Let me respond to questions raised by Barry Salt in his second set of comments on our paper, AEHF (Cutting, DeLong, & Nothelfer (2010) “Attention and the evolution of Hollywood films”, Psychological Science, 21, 440-447, ). In doing so I will move on to some new analyses prompted by them.
Temporal Resolution: Our shot measurements were made to the nearest frame. But given Salt’s query I was prompted to reanalyze several of our films and also a number of those available on the cinemetrics database. I used various shot resolutions. Happily, I discovered that frame accuracy is not necessary for either our power spectrum results or our autoregression results. That is, computational outcomes are essentially identical when one uses shot lengths to the nearest frame, to the nearest 1/10 second, to the nearest second, and in cases for films with ASLs longer than 5 seconds even to the nearest 5 seconds and sometimes longer. In general, it would appear that our kinds of measured results are the same as long as the resolution is in the same ballpark as the ASL. To me this makes sense; it is the pattern of shorter and longer shots, not the variation within shorter or within longer shots, that matters. This is an idea not too far from Salt’s moving average analysis.
I also analyzed 20 films from the cinemetrics database that were also in our AEHF sample. Equally happily, even when the number of cinemetrics shots reported for a film differed from ours by as much as 20% (fewer or more), the power spectrum results were hardly affected. In sum, our analytic tools are strikingly robust against two different kinds of variations – resolution and accuracy.
An Incomplete Evolution Towards 1/f. Salt also asks “Why are films mostly still falling short of the maximum mAR and 1/f slope after 90 years of the use of standard film construction, if the postulated psychological effect is so powerful?” Reasonable question, although actually there is no maximum mAR and the slopes of power spectra can easily exceed 1/f (random walks, of which Brownian motion is a representative, are 1/f 2). But why the evolution is not complete was not the object of our report. Instead, the news was that there was an increasing trend towards 1/f over time. Insofar as we know, there is no evolutionary trend from near randomness towards 1/f that has yet been traced in any domain in any other science, let alone for any other art form.
More on Analyses of Films as Done in AEHF
In his reply Salt works to bring our results into alignment with the more normal measurements done in cinemetrics, and this is good. So let me explain a bit more about our analyses in the hope of bridging the gap even more. I will do this with two films (as measured by Salt) on the Cinemetrics database – The 39 Steps: (6) ASL 8.6 and Sunset Blvd.: (6) ASL 14.9. Salt used these as examples in his second commentary.
In the figure below are six panels, the top three for the former film and the bottom three for the latter. Consider the leftmost panels first. In Salt’s first commentary on our paper he mentioned the Lag 1 correlations for a few films. In my reply I suggested that the full autocorrelation function is more informative, and that Lags 1 alone can be a bit misleading. It seems prudent here to make good on that claim.
Autocorrelation Functions. What is autocorrelation good for? The full autocorrelation function can be used to see the relative strengths of shot length “waves” that course throughout a given film, related (but not identical) to the kinds of waves that Salt has shown with his moving average analyses. Autocorrelation is a process that measures the relationship (correlation, r) of a given shot length with subsequent shot lengths.
Lag 0 autocorrelation values are always 1.0. This is a correlation that compares every shot length against itself. Dull, but an integral part of the procedure. Nonetheless, for display purposes we’ve truncated the ordinates (the vertical axes) in panels a so that they don’t go up to the value of 1.0. This makes the remaining patterns more salient.
Interest begins with the Lag 1 autocorrelation, which Salt (2006, Moving into Pictures, p. 396) was first to report for films. It measures the relationship of the length of each shot with the next one – for example, Shots 1 with 2, 2 with 3, 3 with 4, …., 306 with 307, and so forth. Lag 2 correlates shots separated by one other shot – Shots 1&3, 2&4, 3&5, …, 305&307, and so forth. One continues this procedure at increasing lags. For panels a I truncated the autocorrelation function at Lag 256 for both The 39 Steps (565 shots) and Sunset Boulevard (491 shots).
Notice first the function (the jagged red line) for Sunset Boulevard at the bottom left. It bounces up and down quite noisily and seemingly without any particular pattern. Contrast its turbulence with the slightly more settled and articulated function for The 39 Steps at the top left. I’ve inserted two black arrows above that function. The first arrow (labeled x) is located at about Lag 75 and the second (labeled y) at about Lag 175. These arrows center on larger regions, which I will call “ripples,” in the autocorrelation function. Given that the ASL for The 39 Steps is about 8.6 s the peaks in these ripples occur on average about 10.8 and 25.1 minutes apart. Thus, these ripples represent distant shot correlations across the entire film (not just from the beginning to 10.8 and 25.1 minutes into the film).
The first ripple (x) is negative (correlations below zero) and it has a width that stretches from about Lag 50 to about Lag 100. The extent of the first ripple means that, throughout the course of the 565 shots in The 39 Steps, shots that are between about 50 and 100 shots apart (examples: 1&51, 1&52, through 1&101; 32&82, 32&83, through 32&132; and 225&275, 225&276, through 225&325; etc.) are slightly negatively correlated with one another. That is, where one shot is longer its mate is generally shorter than the average, and vice versa. This negative relation occurs across the length of the film. This ripple represents power (the measured height of the “wave” of shot lengths) in the power function, which I discuss below.
The second, smaller, and positive ripple (correlations above zero) concerns shot pairs separated by between about 160 and 190 other shots (examples: Shots 1&161 through 1&191; 32&192 through 32&212; and 225&385 through 225&415; etc.). These pairs will generally have a slightly positive correlation. That is, a longer shot will be generally matched with another one longer than average, and shorter ones with shorter ones. There are other, yet smaller, ripples in both autocorrelation functions but these two will suffice for my case.
Power Spectra. What is the power spectrum good for? The power spectrum is the Fourier analytic twin of the autocorrelation function. That is, despite apparent differences, both representations show the same relationships in the shot length data. The middle panels (panels b) show the power spectra of the two films. By convention these are plotted on log-log coordinates. On the horizontal axis represents the width of the shot window being considered, stepped out in powers of 2 for reasons explained in my previous response. By convention these are plotted so that the largest traveling window is on the left. On the vertical axis is power (proportional to the square of the amplitude, or “height”) of the different “waves” (both in the raw shot lengths as seen in the cinemetrics data and in the ripples in the autocorrelation function). So, the average heights of the 2-, 4-, and 8-shot windows as incrementally measured across the whole film are quite small; those for the 16-, 32-, and 64-shot windows are greater; and those for the 126- and 256-shot windows are the greatest. How much greater is important. For The 39 Steps the incremental increase in the height of these averaged waves is considerably more than for Sunset Boulevard. In particular, the ripples seen in the autocorrelation function for The 39 Steps cause its power function to be steeper; the relative lack of ripples in the Sunset Boulevard autocorrelation function causes its to be shallower.
The black functions shown in panels b are the raw power functions calculated from the data. Again, these are a bit noisy. The red functions are fits to those data with a model proposed by David Gilden (see D. L. Gilden 2001 “Cognitive emission of 1/f noise” Psychological Review, 108, 33–56). It is from this model that the values of the slopes are determined.
The slopes for the two films taken from the cinemetrics data are 0.87 and 0.28. The negative numbers for the slopes (α) in panels b indicate that the functions descend from left to right. Confusingly, by convention these are often reported as positive values (we did so in AEHF) because they represent the alpha parameter in the function 1/f α. When a positive exponent is in the denominator it is the same thing as a negative exponent in the numerator, and vice versa (1/f α = f -α). In AEHF we reported the values for The 39 Steps and Sunset Boulevard to be 0.93 and 0.26, pretty close to what I found here.
Partial Autocorrelation and mAR Measures. Salt opens his second response with statements about our mAR indices, so let me explain them a bit more too. Partial autocorrelation begins the same as autocorrelation. That is, the Lag 0 data (r = 1.0) and the Lag 1 data are the same. Starting with Lag 2, however, things get more complicated. The Lag 2 partial correlation considers three things – in our case, the relation of the length of Shots 1 and 3, Shots 1 and 2, and Shots 2 and 3. In particular, it determines the Shots 1&3 relationship while factoring out the intermediate Shots 1&2 and 2&3 relationships. The Lag 3 partial autocorrelation grows more complex. It considers the relationship between Shots 1&4 with the relationships between Shots 1&2, 2&3, 3&4, 1&3, and 2&4 factored out. The Lag 4 partial autocorrelation is more so, considering the relationship between Shots 1&5 with those for Shots 1&2, 2&3, 3&4, 4&5, 1&3, 2&4, and 3&5 factored out; and so forth for increasing lags. In a practical sense, this means that the Lag 2, Lag 3, and larger partial autocorrelations are almost always less than the Lag 2, Lag 3, and larger autocorrelations. Thus, the partial autocorrelation function descends towards zero much faster than the autocorrelation function and there are no larger scale ripples in it.
The raw partial autocorrelation functions for the two films are shown in blue in panels c out to Lag 20. These, like the autocorrelation functions and the power functions, are noisy. Certain autoregressive (AR) models measure the number of lags that remain above a threshold. Numerically, the threshold equals 1/(2*sqrtn), where n in our case is average number of shots (1132) in the 150 films that we sampled. To our eyes, the partial autoregressive functions are too noisy to be trusted in this way, so we smoothed them and saw where they intersected this threshold. The arrows in panels c point to intersections of the smoothed fits and the threshold. These are the mAR indices. Their lag values are 3.16 and 1.39 for The 39 Steps and Sunset Boulevard, respectively, for the shot data taken off the cinemetrics website. In AEHF we reported them as 3.22 and 1.63, and these values are also reasonably close.
What are AR and mAR indices good for? Almost all autoregressive (AR) applications lean heavily towards prediction. In economics one might take a year’s worth of closing prices of a stock market variations to predict possible gains in the near future; in climatology one might take many years of temperature variation to predict future climate change; etc. Clearly, and in contrast, we are not interested in predicting the next shot or series of shots in a film. Instead, the usefulness of our mAR approach is to provide an index of the local relatedness of shot lengths aggregated across a whole film. Thus, the local shot lengths of The 39 Steps are more related to one another than those of Sunset Boulevard. This would suggest that the “waves” seen in Salt’s moving averages should be clearer. And Salt’s notes this in his second comment after performing moving averages: “Sunset Blvd. has no flat regions with approximately equal shot lengths, whereas The 39 Steps has several.” This would be a consequence of the different mARs.
In closing, and echoing what I said last time, all of these calculations are intended to extend more typical cinemetrics analyses, and to help inform those interested in the physical structure of film and how that might mesh with the structure of the human mind.