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Results of risk assessment by the ITO3dE model
The results in the I dimension show that, overall, the distribution was high in the west and low in the northeast and the southeast in all three periods (Fig. 3, I, II, III; Table 2), and this tallies with the topography of Chongqing. The northwestern and central regions of Chongqing are mainly hilly and slightly mountainous, while the southeastern and northeastern regions represent the Dabashan Mountain system and the Daloushan Mountain system, respectively. Thus, farmland in Chongqing is mainly distributed in the western regions as well as in regions with extensive flat areas, such as Dianjiang and Liangping. Some regions in Dianjiang, Yongchuan, Dazu, Shapingba, Wansheng, and Jiangbei show relatively high risks, but the risk level is still medium. Hence, it can be concluded that the risk level in the I dimension during 2005–2015 is, overall, not high. Considering there are too many single-factor graphs, we omitted these graphs, but provide the following description: Among the three single factors, I1 has the highest value, and I1 and I2 both present a first increasing and then decreasing trend (the maximum values of I1 in 2005, 2010, and 2015 were 3.38, 4.08, and 2.78, respectively, and those of I2 in 2005, 2010, and 2015 were 2.71, 3.37, and 2.48, respectively). For the I1 results, the risk levels of the regions with higher levels in 2005, such as Yongchuan, Fuling, and Liangping, showed a certain decrease in 2015, but the risk levels of some regions such as Pengshui, Qianjiang, and Xiushan showed an increasing trend. The risk grade of I2 was relatively lower than that of I1, but overall, the spatiotemporal variation trend was consistent with that of I1, except for the increasing trend of the risk level of Qianjiang. Basically, the risk grade of I3 was zero; only the risk level of Bishan was in the medium risk status, while those of Hechuan and Fengdu were low.
Figure 3

Result distribution map of I, T, and O dimensions of Chongqing in 2005, 2010, 2015.

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Table 2 Statistical results of I, T, and O dimensions in 2005–2015.
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Spatially, the results in the T dimension presented, overall, an opposite distribution pattern when compared to the I dimension, that is, with low levels in the western regions and high levels in the northeastern and southeastern regions (Fig. 3, IV, V, VI; Table 2). The annual differences in the T dimension data are mainly determined by the variations in the factors I4 and I7, which showed relatively higher risk levels in all three periods. The values of I4 in the years 2005, 2010, and 2015 were 1.42–5.78, 0.84–6.12, and 0.14–6.93, respectively, while those of I7 were 0, 0–5.38, and 0–5.06, respectively. Because Chongqing is a typical mountainous city with purple soil33, high-risk and extremely high-risk regions, I5 and I6, are widely distributed across the city. In addition, due to the introduction of the factor I8, the water areas had a higher risk level, which is consistent with the actual situation of AGNPS.
The results in the O dimension showed a smaller interannual variation, with a low overall risk level (Fig. 3, VII, VIII, IX; Table 2). The O dimension levels were mainly affected by the spatial changes in the paddy field area. As mentioned above, during the 10 years, the area of paddy fields in Chongqing was nearly reduced by half, which led to the decrease in the spatial distribution of I12 and an increased risk in counties such as Kaizhou, Fengjie, Liangping, and Changshou. Spatially, Yongchuan, Shapingba, Bishan, Dianjiang, Changshou, and Kaizhou showed higher risk levels, and the risk levels of Kaizhou, Fengjie, Wanzhou, Liangping, and Changshou showed a significantly increasing trend. The high risk values of I9 were mainly distributed in Yongchuan, Shapingba, Jiangbei, Changshou, Dianjiang, and Liangping, with Shapingba showing the highest value of 3.75, while Chengkou, Wushan, Fengjie, Shizhu, and Xiushan had lower values. The high risk values of I10 were mainly distributed in the western regions and were below the medium risk levels. The risk values in 2010 were higher than those in 2005 or 2015, but did not surpass 3.0, and the high values were mainly distributed in the western regions as well as in Dianjiang, Wanzhou, and Liangping. The risk values of I11 were all below 3.0, and the highest value of 2.78 was found for Fengjie; higher values were mainly distributed in the northeastern and southeastern counties. The high risk values of I12 were mainly distributed in the northeastern and southeastern counties, which mostly have only small areas of paddy fields.
Figure 4 shows the data on AGNPS risks during 2005–2015 in Chongqing. The risk distribution trends in 2005, 2010, and 2015 were basically consistent and in the ranges of 0.40–2.28, 0.41–2.57, and 0.41–2.28, respectively. The maximum risk values were all below 3.0 for the three periods. Regions with medium levels were mostly distributed in the western regions of Chongqing (Dazu, Jiangjin, etc.) as well as in the counties Dianjiang, Liangping, Kaizhou, Wanzhou, and Zhongxian. Larger spatial differences were observed among different counties or different parts of a certain county; for example, the middle flatland part and the mountain systems at the two sides in Liangping or the northwestern and southeastern parts in Shizhu.
Figure 4

Spatiotemporal distribution graph of the evaluation results of agricultural NPSP risks in Chongqing during 2005–2015: (a) 2005; (b) 2010; (c) 2015.

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Spatiotemporal change results of risk by transition matrix analysis
By assigning no risk, low risk, and medium risk levels with 1, 2, and 3, respectively, in GIS, we can obtain the spatiotemporal transition matrix according to the formula of the transition matrix. Figure 5 shows the spatiotemporal transition situation of the AGNPS risk evaluation in Chongqing. Basically, high levels show no changes, and the proportions of ‘no-risk no-change’, ‘low-risk no-change’, and ‘medium-risk no-change’ situations were 10.86%, 33.42%, and 17.25%, respectively, accounting for 61.53% of the total area of Chongqing. Among these, the ‘no-risk no-change’ situation was mainly distributed in Rongchang, the east of Nanchuan, Shizhu, Pengshui, and Qianjiang; the ‘low-risk no-change’ situation was widely distributed in Wulong, the southeast of Fengdu, the south of Nanchuan, and the northeastern counties of Chongqing, while the ‘medium-risk no-change’ situation was mainly distributed in Shapingba, Yongchuan, Dianjiang, the north of Nanchuan, and Kaizhou.
Figure 5

Spatiotemporal transition situation of agricultural NPSP risks in Chongqing during 2005–2015.

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During 2005–2015, the proportions of risk increase, risk decline, and risk fluctuation were 13.45%, 17.66%, and 7.36%, respectively. Risk increases mainly occurred in central Jiangjin, central Fengdu, Pengshui, Qianjiang, the midwest of Yunyang, central Liangping, Wuxi, Wushan, and Chengkou, while risk declines were mainly observed for the main urban area of Chongqing, northern Tongliang, Dazu, Youyang, and Xiushan. Risk fluctuation was concentrated in Jiangjin, Bishan, Fuling, and Youyang.
Results of risk concentration degree by Kernel density analysis
Figure 6 shows the kernel density analysis results of the medium-risk regions. As seen in the figures, the peak values of the kernel density at these three periods were all around 1,110, suggesting that the maximum gathering degree of medium-risk pattern spots basically showed no changes. The spatial distribution of kernel density at these three periods showed a consistent trend, but the distribution differences at different periods were significant. In 2005, medium-risk regions were mainly concentrated in Shapingba, southern Dazu, central Yongchuan, eastern Beibei, Dianjiang, central Kaizhou, northwestern Shizhu, northern Nanchuan, central Wanzhou, southwestern Zhongxian, and southeastern Xiushan, while in 2010, such regions mainly occurred in Shapingba, eastern Jiangjin, southeastern Beibei, northern Nanchuan, northeastern Changshou, Dianjiang, northern Fuling, northern Fengdu, northeastern Shizhu, northeastern Liangping, central Kaizhou, Wanzhou, northeastern Pengshui, and eastern Xiushan. In 2015, medium-risk regions were mainly concentrated in Shapingba, Yongchuan, central Jiangjin, northwestern Nanchuan, northeastern Beibei, Dianjiang, Liangping, the junction of Fuling and Fengdu, central Kaizhou, northern Yunyang, eastern Pengshui, southeastern Qianjiang, and central Xiushan.
Figure 6

Kernel density graphs of medium-risk areas in Chongqing during 2005–2015: (a) 2005; (b) 2010; (c) 2015.

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To further explore the distribution of regions with the high-risk gathering zones (Table 3), we conducted a separate analysis on the regions with kernel density values higher than 1,000 (the kernel density values of these regions were divided into 10 grades with equal intervals, and the 10th grade had values from 1,000 to 1,110).
Table 3 Distribution of regions with high-risk gathering zones.
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Results of hot and cold spots by Getis-Ord Gi* analysis
Applying Getis-Ord Gi* analysis is helpful to clearly identify high-value hot spots (Hot Spot-99% Confidence) and low-value cold spots (Cold Spot-99% Confidence). Figure 7 shows the Getis-Ord Gi* analysis results; the overall variation trends of high-value hot spots and low-value cold spots were consistent in all periods, with significant distribution differences. The regions located in the high-value hot spot zones in all three periods were Yongchuan, Shapingba, Dianjiang, Liangping, northwestern Fengdu, and Zhongxian, while those located in the low-value cold spot zones were Chengkou, Wuxi, Wushan, Pengshui, and Rongchang. Throughout the 10 years, the high-value hot spot zones showed significant diffusion in Fengjie, Yunyang, Kaizhou, central Qianjiang, and northern Nanchuan, while the low-value cold spot zones showed significant diffusion in some parts of the midwestern counties such as central Fuling and southern Yubei. These high-value hot spots or low-value cold spots were mainly distributed in the above-mentioned regions and their surrounding areas and showed significant “gathering trends”. The spatiotemporal variation trend of the distribution of these high-value hot spots or low-value cold spots can reflect the variation tendencies of hot spots or cold spots in different regions. Over time, the high-value hot spot zones gradually migrated towards the northeastern counties of Chongqing, while the low-value cold spot zones in the midwestern counties presented an obvious diffusion trend. The low-value cold spot zones in the northeastern regions gradually decreased, while those in the southeastern regions tended to become more fragmented. These results indicate that the high-value hot spot zones gradually dominated the northeastern regions, while the low-value cold spot zones gradually dominated the midwestern regions.
Figure 7

Getis-Ord Gi analysis results in Chongqing during 2005–2015.

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Study system and species
This study was performed in Welgevonden Game Reserve (WGR), a privately owned game reserve in the Limpopo province, South Africa (24° 10′ S; 27° 45′ E to 24° 25′ S; 27° 56′ E). The reserve is located in the mountainous Waterberg region. WGR was established on former agricultural lands in the early 1980s and the main occurring vegetation types are Waterberg Mountain Bushveld and Sour Bushveld. The Waterberg region has a temperate climate, with two distinct seasons, characterized by the rainfall regime: a dry season ranging from April to September and a wet season ranging from October to March, with an average annual precipitation in WGR of 634 mm. Our study area is an enclosed breeding camp within WGR, with a size of approximately 1200 ha. Main predator species such as lion, cheetah and spotted hyena were excluded from this study area, as well as elephant and rhino.
WGR equipped 35 impala (Aepyceros melampus), 34 blue wildebeest (Connochaetes taurinus), 35 plains zebra (Equus burchellii) and 34 common eland (Taurotragus oryx) with a GPS and accelerometer sensor equipped custom made collar; an estimated 23% of the individual impalas present in the area, 48% of the eland, 40% of the wildebeest and 40% of the zebra. However, due to malfunctioning and errors made in the sensor development process, only 83 of the sensors yielded data at any point in time, thus lowering the effective density of sentinel animals. During the experimental intrusions (see below), the median number of data-yielding sensors was 47, and minimally 30. The animal movement data were recorded day and night and transmitted wirelessly in near real-time to five long-range low-power LoRa radiocommunication gateways in the study area, from where data packages were routed to an on-line data warehouse via a 3G/4G backhaul. The deployment of these sentinel animals were approved by the board and CEO of WGR as a management action and was performed in accordance with relevant guidelines and regulations (see Supplementary GPS Collaring letter).
Experimental intrusions
Between September 2017 and March 2018, WGR employees performed experimental intrusions (lasting ca. 2 h) on foot and by car through the study area, at varying locations and movement routes through the study area, independent from the locations of the sentinel animals. The movement of the intrusions were tracked by GPS, and the relevant metadata for each intrusion recorded (mode of transport, group size, start time, end time). The intrusions were distributed in a stratified way over the mornings, middays and afternoons (with time slots relative to specific solar positions: sunrise, solar noon and sunset). Furthermore, the intrusions were temporally spread in such a way to avoid a disturbance overflow for the sentinel animals, by performing a maximum of five experiments per week and a maximum of two experiments per day (and then only with one intrusion in the morning and one in the afternoon).
Data gathering
The animal sensors gathered location data via GPS and overall dynamic body accelerations47 (ODBA) via a tri-axial accelerometer (range ± 2 g; sampling frequency 100 Hz, down-sampled to 10 Hz prior to analysis). The GPS was scheduled to record spatial position at irregular intervals depending on the level of activity as gauged by ODBA. All sensors were scheduled to record locations every 15 min in the absence of sufficient activity (given that successive fixes were further than 5 m apart, else a geofence was applied and the new coordinate was omitted to save bandwidth and battery power, thereby assuming that the animal still was at its previous location). The GPS fix rate was increased up to 2- or 10-min intervals (depending on two different sensor settings) when ODBA indicated sufficient activity (after checking for the geofence). ODBA data were sampled continuously and summarized per 15 s window in a mean, maximum and variance value.
The experimentally intruding groups were outfitted with handheld GPS devices that recorded their location every 5 s and these groups logged and timestamped all their pre-defined activities and metadata on a tablet using CyberTracker48 during their intrusion. Most cars traveling through the study area were tracked by GPS as well to filter the animal data for disturbances by cars unrelated to the experimental intrusions.
Weather data (temperature, radiation, precipitation and wind) in the study area were recorded on a 3-min resolution with a weather station in the north of the study area. We assumed the 1200 ha study area to be sufficiently small to assume the weather station data to be representable for the prevailing weather conditions throughout the study area. GIS data of the study area (summarized in Supplementary Table S1) consisted of information on topography, infrastructure (e.g., fences, roads, powerlines, etc.) and vegetation cover (supervised classification of 25 cm resolution aerial imagery into four classes: trees, herbaceous/grass, sand/soil and other/built-up area).
All further data processing and algorithm development was done in the software R3.5.049.
Data pre-processing
To link the animal location data with the intrusion location data, as well as to correct for the substantial level of positional noise present in the animal location data, we modelled the animal location data to regular 1-min resolution trajectories using the following five steps. First, we filtered out large obvious errors (e.g., obvious outliers and irregularities such as locations far outside the study area) from the data. Second, we corrected systematic medium-scale outliers: ‘spikes’ that occurred due to positional outliers. Such spike-like outliers were visible during sensor testing while following known straight-line trajectories along an airstrip, thereby confirming that these spike-like geometries most likely resulted from positional error rather than true animal movement. Points were classified as anomalous spike points when (a) the displacement to and away from this point was high ( > 500 m), (b) when the distance between the locations before and after this point was small, and (c) when the turning angle at this point approached 180 degrees. Therefore, we corrected the locations that were classified as spike-like anomalies by shifting them closer to the straight line between the neighboring points. The extent of this shift was set relative to the degree of spikiness of the points (the spikier the pattern, the larger the shift towards the midpoint of the adjacent coordinates). Third, after filtering and correcting the original locations we smoothed the timeseries of x/y coordinates at each original timepoint with a Kalman smoother using a dynamic linear model. Fourth, we linearly interpolated the locations to a 10 s resolution based on ODBA, where we considered the animal to be stationary between multiple timepoints if the accelerometer signal suggested the animal was not moving. Fifth, we fitted an X-spline through the data, where we gave the linearly ODBA-interpolated locations a smaller weight, and sampled the fitted spline on a regular 1-min resolution. These pre-processing steps resulted in the modelled animal trajectory data, composed of spatial locations every minute, and averaged ODBA statistics per step (i.e., the segments between consecutive coordinates). These data were used as input for the next steps in the analyses. In contrast to the animal data, the raw intrusion data were of a high temporal resolution and spatial accuracy so that we only needed to subset the data in order to acquire 1-min resolution time-synchronized intrusion trajectories.
The first three parts of the data pre-processing were only needed because of firmware issues in our custom-made sensors. Without these issues, a simple denoising technique like a Kalman filter will suffice.
Feature engineering and processing
We computed a plethora of human-engineered features from the animal trajectories, ODBA data, weather data and several GIS layers with environmental data from the study area (summarized in Supplementary Table S1). All features were computed such that they could not directly be linked to specific points in space or time (by computing movement features relative to the environmental variables), so that only behavioral patterns and abnormalities therein could be linked to intrusion presence. After engineering these base features, we transformed certain features (after visual inspection of the histograms) to approximately symmetric distributions using logarithms. Then we truncated the distributions to the lower and upper 0.001 percentile to correct possible outliers. After that, we standardized all computed features to zero mean and unit variance per species. We also computed scaled versions of selected features by subtracting the mean and dividing by the variance of the selected features per reference set to capture deviations from normal behavior: (1) per area (characterized by a 30 by 30 m neighborhood around each grid cell), (2) per time of day (morning, midday, afternoon) in a period of 5 weeks around each intrusion or control, (3) per area per time of day per 5 weeks, and (4) per individual sentinel per time of day per 5 weeks (Supplementary Table S1). Furthermore, after computing and standardizing the features, we computed more features by applying moving window computations (5 min centered, 10 and 20 min lagging, and the difference between these: 5 min centered minus 10 and 20 min lagging) on the standardized features to capture (the change in) the recent history of animal movement descriptors (mean and standard deviation of all features, fitted Mean Squared Displacement exponential function parameters, net-gross distance ratio and variance of log First Passage Times). Finally, we discretized all features to ordinal values to avoid odd-, fat- and heavy-tailed distributions. In total we computed 2117 features describing different aspects of movement geometry of individual trajectories, herd topology and the interactions with landscape variation.
Subsetting and dimensionality reduction
Before analyzing the computed animal movement features, we applied some filtering on the data. We removed all periods with an experimental intrusion during which there were less than 30 active animal sensors in total. We also removed data of both animals and intrusion when they were close to the reserve’s main gate in order to avoid dilution of the data with other known disturbances. This resulted in 57 intrusions that were selected for further analyses. For every intrusion we selected control data of the same period one or 2 days earlier or later during which no intrusion took place, resulting in an approximately balanced intrusion-control dataset. Furthermore, we removed data from animals that were located within 250 m and within 20 min of a vehicle moving through the area that was not part of our experiment.
For each feature, we computed 4 importance metrics based on binary labelled data: records associated to locations within 1 km from the intrusion (subscript 1) versus an equally-sized random selection of data points during control periods (subscript 0): Mahalanobis distance, marginality (computed as (frac{{mu }_{1}-{mu }_{0}}{{sigma }_{0}}), for sample mean (mu) and sample standard deviation (sigma)), specialization (computed as (frac{{sigma }_{1}}{{sigma }_{0}})) and the Mean Decrease Accuracy of a Random Forest classifier (with default hyperparameters). We then ranked the features according to their importance and selected a feature for further analyses if it occurred in the top 125 features for any of the 4 importance measures described above (resulting in a total of 361 selected features). Subsequently, we converted the selected features per main feature class (Supplementary Table S1) to principal components, keeping those principal components that capture the most variation (in total 95%), which resulted in 99 selected components in total. Finally, we transformed these components again via a second principal component analysis, now across all the selected 99 components. In subsequent training of the animal behavior classifier, we optimized the total number of included components as a hyperparameter, which resulted in the first 8 principal components in the best performing classifier.
Labelling
We labelled the sentinel movement data through visual inspection of the animal and intruder trajectories, where we considered the animals’ behavior to be undisturbed when the animal was not near an intrusion, or when the animal was close to an intrusion yet did not visually display a change in behavior. However, when the animal was near the intrusion and displayed a sudden or gradual behavioral change in response to intrusion proximity, we labelled the data as ‘flight’ (changing the movement direction away from the intrusion, possibly with increased speed) or ‘regroup’ (when individuals clustered together). In total, only ca. 1% of the animal data were associated to either flight or regroup behavior (which we will refer to as ‘response’ behavior). A few animals also appeared to exhibit behavior we could label as ‘freeze’, i.e., halting movement in the proximity of the intrusion, yet this class was too underrepresented to be accurately predicted and hence dropped from the final dataset. Furthermore, we assigned a qualitative measure of intensity to each labelled behavioral response (‘low’, ‘medium’, ‘high’) to describe how visually pronounced this response was. Besides the supervised labelling based on visual inspection of behavioral responses via video animations of the trajectories, we also labelled data using an unsupervised k-means nearest neighbor classifier, where we clustered the feature space consisting of the 99 features selected as described above into 25 clusters per species.
Animal behavior classification
We trained an RBF kernel C-classification Support Vector Machine (SVM) with a subsequent moving window over the outputted probabilities to distinguish undisturbed versus response behavior. In the training datasets we only included the data separated by more than 1 km from the intrusion and labelled as ‘undisturbed’, and removed 90% thereof to train algorithms with a more balanced dataset. Furthermore, we only trained and validated on data with intrusions present in the area. We trained another SVM to distinguish the flight response from the regroup response. All computations were done in R 3.5.0 with the e1071 package on the Linux High Performance Cluster of Wageningen University and Research. We optimized the following hyperparameters and model settings during the training phase for the Average Precision via a grid search (with the selected values between brackets):

gamma (undisturbed-response: 10–3.2; flight-regroup: 10–2.0);

cost (undisturbed-response: 10–2.2; flight-regroup: 10–1.5);

number of principal components to include as features (undisturbed-response: 8; flight-regroup: 12);

species-specific models versus one model with species dummy variables included in the features (species-specific models);

specific models for the different times of day versus one model with time of day dummy variables (one model);

response intensities to include in the training data (only medium and high intensities);

weights to assign to the classes (equal weights);

the quantile to be computed of the SVM probabilities by the moving window (100%, i.e., maximum value);

the alignment of the moving window (centered);

the size of the moving window (15 min on both sides).

The best model was selected via a leave-one-intrusion-out cross-validation approach. We summarized the predictive performance by computing the Average Precision of the least occurring class (i.e., ‘response’ for the undisturbed-response model: 46%, Supplementary Fig. S2; and ‘regroup’ for the flight-regroup model: 80%, Supplementary Fig. S3). After having computed these probabilities with an SVM and a temporal window smoother, we tried to improve the predicted performance by including the predicted animal response probabilities of nearby animals. However, this spatial explicit approach hardly improved the predictive performance, indicating that the spatial contextualization of behavioral response was sufficiently captured by the computed features. We therefore did not include this spatial contagion effect of predicted animal response probabilities in the final analysis.
System classification—detection
Based on the predicted SVM response probabilities and feature cluster analysis, we computed summary features per 15 min of each intrusion and control period. These summary features related to the odds ratios of the probability of association of unsupervised clusters with intrusions versus controls, the SVM predicted probabilities of behavioral response, and several features describing the values (and its spatial structure, e.g., clustering or autocorrelation) of these SVM predicted response probabilities. After computing summary features per 15 min, we summarized them even further for the intrusions versus controls using the following eight statistics: mean, standard deviation, minimum, maximum, mean of the lagged differences, standard deviation of the lagged differences, minimum of the lagged differences and maximum of the lagged differences.
After computing the summary features, we build a logistic regression classifier to distinguish intrusions from controls. To create a parsimonious model, we iteratively added features to the model and evaluated its performance after each iteration. We evaluated the performance based on the model accuracy and performed validation through 25 times twofold cross-validation in a stratified way (by 25 times choosing a balanced random sample of intrusions and controls). We determined the sequence of adding features to the model by performing an independent two-sample t-test for each feature between the intrusions and controls. The feature with the largest t-value was then added to the model. After each feature addition, we removed its correlation with the remaining features using linear regressions with the added feature as independent variable and the remaining features as dependent variables, from which we extracted the residuals, standardized them to zero mean and unit variance, and applied the t-tests again. The (original) feature with the largest t-value was then added to the model again. This procedure was repeated until all features were ordered corresponding to their “importance”. We then performed logistic regressions without interactions between the features for an increasing number of features (Supplementary Fig. S4). The model already performed quite accurately with only 7 features (86.1% accuracy ± SD 3.3%, precision 82.6% ± SD 6.9%, recall 89.2% ± SD 5.1%). However, with 20 features and 2-way interactions the model achieved the maximum accuracy (90.9%).
System classification—localization
The data gathered during intrusions that were correctly predicted as such by the detection classifier were used to train the intrusion localization algorithm. The probability surface of the location of the intrusion was fitted relative to that of the sentinel animals using:

$${O}_{i,j} sim frac{{p}_{j}left({f}_{wn}left({theta }_{i,j},{mu }_{j},{rho }_{1}right) {f}_{ln}left({gamma }_{i,j},{mu }_{1},{sigma }_{1}right)right) left(1-{p}_{j}right) left({f}_{wn}left({theta }_{i,j},{mu }_{j},{sigma }_{0}right) {f}_{ln}left({gamma }_{i,j},{mu }_{0},{sigma }_{0}right)right)}{{f}_{wn}left({theta }_{i,j},{mu }_{j},{rho }_{0}right) {f}_{ln}({gamma }_{i,j},{mu }_{0},{sigma }_{0})}$$

where ({O}_{i,j}) is the odds ratio of intrusion presence at location (i) evaluated for individual (j), ({p}_{j}) is the SVM-predicted probability that individual (j) is exhibiting response behavior. The function ({f}_{wn}) is the wrapped normal probability density function, ({theta }_{i,j}) is the direction from location (i) to the location of the focal animal (j), ({mu }_{j}) is the movement direction of individual (j), ({rho }_{1}) and ({rho }_{0}) are the standard deviations of the unwrapped distributions. The function ({f}_{ln}) is the lognormal probability density function, where ({gamma }_{i,j}) is the distance of location (i) to (j), ({mu }_{1}) and ({mu }_{0}) as well as ({sigma }_{1}) and ({sigma }_{0}) are the log-normal distribution parameters (respectively log-mean and log-sd).
The parameters ({mu }_{1}), ({sigma }_{1}) and ({rho }_{1}) capture the geometry of intrusion-animal topology for animals that exhibited a predicted behavioral response to the intrusion. Similarly, ({mu }_{0}), ({sigma }_{0}) and ({rho }_{0}) are the corresponding parameters for animals that were predicted to be undisturbed. The parameters ({mu }_{1}), (mathrm{log}({sigma }_{1})) and (mathrm{log}({rho }_{1})) were fitted to the data assuming a 3rd order polynomial relationship to ({t}_{s}): the time (in minutes) since the start of the predicted behavioral response (using the maximum F1 classification score). Since the behavioral response signature is lost over time, we truncated ({t}_{s}) to 45 min (thus ({t}_{s} >45) min was set to ({t}_{s}=45)). The parameters ({mu }_{0}), ({sigma }_{0}) and ({rho }_{0}) were estimated using the data of the controls and with randomly generated intrusion locations in the study area, in order to correct for the effects of geometry of the study area on the predicted response surfaces. The probability surface ({P}_{i}) was then calculated as:

$${P}_{i}=alpha sum_{j}{O}_{i,j}$$

where (alpha) is a normalization constant so that ({P}_{i}) integrates to 1 over the area covered by the rectangular axis-aligned bounding box around the study area.
To measure the prediction accuracy of each localization surface, we simplified each surface to a point coordinate located at the location of maximum probability, and computed the Euclidian distance to the known true position of the intrusion. We then summarized each experimental intrusion by selecting the 10 prediction surfaces with the most condense highest probability density, i.e., those in which the top 5% probability density is contained in the smallest, most condense, area. The spatial error of the localization prediction associated with these selected predictions was further summarized by taking the average Euclidian distance over the 10 selected predictions. More

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