Many chemicals partition in water and can exert adverse effects on aquatic systems threatening the survival of the members of these ecosystems.
Calculating a standard measure of aquatic toxicity is a long and costly procedure. However, using actual data from a waterbody, we can model this variable using Machine Learning techniques. As a result, we do not need to measure it in the laboratory.
This example is solved with Neural Designer. You can use the free trial to understand how the solution is achieved step by step.
This is an approximation project since the variable to be predicted is continuous.
The fundamental goal here is to model the LC50 (the standard measure of toxicity) as a function of the sample's molecular properties.
The first step is to prepare the data set . This is the source of information for the approximation problem. It is composed of:
The file aquatic-toxicity.csv contains the data for this example. Here the number of variables (columns) is 9, and the number of instances (rows) is 546.
We have the following variables for this analysis:
Our target variable will be the last one, LC50.
The instances are divided into training, selection, and testing subsets. They represent 60%, 20%, and 20% of the original instances, respectively, and are split at random.
Calculating the data distributions helps us check for the correctness of the available information and detect anomalies. The following chart shows the histogram for the power-generated variable:
It is also interesting to look for dependencies between input and target variables. To do that, we can plot an inputs-targets correlations chart.
MLOGP and RDCHI are the most correlated variables because they measure lipophilicity, which is the driving force of narcosis.
In a scatter chart we can visualize how this correlation works.
The second step is to build a neural network that represents the approximation function. For approximation problems, it is usually composed by:
The neural network has 8 inputs (TPSA, SAacc, H-050, MLOGP, RDCHI, GATS1p, nN, C-040) and 1 output (LC50).
The scaling layer contains the statistics of the inputs. We use the automatic setting for this layer to accommodate the best scaling technique for our data.
We use 2 perceptron layers here:
The unscaling layer contains the statistics of the outputs. We use the automatic method as before.
The next graph represents the neural network for this example.
The fourth step is to select an appropriate training strategy. It is composed of two parameters:
The loss index defines what the neural network will learn. It is composed of an error term and a regularization term .
The error term chosen is the normalized squared error . It divides the squared error between the outputs from the neural network and the targets in the data set by its normalization coefficient. If the normalized squared error has a value of 1, then the neural network is predicting the data 'in the mean,' while a value of zero means a perfect prediction of the data. This error term does not have any parameters to set.
The regularization term is the L2 regularization . It is applied to control the complexity of the neural network by reducing the value of the parameters. We use a weak weight for this regularization term.
The optimization algorithm is in charge of searching for the neural network parameters that minimize the loss index. Here we chose the quasi-Newton method as optimization algorithm.
The following chart shows how the training (blue) and selection (orange) errors decrease with the epochs during the training process. The final values are training error = 0.331 NSE and selection error = 0.481 NSE , respectively.
Even though we are getting moderately good results, our model is far from perfect, mainly because of the small size of the Data Set we are working with. This might be one of the biggest issues of Machine Learning.
In this case, Model selection algorithms aren't very useful to improve our model's performance, as having a more complex arquitecture can also broaden the small Data Set problem.
The purpose of the testing analysis is to validate the generalization capabilities of the neural network. We use the testing instances in the data set, which have never been used before.
A standard testing method in approximation applications is to perform a linear regression analysis between the predicted and the real pollutant level values.
For a perfect fit, the correlation coefficient R2 would be 1. As we have R2 = 0.744, the neural network is predicting the testing data quite well, considering our small Data Set issues.
We have achieved a mean error of 8.64%.
In the model deployment phase, the neural network is used to predict outputs for inputs that it has never seen.
We can calculate the neural network outputs for a given set of inputs:
For that purpose, we can use
Response
Optimization
.
The objective of the response optimization algorithm is to exploit the mathematical model to look for
optimal operating conditions.
Indeed, the predictive model allows us to simulate different operating scenarios and adjust the control
variables to improve efficiency.
An example is to minimize LC50 toxicity while maintaining the number of nitrogen atoms equal to the desired value.
The next table resumes the conditions for this problem.
Variable name | Condition | |
---|---|---|
TPSA(Tot) | None | |
SAacc | None | |
H-050 | None | |
MLOGP | None | |
RDCHI | None | |
GATS1p | None | |
nN | Equal to | 2 |
C-040 | None | |
LC50 | Minimize |
The next list shows the optimum values for previous conditions.
Directional outputs plot the neural network outputs through some reference points.
The next list shows the reference point for the plots.
We can see here how MLOGP affects LC50:
The mathematical expression represented by the predictive model is displayed next:
scaled_TPSA(Tot) = TPSA(Tot)*(1+1)/(347.3200073-(0))-0*(1+1)/(347.3200073-0)-1;
scaled_SAacc = SAacc*(1+1)/(571.9520264-(0))-0*(1+1)/(571.9520264-0)-1;
scaled_H-050 = H-050*(1+1)/(18-(0))-0*(1+1)/(18-0)-1;
scaled_MLOGP = MLOGP*(1+1)/(9.147999763-(-6.446000099))+6.446000099*(1+1)/(9.147999763+6.446000099)-1;
scaled_RDCHI = RDCHI*(1+1)/(6.43900013-(1))-1*(1+1)/(6.43900013-1)-1;
scaled_GATS1p = GATS1p*(1+1)/(2.5-(0.2809999883))-0.2809999883*(1+1)/(2.5-0.2809999883)-1;
scaled_nN = nN*(1+1)/(11-(0))-0*(1+1)/(11-0)-1;
scaled_C-040 = C-040*(1+1)/(11-(0))-0*(1+1)/(11-0)-1;
perceptron_layer_output_0 = tanh[ 0.291703 + (scaled_TPSA(Tot)*-0.391549)+ (scaled_SAacc*0.251752)+ (scaled_H-050*0.085857)+ (scaled_MLOGP*-0.277858)+ (scaled_RDCHI*-1.14629)+ (scaled_GATS1p*0.803457)+ (scaled_nN*-0.240904)+ (scaled_C-040*-0.137404) ];
perceptron_layer_output_1 = tanh[ 0.240456 + (scaled_TPSA(Tot)*0.895344)+ (scaled_SAacc*-0.575449)+ (scaled_H-050*0.216825)+ (scaled_MLOGP*0.676507)+ (scaled_RDCHI*0.277635)+ (scaled_GATS1p*-0.115849)+ (scaled_nN*-0.0410888)+ (scaled_C-040*-0.239895) ];
perceptron_layer_output_2 = tanh[ 1.35559 + (scaled_TPSA(Tot)*0.868684)+ (scaled_SAacc*-0.00789895)+ (scaled_H-050*0.878771)+ (scaled_MLOGP*0.816033)+ (scaled_RDCHI*-2.5133)+ (scaled_GATS1p*2.02157)+ (scaled_nN*-0.148986)+ (scaled_C-040*-1.02231) ];
perceptron_layer_output_0 = [ -0.309113 + (perceptron_layer_output_0*2.07224)+ (perceptron_layer_output_1*2.20069)+ (perceptron_layer_output_2*-1.55368) ];
unscaling_layer_output_0 = perceptron_layer_output_0*(10.04699993-0.1220000014)/(1+1)+0.1220000014+1*(10.04699993-0.1220000014)/(1+1);