Film style and narration in Rashomon

UPDATE: 13 April 2014: The revised version of this article has now been published as Film Style and Narration in Rashomon, Journal of Japanese and Korean Cinema 5 (1-2) 2013: 21-36. DOI: 10.1386/jjkc.5.1-2.21_1.

A post-print of the article can be downloaded here: Nick_Redfern – Film style and narration in Rashomon (post print)

And so after a long (and much enjoyed break) I return to the blogosphere with the first draft of paper on film style and narration in Rashomon. This paper is different to other statistical analyses of film style I have published on this site and to all other studies of film style and narration because it uses multivariate analysis to look at several different aspects of film style together. The method used is multiple correspondence analysis, and you can find a good introductory chapter on MCA here. The software I used is FactoMineR for R, and the website explaining how to do the analysis can be found here.

Multivariate analysis has been used in the quantitative study of literature for some time (see the links below the abstract), but this is the first time multivariate analysis has been applied to film style and it appears to work very well. I am currently looking at some other applications, particularly in distinguishing between the different parts of portmanteau horror films (which is a proper scholarly endeavour and not simply an excuse to watch lots of portmanteau horror films).

An Excel file contain the data used in the analysis can be accessed here: Nick Redfern – Rashomon. This file contains two worksheets: the first is the shot length data for the film, and the second is that data used in the multiple correspondence analysis.

Abstract

This article analyses the use of film style in Rashomon (1950) to determine if the different accounts of the rape and murder provided by the bandit, the wife, the husband, and the woodcutter are formally distinct by comparing shot length data and using multiple correspondence analysis to look for relationships between shot scale, camera movement, camera angle, and the use of point-of-view shots, reverse-angle cuts, and axial cuts. The results show that the four accounts of the rape and the murder in Rashomon differ not only in their content but also in the way they are narrated. The editing pace varies so that although the action of the film is repeated the presentation of events to the viewer is different each time. There is a distinction between presentational (shot scale and camera movement) and perspectival (shot types) aspects of style depending on their function within the film, while other elements (camera angle) fulfil both these functions. Different types of shot are used to create the narrative perspectives of the bandit, the wife, and the husband that marks them out as either active or passive narrators reflecting their level of narrative agency within the film, while the woodcutter’s account exhibits both active and passive aspects to create an ambiguous mode of narration. Rashomon is a deliberately and precisely constructed artwork in which form and content work together to create an epistemological puzzle for the viewer.

On the multivariate analysis of literature see the following:

Hoover DL 2003 Multivariate analysis and the study of style variation, Literary and Linguistic Computing 18 (4): 341-360.

Stewart LL 2003 Charles Brockden Brown: quantitative analysis and literary style, Literary and Linguistic Computing 18 (2): 129-138.

Tabata T 1995 Narrative style and the frequencies of very common words: a corpus-based approach to Dickens’s first person and third person narratives, English Corpus Studies 2: 91-109.

The mAR index of Hollywood films

UPDATE (March 2015): A revised version of this paper has now been published as Robust estimation of the mAR index of high grossing films at the US box office, 1935 to 2005, Journal of Data Science 12 (2) 2014: 277-291.  [The pdf of this article can be accessed here: 4.JDS-1181_final-1].

UPDATE: reviewing the methodology of the mAR index in general, Mike Baxter noted an error in the data whereby I had reported the exponent of the negative exponential function instead of the mAR index for films from the 1960s. I have now corrected this and redone the analysis and the graphs (which are still cool). This mainly effects the conclusions regarding differences between genres. Overall, it turns out that, as a result of this error, I had actually underestimated the difference between the classical and rank mAR indices. If anyone finds any other errors then feel free to add a comment to this post and I’ll try to correct it as soon as possible.

And so to finish the month as we started, looking at robust estimates of the mAR index of film style. Below is the first draft of a paper comparing the mAR index  based on the methods used by James Cutting, Jordan De Long and Christine Nothelfer to describe the clustering of shots in motion picture with a rank-based alternative that is resistant to outliers. Naturally, it features some pretty cool graphs.

Robust estimation of the modified autoregressive index for high grossing films at the US box office, 1935 to 2005 The modified autoregressive (mAR) index describes the clustering of shots of similar duration in a motion picture. In this paper we derive robust estimates of the mAR index for high grossing films at the US box office using a rank-based autocorrelation function resistant to the influence of outliers and compare this to estimates obtained using the classical, moment-based autocorrelation function. The results show that (1) The classical mAR function underestimates both the level of shot clustering and the variation in style among the films in the sample.; (2) there is a decline in shot clustering from 1935 to the 1950s followed by an increase from the 1960s to the 1980s and a levelling off thereafter rather than the monotonic trend indicated by the classical index, and this is mirrored in the trend of the median shot lengths and interquartile range; and (3) the rank mAR index indentifies differences between genres missed by the classical index.

 

Robust estimation of the modified autoregressive index of film style

Earlier this I looked at the time series structure ITV news bulletins using robust methods of autocorrelation. This post follows on from that earlier study, this time looking at BBC news bulletins. This paper was written with three goals in mind. First, I wanted to improve on the method used before. Second, I wanted to try the rank based method of estimating the mAR index. Third, I wanted to apply these methods to a different cluster of data sets to see if I would come up with similar results.

The paper can be accessed as a pdf file here: Nick Redfern – Robust estimation of the modified autoregressive index of film style

Abstract

The modified autoregressive index (mAR) describes the tendency of shots of similar length to cluster together in a motion picture but is not resistant to the influence of outliers if derived from the classical moment-based partial autocorrelation function. In this paper we calculate robust estimates of the modified autoregressive index based on outlier-resistant partial autocorrelation function based on the ranks of the shot length data and robust measure of scale. The classical, rank, and robust methods of determining mAR are compared for a sample of BBC news bulletins.

Some notes on cinemetrics V

This post addresses some issues raised by Mike Baxter as part of the ‘cinemetrics conversation’ at the Cinemetrics website (and is the post I would have produced last week had I been able to remember which bit of software had the right command to create the necessary graph). You can find an introduction to the conversation here and my first response to some of the issues raised here.

I want to address two issues: first, the nature of outliers in shot length distributions and better methods of representing such distributions than I have used up to now; and, second, the straw-man the median shot length has become in Baxter’s comments.

Baxter’s comments in response to the earlier can be found in the second tab under his name here. In section 2 Baxter questions my use of the term ‘outlier’ and the definition used to identify such shots.  This is fair enough – we wouldn’t get very far if such definitions weren’t questioned. In the examples of Lights of New York and Scarlett Empress, Baxter argues there is no evidence of outliers since

it’s difficult to identify any point at which ‘extremes’ begin, or discontinuities in the distribution of the kind I think are needed to assert, with any confidence, that you are dealing with ‘outliers.’

Baxter never defines what such a discontinuity would look like and so his argument is vague. (Arguably this is the semantic version of a slippery slope).

Figure 1 shows the kernel density and box plot of Lights of New York. There is a 12.2 second gap between the five shots of longest duration and the sixth longest, presumably the sort of discontinuity Baxter refers to and he does concede he might be prepared to accept five shot lengths as extreme values (though he does not say on what basis). From Figure 1 we can see there are in fact several such discontinuities, and that the kernel density is zero at several points in the upper tail (indicating the kernels do not overlap), particularly above 30 seconds (which corresponds to the 22 extreme outliers identified using this type of box plot). However, a limitation of this boxplot is that it does not take into account the skew of the distribution and so over identification of outliers is a problem.

Figure 1 Kernel density and boxplot of shot lengths in Lights of New York (1928)

Figure 2 presents the same data using an adjusted boxplot that takes into account the skewed nature of the data. This method uses the med-couple, a robust measure of skewness, to identify outliers. The adjusted boxplot can be generated using the adjbox() command in the R package robustbase.

The number of outliers in Figure 2 is much less than in the original boxplot: in the upper tail 10 shots greater than 55 seconds are identified as outliers (or 3% of the total).  Nonetheless, there are still some values which are sufficiently removed from the rest of the data to be classed as outliers even when accounting for the asymmetry of the distribution. Whether or not Baxter would accept this definition would depend on the interpretation of his use of the term ‘discontinuity,’ which he does not define.

Surprisingly, this method identifies three outliers in the lower tail of the distribution (which I wasn’t expecting and will have to think about more).

Figure 2 Kernel density and adjusted boxplot of shot lengths in Lights of New York (1928)

The following article describes the adjusted boxplot and its calculation:

Vandervieren E and Hubert M 2008 An adjusted boxplot for skewed distributions, Computational Statistics and Data Analysis 52 (12): 5186-5201. An ungated, earlier version of this paper can be accessed here.

Even if we accept Baxter’s argument that there are no outliers in Lights of New York it remains necessary to be aware of the problems caused by outliers in data sets and to check the distribution of shot lengths so that we  are not be fooled by non-robust statistics. Certainly more effort will have to be devoted to defining what is or is not an outlier (in either statistical or filmic terms) in research if this type. (But it is much easier when you remember which bit of software to use).

Finally, I wish to address a misrepresentation that has taken a hold at this early stage in the ‘cinemetrics conversation.’

Baxter writes

the use of either the ASL or median as the statistic for attempting to summarise ‘style’ doesn’t make much sense (as Salt observes) [original emphasis].

This argument is a straw-man.

I have never stated that the median shot length is the statistic for describing film style. I have argued that the median shot length is better than the mean shot length for describing film style, and should therefore be preferred for the following reasons:

• the median is conceptually simple and easy to calculate, and is certainly no more difficult than the mean.
• the median shot length has a clearly defined meaning and the difference between two median shot lengths is also meaningful, whereas the meaning of the mean the difference between two mean shot lengths is not clear in either case (and seem to change every time I raise an objection against them).
• the median shot length is not affected by a monotone transformation (the median of a data set is the same as the median of the logarithmic transformation of a data set), while the possibilities for confusing the arithmetic and geometric means are endless.
• the median locates the centre of a distribution irrespective of its shape, whereas this is not true of the mean.
• the median is not affected by outliers or extreme values (however you choose to define them), whereas this is not true of the mean.
• interpretations of film style based on the median shot length are consistent with graphical methods and (it turns out) with dominance statistics (Cliff’s d, HLΔ), while those based on the mean shot length are not.

But I have always argued that it is important use a range of statistical methods to get a full understanding of the nature of film style.

As far as I am aware I am the only person writing about film style to even consider the dispersion of shot lengths in a motion picture and the appropriate methods to use this. I am also the only person to use a range of graphical methods (probability plots, boxplots, empirical cumulative distribution functions, kernel densities, order structure matrices, running Mann-Whitney Z statistics, rank-frequency plots) to describe film style. I am the only person in film studies to employ confidence intervals, statistical hypothesis tests, effect sizes, or even to describe the methodologies I use in studying film style. (Others working outside films studies in disciplines where quantitative methods are commonplace also use such tools as a matter of routine, and those within film studies would do well learn by their example).

I am also the only person who has attempted to describe these methods so that others may try to analyse film style for themselves. I am the only person who has brought to the attention of researchers in film studies the availability of free learning resources and software for statistics. I am the only person to look outside film studies for empirical research on film style and to bring it to the attention of film scholars. I am the only person to address the issue of statistical literacy in film studies (here and here).

Baxter writes that

the accessibilty of computational power, and essential simplicity of important statistical ideas (however mathematically complex) is a hobby-horse of sorts.

I am glad to hear this, because it means that if someone else is prepared to devote some time and effort to explaining statistical concepts and methods to film scholars then I won’t have to do it on my own.

However, as Baxter presents the argument I am interested in the median shot length only while Barry Salt apparently does not have a narrow attachment to a particular statistic of film style and embraces a pluralistic approach. However, I am not aware of any forum in which Salt has made any concession to his view that the mean shot length is the only appropriate statistic of film style. In fact, I am unaware of any other statistics of film style used by Salt besides the average shot length and the histogram (while his odd comments on the calculation of kernel density estimates indicates he may not properly understand other methods).

Baxter has his argument back to front here: you won’t find methodological ecumenism in the statistical analysis of film style in the work of Barry Salt.

Analysing film style using dominance statistics

UPDATE: An article using the ideas introduced in this post has now been published as Comparing the Shot Length Distributions of Motion Pictures using Dominance Statistics, Empirical Studies of the Arts 32 (2) 2014: 257-273. DOI: 10.2190/EM.32.2.g. It can be found here.

Statistical comparisons of film style have been based on the average shot length (either the mean or the median), so that, for example, given the ASLs of two films the one with the greater average is said to be edited more slowly.

In his first contribution to the Cinemetrics conversation, Mike Baxter argued that in some circumstances neither the mean nor the median were useful statistics of film style. In this post I look at how we might compare the shot length distributions of two films or two groups of films beginning with an average shot length. The methods used are Cliff’s d statistic, which measures the stochastic dominance of one sample over another, and the Hodges-Lehman median difference, which measures the average distance between. Results produced by these methods are then compared to the interpretation of film style using average shot lengths, measures of dispersion, and graphical methods. This will also provide us with an opportunity to consider Baxter’s further claim that it makes little difference which average was used since either would lead to the same interpretation of film style.

Cliff’s d statistic

Cliff (1993, 1996) introduced the stochastic difference

d = P(X >Y) – P(X<Y)

as a nonparametric method of measuring the extent to which two samples (X and Y) overlap. This means we find the probability that an observation in the sample is X is greater than an observation in sample Y, and from this we subtract the probability that an observation in Y is greater than an observation in X. Ties are not included in the calculation. Cliff’s d statistic can be calculated as a linear transformation of the probability of superiority:

d = 2PS – 1

where PS is equal to the Mann-Whitney U test statistic divided by the product of the sample sizes (PS = U/nm) (see Delaney & Vargha 2002). Since PS = P(X > Y) + 0.5P(X = Y), ties are accounted for. The value of d ranges from -1 (when every observation in X is less than every observation in Y) to 1 (when every observation in X is greater than every observation in Y); and stochastic equality occurs at 0 (when there is complete overlap between the distributions).

This statistic has several advantages for comparing two distributions:

• It is not based on any assumptions about the data
• it is robust against outliers and unequal variances
• it is invariant under monotonic transformation
• it provides a more direct answer to the sort of questions researchers often wish to ask of data: ‘if one’s primary interest is in a quantification of the statement “Xs tend to be higher than Ys,” then [d] provides an unambiguous description of the extent to which this is so’ (Cliff 1996: 125).

The stochastic dominance of one sample over another can be visualised graphically since d measures the extent to which one population distribution lies to the right of another.

Hodges-Lehmann median difference

Although we can use Cliff’s d to discover if the shots in one film tend to be shorter than the shots in another it cannot tell us how much shorter those shots tend be. For this we need another statistic. The Hodges-Lehmann median difference (HLΔ) for two samples is the median of all the pairwise differences between every observation in X and every observation in Y:

HLΔ = med{x_i – y_j}

In other words, subtract the length of every shot in film A from every shot in film B and then find the median of the n × m differences. HLΔ is a measure of the average distance between observations in and X and observations in Y.

Comparing the style of two films

As a first example let’s use the example of Lights of New York and Scarlet Empress I used in my own contribution to the Cinemetrics conversation. Basing our interpretation on the median shot lengths we see that Lights of New York has a median of 5.1 seconds and that Scarlet Empress has a median of 6.5 seconds, indicating that the former is edited more quickly than the latter. In contrast, an interpretation based on the mean shot length implies that both films are cut equally quickly since each film has a mean shot length of 9.9 seconds.

To calculate d we first need to perform the Mann-Whitney U test, which gives us U = 88188, and then we derive the probability of superiority by dividing by the product of the sample sizes (338 and 601):

PS = U/nm = 88188/(338 × 601) = 0.4341.

From this we can calculate the stochastic dominance between the two distributions:

d = 2PS – 1 = (2*0.4341) – 1 = –0.1318.

Therefore, we conclude that shots in Lights of New York tend to be of shorter duration than those of Scarlet Empress. This can be clearly seen in Figure 1, which shows the empirical cumulative distribution functions of the two films.

Figure 1 The empirical cumulative distributions of Lights of New York and Scarlet Empress (KS Test: D = 0.12, p = <0.01)

The function of Scarlet Empress tends to lie to the right of that of Lights of New York indicating that it has shots of longer duration, except for the very upper tail where the presence of a few unusually long takes in Lights of New York, which account for only ~7% of the shots in this film. It is this handful of shots that pulls the mean away from the mass of the data, and if we remove the 24 longest shots from the distribution of Lights of New York we see that the mean shot length falls to 6.4 seconds. This is clearly a very influential group of outliers as just this 7% of the total number of shots leads to a 33% difference in the mean equivalent to a 3.5 second increase. It takes an act of wilful perversity to claim that there are no outliers present in this data, that the mean of not greatly influenced by those outliers, and that the mean shot length is an accurate description of the style of this film.

For these two films HLΔ = -1.0 (95% CI: -1.6, -0.4), which means that on average a shot in Lights of New York is 1 second shorter in duration than a shot in Scarlet Empress.

The interpretation of the difference in the style of these films based on Cliff’s d and HLΔ is consistent with that based on the median shot length but not with the conclusion derived from the mean shot length. The difference in these statistics indicates that far from leading to the same conclusion they lead to contrary and incompatible conclusions, and so Baxter’s argument that the choice of statistic is irrelevant does not hold in this case.

Comparing the style of two groups of films

Comparing the style of two groups of films we use the same methods described above and calculate the pairwise statistics for all the films in both samples. We can then take the median value of the n × m d statistics and of the n × m HLΔ statistics as estimates of the differences of the

To illustrate this I use the example of the Laurel and Hardy short films I discussed in an earlier paper. In this study I compared the median shot lengths of a sample of silent films and a sample of sound films starring Laurel and Hardy produced between 1927 and 1933, and concluded that there was a statistically significant difference between the two samples of medians but that it was a small difference reflecting the continuity of a mode of production, of creative personnel, and of a style of comedy with the introduction of sound technology. The difference in the median shot lengths was estimated to be HLΔ = 0.5 seconds (95% CI: 0.1, 1.1) and PS = 0.2333. (I also compared statistics of the dispersion of shot lengths in these films but I won’t discuss these here).

If this analysis had been conducted using the mean shot length then I would have reached a different conclusion, with HLΔ = 1.5 seconds (95% CI: 0.8, 2.3) and PS = 0.1188. This result would appear to indicate that the introduction of sound technology had a large impact on the style of Laurel and Hardy films and would lead us to conclude there is no continuity from the silent to the sound era. Again, there is a difference in the interpretation of the style of these films indicated by the different statistics: the estimate of the impact of sound technology based on the means is 300% greater than that based on the medians. Again, Baxter’s argument that the choice of statistic does not matter simply doesn’t hold water.

What conclusion do the dominance statistics lead to? As we have a sample of 12 silent films and a sample of 20 sound films we need to perform a total of 12 × 20 = 240 calculations. Table 1 presents the pairwise comparisons for Cliff’s d, while the pairwise HLΔ statistics are in Table 2.

The median of the pairwise Cliff’s d statistics is -0.0957 (95% CI: -0.1192, -0.0723). This indicates that shots in the silent films of Laurel and Hardy tend to be of shorter duration than those of their sound films, and that this effect is relatively small.

Table 1 Pairwise Cliff’s d statistics for silent and sound films of Laurel and Hardy. (This table is very large so click on it to see it full size).

The median of the pairwise HLΔ statistics is 0.4s (95% CI: 0.3, 0.5), which again indicates a significant if small difference between the samples with the shots in the soundtending to be of slightly longer duration on average than those of the silent films.

Table 2 Pairwise HLΔ statistics for silent and sound films of Laurel and Hardy. (This table is very large so click on it to see it full size).

Both these results are consistent with my analysis based on the mean shot length. Neither of these statistics is compatible with the interpretation based on the mean shot lengths.

A problem with applying Cliff’s d and HLΔ in this way is that as the sample sizes grow the number of pairwise comparisons becomes very large. For example, if we wanted to compare the style of two groups of films with 100 films in each sample we would have to perform 100 × 100 comparisons. That’s a total of 10,000 Mann-Whitney U tests, and while we are interested in film style I don’t think we’re that interested. It is here that the consistency of Cliff’s d and HLΔ with the median shot length is valuable. It is quick and easy to perform even a very large number of pairwise comparisons of median shot lengths simply by copying formulas across a range of cells in an Excel spreadsheet, for example. We can use the median shot length in the place of the dominance statistics thereby greatly speeding up the analytical process while allowing us to remain secure in our interpretation of the data. We cannot use the mean shot length in the same way since this method is not consistent with any of the others.

Conclusion

Based on the above discussion we can arrive at the following conclusions:

• The claim that it does not matter which statistic of film style we use since using either the mean or the median will lead to the same interpretation is clearly not true and the choice of statistic will affect the size of any effect. In turn, this will have a direct impact on our conclusions about the nature of film style.
• We can analyse the style of films using dominance statistics that do not require any average shot length. Cliff’s d and HLΔ are. The meaning and interpretation of these statistics may correspond more closely to questions we wish to ask of film style than using average shot lengths (though we still need descriptive statistics and graphs to provide information about the shot length distribution).
• It may not be practical to use dominance statistics for comparing large samples of films due to the very large number of pairwise comparisons required. Mike Baxter indicated that an average shot length could be thought of as a ‘proxy statistic’ of film style, and the median shot length can certainly be used in this sense by virtue of its consistency with Cliff’s d and HLΔ.
• The mean shot length is not robust in the presence of outliers and leads to fundamentally flawed interpretations of film style. It is not consistent with either  Cliff’s d or HLΔ, and cannot be used to answer the question ‘do the shots in film A tend to be longer than the shots in film B.’

References

Cliff N 1993 Dominance statistics: ordinal analyses to answer ordinal questions, Psychological Bulletin 114 (3): 494-509.

Cliff N 1996 Ordinal Methods for Behavioural Data Analysis. Mahwah, NJ: Lawrence Erlbaum Associates Inc.

Delaney HD and Vargha A 2002 Comparing several robust tests of stochastic equality with ordinally scaled variables and small to moderate sample sizes, Psychological Methods 7 (4): 485-503.

The Cinemetrics Conversation I

Over the past few months Yuri Tsivian at the Cinemetrics Database has been organizing myself and various other people interested in statistics into producing some short (and some long) pieces on this topic. (No mean task on his part I think you’ll agree). From this week they have started to appear on the Cinemetrics website, and you can access them here. I reproduce Yuri’s introduction to the area below so you can get an inkling of what has been going on.

This conversation brings together statistical scientists and scholars that study film. What gathers us are two things. First, we are driven by mutual curiosity about cinemetrics as a field. What can numbers tell us about films and how do films fit in with what we know about numbers? Another thing we hope to find out has to do with Cinemetrics as a site. What variables should Cinemetrics make available to its users and which statistical tools need to be added to Cinemetrics labs? We plan to tackle these questions in a series of notes posted here starting from now through spring 2013.

Let me start off by introducing the team. My name is Yuri Tsivian, I study film, teach it at the University of Chicago and, in tandem with computer scientist Gunars Civjans, run the site that hosts this conversation. Beside me are two film scholars, Barry Salt of London Film School who pioneered the discipline of film statistics in 1974 and whose personal database and multiple essays are found elsewhere on this website, and Nick Redfern whose own website features over 50 cinemetrics studies and reflections. On the other side are two academic statisticians, Mike Baxter of Nottingham Trent University who has been publishing in statistical archaeology and quantitative geography since late 1970s and whose more recent interest in film statistics resulted in 3 essays on the subject, and Vanja Dukic of the University of Colorado at Boulder who happened to be around when Cinemetrics was born in 2005 and to whose expertise this site owes its first statistical steps.

The way I would like this conversation to evolve is round by round. To give it a sense (or semblance) of direction I will start each round by posing a question about this or that aspect of statistical films studies which our four experts might use as a starting point. Here is an approximate plot which is quite likely to change as new questions arise in the course of the conversation.  My first question (of which more later) is about the role of ASLs, medians and outliers. This subject may well lead us to questions about log-normality tests which will ring in the second round. We may go on from there to the 3rd question which would relate to whether parametric or non-parametric statistics works better for films. The 4rth question might be about autocorrelation or other possible methods to establish cases in which shots tend to cluster, and if there is periodicity to this.  We may then want to discuss the uses of descriptive, inferential and experimental statistics in film studies; I would also be interested in learning more about best ways to establish possible correlations between different variables of film style. We might then go on to the question of how to visualize data, for instance, whether old good bar plots work well enough to represent the shot scale profile of a motion picture. Again, all this is just a scheme which we may either flesh out or send the way of all flesh.

So head on over there to find out what is going on. There will soon be comments boxes appended to the essays so you can join in the process.

Using box plots to analyse film style

Numerical descriptions of film style are valuable but it is often simpler and more informative to use graphical representations of shot length data to aid us in analysing film style. Following on from earlier posts on using kernel densities (here) and cumulative distribution functions (here) this post rounds out this short series by looking at box plots and vioplots. Potter (2006) provides a detailed survey of the methodology of constructing and interpreting box plots and a discussion of extensions and alternatives.

Box-plots are an excellent method for conveying a large amount of information about a data set quickly and clearly, and do not require any prior assumptions about the distribution of the data. Analysing the box-plots of shot lengths in motion pictures we compare the centre and variation of the data, and identify the skew and the presence of outliers. They are also an efficient method of comparing multiple data sets, and placing the box-plots for two or more films side-by-side allows us to directly compare the centre and variation of shot length distributions in intuitively.

The box plot provides a graphical representation of the five-number summary, which includes the minimum value, the lower quartile, the median, the upper quartile, and the maximum value of a data set. The core of the data is defined by the box, which covers the distance between the lower and upper quartiles (i.e. the IQR), and the horizontal line within the box represents the median value of the data. The inner fences are marked by error bars extending from the box, and data points beyond these limits are classed as outliers. An outlier is defined as greater than Q3 + (IQR × 1.5) and an extreme outlier as greater than Q3 + (IQR × 3). Typically, there are no outliers at the low-end of a shot length distribution, and the error bar descends to the value of the shortest shot in a film.

To illustrate, Table 1 presents the descriptive statistics for the three main ITV news bulletins broadcast on 10 August 2011. There is nothing wrong with this information, and we can see immediately that these bulletins have similar styles. They have similar medians indicating they are cut equally quickly, whilst the lunchtime bulletin has slightly more variation of the middle 50 per cent than the other two bulletins. We can also see that the distributions of shot lengths in these films are asymmetric and that the maximum values are much longer than other shots. However, we cannot tell if these maximums are isolated outliers or if there are a large number of such values.

Table 1 Descriptive statistics of ITV news bulletins broadcast on 10 August 2011

Figure 1 presents the box plots of these bulletins, and gives us some of the detail we are looking for. We can see the same information we get in Table 1, but it is easier to make the comparisons across a single scale than to try to imagine the distribution froma set of numbers. We can also see that these bulletins share some other features – the error bars extend a similar distance from the upper quartile with shots in this range (10-18 seconds) associated with short interviews with members of the public, while the clusters of outliers that can be seen for each bulletin in the range 18-30 seconds are associated with the news kernel that begins each item and longer interviews as part of a news report. Longer takes occur when a reporter is speaking directly to camera, typically as part of a two-way interview. We can therefore see that similar events in the discourse structure of these news bulletins occupy a similar amount of screen time within the same bulletin and across the bulletins broadcast on the same day. You cannot tell that from the five-number summary. This is a crucial advantage of using graphical methods alongside numerical summaries – they can be used analytically as well as descriptively. You can learn more from a Figure 1 than you can from Table 1, though it would be best to include both in a piece of research since knowing the actual values of the descriptive statistics is useful to the reader.

Figure 1 Box plots of three ITV news bulletins

By using a box plot we can see some of the structure of the data obscured by the five-number summaries. However, one of the problems with box plots is that they flatten out the detail of the distribution in the box and between the box and the ends of the error bars. This can be remedied by combining box plots with a kernel density to produce a vioplot. This has the advantage of making all the information available from these two types of plots in a single figure. Figure 2 presents the vioplots of these bulletins.

Figure 2 Box plots of three ITV news bulletins

From Figure 2 we can see all the detail from the box plots AND we can see that the density of shot lengths in those areas where the box plot provides no detail. For example, the similarities in the 10-18 second range are more apparent in Figure 2 than Figure 1. For an alternative way of combining box plots and kernel densities to describe these data sets see here.

It has become increasingly common for film scholars to cite average shot lengths, but this information is rarely useful to the reader. It is usually the wrong average, is unaccompanied by a measure of dispersion, and simply does not provide enough information for anyone to make a sensible judgement about the nature of a film’s style. If you do want to use statistics to make a point about film style then please include kernel densities, cumulative distribution functions, or box/vioplots so that we can see what you are talking about. This should be standard practice in research and publishing in film studies.

References

Potter K 2006 Methods for presenting statistical information: the box plot, in H Hagen, A Kerren, and P Dannenmann (eds.) Visualization of Large and Unstructured Data: Lecture Notes in Informatics GI-Edition S-4: 97–106.

Exploratory data analysis and film form

Following on from my earlier posts on the editing structure of slasher films, this week I have a draft of a paper that combines my early observations (much re-written) along with an analysis of the relationship between editing and the narrative structure of Friday the Thirteenth (1980)

Exploratory data analysis and film form: The editing structure of slasher films

We analyse the dynamic editing structure of four slasher films released between 1978 and 1983 with simple ordinal time series methods. We show the order structure matrix is a useful exploratory data analytical method for revealing the editing structure of motion pictures without requiring a priori assumptions about the objectives of a film. Comparing the order structure matrices of the four films, we find slasher films share a common editing pattern closely comprising multiple editing regimes with change points between editing patterns occur with large changes in mood and localised clusters of shorter and longer takes are associated with specific narrative events. The multiple editing regimes create different types of frightening experiences for the viewer with slower edited passages creating a pervading sense of foreboding and rapid editing linked to the frenzied violence of body horror, while the interaction of these two modes of expression intensifies the emotional experience of watching a slasher film.

The paper can be accessed here: Nick Redfern – The Editing Structure of Slasher Films.

The shot length data for all four films can be accessed as a single Excel file: Nick Redfern – Slasher Films.

Analysing the editing structure of these slasher films is only part of this paper. Another goal was to outline exploratory data analysis as a data-driven approach to understanding film style that avoids a specific problem of existing ways of thinking about film style.

Existing methods of analysing film style make a priori assumptions about the functions of style and then provide examples to support this assertion. This runs the risk of begging the question and circulus in probando, in which the researcher’s original assumption is used as a basis for selecting the pertinent relations of film style which are then used to justify the basis for making assumptions about the functions of film style. We would like to avoid such logically flawed reasoning whilst also minimising the risk that we will miss pertinent relations that did not initially occur to us. By adopting a data-driven approach we can derive the functions of film style by studying the elements themselves without the need to make any such a priori assumptions. Exploratory data analysis (EDA) allows us to do this by forcing us to attend to the data on its own terms.

Although this is a method developed within statistics, EDA can be applied not just to numerical data but to any situation where we need to understand the phenomenon before us. For example, I had not noticed that the number of scenes between hallucinations in Videodrome reduces by constant factor until I sat down and wrote out the narrative structure of the film (see here).

Two very useful references are:

Behrens JT 1997 Principles and practices of exploratory data analysis, Psychological Methods 2 (2): 131-160.

Ellison AM 1993 Exploratory data analysis and graphic display, in SM Scheiner and J Gurevitch (eds.) Design and Analysis of Ecological Experiments. New York: Chapman & Hall: 14-45.

In this paper I discuss some relations between editing and the emotional experience of watching slasher films, and below are listed some interesting references that follow on from last week’s collection of paper on neuroscience and the cinema:

Bradley MM, Codispoti M, Cuthbert BN, and Lang PJ 2001 Emotion and motivation I: defensive and appetitive reactions in picture processing, Emotion 1 (3): 276-298.

Bradley MM, Lang PJ, and Cuthbert BN 1993 Emotion, novelty, and the startle reflex: habituation in humans, Behavioural Neuroscience 107 (6): 970-980.

Lang PJ, Bradley MM, and Cuthbert BN 1998 Emotion, motivation, and anxiety: brain mechanisms and psychophysiology, Biological Psychiatry 44 (12): 1248-1263.

Lang PJ, Davis M, and Öhman A 2000 Fear and anxiety: animal models and human cognitive psychophysiology, Journal of Affective Disorders 61 (3): 137-159.

Willems RM, Clevis K, and Hagoort P 2011 Add a picture for suspense: neural correlates of the interaction between language and visual information in the perception of fear, Social Cognition and Affective Neuroscience 6 (4): 404-416.

The editing structure of The House on Sorority Row (1983)

Following on from earlier posts on the editing structure of Halloween (here) and Slumber Party Massacre (here), this week I look at the editing in The House on Sorority Row (1983). The shot length data can be accessed here: Nick Redfern – The House on Sorority Row. The shot length data has been corrected by a factor of 1.0416, and includes the opening credits since these are shown over footage of the characters and locations and are therefore relevant to the narrative.

As before I’m using the order structure matrix to visualise the time series of the data for this film, but to make clearer how the matrix relates to observed data values I’ve included two run charts in Figure 1 showing the shot lengths (bottom) and the ranks of the shot lengths (middle).

Figure 1 Order structure matrix (top), ranks (middle), and shot length data (bottom) for The House on Sorority Row (1983)

With a median shot length of 3.0s and interquartile range of 3.7s The House on Sorority Row is edited more quickly than Halloween (median = 4.2s, IQR = 5.7s) but is similar to Slumber Party Massacre (median = 3.2s, IQR = 4.5s). There is no clear trend in shot lengths across the whole film and there are no clear distinctions between different narrative sections similar to the very abrupt shift we see in the final third of Halloween. Nonetheless, this film follows the general formal pattern set out in the earliest films of this sub-genre, with a number of clusters of longer and shorter takes associated with the same types of narrative events as in the other films. The replication of narrative events, character types, themes, and actions in the slasher film has been extensively analysed, and looking at their editing structure in detail it becomes very clear just how quickly a single style of editing became established in this type of film. There are only a few years between them, but the only major difference between Halloween, Slumber Party Massacre, and The House on Sorority Row is that the latter two films are cut more quickly.

The main feature in Figure 1 is the confrontation between and the girls that begins at shot 302 and runs until shot 440. This sequence is edited very quickly (Σ = 362.2s, median = 2.0s, IQR = 2.1s), but it is clear from Figure 1 that from shot 302 to shot 366 the length of the shots actually get shorter as the scene reaches its peak: the girls force Mrs. Slater into the swimming pool at gun point and the moment of greatest tension – as one of the girls fires a shot into the pool – is the point at which editing is fastest. From shot 367 the sequence slows down using longer shots, and this can be clearly seen in the order structure matrix and the run chart of the ranks. Of course, longer is a relative term, and the ‘slowing down’ of the editing in the second part of this scene means a shift from shots less than 0.5 seconds to shots between 1.5 and 5 seconds (though there are few longer than 10 seconds). (The editing in this sequence is related to the cluster of short shots that can be seen as the white column at shots 89 – 102, and which features Vicki practising with the gun). There is clearly a relationship between the way in which this scene is edited and the way in which the emotional impact of the scene is generated; and, while it is clear from watching the film that it is edited very quickly, it is easier to appreciate how this scene is structured by looking at the time series given the difference between shorter and longer shots may only be a couple of seconds.

The other clusters of shorter takes serve a different function but are also related to moments of intense emotion. The cluster beginning 165 is part of a sequence of photographs of Mrs. Slater’s old sorority classes that begins quite slowly as the camera pans across the photos; but from shot 165 there is a change to rapid editing (accompanied by a change in the music and the use of whip pans) as Mrs. Slater tears up the pictures and burns them. Again, the change in editing style is associated with a change in the mood of the scene. The cluster of short shots from shot 855 to shot 874 is typical of the rapid editing in the latter stages of a slasher film, and is associated with the killing of Vicki and Liz as they dispose of a body. The intensity of the violence is reflected in the intensity of the editing.

This last cluster sits between two sequences edited much more slowly. The dark column in the matrix between shots 797 and shot 854 focuses on Katherine’s attempt to raise help by calling Dr. Beck, and his subsequent arrival and explanation of the night’s events. It also includes the scenes in the graveyard and the attempts to dispose of a body that we know results in disaster. This sequence is heavy on plot since it explains much if the background about Mrs. Slater and her son, Eric (i.e. the killer). The sequence that follows on from the deaths of in the graveyard (shots 875-897) shifts us back to Katherine and Dr. Beck, and is again lacking action while setting up the film’s finale.

The earlier clusters of longer takes slow down the pace of the film in order to create a pervasive sense of foreboding that de-accentuates the violence of the killings and which seek to put the viewer on edge. Shots 480-540 focus on the girls at the party and their anxiety that the body of Mrs. Slater might be discovered. This is framed as a series of long takes as Katherine meets Peter and resists his attempts to make her enjoy the party; and is notable for an elaborate tracking shot as the girls exchange glances across the dance floor. This cluster also includes the scene in which makes the rookie mistake of going down to a darkened cellar by herself to check the fuse box, and again uses a slow editing pattern to build tension before she is finally dispatched. Similarly, shots 655-692 follow Katherine as she tries to find the girls who have gone missing from the party and explores the attic room of the Mrs. Slater’s murderous son. These scenes are again important for establishing plot points and Katherine finds important symbolic objects (e.g. the jack-in-the-box), but their main purpose is to build up a state of nervous apprehension in the viewer. Interestingly, this is achieved by using slow panning shots from Katherine’s point-of-view whereas such shots in slasher films are typically used to represent the killer’s stalking of his victims. This sequence also includes the other members of the sorority trying to dispose of Mrs. Slater’s body only to run into a policeman. These sequences and the various narrative threads they present serve to create an emotionally tense atmosphere for the viewer but unlike the aggressive tensity of the rapidly cut sections this mood is one of foreboding.

This use of two different editing patterns to create two different moods for the viewer is characteristic of the slasher film and can also be seen in the time series of Halloween and Slumber Party Massacre. We tend to speak of the style of a film in singular terms as though it definitely has one – and only one – mode of expression; but since the slasher film uses different editing patterns to create different effects it would make more sense to talk of the styles of these films. This can also be seen in the time series of RKO musicals (see here, here, and here).

The ‘final girl’ sequence begins at shot 985 (Σ = 434.4s, median = 2.7s, IQR = 2.1s). Here The House on Sorority Row does show some (minor) differences to Halloween and Slumber Party Massacre. In this film we have a progressive increase in the cutting rate, and the shift to shorter shots is particularly marked in the run chart of the shot ranks. The first part of this sequence is edited relatively slowly as Katherine makes her way through the sorority house to the attic, and this can be seen in the dark column at this point in the matrix in Figure 1. This is different to the other films in which this corresponding sequence begins when the killer attacks the final girl (as can clearly be seen at shot 437 in the matrix for Halloween). In The House on Sorority Row the final girl goes looking for the killer. Once the struggle between Katherine and the killer begins (shot 1063) we see the same rapid editing observed in the Halloween and Slumber Party Massacre, but we do not see the same fast-slow-fast pattern noted in the other films as the struggle between the killer and the final girl is temporarily suspended. This is due to the postponement of the killer’s return once we think he has been killed. The last shot of the film is a close-up of the eye as we discover Katherine has not defeated him and assume their struggle to the death will continue. The House on Sorority Row presents the same final girl sequence as the other slasher films I have looked at but cuts the narrative (and therefore the editing pattern) off before it reaches its ‘natural’ conclusion.

Like Halloween, The House on Sorority Row was remade in 2009 and a future post will look at the similarities and the differences between the original version of these films and their later reinvention.

Using the ECDF to analyse film style

Last month I looked at using kernel densities to analyse film style, and to follow-up this week’s post will focus on another simple graphical method for understanding film style: the empirical cumulative distribution function (ECDF).

Although it has a grand sounding name this is a very simple method for getting a lot of information very quickly. Most statistical software packages will calculate the ECDF for you and draw you a graph, but it is very simple to create an EXCEL or CALC spreadsheet to do this since it does not require any special knowledge.

The ECDF gives a complete description of a data set, and  is simply the fraction of a data set less than or equal to some specified value. Several plotting positions for the ECDF have been suggested, but here we use the simplest method:

which means that you count the number of shots (x) less than or equal to some value (X), and then divide by the sample size (N). Do this for every value of x in your data set and you have the ECDF. We can interpret this fraction in several ways: we can think of it as the probability of randomly selecting an x less than or equal to X (P[x ≤ X]); or we can think of it as the proportion of values less than or equal to X; or, if we multiply by 100, the percentage of values in a data set less than or equal to X.

For example, using the data set for Easy Virtue (1928) from the Cinemetrics database available here we can calculate the ECDF as illustrated in Table 1.

Table 1 Calculating the ECDF for Easy Virtue (1928) (N = 706)

To start, look at the value of X in the first column and then count the number of shots in the film with length less than or equal to that value. The first value is 0.9 but there are no shots this short in the film and so the frequency is zero. Divide this zero by the number of shots in the film (i.e. 706) and you have the ECDF when X = 0.9, which is 0 (because 0 divided by any number is always 0). Next, X = 1.0 seconds and there is 1 shot less than or equal to this value and so the ECDF at X = 1.0 is 1/706 = 0.0014. Turning to X = 1.1 we see there are three shots that are 1.1 seconds long AND there is one shot that is shorter in length (i.e. the one at 1.0s), and so the ECDF at X = 1.1 is 4/706 = 0.0057. This is equal to the frequency of 1.0 second long shots divided by N (0.0014) PLUS the frequency of shots that are 1.1 seconds long (3/706 = 0.0042) – and that is why it’s called the cumulative distribution function. From this point you keep going until to reach the end: the longest shot in the film is given as 66.6 seconds long and so all 706 shots must be less than or equal to 66.6 seconds and so at this value of X the ECDF = 706/706 = 1.0. The ECDF is 1.0 for any value of X greater than the maximum x in the data set.

It really is this easy. And you can get a simple graph of F(x) by plotting x on the x-axis and the ECDF on the y-axis. More usefully, you can plot the ECDFs of two or more films on the same graph so that you can compare their shot length distributions. Figure 1 shows the empirical cumulative distribution functions of Easy Virtue and The Skin Game (1931 – access the data here).

Figure 1 The empirical cumulative distribution functions of Easy Virtue (1928) and The Skin Game (1931)

Now clearly there is a problem with this graph: because the shot length distribution of a film is positively skewed all the shots are bunched up on the left-hand side of the plot and you cannot see any detail. This can be resolved by redrawing the x-axis on a logarithmic scale, which stretches out the bottom end of the data which has all the detail and squashing the top end which has only a few data points. This can be seen in Figure 2.

Figure 2 The empirical cumulative distribution functions of Easy Virtue (1928) and The Skin Game (1931) on a log-10 scale

These two graphs present exactly the same information, but at least in Figure 2 we can find the information we want. In transforming the x-axis we have not assumed the shot length distribution of either film follows a lognormal distribution – which is just as well because this is obviously not true for either film.

Now what can we discover about the editing in these two films?

First, it is clear that these two films have same median shot length because the probability of randomly selecting a shot less than or equal to 5.0 seconds is 0.5 in both films. The definition of the median shot length is the value that divides a data set in two so that half are less than or equal to x and greater than or equal to x (i.e P(x ≤ X) = 0.5. We might therefore conclude that they have the same style. However, these two films clearly have different shot length distributions and it is easier to appreciate this when we combine numerical descriptions with a plot of the actual distributions.

A basic rule for interpreting the plot of ECDFs for two films is that if the plot for film A lies to the right of the plot for film B then film A is edited more slowly. Obviously this is not so clear cut in Figure 2.

Below the median shot length, the ECDF of The Skin Game lies to the left of that of Easy Virtue indicating that at those shot lengths it has a greater proportion of shots at the low-end of the distribution: for example, 25% of the shots in The Skin Game are less than or equal to 2.0 seconds in length compared to just 6% of the shots in Easy Virtue. This would seem to indicate that The Skin Game is edited more quickly than Easy Virtue. At the same time we see that above the median shot length that the ECDF of The Skin Game lies to the right of that of Easy Virtue indicating that it has a lower proportion of shots at the high-end of the distribution: for example, 75% of the shots in Easy Virtue are less than or equal to 8.3 seconds compared to 66% of the shots in The Skin Game. This would appear to suggest that The Skin Game is edited more slowly than Easy Virtue. Clearly there is something more interesting going on than indicated by the equality of the medians, and the answer lies in how spread out the shot lengths of these two films. The ECDF of Easy Virtue is very steep and covers only a limited range of values, where as the ECDF of The Skin Game covers a much wider range of shot lengths. The interquartile range of Easy Virtue is 5.2 seconds (Q1 = 3.1s, Q2 = 8.3s) indicating the shot lengths of this film are not widely dispersed; while the IQR of The Skin Game is 12.7s (Q1 = 2.0s, Q3 = 14.7s).

This example is an excellent demonstration of why it is important to always provide a measure of the dispersion of a data set when describing film style. It is not enough to only provide the average shot length since two films may have the same median shot length and completely different editing styles. See here for a discussion of appropriate measures of scale that can be used. It should be standard practice that an appropriate measure of dispersion is cited along with the median shot length for a film by any researcher who wants to do statistical analysis of film style, and journal editors and/or book publishers who receive work where this is not the case should send it back immediately with a note asking for a proper description of a film’s style. If you don’t include any description – either numerical or graphical – of the dispersion of shot lengths in a film then you haven’t described your data properly.

We can also use the ECDFs for two films to perform a statistical test of the null hypothesis that they have the same distribution. This is called the Kolmogorov-Smirnov (KS) test, and the test statistic is simply the maximum value of the absolute differences between the ECDF of one film (F(x)) and the ECDF of another film (G(x)) for every value of x. The ‘absolute difference’ means that you subtract one from the other and then take only size of the answer and ignore the sign (i.e. ignore if its positive or negative):

Table 2 shows this process for the two films in Figures 1 and 2.

Table 2 Calculating the Kolmogorov-Smirnov test statistic for the ECDFs of Easy Virtue (1928) and The Skin Game (1931)

In the first column in Table 2 we have the lengths of the shots from the smallest in the two films (0.6 seconds) to the longest (174.7 seconds), and then in columns two and three we have the ECDF of each film. Column four is the difference between the ECDFs of the two films, subtracting the ECDF of The Skin Game from the ECDF of Easy Virtue for every x: so when x = 0.6, we have 0-0.0037 = -0.0037. The final column is the absolute difference, which is just the size of the value in the fourth column and the sign is ignored: the absolute value of -0.0037 is 0.0037. Do this for every value of x and find the largest value in the final column.

In the case of these two films the maximum absolute difference occurs when x = 2.0 and is statistically significant (p < 0.01). Therefore we conclude these two films have different shot length distributions. (You may find that different statistics software give slightly different answers to this depending on the plotting position used).

An online calculator for the KS-test that will also draw a plot of the ECDFs can be accessed here, and is accompanied by a very useful explanation. (NB: this only works for data sets up to N = 1024). Rescaling the x-axis of our plot of the two ECDFs does not affect the KS-test since the ECDFs are on the y-axis and D column in Table 2 is the vertical difference between them.

(There is also a one-sample of the KS-test for comparing a single distribution to a theoretical distribution to determine goodness-of-fit, but there are so many other methods that do exactly the same thing better that it’s not worth bothering with).

The ECDF is very easy to calculate, the graph is very easy to produce and provides a lot of information about a data set for every little effort, and the KS-test is also a very simple way of comparing two data sets. There is no bewildering mathematics involved: just count, divide, add, subtract, and ignore. The statistical analysis of film style really is this easy.