated using signed errors (Kononowicz & van Wassenhove, 2019). We tested the typical confidence ratings for all estimations, and for duration estimation, we also included the signed error estimation.

In our study, participants could access the magnitude of their duration and instantaneous speed errors, with a negative relationship between confidence reports and absolute relative errors, confirming that the larger the errors, the less confident participants were in their estimates. Strangely, no such relationship was found for distance and average speed errors. Participants could also accurately report the direction of their temporal errors in the duration estimation, supporting the fact that participants do have a timing error mechanism monitoring even for long retrospective durations (Öztel et al., 2021). These results further strengthen the idea that time and possibly instantaneous speed are favored primary representations in ecological situations, even when participants are not in self‐motion per se, but passively transported. Under this experimental context, and contrary to the expectations of a common magnitude system and a unique error monitoring system, our results suggest that duration, but not distance, could be used for metacognitive evaluation. One possibility is that in the ecological context tested here, time was a favored dimension whose estimation was more natural than distance estimation. Further explorations would be needed to be more conclusive on these aspects, considering the originality of the paradigm tested here compared to typical experiments in the laboratory.

5. Conclusions

To the best of our knowledge, this is the first study investigating magnitudes and speed perception in the context of a train journey (passive linear vection). We report several measures suggesting that the temporal dimension of passengers’ experience is an essential primitive onto which the perception of distance may be built. Our ecological approach provided novel results, such as the scaling of the central tendency to the midway of the journey for both magnitudes. This observation suggests a cognitive heuristic based on landmarks. Additionally, our results are congruent with the notion that time and speed are primitive computations that may mediate the estimation of spatial distances, as has been proposed during self‐motion and vector computation.

Funding

This work was supported by the Projet‐ANR‐18‐CE22‐0016 to V.v.W.

Conflicts of interest

The authors declare no conflicts of interest.

Ethics approval statement

The study was validated by the Ethics Committee for Research (CER) of Paris‐Saclay University (CER‐Paris‐Saclay‐2018‐034‐A2).

Supporting information

Supporting Information

Data availability statement

All data, code, analyses, and linguistic materials are accessible at https://doi.org/10.5281/zenodo.14895063.