In this post I outline how count data may be modelled using a negative binomial distribution in order to more accurately present trends in time series count data than using linear methods. I also show how to use ANOVA to identify the point at which one model gains explanatory power, and how confidence intervals may be calculated and plotted around the predicted values. The resulting illustration gives a robust visualisation of how the Beslan Hostage crisis has taken on features of a memory event
Recently I wrote up a piece about quantifying memory and news, and proposed that two distinct linear models might be the way to go about it. However, the problem with linear models is they by their nature don’t take into account the ways in which trends may be non-linear.They also lead to nonsense predictions, such as negative values.
Generally, then, linear models should be avoided when mapping count data. What are the alternatives? Typically, a Poisson distribution would be ideal way to capture the probability of a clustering of observations being non-random. A feature of the Poisson distribution is that it assumes the sample mean equals the sample variance; this is very frequently violated when dealing with news data, as a story will have a small number of large values, followed by a large number of small values, resulting in a low mean and a high variance. Instead a negative binomial distribution may be used, which takes a value theta specifying the degree to which the variance and mean are unequal. Estimates provided by a negative binomial model are the same as Poisson estimates, but probability values tend to be more conservative.
One strength of R is its ability to model so called generalised linear models. The negative binomial distribution comes from the package MASS; theta may be calculated using glm.nb:

library(MASS)
fmla <- as.formula("count~date+e_news+a1+a2+elections")
theta <- glm.nb(fmla, data = mdb)$theta
results <- glm(fmla, data = mdb, family = negative.binomial(theta))

The above results may be considered in ANOVA to identify which variables contribute significantly to the model.

anova(results)

Analysis of Deviance Table

Model: Negative Binomial(10.85), link: log

Response: count

Terms added sequentially (first to last)

          Df Deviance Resid. Df Resid. Dev
NULL                         96        994
date       1      731        95        263
e_news     1       69        94        194
a1         1       52        93        142
a2         1        1        92        141
elections  1       12        91        129

Details about the coding of the variables, and the logic behind models contrasting stories as news or memory events may be found here.
From the ANOVA results I can identify which group of variables are contributing the most substantially to describing the data distribution: memory variables, or news variables. As I am interested in distributions where memory effects are apparent, and these develop only over time, I loop through the data deleting the first month's values, until such a time as there is no data left, or the memory variables have greater explanatory power than the news estimator:

mdb2 <- mdb  #copy the data
n <- 0
news <- 0
memory <- 0
while (n == 0) {
    aov3 <- glm(fmla, data = mdb2, family = negative.binomial(glm.nb(fmla, data = mdb2)$theta))
    anova(aov3)
    t <- data.frame(t(anova(aov3)[2]))
    news <- sum(t[, grep("date|news", colnames(t))])
    memory <- sum(t[, grep("a1|a1", colnames(t))])
    if (news > memory) {
        n <- 0
        mdb2 <- mdb2[mdb2$date > min(mdb2$date), ]
    } else (n <- 1)
}

From the negative binomial model predictions and confidence intervals for the period identified as of potential memory significance may be created. The 95% confidence interval either side of the predicted value is calculated by multiplying the standard error by 1.96. I want to plot the whole data, but only predicted values for the period identified as significant, so next I removed predictions for the data up until the period with memory potential. Finally I created a data frame containing the interval for which no estimates were calculated (this will be used to blur out data in ggplot).

estimate <- (predict.glm(aov3, newdata = mdb, type = "response", se.fit = T))
mdb$estimate <- estimate$fit
mdb$se <- estimate$se.fit
mdb$estimate[mdb$date < min(mdb2$date)] <- NA
mdb$se[mdb$date < min(mdb2$date)] <- NA
mdb$date <- as.Date(mdb$date)
mdb$upper <- mdb$estimate + (1.96 * mdb$se)
mdb$lower <- mdb$estimate - (1.96 * mdb$se)
mdb$lower[mdb$lower < 0] <- 0
rect <- data.frame(min(mdb$date) - months(1), min(as.Date(mdb2$date)))
colnames(rect) <- c("one", "two")

In the plot below I visualise the square root of articles about the Beslan hostage tragedy in the Russian press. The square root is chosen to prevent the high initial interest from obscuring the trend that emerged over time. To create the plot I

• add a ribbon representing the confidence interval
• plot the observed values
• add a dotted line representing the fitted values
• edit the formatting and add a title
• add a shaded rectangle over the area I wish to ignore:

ggplot(mdb, aes(date, sqrt(count), ymax = sqrt(upper), ymin = sqrt(lower)), 
    environment = environment()) + geom_ribbon(colour = "red", fill = "light grey", 
    alpha = 0.4, linetype = 2) + geom_point(colour = "dark green", size = 3, 
    alpha = 0.8) + geom_line(aes(date, sqrt(estimate))) + theme_bw() + ggtitle(paste0("Regression graph for the Beslan Hostage crisis, exhibiting possible features of memory event since ", 
    as.Date(min(mdb2$date)))) + geom_rect(aes(xmin = rect$one, xmax = rect$two, 
    ymin = -Inf, ymax = +Inf), fill = "light grey", colour = "grey", linetype = 2, 
    alpha = 0.015)

anova(aov3)

Analysis of Deviance Table

Model: Negative Binomial(6.449), link: log

Response: count

Terms added sequentially (first to last)

          Df Deviance Resid. Df Resid. Dev
NULL                         70      135.3
date       1    19.18        69      116.1
e_news     1     0.73        68      115.4
a1         1    24.54        67       90.9
a2         1     0.65        66       90.2
elections  1     0.29        65       89.9

Notice in the above table how the anniversaries variables exceed the explanatory power of the news and date variables. This indicates that by the end of 2006 Beslan was increasingly featuring as a memory event and less as a news story. Also notice how the remaining deviance is quite large - this model apparently fits the data less well than the model for the entire data (it explained 85% of the deviance), but this is due to the original estimate being biased by the accurate prediction of a few outliers.
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