Extended Data Fig. 3: Repertoire dating and the direction of slow change.

a, The four nearest neighbours for example vocalizations (bird 4, syllable b, from Fig. 1a). Production times of nearest neighbours (numbers) need not equal that of the corresponding example rendition. b, Neighbourhood production times for three renditions from day 70 (analogous to Fig. 2a, inset). Rendition 2 is ‘typical’ for day 70 (most neighbours lie in the same or adjacent days); renditions 1 and 3 are a ‘regression’ and an ‘anticipation’ (with neighbours predominantly produced in the past or future, respectively). c, All renditions from day 70 (a subset of the points in Fig. 2a). Colours correspond to repertoire time (50th percentile in Fig. 2d). Anticipations (repertoire times greater than 70) and regressions (repertoire times less than 70) occur at locations corresponding to vocalizations typical of later and earlier development (compare with Fig. 2a). Numbers 1–3 mark the approximate locations of the example renditions in b. d, Mixing matrices for additional birds (analogous to Fig. 2e, using the same birds as in Extended Data Fig. 2d). Bird 3 produced only a very few vocalizations (mostly calls) before tutoring onset (black arrows). The mixing matrices consistently show a period of gradual change starting after tutor onset and lasting several weeks. This gradual change typically slows down (resulting in larger mixing values far from the diagonal) at the end of the developmental period considered here (day 90 post-hatch; later periods are in Extended Data Fig. 6). Grey values correspond to the base-2 logarithm of the mixing ratio (LMR), that is, histograms over the pooled neighbourhood times (Fig. 2c) normalized by a null hypothesis obtained from a random distribution of production times (see Supplementary Methods). For example, an LMR value of 5 implies that renditions from the corresponding pair of production times are 2⁵ = 32 times more mixed at the level of local neighbourhoods than would be expected by chance (that is, there is a random distribution of production times across renditions). e, As in d, bird 2, but after shuffling production times among all data points. Effects under this null hypothesis are small (the maximal observed mixing ratio is 2^0.06 or approximately 1.042). Similar, small effects under the null hypothesis are obtained for the other mixing matrices discussed throughout the text. f–h, Properties of the behavioural trajectory inferred from the mixing matrix in Fig. 2f. f, Pairwise distances between points along the inferred behavioural trajectory (x axis), plotted against measured disparities (y axis). Disparities are obtained by rescaling and inverting the similarities in Fig. 2f (see Supplementary Methods). The points on the trajectory are inferred with ten-dimensional non-metric MDS on the measured disparities. Importantly, the pairwise distances between inferred points faithfully represent the corresponding, measured disparities (all points lie close to the diagonal; MDS stress = 0.0002). g, h, Structure of low-dimensional projections of the behavioural trajectory. We applied principle-component analysis to the ten-dimensional arrangement of points inferred with MDS and retained an increasing number of dimensions (number of dimensions indicated by greyscale). For example, the projection onto the first two principle components is shown in Fig. 2h (MDS dimension 2 in g, h). The first two principle components explain 75% of the variance in the full ten-dimensional trajectory. g, Measured (true) disparity (thick grey curve) and distances along the inferred trajectories (points and thin curves) as a function of the day gap (δ) between points. For any choice of projection dimensionality and δ, we computed the Euclidean distances between any two points separated by δ and averaged across pairs of points. The measured (true) disparities increase rapidly between subsequent and nearby days, but only slowly between far apart days (thick grey curve). Low-dimensional projections of the trajectory (for example, MDS dimension 2) underestimate the initial increase in disparities. h, Angle between the reconstructed direction of across-day change for inferred behavioural trajectories, as a function of the day gap between points. Same conventions and legend as in g. For the one- and two-dimensional trajectories, the direction of across-day change varies little or not at all from day to day (see inset; the arrow indicates the angle of across-day change). On the other hand, the direction of across-day change along the full, ten-dimensional behavioural trajectory is almost orthogonal for subsequent days. Data shown in g, h suggest that the full behavioural trajectory is more ‘rugged’ than indicated by the two-dimensional projection in Fig. 2h. This structure is consistent with the finding that across-day change includes a large component that is orthogonal to the directions of slow change and of within-day change (Fig. 3j). Note that a shows 200-ms spectrogram segments, whereas b–h are based on 68-ms segments (as are most of the analyses).