Artificial neutral network A neural network consists of input nodes, a hidden layer, and output nodes. For example, these input nodes may represent three measurements that we do to determine if someone has prostate cancer or not. Suppose that the person enters the hospital; we check the age, the concentration of the prostate-specific antigen from a blood sample, and collect a score between 1 and 5 from an MRI scan of the prostate. Then, we plug in the corresponding values in the input nodes and use the network to tell if the person has prostate cancer or not. To understand how neural nets work, we will here have a look at the super simple example. At the end of this video, we'll try to understand what happens if you include a hidden layer. A look at some simple R code if you’d like to reproduce the first example shown. We will here see how to train a neural network to predict if someone has prostate cancer based on the PSA level. Our training data consists of seven patients that we know have prostate cancer and seven individuals that we know are healthy. Based on a blood sample, we have determined the concentration of PSA, or the prostate-specific antigen, in the blood of these individuals. Note that this data has been simulated for the purpose of this video. We'll here use the simplest possible neural network model. Since we only have one measured variable, the PSA concentration, the network will only have just one input node. Note that we do not have any hidden layer because we'd like to use the simplest possible network. The network has two output nodes because we would like to use the network to predict if someone has prostate cancer or not. This is called a bias, which is used to modify the activation function. There are several different activation functions to select from. We will here use the sigmoid function, which is the same function that is used in logistic regression. This will later allow us to compare the neural network with logistic regression. Let's make a plot of this data, where these points represent the cancer patients that are hard-coded as ones, while these points represent the healthy individuals that are coded as zeros. For example, this healthy individual has a PSA level of 2.5, whereas this cancer patient has a PSA level of 2.1. Training a neural network means that we find the optimal values of the weights and the weight associated with the bias, so that the activation function, in this case, generates a sigmoid curve that can be used to predict the outcome in an optimal way. Note that the logistic activation function generates a value between 0 and 1, and that *e* is Euler's number. If you train this simple network, you will obtain the following weights. Let's use this network to see if this healthy person is correctly predicted as being healthy by the network. We plug in its PSA level of 2.0 into the input node to calculate the probability that the person has cancer. We use the weights associated with the output node in this equation, where *W_cr* is the bias weight, and this is the weight for the input signal, while this is the value of the input node. Since we only have one weight that is associated with the output node, *n* is equal to 1. Let's plug in the weights into the equation and do the math. We now plug in this value into the logistic activation function and calculate it. This value is the value for the output node (cancer), which can, in this case, be seen as the probability that the person has prostate cancer. We'll now calculate the value of this output node, where we plug in the corresponding values in the equation and do the math. Let's place the corresponding value here, which can be seen as the probability that the person is healthy. We can now use some kind of threshold to determine if you should classify the individual to have prostate cancer or not based on these two values. A common kind of value to use in binary classification is 0.5 because it is the value between 0 and 1. Based on this cutoff value, the network will predict that the person with the PSA level of 2 is healthy or the output value associated with the healthy class is greater than 0.5. Note that this value is exactly the same value as if we would draw a vertical line from 2 to the logistic curve and then the horizontal line from the curve to the y-axis. The height of the activation function is therefore equal to 0.438. This value is then simply just 1 minus 0.438. This value is equal to the height of the flipped curve because the slope of the curve should be negative if you go this way in the network. Let’s see how well this neural network can predict the class of our training data. This is the data that we used to train the network, and these are the values generated by the output node for cancer, which can be obtained if you plug in these values in the equation and the weights, and then compute the corresponding values by the activation function. For example, this is the value that we calculated by hand previously based on the PSA level of 2.0, which corresponds to the height of the activation function. We can now use the output values and the cutoff value of 0.5 to see how well the model predicts the training data. Since these individuals had values greater than 0.5, they are predicted to have prostate cancer. Since we know that these individuals actually have prostate cancer, we know that the neural network has made the correct predictions for these individuals. However, the network incorrectly predicts that this cancer patient is healthy. These individuals have output values that are less than 0.5, which means that they are predicted to be healthy. Since we know that these individuals are healthy, we know that they have been correctly classified by the network. This healthy person is incorrectly classified as having prostate cancer because it has a relatively high PSA level. In total, the network makes 12 correct predictions out of 14 possible, which gives an accuracy of about 86%.