Snjezana Soltic
Manukau Institute of Technology, New Zealand
ssoltic@manukau.ac.nz

Lora Peacock
Ministry of Agriculture and Forestry, New Zealand
lora.peacock@maf.govt.nz
 

 Soltic, S. & Peacock, L. (2006, October), A Comparison of Inductive and Transductive Models for Predicting the Establishment Potential of the Exotic Scale, Aspidiella Hartii (Cockerell) in New Zealand. Bulletin of Applied Computing and Information Technology Vol. 4, Issue 2. ISSN 1176-4120. Retrieved from

ABSTRACT

The purpose of this investigation is to assess the applicability of transductive reasoning for building predictive models in environmental studies. We used inductive and transductive reasoning to prepare predictive models to assess the establishment potential of Aspidiella hartii (Cockerell), an insect pest of potential economic importance in New Zealand. We built a global, a local and an individualised model, and assessed their performance using leave-one-out cross validation and kappa statistics. The prediction models were also built (and tested) for predicting the insect pest’s establishment potential at two locations, Bombay (India) and Wellington airport (New Zealand), where the pest has been established and is considered absent, respectively.

The inductive and transductive approaches to predicting pest establishment potential are compared and their utility in predicting and interpreting pest distribution are discussed. Based on our results, we conclude that transductive reasoning has a future in predictive modelling of pest insect distributions.

Keywords

Transductive reasoning, inductive reasoning, establishment potential, kappa statistics, leave-one-out cross validation

1. INTRODUCTION

By increasing and diversifying the volume of trade we also increase the possibility of the introduction of new species, the impact of which might be damaging to our agricultural and indigenous systems. Polyphagous insect pests are frequently intercepted on fresh produce coming into New Zealand from various overseas locations and the trend in these arrivals is continuously increasing.

Studies have been conducted to identify which characteristics make some insect species more invasive than others. For a species to establish in a new location the conditions for establishment must be fulfilled. Firstly, the species has to reach the location, preferably but not exclusively in superior numbers (Williamson & Fitter, 1996; Worner, 2002). Secondly, the species biological characteristics and the environment of the location being ‘invaded’ must be favourable to species establishment (Baker, 2002). The location’s climatic suitability is considered a very important factor that will determine whether a pest insect is likely to establish in new locations (Cook, 1931; Royer & Yang, 1991; Sutherst et al., 1991; Cohen, 1998; Dobesberger, 2000; Baker, 2002; Dentener et al., 2002; Dobesberger, 2002; Rafoss, 2003; Eyre et al., 2005). In particular, temperature, relative humidity, soil moisture, and their combined effects are believed to play a significant role in the likelihood that insect species will be able to establish a viable population (Baker, 2002; Dobesberger, 2002). Therefore, climate-matching techniques that compare the climate/environmental characteristics of the areas where species are naturally distributed to new areas where they could potentially establish have often been used to predict the locations that may be invaded (Manel et al., 2001; Peterson & Verglais, 2001; Baker, 2002; Worner, 2002).

Invasions of insect species can potentially result in serious impacts on a country’s economic development and environmental sustainability. Therefore, predictive modelling of species’ potential distribution is becoming increasingly important. A variety of methods and models have been introduced and used. Examples include the use of generalized regressions (Guisan et al., 2002; Lehmann et al., 2002; Zaniewski et al., 2002), artificial neural networks (Lek et al, 1996; Lek & Guégan, 1999; Manel et al., 1999; Peacock, 2005) and models incorporating a geographic information system (Peterson & Vieglais, 2001; Austin, 2002; Gibson et al., 2004). A useful overview of techniques for predicting pest distribution can be found in Guisan and Zimmermann (2000).

Aspidiella hartii, (Cockerell) (yam scale), has a status of unwanted pest in New Zealand, where it has been a pest of potential economic importance since 1998 (NZ Ministry of Agriculture and Forestry). This insect is not yet present in New Zealand, but it is often ‘intercepted’ at New Zealand borders. Yam scale is an armoured scale that feeds on plant juices and it is associated with yams but also occurs on other root crops such as ginger, turmeric and taro. Damage caused by an individual scale is small, but in large populations, the insect can cause significant damage. Yam scale is a widely distributed species, occurring in Oceania, including Australia, Asia, Africa and the Caribbean (CABI crop compendium). According to Heu (2002) it has been eradicated from Hawaii. Clearly, if we can predict the suitability of locations within New Zealand as possible habitats of this scale, then negative economic impacts could be minimised or prevented.

Most predictive models in ecological modelling apply inductive inference where a model is created from all available data and then it is applied on new, unseen data. According to Vapnik (1998), inducing a model based on all data first might be needless. He proposes the use of a paradigm referred to as transductive reasoning and transductive inference, where the focus is on one single point in the problem space. Transduction has been proposed to suit medical applications of learning systems, where the focus is on the individual patient data rather than on a group of people (Kasabov & Pang, 2003; Kukar, 2004; Song & Kasabov, 2004; Kukar & Grošelj, 2005). Here, we investigate the suitability of using transductive reasoning in modelling the establishment potential of Aspidiella hartii. We explore whether the transductive approach could be more effective than inductive modelling for ecological modelling of insect species distributions.

The difference between inductive and transductive modelling is dealt with in Section 2. Modelling of the establishment potential of A. hartii is covered in Section 3. The results and the findings are discussed in Section 4.

2. INDUCTIVE VERSUS TRANSDUCTIVE MODELLING

Inductive reasoning is the most often used inference tool in machine learning. Inductive methods estimate a general model (a rule, a function) from a set of observed data samples from input space. This model is then further used to predict outcome values for an unseen data vector from the same space (deduction). Regression analyses, largely used in ecological modelling, are an example of inductive reasoning. Given a data set: , where   and   are attributes of , and   is the vector under estimation. The target is to predict   in terms of   by modelling an estimation function . Once the function is obtained, it is used to predict the outcome for the new example.

Transductive modelling (Vapnik, 1998) focuses on single data units in the input space. This modelling results in a number of individual models, each tailored to best describe only one data vector. A model is prepared based on the characteristics of the sample’s closest data points, where the measure of closeness can be measured by Euclidean distance (Vapnik, 1998). When a simple K Nearest Neighbour method (KNN) is used to select the data points that seem most relevant for the modelling task, the output value for the new data sample is calculated as the average of the output values of the all most relevant samples. Song and Kasabov (2005) used the weighted KNN, and calculated the output as where is the output value for the sample   from   and   are its weights measured as .  The vector   is defined as the distance from the new sample and   nearest neighbours   for . The neighbouring sample closest to the new sample   has   and the sample furthest to   is assigned .

3. MODELLING THE ESTABLISHMENT POTENTIAL OF A. HARTII

3.1 Data Characteristics

In this experiment, meteorological data compiled from 74 worldwide sites where A. hartii has been recorded as either present (35 sites) or considered absent (41 sites) were used (Peacock, 2005). The presence and absence of the insect (the response variable) are related to the explanatory variables consisting of maximum summer air temperature (C°), minimum winter air temperate (C°) and annual potential evapotranspiration (mm/yr). Each site (a sample location) was given a class label presence/absence (1/0).

3.2 Models

Here, we present three group-discrimination models used to predict the establishment potential of A. hartii under current climatic conditions, two of which are result of inductive reasoning and one is the result of transductive reasoning. The likelihood of the presence/absence was modelled as a function of the three explanatory variables mentioned above.

The first model (Model 1) is a static model fitted using linear regression and inductive reasoning, covering the whole problem space (a global model).

The second model (Model 2) creates a number of local models that collectively cover the whole problem space. Model 2 is based on an evolving neuro-fuzzy inference system called DENFIS (Kasabov, 2002). In DENFIS the first-order Takagi-Sugeno type  fuzzy rules are used and the linear functions are created and updated through learning from data by using the linear least square estimator (Kasabov & Song, 2002). The rules are as follows: if x₁ is R_i1 and x₂ is R_i2 and … and x_q is R_iq then y_i is f_i(x₁, x₂,…,x_q), where f_i is a linear function in a form f_i(x) = β_i0 + β_i1 x₁+ … β_iq x_q and R_ij are fuzzy sets defined by their fuzzy membership functions μ_Rij. Here the Gaussian membership functions were used for the antecedent part. In DENFIS the partitioning of the input space is implemented using the evolving clustering method (ECM). The cluster centres and radii are correspondingly taken as initial values of the centres and widths (σ) of the Gaussian membership functions. Model 2 is suitable for modelling various climate influences on the distribution of the target species (A. Hartii) due to its dynamic evolving structure.

The third model (Model 3) builds an individual model for each site in  the test. Here, a weighted KNN method is used where the output for a site x_i is calculated based on output of k-nearest samples and their distance from x_i. The accuracy of this model depends on the number of nearest samples used to build the model.

3.3 Model Evaluation

Validation is an essential step in model development (Guisan & Zimmermann, 2000; Pearce & Ferrier, 2000; Robinson & Froese, 2004), where the suitability of a model for the specific application is evaluated, and the model performance is, preferably, compared to the performance of different techniques and models. In this step the ability of the model to predict independent data is tested as well. Here, we use leave-one-out cross validation and kappa (κ) statistics (Landis & Koch, 1977; Fielding & Bell, 1997).

In leave-one-out cross validation (also known as jackknife sampling), a model is fitted to the n-1 data samples from the problem space, where n is the total number of data, and this model is then used to predict the target variable for the data sample excluded from the fitted model. In our case that means we created 74 models leaving one site out of the modelling each time and then the target value was calculated for that particular site.

The kappa statistics is a threshold-dependent measure based on the confusion matrix (Table 1), where P_T + P_F + A_T + A_F = n and n is the total number of samples, here the total number of sites. P_T is the number of present sites correctly predicted by the model, A_T is the number of absent sites correctly predicted by the model, while P_F (false negatives) and A_F (false positives) are the numbers of incorrect predicted presences and absences, respectively. Kappa values (κ) are calculated using equation 1 (Farber and Kadmon, 2003).

                               (1)

Table 1. Confusion matrix

The kappa value of 1 denotes a perfect agreement between the model’s predictions and observations, while a zero value is an agreement approximately the same as would be expected by chance (Monserud & Leemans, 1992). The models’ performance was ranked using Table 2 (Monserud & Leemans, 1992). We also calculated the models’ sensitivity (sen), specificity (spe) and overall accuracy (a) (Table 3). Sensitivity and specificity are measures of a model correctly predicting true presence and true absence, respectively.

Table 2. The level of agreement between observed and predicted values (Monserud & Leemans, 1992)

Table 3. Equations used to calculate sensitivity, specificity and overall accuracy

4. RESULTS

4.1 Performance of the Models

To begin with, we examine the graphs shown in Figures  1, 2 and 3, in which the effect of threshold variation on the performance of each model is illustrated by plotting sensitivity (sen) , specificity (spe), kappa (κ) and overall accuracy (a) achieved by Model 1, Model 2 and Model 3, respectively. Note that all three models generate the predictions that have a value within the range [0, 1]. Predictions above or equal to a threshold value are treated as presences, while predictions below the threshold values are treated as absences. When the threshold value is altered, the values in the confusion matrix change, consecutively affecting the sensitivity, specificity, kappa and overall accuracy values. It is possible to deduce from the graphs shown in Figures 1, 2 and 3 the extent to which the performance of each model is dependent on the threshold value. Each plot includes the range of threshold values investigated, from 0.3 to 0.9 – yielding seven plotted points for each model. As stated earlier, the performance of Model 3 is dependant on the number of neighbouring samples used during the modelling stage. Here, we present results achieved when 35% neighbouring samples were used. Increasing and decreasing the number of neighbours decreased the model’s accuracy. We evaluated the predictive power of models using the leave-one-out cross validation approach as explained in Section 3.3.

The graphs shown in Figures 1, 2 and 3 follow similar trends. The sensitivity (sen) firstly increases with the threshold, reaches a maximum value, and then starts decreasing. The specificity (spe) starts with a maximum value and then decreases at higher threshold values. Clearly, the overall accuracy stays in the middle between sen and spe. In spite of the plots’ similarities, they do differ from one another. The overall accuracy (a) of Model 1 reaches its max for thresholds in the range 0.5 - 0.6 while a of Model 2 and Model 3 is maximum for thresholds in the range 0.6 – 0.7. It is interesting to observe that, a and κ exhibit larger changes in Model 1 and Model 2 than in Model 3.

Figure 1 shows the effect of different thresholds on the performance of Model 1. The overall accuracy (a) of the model is in the range [0.53 - 0.90] with a maximum at the threshold of 0.6. The different threshold values lead to differences in κ. At higher thresholds (0.8 and 0.9) κ shows ‘poor’ or ‘no agreement’ between observed and predicted outcome values. The highest κ (‘very good’ agreement) is achieved at threshold values of 0.5 and 0.6. The highest sensitivity (sen = 0.85) of this model is achieved at the threshold of 0.6, and the highest specificity (spe = 0.97) at 0.5, and then both values deteriorate down to 0.33 (sen) and 0.53 (spe) at 0.9. This deterioration pulls the overall accuracy from 0.9 at 0.6 down to 0.53 at 0.9.

Figure 1. Effect of the threshold on the performance of Model 1 expressed in terms of sensitivity (sen), specificity (spe), kappa (κ) and overall accuracy (a)

Somewhat different results are obtained with Model 2. This model exhibits ‘very poor’ and ‘poor’ κ at low thresholds values [0.3-0.5], and κ of 0.72 and 0.77 (‘very good’ agreement) is achieved at thresholds of 0.6 and 0.7, respectively. The highest overall accuracy (a) of Model 2 is in the range [0.53- 0.88] with maximum at 0.7. Sensitivity (sen) is in the range [0.49- 0.82]. This model predicts absence with a high accuracy, namely specificity (spe) is in the range [0.80 -1]. At the thresholds above 0.6, all values but kappa stay above 0.7. κ exhibits a sharp decline, from 0.77 at 0.7 to 0.47 at 0.9.

The graphs for Model 3 (Figure 3) show a smaller variations of all four performance indicators at lower thresholds [0.3 0.7]. In this range, sensitivity, specificity, kappa and overall accuracy exhibit high values. Sensitivity (sen) is 0.88 at 0.7, specificity (spe)  is 0.97, kappa (κ) is 0.85 and overall accuracy (a)  is 0.92. In this range κ shows ‘very good’ agreement between observed and predicted outcome values. For thresholds above 0.7, specificity, kappa and overall accuracy of this model decrease. In contrast, sensitivity firstly decreases to 0.85 at 0.8 and then bounces back to 0.88 at 0.9.

Figure 2. Effect of the threshold on the performance of Model 2 expressed in terms of sensitivity (sen), specificity (spe), kappa (κ) and overall accuracy (a)

Figure 3. Effect of the threshold on the performance of Model 3 expressed in terms of sensitivity (sen), specificity (spe), kappa (κ) and overall accuracy (a)

Figure 4 shows the kappa statistics for the studied models. Close inspection of the three plots in Figure 4 shows that Model 3’s  performance, when measured in terms of κ, is higher than the performance of the other two models. In particular, at the thresholds of 0.5, 0.6 and 0.7, Model 3 kappa is equal to 0.82, 0.85 and 0.85, respectively. In the same range the Model 1 and Model 2 kappa values are: 0.80, 0.80 and 0.61 for Model 1 and 0.32, 0.72 and 0.77 for Model 2. At the threshold of 0.6, Model 3 outperformed the other two models (κ = 0.85).  Model 1 was  the second best (κ=0.80) and Model 2 had the lowest κ (κ = 0.72).

Figure 4. Kappa statistics (κ). Values above 0.85 indicate ‘excellent’ agreement, values in the range [0.7 0.85] ‘very good’ agreement, values in the range [0.55 0.70] ‘good’ agreement and values below 0.55 ‘fair’, ‘poor’ and ‘no agreement’

4.2 Prediction of Establishment Potential at Two Sites, Bombay and Wellington Airport

We also used the inductive and transductive techniques to assess the suitability of two locations for establishment by A. hartii, at Bombay (India) and Wellington airport (New Zealand). The pest insect is known to be present in Bombay, and it is considered absent at Wellington airport. The locations were removed from the original data set. Using the reduced data set three models were prepared. Then the models were applied on the removed locations.

All three models predicted low likelihood of this insect’s establishment at the New Zealand location, where the scale is observed to be absent, and high likelihood of establishment at Bombay, where A. hartii is already present.

Model 1 resulted in a global model: y = 0.12 + 0.00057*PE – 0.02*t_S + 0.01*t_W. According to this model, the establishment potential of this insect y increases with annual potential evapotranspiration (PE) and winter air temperature (t_W), but decreases with summer air temperature (t_S). Influence of PE on the establishment potential is negligible comparing to the influences of summer and winter air temperatures.

Model 2 delivered two local models (L_m1 and L_m2). Local model 1 (L_m1) is for warmer locations with average PE values: y = 0.14 + 0.26*PE + 0.08* t_S + 1.57* t_W. At those locations, the likelihood of the insect’s establishment increases with higher values of explanatory variables. Local model 2 (L_m2) describes the establishment potential at cooler locations with lower PE values: y = 1.00 - 0.03*PE - 0.05* t_S + 0.21* t_W, at which the likelihood of the insect’s establishment increases with warmer winter air temperature, but decreases with summer air temperature and PE. It is interesting to observe that, L_m1 and L_m2 found that this insect’s establishment potential is positively dependent on winter temperature.

As mentioned earlier, the performance of Model 3 is dependent on the number of neighbouring samples used during the modelling stage. This influence is shown in Figure 5, where the likelihood of the insect establishment at two localities (Bombay and Wellington airport) is plotted against the number of neighbouring samples (k) used during the modelling. k is expressed as a percentage value of the total number of samples available for modelling. Each plot includes the range of k values investigated, from 5% to 45%. Note that there are nine points for each locality (Bombay (x) and Wellington airport (○)). Model 3 predicts the presence of the insect in Bombay correctly. Note that 1 means presence of the insect and 0 indicates absence of the insect. Although, the predicted establishment potential varies with k (min = 0.84, max = 0.97), these variations are not significant. Model 3 predicts that it is unlikely for Aspidiella harii to establish at Wellington airport. The establishment potential of this species is found to be 0 for all ks from 5% to 40% and 0.05 for k = 45%.

Figure 5. Establishment potential of A. hartii calculated using transductive reasoning

5. CONCLUSION

We have made use of inductive and transductive reasoning to assess the applicability of these approaches for assessing the establishment potential of invasive pest insects. The inductive reasoning was used to prepare a global model based on a regression analysis (Model 1) and to build local models (Model 2) based on a dynamic evolving fuzzy neural network. Individualised models (Model 3) were created for each location from the problem space using a weighted KNN method. The leave-one-out cross validation and kappa values were used to compare the accuracy of the models over the range of threshold values, from 0.3 to 0.9.

It can be seen that Model 3 scored the most ‘very good’ or ‘good’ performance marks over the range of thresholds. Its performance is superior to the other two inductive models at the threshold values in the range from 0.3 to 0.8 and performed worse than Model 2 but much better than Model 1 at the threshold of 0.9. This model also achieved the highest overall accuracy (0.92) compared to the overall accuracies of Model 1 (0.90) and Model 2 (0.88). Based on these results, we conclude that transductive reasoning could be used as a tool for the modelling of insect pest distributions, particularly when distributional data are limited on each species.

We also assessed the models’ explanatory capabilities at each geographic site. Model 1 is a static model fitted using linear regression and inductive reasoning and therefore it delivered one model for all sites. A possible criticism of this model is that it is highly unlikely that one model can accurately explain climate variation at 74 sites worldwide. Model 2 resulted in two local models, L_m1 for warmer sites and L_m2 for cooler sites. This is a better solution to using one global model. In the case of Aspidiella hartti both models (L_m1 and L_m2), predicted that winter temperature was influential in establishment. This is because Aspidiella hartti is restricted to a sub-tropical distribution, thus cold winter temperature would be a limiting factor for the potential establishment of this species in temperate climates. The explanatory capabilities of  Model 3 are less transparent since the establishment potential at each location is calculated based on the average of weighted output values of the k nearest samples to the particular location. To better understand how this value is influenced by the climatic characteristics of that site one needs to study the characteristics of the k nearest samples to that particular location. However it is clear that transductive techniques can be used to isolate climate and abiotic factors that may influence site establishment by insect pest species. Such information is certainly beneficial to bio-security agencies who need to be alerted to areas where new  insect pest invasions could occur.

6. ACKNOWLEDGEMENTS

The authors wishe  to acknowledge the support of this work by the Research Committee of the Electrical and Computer Engineering Department at the Manukau Institute of Technology, through the Departmental Research Fund and to thank Sue Worner and Nikola Kasabov for their invaluable guidance during all phases of this work, as well as, Ilkka Havukkala for commenting on an early draft of the manuscript.

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