Ecological interactions and the Netflix problem

Data

The first data-set was obtained from the study of Digel et al. (2014), who documented the presence and absence of interactions among 881 species from 48 forest soil food webs, details of which are provided in the original publication. A total of 34,193 unique interactions were observed across the 48 food webs, and a total of 215,418 absence of interactions. In order to improve representation of interactions involving low trophic levels species that were not identified at the species level in the first data-set, we compiled a second data-set from a review of the literature. We selected all articles involving interactions of terrestrial invertebrate species for a total of 126 studies, across these, a total of 1,439 interactions were recorded between 648 species. Only 88 absences of interactions were found. We selected traits based on to their potential role in consumption interactions (Table 2). For each species or taxa, these traits were documented based on a literature review or from visual assessment of pictures. In addition to these traits, we included two proxies for hard-to-measure traits: feeding guild and taxonomy. The traits were chosen for their potential relevance for species interactions and their availability, see (Laigle et al., in press) for greater details on the data-set.

K-nearest neighbour

Our recommender uses the K-nearest neighbour (KNN) algorithm (Murphy, 2012). The KNN algorithm is an instance-based method, it does not build a general internal model of the data but instead bases predictions on the K nearest (i.e., most similar) entries given some distance metrics. In the case of recommendation, each species is described by a set of traits and a set of preys, and the algorithm will recommend new preys to the species based on the preys of its K nearest neighbours. For example, if K = 3, we take the set of preys of the three most similar species to decide which prey to recommend. If species A is found twice and B once in the set of preys of the most similar species, we will recommend A first (assuming, of course, that the species does not already have this prey). See Table 3 for a complete example of recommendation. In the “Netflix” problem, this is equivalent to recommending new TV series/movies to a user by searching for the users with the most similar taste and using what they liked as recommendation. It is also possible to tackle the reverse problem: Amazon uses item-based recommendations, in which case we are looking for similar items instead of similar users to base our recommendations (Aggarwal, 2016).

Choosing the right value for K is tricky. Low values give high importance to the most similar entries, while high values provide a larger set of examples. Fortunately, the most computationally intensive task is to compute the distances between all pairs, a step that is independent of K. As a consequence, once the distances are computed, we can quickly run the algorithm with different values of K.

Different distance measures can be used. We will use the Tanimoto coefficient for recommendations. The Tanimoto (or Jaccard) similarity measure is defined as the size of the intersection of two sets divided by their union, or: (1)${\displaystyle tanimoto\left(x,y\right)=\frac{\left|x\cap y\right|}{\left|x\cup y\right|},}$

Since it is a similarity measure in the [0, 1] range, we can transform it into a distance function with 1 − tanimoto(x, y). The distance function uses two types of information: the set of traits of the species (see Table 2) and their set of preys. We define the distance function with traits as: (2)${\displaystyle tanimot{o}_d\left(x,y,w_t\right)=w_t\left(1-tanimoto\left(x_t,y_t\right)\right)+\left(1-w_t\right)\left(1-tanimoto\left(x_i,y_i\right)\right),}$

where w_t is the weight given to traits, x_t and y_t are the sets of traits for species x and y, and x_i, y_i are their sets of preys. Thus, when w_t = 0, only interactions are used to compute the distance, and when w_t = 1, only traits are used. See Table 3 for an example.

The data is the set of preys and binary traits for each species (Table 2). To test the approach, we randomly remove an interaction for each species and ask the algorithm to recommend up to 10 preys for the species with the missing interaction. Interactions are removed one-at-a-time and similarity is computed before the interaction is removed. The code for computing similarities after the interaction is removed is available in the code repository, but it has little effect on the results while making the program much slower to run since the similarity matrix must be computed for each trial. We count how many recommendations are required to retrieve the missing interactions and compute the top1, top5, and top10 success rates, which are defined as the probabilities to retrieve the missing interaction with 1, 5, or 10 recommendations. We repeat this process 10 times for each species with at least 2 preys, totally 7,200 attempts. We test all odd values of K from 1 to 19, and w_t = {0, 0.2, 0.4, 0.6, 0.8, 1}. We also divided species in groups according to the number of preys they have to see if it is easier to find the missing interaction for species with fewer preys.

Supervised learning

We also do a simple test with random forests to see if it is possible to predict interactions in this data-set using only the traits (Breiman, 2001). In this case, the random forests perform supervised learning: we are trying to predict y (interaction) from the vector of traits x by first learning a model on the training set, and testing the learned model on a testing set. We keep 5% of the data for testing. We perform grid search to find the optimal parameters for the random forests.

For our predictions, we count the number of true positives (tp), true negatives (tn), false positives (fp) and false negatives (fn). The score for predicting interactions (Score_y), non-interactions (Score_¬y) and the accuracy are defined as

(3)${\displaystyle Scor{e}_y=\frac{tp}{tp+fp},}$ (4)${\displaystyle Scor{e}_{\neg y}=\frac{tn}{tn+fn},}$ (5)${\displaystyle Accuracy=\frac{Scor{e}_{y}34193+Scor{e}_{\neg y}741968}{881^2},}$

with 34,193 and 74,1968 being the number of observed interactions and non-interactions in the 881 by 881 matrix. We then use the True Skill Statistics (TSS) to measure how accurate the random forest is, defined a (6)${\displaystyle TSS=\frac{\left(tp\times tn\right)-\left(fp\times fn\right)}{\left(tp+fn\right)\left(fp+tn\right)}.}$

The TSS ranges from −1 to 1.

Code and data

Since several machine learning algorithms depends on computing distances (or similarities) for all pairs, many data structures have been designed to compute them efficiently from kd-trees discovered more than thirty years ago (Friedman, Bentley & Finkel, 1977) to ball trees, metric skip lists, navigating nets (Izbicki & Shelton, 2015), and cover trees (Beygelzimer, Kakade & Langford, 2006; Izbicki & Shelton, 2015). We use an exact but naive approach that works well with small data-sets. Since distance(x, y) = 0 if x = y and distance(x, y) = distance(y, x), our C++ implementation stores the distances in a lower triangular matrix without the diagonal, yielding n(n − 1)∕2 distances to compute. A linear scan is then used to find the most similar species. Computing the distance matrix and testing the predictions 7,000 times for a set of parameters takes less than a second. We used Scikit for random forests (Pedregosa et al., 2011). The C++11 code for the KNN algorithm, Python scripts for random forests, and all data-sets used are available at https://github.com/PhDP/EcoInter (also stored on Zenodo with a DOI: Desjardins-Proulx, 2016).

Recommendation

While matrix imputation has a 50% change of success by random, the Tanimoto KNN needs to pick the right prey among up to 881 possibilities. Yet, it succeeds on its first recommendation around 50% of the times. When the first recommendation fails, the next nine recommendations only retrieve the right species around 15% of the times so the top5 and top10 success rates are fairly close to the top1 success rate (see Fig. 1). The Tanimoto measure is particularly effective for species with fewer preys, achieving more than 80% success rate for species with 10 or fewer preys (Fig. 2).

The highest first-try success rates (the probability to pick the missing interaction on the first recommendation) are found with K = 7 and no weights to traits, and with K = 17 and a small weight of 0.2 to traits (Fig. 3). Overall, the value of K had little effect on predictive ability.

Supervised learning

Random forests predict correctly 99.55% of the non-interactions and 96.81% of the interactions, for a TSS of 0.96. Much of this accuracy is due to the three real-valued traits (body mass, Ph₀, Ph₁). Without them, too many entries have the same feature vector x, making it impossible for the algorithm to classify them correctly. Removing the binary traits has little effect on the model. With only body mass, Ph₀, Ph₁, the TSS of the random forests is 0.94.