Neural Network

1. Neural networks offer modelling procedures for nonlinear data, which enables you to discover more complex relationships in your data. You can thus develop more accurate and effective predictive models. The introductory section of these slides is loosely based on two notes by A.T.S. Carneiro Note 1 and Note 2. Neural Network

2. The scientific literature presents several regression algorithms, several of which are implemented in the SPSS Modeller. The multilayer perceptron neural network algorithm (MLP) was selected for these examples. The MLP neural network algorithm is based on the functional principles of biological neural structures, as indicated in the figure (below). Multilayer Perceptron Neural Network Algorithm (MLP)

3. Neural computing researches attempt to organise mathematical models similarly to the structures and organization of neurons in biological brains to achieve similar processing abilities, in addition to the inherent capacities of biological brains, such as learning based on examples, trial and error, knowledge generalization, among many others. Multilayer Perceptron Neural Network Algorithm (MLP)

4. Biological Neural Structure

5. Based on this analogy to biological neurons, the MLP neural network algorithm implements a neural network that is composed of layers of artificial neurons that are stimulated by input signals, which are transmitted through the network via synapses connecting neurons in different layers. The figure presents an artificial perceptron neuron model with n inputs {x1, x2, …, xn}, in which each input xi has an associated synapse wi, and an output y. Multilayer Perceptron Neural Network Algorithm (MLP)

6. Artificial Neuron Model

7. There is also an additional neuron parameter, named w0, known as bias that can be interpreted as a synapse associated to an input x0 = -1. The output of the neuron y is based on the product between input vector x (x0, x1, x2, …, xn) and vector w (w0,w1, w2, …, wn) composed of synapses, including the bias (w0), Multilayer Perceptron Neural Network Algorithm (MLP)

8. The neuron output is then obtained through the activation function of neuron , in which a hyperbolic tangent function is usually adopted (sigmoid-nature function), defined by for a generic value a; however, it is convenient to use other activation functions in certain scenarios. Multilayer Perceptron Neural Network Algorithm (MLP)

9. The artificial neuron model is feed forward, that is, connections are directed from inputs (x0, x1, …, xn) to output y of the neuron. The figure presents the layout of perceptron neurons in an MLP neural network, in which there are two neuron layers, one hidden and one output. Regarding the neural network that is presented in the figure, each neuron of the hidden layer is connected to each neuron in the output layer. Therefore, inputs of the output layer neurons correspond to the outputs of hidden layer neurons. Multilayer Perceptron Neural Network Algorithm (MLP)

10. The analyst that uses the artificial neural network algorithm must choose how many neurons to use in the hidden layer, considering the set of input data, since with a low number of neurons in the hidden layer; the neural network is not able to generalize each class's data. However, a high number of neurons in the hidden layer prompts the over fitting phenomenon, in which the neural network exclusively learns training data, and does not generalize learning for data classes. Multilayer Perceptron Neural Network Algorithm (MLP)

11. Example Of A Neural Network With A Hidden Layer

12. The neural network training process is conducted based on a back propagation algorithm, with the purpose of adjusting values that are associated to synapses to allow the neural network to map an input space and output space, in which the input vectors x are samples of the input space and each input vector is associated to an output z, which can be represented by a vector z (z1, z2, …, zn), based on a scalable value or a symbolic value. Specifically, for symbolic values, a neuron in the output layer corresponds to each of the possible symbols that are associated to the input vector. Multilayer Perceptron Neural Network Algorithm (MLP)

13. During the neural network training process, a set of input data is initially determined, to which the associated output is known, and random values are attributed to each synapse in the neural network. The data is presented to the neural network and the supplied output is compared to the actual output, generating an error value. The error value is then employed to adjust reverse neural network synapses, from output to inputs (back propagated). Multilayer Perceptron Neural Network Algorithm (MLP)

14. The process of adjusting synapse values is repeated until an interruption criterion is established, for example, a fixed number of repetitions or a minimum error. Thus, in each repetition, the outputs that are provided by the neural network get closer to the actual output. The synapse value correction equations minimize errors between the output that is provided by the neural network and the actual output. Multilayer Perceptron Neural Network Algorithm (MLP)

15. For conventional regression, the methodology employed consists in using neurons with linear activation function and assigning an output neuron to map each of the output vector's components. In cases where the output is the only scalable value, the neural network is designed with a single neuron in the output layer. Two examples are considered. Multilayer Perceptron Neural Network Algorithm (MLP)

16. For conventional regression, the methodology employed consists in using neurons with linear activation function and assigning an output neuron to map each of the output vector's components. In cases where the output is the only scalable value, the neural network is designed with a single neuron in the output layer. Two examples are considered. Multilayer Perceptron Neural Network Algorithm (MLP)

17. This data set was downloaded from archive or more directly example. Data Set A

18. It describes variables affecting concrete compressive strength. Concrete is the most important material in civil engineering. The concrete compressive strength is a highly nonlinear function of age and ingredients. These ingredients include cement, blast furnace slag, fly ash, water, super-plasticizer, coarse aggregate, and fine aggregate. Data Set A

19. It describes variables affecting concrete compressive strength. Concrete is the most important material in civil engineering. The concrete compressive strength is a highly nonlinear function of age and ingredients. These ingredients include cement, blast furnace slag, fly ash, water, super-plasticizer, coarse aggregate, and fine aggregate. Number of instances (observations): 1030 Number of Attributes: 9 Attribute breakdown: 8 quantitative input variables, and 1 quantitative output variable Missing Attribute Values: None Data Set A

20. The file that is used is organised in nine columns, in which each line represents data that is collected from a concrete mixture analysed in a lab. The first seven columns correspond to data about concentration of elements in the mixture, in kg by m3 of concrete; the following column corresponds to the age of the concrete, in days; and the last column corresponds to the sturdiness of the concrete, which is measured in MPa (mega Pascal, pressure measurement unit). Data Set A

21. Variable Information

22. All of these attributes are numerical variables whose values correspond to the measurement unit; thus, the neural network that is used is designed to solve a regression type problem in which the input space comprises the first eight columns of the file (cement concentration to age) and the output space corresponds to the ninth column of the file (concrete sturdiness). I-Cheng Yeh, "Modeling of strength of high performance concrete using artificial neural networks," Cement and Concrete Research, Vol. 28, No. 12, pp. 1797-1808 (1998). Data Set A

23. What Does It Look Like?

24. SPSS Commands

25. SPSS Commands Note that the covariates have been standardised, so that none are numerically dominant.

26. SPSS Commands As a first trial a small hidden layer is adopted, this may be increased in succeeding applications of the procedure.

27. SPSS Commands

28. SPSS Commands The outputs are saved so that further analysis may be undertaken.

29. Neural Network With Optimal Architecture

30. The figure presents a dispersion chart containing actual concrete resistance values on the horizontal axis and values that are estimated by the neural network with two hidden neurons in the vertical axis. Results Of The Fitting Of The Data

31. The identity line (diagonal), which represents the ideal result, in which each concrete sample would present the exact same resistance as estimated by the neural network. Results Of The Fitting Of The Data

32. In this case, the high concentration of points next to the identity line is visually evident, which is confirmed by correlation and error, reported in the final table, evidencing the efficiency of the proposed methodology. Results Of The Fitting Of The Data

33. Residuals Of The Results Of The Fitting Of The Data

34. Input Variables And Their Respective Relevance As previously mentioned, each neural network input element has an associated synapse, which is represented by a numerical value that controls input; the higher the synapse value, the more relevant is the input for the result that is generated by the neural network.

35. Input Variables And Their Respective Relevance The figure presents a relevance chart for each input field to obtain fitting results, information that becomes available after the creation of the node corresponding to the model generated.

36. Metrics are applied to the results represent the linear correlation coefficient, which is based on the equation and absolute average error. Which is based on the equation for both generic data sets, in which the correlation coefficient measures the influence between data sets, and can also be interpreted as a coefficient of similarity between two data sets; and the absolute average error measures inconsistencies between two data sets.. Additional Analysis

37. Additional Analysis

38. Additional Analysis Also

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43. Additional Analysis

44. The commands are extensive, if interested refer to the printer friendly version of these notes. Syntax

45. Any solution will depend on the random number seed employed, so values will only be broadly similar with those presented here). Summary

46. Summary By analysing results that are obtained by the neural network with two, four, and six hidden layer neurons, it is concluded that the best neural network configuration is probably four hidden neurons, which presents both a higher correlation and decreased error between actual data and estimated values. The goal is to be parsimonious, combining a good fit with as few neurons as possible.

47. The second example is possibly closer to those you might encounter. The SPSS commands are identical and the secondary analysis is simply reduces to a comparative table. Data Set B

48. This data set was downloaded from archive or more directly example. This breast cancer database was obtained from the University of Wisconsin Hospitals, Madison from Dr. William H. Wolberg. O. L. Mangasarian and W. H. Wolberg: "Cancer diagnosis via linear programming", SIAM News, Volume 23, Number 5, September 1990, pp 1 & 18. Data Set B

49. Samples arrive periodically as Dr. Wolberg reports his clinical cases. The database therefore reflects this chronological grouping of the data. This grouping information appears immediately below, having been removed from the data itself: Number of Instances: 699 (as of 15 July 1992) Number of Attributes: 10 plus the class attribute There are 16 instances in Groups 1 to 6 that contain a single missing (i.e., unavailable) attribute value. Class distribution Benign: 458 (65.5%) Malignant: 241 (34.5%). Data Set B

50. Attribute Information