---
title: "Introduction to scanstatistics"
author: "Benjamin Allévius"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
bibliography: references.bib
vignette: >
  %\VignetteIndexEntry{Introduction to scanstatistics}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, eval=TRUE, echo=FALSE}
knitr::opts_chunk$set(fig.width=7.15, fig.height=5)
```

## What are scan statistics?
Scan statistics are used to detect anomalous clusters in spatial or space-time 
data. The gist of the methodology, at least in this package, is this:

1. Monitor one or more data streams at multiple _locations_ over intervals of 
   time. 
2. Form a set of space-time _clusters_, each consisting of (1) a collection of 
   locations, and (2) an interval of time stretching from the present to some 
   number of time periods in the past.
3. For each cluster, compute a statistic based on both the observed and the 
   expected responses. Report the clusters with the largest statistics.
   
## Main functions

### Scan statistics

- __`scan_eb_poisson`__: computes the expectation-based Poisson scan 
  statistic [@Neill2005].
- __`scan_pb_poisson`__: computes the (population-based) space-time scan
  statistic [@Kulldorff2001].
- __`scan_eb_negbin`__: computes the expectation-based negative binomial scan 
  statistic [@Tango2011].
- __`scan_eb_zip`__: computes the expectation-based zero-inflated Poisson
  scan statistic [@Allevius2017].
- __`scan_permutation`__: computes the space-time permutation scan statistic 
  [@Kulldorff2005].
- __`scan_bayes_negbin`__: computes the Bayesian Spatial scan statistic 
  [@Neill2006], extended to a space-time setting.

### Zone creation                  

- __`knn_zones`__: Creates a set of spatial _zones_ (groups of locations) to
                   scan for anomalies. Input is a matrix in which rows are the
                   enumerated locations, and columns the $k$ nearest neighbors.
                   To create such a matrix, the following two functions are
                   useful:
    + __`coords_to_knn`__: use `stats::dist` to get the $k$ nearest neighbors
                           of each location into a format usable by `knn_zones`.
    + __`dist_to_knn`__: use an already computed distance matrix to get the
                         $k$ nearest neighbors of each location into a format 
                         usable by `knn_zones`.
- __`flexible_zones`__: An alternative to `knn_zones` that uses the adjacency
                        structure of locations to create a richer set of zones.
                        The additional input is an adjacency matrix, but 
                        otherwise works as `knn_zones`.

### Miscellaneous                        

- __`score_locations`__: Score each location by how likely it is to have an 
                         ongoing anomaly in it. This score is heuristically 
                         motivated.
- __`top_clusters`__: Get the top $k$ space-time clusters, either overlapping or
                      non-overlapping in the spatial dimension.
- __`df_to_matrix`__: Convert a data frame with data for each location and time
                      point to a matrix with locations along the column 
                      dimension and time along the row dimension, with the 
                      selected data as values.
                      
## Example: Brain cancer in New Mexico
To demonstrate the scan statistics in this package, we will use a dataset of the 
annual number of brain cancer cases in the counties of New Mexico, for the years 
1973-1991. This data was studied by @Kulldorff1998, who detected a cluster of 
cancer cases in the counties Los Alamos and Santa Fe during the years 
1986-1989, though the excess of brain cancer in this cluster was not deemed
statistically significant. The data originally comes from the package _rsatscan_ 
[@rsatscan], which provides an interface to the program 
[SaTScan](http://www.satscan.org), but it has been aggregated and extended for 
the _scanstatistics_ package.

To get familiar with the counties of New Mexico, we begin by plotting them on a 
map using the data frames `NM_map` and `NM_geo` supplied by the _scanstatistics_ 
package:
```{r newmexico_map, eval=TRUE}
library(scanstatistics)
library(ggplot2)

# Load map data
data(NM_map)
data(NM_geo)

# Plot map with labels at centroids
ggplot() + 
  geom_polygon(data = NM_map,
               mapping = aes(x = long, y = lat, group = group),
               color = "grey", fill = "white") +
  geom_text(data = NM_geo, 
            mapping = aes(x = center_long, y = center_lat, label = county)) +
  ggtitle("Counties of New Mexico")
```

We can further obtain the yearly number of cases and the population for each
country for the years 1973-1991 from the data table `NM_popcas` provided
by the package:
```{r load_count_data}
data(NM_popcas)
head(NM_popcas)
```

It should be noted that Cibola county was split from Valencia county in 1981, 
and cases in Cibola have been counted to Valencia in the data.

### A scan statistic for Poisson data
The Poisson distribution is a natural first option when dealing with count data. 
The _scanstatistics_ package provides the two functions `scan_eb_poisson` and
`scan_pb_poisson` with this distributional assumption. The first is an
expectation-based[^1] scan statistic for univariate Poisson-distributed data 
proposed by @Neill2005, and we focus on this one in the example below. 
The second scan statistic is the population-based scan statistic proposed by
@Kulldorff2001.

[^1]: Expectation-based scan statistics use past non-anomalous data to estimate 
distribution parameters, and then compares observed cluster counts from the time 
period of interest to these estimates. In contrast, _population-based_ scan 
statistics compare counts in a cluster to those outside, only using data from 
the period of interest, and does so conditional on the observed total count.

#### Theoretical motivation
For the expectation-based Poisson scan statistic, the null hypothesis of no 
anomaly states that at each location $i$ and duration $t$, the observed count is 
Poisson-distributed with expected value $\mu_{it}$:
$$
 H_0 \! : Y_{it} \sim \textrm{Poisson}(\mu_{it}),
$$
for locations $i=1,\ldots,m$ and durations $t=1,\ldots,T$, with $T$ being the 
maximum duration considered. Under the alternative hypothesis, there is a 
space-time cluster $W$ consisting of a spatial zone $Z \subset \{1,\ldots,m\}$ 
and a time window $D = \{1, 2, \ldots, d\} \subseteq \{1,2,\ldots,T\}$ such that 
the counts in $W$ have their expected values inflated by a factor $q_W > 1$ 
compared to the null hypothesis:
$$
H_1 \! : Y_{it} \sim \textrm{Poisson}(q_W \mu_{it}), ~~(i,t) \in W.
$$
For locations and durations outside of this window, counts are assumed to be 
distributed as under the null hypothesis. 
Calculating the scan statistic then involves three steps:

* For each space-time window $W$, find the maximum likelihood estimate of
  $q_W$, treating all $\mu_{it}$'s as constants. 
* Plug the estimated $q_W$ into (the logarithm of) a likelihood ratio with 
  the likelihood of the alternative hypothesis in the numerator and the 
  likelihood under the null hypothesis (in which $q_W=1$) in the denominator, 
  again for each $W$.
* Take the scan statistic as the maximum of these likelihood ratios, and the
  corresponding window $W^*$ as the most likely cluster (MLC).

#### Using the Poisson scan statistic
The first argument to any of the scan statistics in this package should be a 
matrix (or array) of observed counts, whether they be integer counts or 
real-valued "counts". In such a matrix, the columns should represent locations 
and the rows the time intervals, ordered chronologically from the earliest 
interval in the first row to the most recent in the last. In this example we 
would like to detect a potential cluster of 
brain cancer in the counties of New Mexico during the years 1986-1989, so we
begin by retrieving the count and population data from that period and
reshaping them to a matrix using the helper function `df_to_matrix`:
```{r get_counts}
library(dplyr)
counts <- NM_popcas %>% 
  filter(year >= 1986 & year < 1990) %>%
  df_to_matrix(time_col = "year", location_col = "county", value_col = "count")
```

#### Spatial zones
The second argument to `scan_eb_poisson` should be a list of integer vectors,
each such vector being a _zone_, which is the name for the spatial component of
a potential outbreak cluster. Such a zone consists of one or more locations 
grouped together according to their similarity across features, and each 
location is numbered as the corresponding column index of the `counts` matrix 
above (indexing starts at 1). 

In this example, the locations are the counties of New Mexico and the
features are the coordinates of the county seats. These are made available in 
the data table `NM_geo`. Similarity will be measured using the geographical 
distance between the seats of the counties, taking into account the curvature
of the earth. A distance matrix is calculated using the `spDists` function from 
the _sp_ package, which is then passed to `dist_to_knn` (with $k=15$ neighbors) 
and on to `knn_zones`:
```{r get_zones, eval=TRUE}
library(sp)
library(magrittr)

# Remove Cibola since cases have been counted towards Valencia. Ideally, this
# should be accounted for when creating the zones.
zones <- NM_geo %>%
  filter(county != "cibola") %>%
  select(seat_long, seat_lat) %>%
  as.matrix %>%
  spDists(x = ., y = ., longlat = TRUE) %>%
  dist_to_knn(k = 15) %>%
  knn_zones
```

#### Baselines
The advantage of expectation-based scan statistics is that parameters such as 
the expected value can be modelled and estimated from past data e.g. by some 
form of regression. For the expectation-based Poisson scan statistic, we can use
a (very simple) Poisson GLM to estimate the expected value of the count in each 
county and year, accounting for the different populations in each region. 
Similar to the `counts` argument, the expected values should be passed as a 
matrix to the `scan_eb_poisson` function:

```{r fit_baselines, eval=TRUE}
mod <- glm(count ~ offset(log(population)) + 1 + I(year - 1985),
           family = poisson(link = "log"),
           data = NM_popcas %>% filter(year < 1986))

ebp_baselines <- NM_popcas %>% 
  filter(year >= 1986 & year < 1990) %>%
  mutate(mu = predict(mod, newdata = ., type = "response")) %>%
  df_to_matrix(value_col = "mu")
```

Note that the population numbers are
(perhaps poorly) interpolated from the censuses conducted in 1973, 1982, and 
1991.

#### Calculation
We can now calculate the Poisson scan statistic. To give us more confidence 
in our detection results, we will perform 999 Monte Carlo replications, by which 
data is generated using the parameters from the null hypothesis and a new scan
statistic calculated. This is then summarized in a $P$-value, calculated as
the proportion of times the replicated scan statistics exceeded the observed 
one. The output of `scan_poisson` is an object of class "scanstatistic", which
comes with the print method seen below.

```{r run_ebp, eval=TRUE}
set.seed(1)
poisson_result <- scan_eb_poisson(counts = counts, 
                                  zones = zones, 
                                  baselines = ebp_baselines,
                                  n_mcsim = 999)
print(poisson_result)
```

As we can see, the most likely cluster for an anomaly stretches from 1986-1989
and involves the locations numbered 15 and 26, which correspond to the counties
```{r show_MLC, eval=TRUE, comment=NA}
counties <- as.character(NM_geo$county)
counties[c(15, 26)]
```

These are the same counties detected by @Kulldorff1998, though their analysis
was retrospective rather than prospective as ours was. Ours was also data 
dredging as we used the same study period with hopes of detecting
the same cluster.

#### A heuristic score for locations
We can score each county according to how likely it is to be part of a cluster
in a heuristic fashion using the function `score_locations`, and visualize the
results on a heatmap as follows:
```{r county_scores, eval=TRUE}
# Calculate scores and add column with county names
county_scores <- score_locations(poisson_result, zones)
county_scores %<>% mutate(county = factor(counties[-length(counties)], 
                                          levels = levels(NM_geo$county)))

# Create a table for plotting
score_map_df <- merge(NM_map, county_scores, by = "county", all.x = TRUE) %>%
  arrange(group, order)

# As noted before, Cibola county counts have been attributed to Valencia county
score_map_df[score_map_df$subregion == "cibola", ] %<>%
  mutate(relative_score = score_map_df %>% 
                          filter(subregion == "valencia") %>% 
                          select(relative_score) %>% 
                          .[[1]] %>% .[1])

ggplot() + 
  geom_polygon(data = score_map_df,
               mapping = aes(x = long, y = lat, group = group, 
                             fill = relative_score),
               color = "grey") +
  scale_fill_gradient(low = "#e5f5f9", high = "darkgreen",
                      guide = guide_colorbar(title = "Relative\nScore")) +
  geom_text(data = NM_geo, 
            mapping = aes(x = center_long, y = center_lat, label = county),
            alpha = 0.5) +
  ggtitle("County scores")
```

A warning though: the `score_locations` function can be quite slow for large
data sets. This might change in future versions of the package.

#### Finding the top-scoring clusters
Finally, if we want to know not just the most likely cluster, but say the five
top-scoring space-time clusters, we can use the function `top_clusters`.
The clusters returned can either be overlapping or non-overlapping in the 
spatial dimension, according to our liking.
```{r top_counties, eval=TRUE}
top5 <- top_clusters(poisson_result, zones, k = 5, overlapping = FALSE)

# Find the counties corresponding to the spatial zones of the 5 clusters.
top5_counties <- top5$zone %>%
  purrr::map(get_zone, zones = zones) %>%
  purrr::map(function(x) counties[x])

# Add the counties corresponding to the zones as a column
top5 %<>% mutate(counties = top5_counties)
```

The `top_clusters` function includes Monte Carlo and Gumbel $P$-values for each
cluster. These $P$-values are conservative, since secondary 
clusters from the original data are compared to the most likely clusters from
the replicate data sets.

## Concluding remarks
Other univariate scan statistics can be calculated practically in the same way
as above, though the distribution parameters need to be adapted for each scan
statistic. 

# Feedback
If you think this package lacks some functionality, or that something needs 
better documentation, I happily accept feedback either here at GitHub or via 
email at benjak@math.su.se. I'm also very interested in applying the methods in 
this package (current and future) to new problems, so if you know of any 
suitable public datasets, please tell me! A dataset with a multivariate 
response (e.g. multiple counter variables) would be of particular interest for 
some of the scan statistics that will appear in future versions of the package.

# References
