Univariate linear regression

Univariate linear regression is a supervised learning algorithm used to solve regression problems. It fits the data points in a given dataset using a straight line and uses this model to predict values ​​that are not in the dataset.

Principle of univariate linear regression

The principle of univariate linear regression is to use the relationship between an independent variable and a dependent variable, through Fit a straight line to describe the relationship between them. Through methods such as the least squares method, the sum of squares of the vertical distances from all data points to this fitting straight line is minimized, thereby obtaining the parameters of the regression line, and then predicting the dependent variable value of the new data point.

The general form of the univariate linear regression model is y=ax b, where a is the slope and b is the intercept. Through the least squares method, estimates of a and b can be obtained to minimize the gap between the actual data points and the fitted straight line.

Univariate linear regression has the following advantages: fast operation speed, strong interpretability, and good at discovering linear relationships in data sets. However, when the data is non-linear or there is correlation between features, univariate linear regression may not model and express complex data well.

Simply put, univariate linear regression is a linear regression model with only one independent variable.

Advantages and disadvantages of univariate linear regression

The advantages of univariate linear regression include:

• Fast operation speed: Because the algorithm is simple and conforms to mathematical principles, the modeling and prediction speed of the univariate linear regression algorithm is very fast.
• Very interpretable: Finally, a mathematical function expression can be obtained, and the influence of each variable can be clarified based on the calculated coefficients.
• Good at obtaining linear relationships in data sets.

The disadvantages of univariate linear regression include:

• For nonlinear data or when there is correlation between data features , univariate linear regression can be difficult to model.
• It is difficult to express highly complex data well.

In univariate linear regression, how is the squared error loss function calculated?

In univariate linear regression, we usually use the squared error loss function to measure the prediction error of the model.

The calculation formula of the squared error loss function is:

L(θ0,θ1)=12n∑i=1n(y_i−(θ0 θ1x_i))2

where:

• n is the number of samples
• y_i is the i-th sample Actual values ​​
• θ0 and θ1 are model parameters
• x_i is the independent variable value of the i-th sample

In univariate linear regression , we assume that there is a linear relationship between y and x, that is, y=θ0 θ1x. Therefore, the predicted value can be obtained by substituting the independent variable x into the model, that is, y_pred=θ0 θ1x_i.

The smaller the value of the loss function L, the smaller the prediction error of the model and the better the performance of the model. Therefore, we can get the optimal model parameters by minimizing the loss function.

In the gradient descent method, we gradually approach the optimal solution by iteratively updating the values ​​of parameters. At each iteration, the value of the parameter is updated according to the gradient of the loss function, that is:

θ=θ-α*∂L(θ0,θ1)/∂θ

Among them, α is the learning rate, which controls the change of parameters in each iteration.

Conditions and steps for univariate linear regression using gradient descent method

The conditions for using gradient descent method to perform univariate linear regression include:

1) The objective function is differentiable. In univariate linear regression, the loss function usually uses squared error loss, which is a differentiable function.

2) There is a global minimum. For the squared error loss function, there is a global minimum, which is also a condition for univariate linear regression using gradient descent.

The steps for using gradient descent method to perform univariate linear regression are as follows:

1. Initialize parameters. Choose an initial value, usually 0, as the initial value for the parameter.

2. Calculate the gradient of the loss function. According to the relationship between the loss function and the parameters, the gradient of the loss function with respect to the parameters is calculated. In univariate linear regression, the loss function is usually the squared error loss, and its gradient calculation formula is: θ−y(x)x.

3. Update parameters. According to the gradient descent algorithm, update the value of the parameter, namely: θ=θ−αθ−y(x)x. Among them, α is the learning rate (step size), which controls the change of parameters in each iteration.

4. Repeat steps 2 and 3 until the stopping condition is met. The stopping condition can be that the number of iterations reaches a preset value, the value of the loss function is less than a preset threshold, or other appropriate conditions.

The above steps are the basic process of using the gradient descent method to perform univariate linear regression. It should be noted that the choice of learning rate in the gradient descent algorithm will affect the convergence speed of the algorithm and the quality of the results, so it needs to be adjusted according to the specific situation.

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