Temporal fractals in movies and mind

Abstract

Fractal patterns are seemingly everywhere. They can be analyzed through Fourier and power analyses, and other methods. Cutting, DeLong, and Nothelfer (2010) analyzed as time-series data the fluctuations of shot durations in 150 popular movies released over 70 years. They found that these patterns had become increasingly fractal-like and concluded that they might be linked to those found in the results of psychological tasks involving attention. To explore this possibility further, we began by analyzing the shot patterns of almost twice as many movies released over a century. The increasing fractal-like nature of shot patterns is affirmed, as determined by both a slope measure and a long-range dependence measure, neither of which is sensitive to the vector lengths of their inputs within the ranges explored here. But the main reason for increased long-range dependence is related to, but not caused by, the increasing vector length of the shot-series samples. It appears that, in generating increasingly fractal-like patterns, filmmakers have systematically explored dimensions that are important for holding our attention-shot durations, scene durations, motion, and sound amplitude-and have crafted fluctuations in them like those of our endogenous attention patterns. Other dimensions-luminance, clutter, and shot scale-are important to film style but their variations seem not to be important to holding viewers' moment-to-moment attention and have not changed in their fractional dimension over time.

Fig. 1

Three types of temporal noise. The top panels show 512-element samples of white (or random) noise, pink (or fractal) noise, and brown (or Brownian) noise. Slope values refer to the exponent (alpha) in the expression 1/f ^α and Whittle values refer to the exact local Whittle estimator of long-range dependency in the data (Shimotsu & Phillips, 2005). See the text for explanations of both. The bottom panels show the power spectra for each patch of noise for wavelengths (traveling windows along the time-series vector) between 2⁸ to 2¹ shots

Fig. 2

Waveforms for six dimensions of movies – a shot duration, b motion, c sound amplitude, d luminance, e clutter, and f shot scale – taken from the first 512 shots in six movies. The first and waveforms like it are the focus of Studies 1, 2, and 4; the latter five and waveforms like them from other movies are discussed in Studies 5 and 6. Slope = the value of alpha in 1/f ^α; Whittle = a fractional estimate of vector complexity. Both are discussed in the text

Fig. 3

Power spectra for the data (in blue) of and model fits (in red) to the shot-duration fluctuations of nine movies. These reflect a power analysis on the normalized shot vectors for each movie. Notice three trends: steeper-sloped movies tend to be more recent, more recent movies tend to have more shots, and model fits tend to be better for movies with more shots. These parallel trends form the focus of Studies 1–4

Fig. 4

Results of Studies 1 and 2 plotting values of fractal measurements per movie against release years and against number of shots per movie. Shown are four scatterplot results for movies released between 1915 and 2015. a Alpha values (slopes) of 263 movies as a quadratic function of release year (Study 1). Only the right half of that function fits the data well. b The exact Whittle estimate values for 295 movies as a linear function of release year (Study 2). c The slopes as a function of the number of shots in 263 movies (Study 1). d The Whittle estimates for 295 movies as a function of the number of shots (Study 2). Colored areas are 95% confidence intervals on the regressed fits

Fig. 5

Results of Study 3 investigating the relation between length of vectors and their fractal dimension. The panels show means (as points) and standard deviations (not confidence intervals) as shaded areas of noise simulations. Each point represents the mean of 1000 simulation trials. Noises were generated by algorithm (Little et al., 2007) and fit by the models used in Study 1 (measuring slope) and in Study 2 (measuring Whittle estimators by the method of Shimotsu & Phillips, 2005). In each panel, noises were generated with intended slopes of 0.0 to 2.0 in intervals of 0.25. Whittle values average about 43% of slope values in these simulations

Fig. 6

Results of Study 4 where the shot vectors of movies were doubled, concatenated end to end. Slopes (left) and Whittle values (right) for the doubled shot vectors are plotted against the undoubled results of Studies 1 and 2. The diagonal lines represent equal values in both measures; they are not regression lines

Fig. 7

Results of Study 5 plotting fractal dimension against release years for three movie variables. Left: A scatterplot of the increase in long-range dependence, measured by the exact local Whittle estimator and of the scene-duration vectors of 24 movies from 1940 to 2010. The regression line and 95% confidence intervals are also shown. The small error bars in the left panel indicate the standard deviations in generating 1000 pseudo-random sequences with the measured Whittle value and number of scenes for the given movie. Middle: The scatterplots of the fractal-like measure of motion in each shot for the shot vector in 180 movies. Right: A decline in long-range dependence for sound amplitude in sample vectors of 48 movies. All panels show a reliable change in long-range dependence over release years. The upper horizontal green line represents the approximate fractal value of 1/f ¹ (pink noise) as determined in Study 3; the lower line represents 1/f ⁰ (white noise)

Fig. 8

Results of Study 6 for Whittle estimates of three dimensions in the shots of movies. Left: The unchanging Whittle values for shot-luminance vectors across release year for 180 movies. Middle: The same measure for shot-clutter vectors for 180 movies. Right: Shot-scale vectors in 24 movies. The upper horizontal green line represents the approximate value of 1/f ² (brown noise) as determined in Study 3, the middle line represents the approximate value of 1/f ¹ (pink noise), and the lower line the value of 1/f ⁰ (white noise)