Let me respond to questions raised by Barry Salt in his 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 .
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
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
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 on our paper he mentioned the Lag 1 correlations for a few films. In I suggested that the full autocorrelation function is more informative, and that Lags 1 alone can
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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,
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
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
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,
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
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
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
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
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
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.