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# Kernel method

## What is the Kernel method?

Kernel methods are types of algorithms that are used for pattern analysis. These methods involve using linear classifiers to solve nonlinear problems.

Kernels are used in Support Vector Machines (SVMs) to solve regression and classification problems. Support Vector Machines use the Kernel Trick to transform linearly inseparable data into linearly separable data, thus finding an optimal boundary for possible outputs.

Kernel functions are applied to every data instance for the purpose of mapping the original nonlinear observations into a higher-dimensional space. These observations become separable in the higher-dimensional space.

## What are the types of Kernels in SVM models?

Support vector machines use various kinds of kernels. Here are a few of them:

### 1. Linear Kernel

If there are two kernels named x1 and x2, the linear kernel can be defined by the dot product of the two vectors:

K(x1, x2) = x1 . x2

### 2. Polynomial Kernel

We can define a polynomial kernel with this equation:

K(x1, x2) = (x1 . x2 + 1)d

Here, x1 and x2 are vectors and d represents the degree of the polynomial.

### 3. Gaussian Kernel

The Gaussian kernel is an example of a radial basis function kernel. It can be represented with this equation:

k(xi, xj) = exp(-𝛾||xi - xj||2)

The given sigma has a vital role in the performance of the Gaussian kernel. It should be carefully tuned according to the problem, neither overestimated and nor underestimated.

### 4. Exponential Kernel

Exponential kernels are closely related to Gaussian kernels. These are also radial basis kernel functions. The difference between these two types of kernels is that the square of the norm is removed in Exponential kernels.

The function of an exponential function is:

k(x, y) =exp(-||x -y||22)

### 5. Laplacian Kernel

A Laplacian kernel is less prone to changes. It is equal to an exponential kernel.

The equation of a Laplacian kernel is

k(x, y) = exp(- ||x - y||)

### 6. Hyperbolic or the Sigmoid Kernel

Hyperbolic or Sigmoid kernels are used in neural networks. These kernels use a bipolar sigmoid activation function.

The hyperbolic kernel can be represented with this equation:

k(x, y) = tanh(xTy + c)

### 7. Anova radial basis kernel

This is another type of radial basis kernel function. Anova radial basis kernels work rather well in multidimensional regression problems.

An Anova radial basis kernel can be represented with this equation:

k(x, y) = k=1nexp(-(xk -yk)2)d

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# Kernel method

October 14, 2020

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## What is the Kernel method?

Kernel methods are types of algorithms that are used for pattern analysis. These methods involve using linear classifiers to solve nonlinear problems.

Kernels are used in Support Vector Machines (SVMs) to solve regression and classification problems. Support Vector Machines use the Kernel Trick to transform linearly inseparable data into linearly separable data, thus finding an optimal boundary for possible outputs.

Kernel functions are applied to every data instance for the purpose of mapping the original nonlinear observations into a higher-dimensional space. These observations become separable in the higher-dimensional space.

## What are the types of Kernels in SVM models?

Support vector machines use various kinds of kernels. Here are a few of them:

### 1. Linear Kernel

If there are two kernels named x1 and x2, the linear kernel can be defined by the dot product of the two vectors:

K(x1, x2) = x1 . x2

### 2. Polynomial Kernel

We can define a polynomial kernel with this equation:

K(x1, x2) = (x1 . x2 + 1)d

Here, x1 and x2 are vectors and d represents the degree of the polynomial.

### 3. Gaussian Kernel

The Gaussian kernel is an example of a radial basis function kernel. It can be represented with this equation:

k(xi, xj) = exp(-𝛾||xi - xj||2)

The given sigma has a vital role in the performance of the Gaussian kernel. It should be carefully tuned according to the problem, neither overestimated and nor underestimated.

### 4. Exponential Kernel

Exponential kernels are closely related to Gaussian kernels. These are also radial basis kernel functions. The difference between these two types of kernels is that the square of the norm is removed in Exponential kernels.

The function of an exponential function is:

k(x, y) =exp(-||x -y||22)

### 5. Laplacian Kernel

A Laplacian kernel is less prone to changes. It is equal to an exponential kernel.

The equation of a Laplacian kernel is

k(x, y) = exp(- ||x - y||)

### 6. Hyperbolic or the Sigmoid Kernel

Hyperbolic or Sigmoid kernels are used in neural networks. These kernels use a bipolar sigmoid activation function.

The hyperbolic kernel can be represented with this equation:

k(x, y) = tanh(xTy + c)

### 7. Anova radial basis kernel

This is another type of radial basis kernel function. Anova radial basis kernels work rather well in multidimensional regression problems.

An Anova radial basis kernel can be represented with this equation:

k(x, y) = k=1nexp(-(xk -yk)2)d