[MUSIC] In this video, we'll discuss linear classifiers. Linear classifiers are widely used in classification, and they are the building blocks of more advanced classification methods. In this video, we will discuss the two-class case we gave the images labels y equals zero for cat, y equals one for dog. We will concatenate the three-channel images to a vector, or you can do the same for grayscale images. In this video we will show you how a simple function can take an image as an input and output the image class using simple algebra. We will also show you how a similar function will take in images and input, and output a probability of how likely that image belongs to a class. For example, here the function gives a 92 percent chance, it's a dog. The equation of a line in one dimension is given by the following. Here w represents the weight term and b represents the bias term. These are called learnable parameters, and we will use them to classify the image. For arbitrary dimensions, this equation generalizes to a hyperplane. You can represent this equation as a dot product of row vector w and image x. This is called the decision plane. If you're not familiar with vectors, this is just a compact way to express the equation of a line in dots of dimensions. In this video, we will use two dimensions for visualization. Note when the letter x are not bold, it's just simple algebra, not a sample. We will set the weights w_1 to one w_2 to minus one and the bias to one. In two dimensions we can plot the equation as a plane. We can plot the plane, z equals zero. We can see the line where the plane intersects with a plane at z equals zero. If we look at it from above, we get the following image. This is the plane where z equals zero, the line is where the decision plane intersects with the planes z equals zero. This line is called the decision boundary. We can overlay our sample images. Anything on the left side of the line is a dog. Anything on the right side of the line is a cat. Let's see how we can use the value of z to determine if it's a dog or a cat. Let's look at the following sample. Consider the following values for x_1. We can plug those values into an equation and get a value of z equals two. We see z as positive. Similarly, for x_2 and x_3, every point on the left side of the line will give you a positive value. The label for x_4 is a cat and the point lies on the right side of the line. Plugging the value of x into the equation, we see z is negative. Similarly, for sample x_4 and x_6, every point on the right side of the line will output a negative value. If we use z to calculate the class of the points, it always returns real numbers such as negative one, three, negative two, and so on. But we need a class between zero and one, so how do we convert these numbers? We'll use something called a threshold function pictured here. If z is greater than zero, it will return a one, and if z is less than zero, it will return to zero. For the following image, the output is one corresponding to dog. For this input, the value is zero corresponding to cat. Every sample on this side of the plane will be classified as a cat, every sample on this side of the plane will be classified as a dog. A plane can't always separate the data. The sample is misclassified. In this case, the data is not linearly separable. The logistic function resembles the threshold function and is given by the following expression known as the sigmoid function. It will give us a probability of how likely our estimate is. In addition, it has better performance than the threshold function, for reasons we will discuss later. If the value of z is a very large negative number, the expression is approximately zero and for a very large positive value of z, the expression is approximately one. For everything in the middle, the value is between zero and one. To determine y hat as a discrete class, we use a threshold shown by the line. If the output of the logistic function is larger than 0.5, we set the prediction value y hat to one. If it's less, we set y hat to zero. Let's try out some values of x that we used previously with the linear classification. If we set x equal to zero 0.32 into the equation, we get the value of z as two, we pass the value through the sigmoid function. Since the value of the sigmoid function is greater than 0.5, we set y hat to one. Similarly, if we plug in the value of x as minus two, minus three, the value of z is minus two. We pass the value through the sigmoid function. We see the result is less than 0.5. We set y hat to zero. We can also represent the logistic function as a probability. We can find the probability of the image of being dog that is y hat equals one. We can find the probability of being a cat y hat equals zero by using the following we see the image is more likely a dog. Given the following image, we can find the probability of being a dog y hat equals one. We find the probability of it being cat, that is y hat equals zero. We see the image is more likely a cat. If you have the learnable parameters, you can use a linear classifier, you take the photo. Under the hood, you app will use the linear classifier to make a prediction and output the class as a string. [MUSIC]
