# Time Series Analysis of Top Hat (1935)

The editor Millie Moore (*Johnny Got His Gun*, *Go Tell The Spartans*), said

… one of the most important jobs of the picture editor is to control the tempo and pace of the story (Yewdall 2007: 156).

The ebb and flow of pace and tempo determines the dramatic form of a film, and it is through editing (along with camera motion and sound energy) that the viewer’s attention is structured. Dorai and Venkatesh (2001) observed that in Hollywood narrative cinema, large changes of pace occur at the boundaries of story segments (e.g. transitions between scenes), while smaller changes in pace are identified with local narrative events of high dramatic import. Similarly, Cutting *et al*. (2011) noted that within each quarter and possibly each act of a Hollywood film there is a pattern of general shortening and then lengthening of shots reflecting a fluctuating intensification of continuity. Different emotional states are associated with different editing styles (Kang 2002). In the television schedule, adverts are edited more quickly than the programmes around them in order to attract the viewer’s attention and to improve product recall (Young 2007).

It would seem natural that the methods of time series analysis could help us to describe the evolution of the tempo and pace over the course of a film and thereby to understand how and why this element of film style changes.

However, there are a number of problems:

*Time is not an independent variable*: typically we apply time series methods to understand how some variable (e.g. stock prices, animal populations, etc) changes as a function of time, but here the variable of interest is time itself (i.e. the amount of time between two edits). This does not make time series analysis impossible, but it does require careful interpretation of the results: for example, spectral analysis will be event-based rather than time-based, and will show the number of events per cycle rather than the duration of the cycle in some unit of time. Treating this data as a ‘standard’ time series may lead to incorrect interpretation of the style of a film.*Shot length data is typically positively skewed with a number of outliers*: many common methods of time series analysis (e.g. running means, autocorrelation functions) assume that the data is normally distributed, but this is not the case for the shot lengths in a motion picture; and failing to take this into account can lead to flawed conclusions and erroneous estimations of parameters for time series models.*Shot length data may exhibit nonlinear characteristics*: many time series methods assume that the data is linear, but we may find that the style of a film exhibits conditional heteroscedasticity (e.g. the variance of shot lengths in a rapidly edited action sequence will be lower than in slower dialogue sequences), that any cycles present may be asymmetric, or that there are abrupt changepoints in style as one scene ends and another begins. Other nonlinear features may also be apparent.

These problems can be overcome by using ordinal or rank-based methods that make fewer assumptions about the distribution of the data and allows us to conduct exploratory data analysis before deciding on how to model the evolution of style in a film. Crucially, we need not be concerned that time is the variable of interest as these methods require only that the data is ordered – which in this case means the order in which they occurred (shot 1 is the first shot, shot 2 is the second, …). Two methods are illustrated here: running Mann Whitney Z statistics and the order structure matrix. The data set used here is for *Top Hat* (1935), and can be accessed here as an Excel file: Nick Redfern – Top Hat.

#### The running Mann-Whitney Z statistic

The Mann-Whitney U test is a nonparamteric test of the null hypothesis that two random variables are stochastically equal. For an introduction to the Mann Whitney U Test see here. Steve Mauget (2003, 2011) has applied the Mann-Whitney U test to time series analysis of climate data by using moving windows to sample the ranks of shots in order to identify regimes of high and low ranking data points. This method can be used to identify trends in the time-ordered data, to identify any intermittent cyclical regimes, and to identify changepoints in the series as the style of a film evolves. This method is akin to using a moving average, but instead of looking at the level in successive windows, we are looking at the ranks of the data.

The first step in generating a time series is to rank the *N* shots in a film from the smallest to the largest, with tied values assigned the average of the ranks they would have been assigned if there were no ties: if x_{2} and x_{3} have the same value they are assigned an average rank of (2+3)/2 = 2.5. The ranks are then sampled using a window of size n_{1} , and the sum of the ranks of the shots in this window (R_{1}) calculated. The values of n_{1} and R_{1} are used to calculate a U statistic by

and, if the sample is sufficiently large (n_{1} ≥ 10), then this can be transformed to a Z statistic by

If we plot the set of Z statistics produced by applying this method to *Top Hat* we get the time series in Figure 1, which was constructed using a sampling window of 20 shots.The significance of the Z statistic can be determined with reference to a standard normal distribution. Thus if α = 0.05, the critical z-value is ± 1.96; and so when Z ≥ 1.96 we will identify a significant cluster of high-ranking shots (i.e. long takes) and when Z ≤ 1.96 we will identify a significant cluster of low ranking shots (i.e. short takes).

The series in Figure 1 contains a lot of redundant information because consecutive windows overlap the same shots (i.e. if n_{1} = 20 then nineteen of the shots in window 1 will also appear in window 2), and so the windows we are interested in are the most-significant non-overlapping windows.

**Figure 1** Running Mann-Whitney Z statistics for *Top Hat* (1935) using a 20 shot window , with significance at Z = ± 1.96

From Figure 1, we can see that *Top Hat* has a number of peaks and troughs corresponding to clusters of longer and shorter shots.

The first peak (A) includes the meeting between Jerry and Horace at the beginning of the film that sets up the story and the first musical number ‘No Strings (I’m Fancy Free).’ Jerry’s performance of this number is interrupted by Dale, whom he has woken with his dancing, and there is a sequence of a more rapidly edited shot-reverse shot pattern that occurs at the first trough (1). After Dale returns to her room, Jerry decides to cover the floor with sand and dances to a reprise of ‘No Strings,’ and this can be seen in the second peak (B). The following morning, Jerry takes the place of a Hansome cab driver and escorts Dale to the stables, and this is a second quickly edited dialogue scene that occurs at 2. The peak at C occurs with the second musical number, ‘Isn’t This a Lovely Day (to be Caught in the Rain).’ The troughs at 3 and 4 coincide with Dale mistaking Jerry for Horace in the hotel lobby, and a subsequent sequence which cross-cuts between Dale and Jerry in different hotel rooms after the former has slapped the latter. The peak Z statistic occurs at D, which is the sequence at the theatre in which Jerry and Horace talk in the dressing room before Jerry goes on stage to perform ‘Top Hat, White Tie, and Tails.’ This first half of the films takes place in London, and the peaks and troughs are associated with particular aspects of the musical comedy: the peaks (i.e. the clusters of higher ranked and – therefore longer – shots) are associated with the musical numbers, while the troughs are associated with the comedy story line of the mix up in the romance of Dale and Jerry.

As the action moves to Italy, we get trough at 5, which is the sequence in which Dale and Madge chat next to the canal, and 6, which is another sequence cross-cut between locations as Dale and Jerry speak on the phone. The peak at E occurs at the third musical number, ‘Cheek to Cheek,’ and its extended dance sequence. So far we have the same pattern that we saw in the London sequences: comedy is quick and musical slow. However, the peak at F is not associated with a musical number and spans the three scenes. This peak includes the end of the sequence in which Jerry and Horace talk (after Madge has given her husband a black eye), the long static takes in which Dale accepts Beddini’s proposal (as the melody for ‘Cheek to Cheek’ is played in the background), and the beginning of the next sequence as Jerry and Horace are asked by the hotel to vacate the bridal suite and they go onto to talk to Madge. This peak (F) is the only case of the narrative being characterised by a cluster of long shots in the film.

The next trough (7) is the sequence in which Jerry meets Beddini in the bedroom of the bridal suite before he takes Dale out on the canal. The trough at 8 is a cluster of short shots in which the narrative of mistaken identity is resolved between Beddini, Horace, and Madge (but not Dale and Jerry), and is followed by the final peak G, which is the last of the musical numbers, ‘The Piccolino,’ and the carnival dance sequence.

Overall, we can see from the editing structure revealed by using the running Mann-Whitney Z statistic that *Top Hat* is characterised by alternating clusters of longer and shorter takes, in which the former are typically associated with the musical parts of the film and the latter with the comedy-romance narrative.

This method can also be used to compare different films side by side, and in a few weeks I’ll post a paper using this method to analyse the time series of 15 BBC News bulletins that places this data into a single frame of reference so similarities and differences can be identified.

#### The order structure matrix

The same information we obtained from the running Mann-Whitney Z statistic can be seen in the order structure matrix for *Top Hat* in Figure 2, based on whether a shot is greater than or less than the shot that comes after it (Brandt 2005). To construct the matrix we assign a value of 1 when x_{s} ≥ x_{t} and a value of 0 when x_{s} < x_{t}. To make this easier to visualise we assign a colour to each value (1 = black, 0 = white) and plot the matrix in a grid. The dark patches in Figure 2 correspond to the peaks in Figure 1 and exhibit clustering of longer shots in the films, while the light patches correspond to the troughs of Figure 1 and show where the clusters of shorter shots are to be found. Although this plot looks complicated, once you get used to the method and are familiar with the events of the film you can simply read the changes in cutting style from left to right.

**Figure 2** Order structure matrix for *Top Hat* (1935)

Figure 2 was produced by first calculating the matrix in Microsoft Excel; and then cutting and pasting the resulting array of 1s and 0s into the latest version of PAST (which you can download for free here), selecting the whole spreadsheet, and then choosing MATRIX from the PLOT menu.

Alternatively, you can produce Figure 2 by applying the filled.contour command in R to the matrix (see here for an explanation).

This method has a particular limitation: it is only really effective with large data sets, and it can be quite difficult to make out distinct patterns even when there are as many as 250 shots in a film. If, however, you have 500 or more shots, then it is an excellent place to start your exploration of the shot length data for a film.

#### References

**Brandt C** 2005 Ordinal time series analysis, *Ecological Modelling* 182: 229-238. [There is an online version of this paper that can be downloaded for free, but there is no URL associated with it. Search for the title and you’ll find it].

**Cutting JE, Brunik KL, and DeLong JE** 2011 How act structure sculpts shot lengths and shot transitions in Hollywood film, *Projections* 5 (1): 1-16.

**Dorai C and Venkatesh S** 2001 Bridging the semantic gap in content management systems: computational media aesthetics, in *Proceedings 2001 International Conference on Computational Semiotics in Games and New Media*. 10-12 September 2001, Amsterdam: 94-99.

**Kang H-B** 2002 Analysis of scene context related with emotional events, in *Proceedings of the 10th ACM International Conference on Multimedia*. 1-6 December 2002, Juan les Pins, France: 311-314**.**

**Mauget SA** 2003 Intra- to multidecadal climate variability over the continental United States: 1932–99, *Journal of Climate* 16: 3905–3916.

**Mauget SA** 2011 Time series analysis based on running Mann-Whitney Z statistics, *Journal of Time Series Analysis* 32 (1): 47–53.

**Yewdall DL** 2007 *Practical Art of Motion Picture Sound*, third edition. Burlington, MA: Focus Press.

**Young C** 2007 Fast editing speed and commercial performance, *Admap* 483: 30-33.

Posted on June 16, 2011, in Cinemetrics, Film Analysis, Film Studies, Film Style, Film Theory, Statistics, Time Series Analysis, Top Hat and tagged Cinemetrics, Film Analysis, Film Studies, Film Style, Film Theory, Statistics, Time Series Analysis, Top Hat. Bookmark the permalink. 5 Comments.

As one might expect, a centered rolling average on shot number against length with a window of 20 shots produces similar results, with less mathematics. A window of 10 shots separates out the final two musical numbers.

As you say, it is difficult to see what is going on in the second graph, though our Programming Officer is thinking of having a skirt pattern made from it.

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