[AI] Linear Regression - BYU AI Club

• Brain (Neurons)
  • More think about them >> the memory gets stronger (synapsets)
  • Less time with >> memory gets weaker
• Neural Network
  • Shows the probabilitiy (this is the right classification)
• TensorFlow
  • Shows the shape with neurons
• Training
  • When validation set or test set come, we want our machine to learn it well that we don't want to make a perfect fit by doing the machine learning too much
• Best Performing Model
  • To predict better
• Docker
  • Virtual Machine >> callable API
  • it's callable and usable.
• Machine Learning Engineers 
  • fit models to data and make them to predict good.

 

Linear Regression Learning the Model

Learning a linear regression model means estimating the values of the coefficients used in the representation with the data that we have available.

In this section we will take a brief look at four techniques to prepare a linear regression model. This is not enough information to implement them from scratch, but enough to get a flavor of the computation and trade-offs involved.

There are many more techniques because the model is so well studied. Take note of Ordinary Least Squares because it is the most common method used in general. Also take note of Gradient Descent as it is the most common technique taught in machine learning classes.

1. Simple Linear Regression

With simple linear regression when we have a single input, we can use statistics to estimate the coefficients.

This requires that you calculate statistical properties from the data such as means, standard deviations, correlations and covariance. All of the data must be available to traverse and calculate statistics.

This is fun as an exercise in excel, but not really useful in practice.

2. Ordinary Least Squares

When we have more than one input we can use Ordinary Least Squares to estimate the values of the coefficients.

The Ordinary Least Squares procedure seeks to minimize the sum of the squared residuals. This means that given a regression line through the data we calculate the distance from each data point to the regression line, square it, and sum all of the squared errors together. This is the quantity that ordinary least squares seeks to minimize.

This approach treats the data as a matrix and uses linear algebra operations to estimate the optimal values for the coefficients. It means that all of the data must be available and you must have enough memory to fit the data and perform matrix operations.

It is unusual to implement the Ordinary Least Squares procedure yourself unless as an exercise in linear algebra. It is more likely that you will call a procedure in a linear algebra library. This procedure is very fast to calculate.

3. Gradient Descent

When there are one or more inputs you can use a process of optimizing the values of the coefficients by iteratively minimizing the error of the model on your training data.

This operation is called Gradient Descent and works by starting with random values for each coefficient. The sum of the squared errors are calculated for each pair of input and output values. A learning rate is used as a scale factor and the coefficients are updated in the direction towards minimizing the error. The process is repeated until a minimum sum squared error is achieved or no further improvement is possible.

When using this method, you must select a learning rate (alpha) parameter that determines the size of the improvement step to take on each iteration of the procedure.

Gradient descent is often taught using a linear regression model because it is relatively straightforward to understand. In practice, it is useful when you have a very large dataset either in the number of rows or the number of columns that may not fit into memory.

4. Regularization

There are extensions of the training of the linear model called regularization methods. These seek to both minimize the sum of the squared error of the model on the training data (using ordinary least squares) but also to reduce the complexity of the model (like the number or absolute size of the sum of all coefficients in the model).

Two popular examples of regularization procedures for linear regression are:

• Lasso Regression: where Ordinary Least Squares is modified to also minimize the absolute sum of the coefficients (called L1 regularization).
• Ridge Regression: where Ordinary Least Squares is modified to also minimize the squared absolute sum of the coefficients (called L2 regularization).

These methods are effective to use when there is collinearity in your input values and ordinary least squares would overfit the training data.

Now that you know some techniques to learn the coefficients in a linear regression model, let’s look at how we can use a model to make predictions on new data.

Making Predictions with Linear Regression

Given the representation is a linear equation, making predictions is as simple as solving the equation for a specific set of inputs.

Let’s make this concrete with an example. Imagine we are predicting weight (y) from height (x). Our linear regression model representation for this problem would be:

y = B0 + B1 * x1

or

weight =B0 +B1 * height

Where B0 is the bias coefficient and B1 is the coefficient for the height column. We use a learning technique to find a good set of coefficient values. Once found, we can plug in different height values to predict the weight.

For example, lets use B0 = 0.1 and B1 = 0.5. Let’s plug them in and calculate the weight (in kilograms) for a person with the height of 182 centimeters.

weight = 0.1 + 0.5 * 182

weight = 91.1

You can see that the above equation could be plotted as a line in two-dimensions. The B0 is our starting point regardless of what height we have. We can run through a bunch of heights from 100 to 250 centimeters and plug them to the equation and get weight values, creating our line.

Sample Height vs Weight Linear Regression

Now that we know how to make predictions given a learned linear regression model, let’s look at some rules of thumb for preparing our data to make the most of this type of model.

Preparing Data For Linear Regression

Linear regression is been studied at great length, and there is a lot of literature on how your data must be structured to make best use of the model.

As such, there is a lot of sophistication when talking about these requirements and expectations which can be intimidating. In practice, you can uses these rules more as rules of thumb when using Ordinary Least Squares Regression, the most common implementation of linear regression.

Try different preparations of your data using these heuristics and see what works best for your problem.

• Linear Assumption. Linear regression assumes that the relationship between your input and output is linear. It does not support anything else. This may be obvious, but it is good to remember when you have a lot of attributes. You may need to transform data to make the relationship linear (e.g. log transform for an exponential relationship).
• Remove Noise. Linear regression assumes that your input and output variables are not noisy. Consider using data cleaning operations that let you better expose and clarify the signal in your data. This is most important for the output variable and you want to remove outliers in the output variable (y) if possible.
• Remove Collinearity. Linear regression will over-fit your data when you have highly correlated input variables. Consider calculating pairwise correlations for your input data and removing the most correlated.
• Gaussian Distributions. Linear regression will make more reliable predictions if your input and output variables have a Gaussian distribution. You may get some benefit using transforms (e.g. log or BoxCox) on you variables to make their distribution more Gaussian looking.
• Rescale Inputs: Linear regression will often make more reliable predictions if you rescale input variables using standardization or normalization.

See the Wikipedia article on Linear Regression for an excellent list of the assumptions made by the model. There’s also a great list of assumptions on the Ordinary Least Squares Wikipedia article.

Further Reading

There’s plenty more out there to read on linear regression. Start using it before you do more reading, but when you want to dive deeper, below are some references you could use.

Machine Learning Books that Mention Linear Regression

These are some machine learning books that you might own or have access to that describe linear regression in the context of machine learning.

• A First Course in Machine Learning, Chapter 1.
• An Introduction to Statistical Learning: with Applications in R, Chapter 3.
• Applied Predictive Modeling, Chapter 6.
• Machine Learning in Action, Chapter 8.
• The Elements of Statistical Learning: Data Mining, Inference, and Prediction, Chapter 3.

Posts on Linear Regression

Below are some interesting essays and blog posts on linear regression that I have come across.

• Ordinary Least Squares Regression: Explained Visually
• Ordinary Least Squares Linear Regression: Flaws, Problems and Pitfalls
• Introduction to linear regression analysis
• Four Assumptions Of Multiple Regression That Researchers Should Always Test

Know any more good references on linear regression with a bent towards machine learning and predictive modeling? Leave a comment and let me know.

Summary

In this post you discovered the linear regression algorithm for machine learning.

You covered a lot of ground including:

• The common names used when describing linear regression models.
• The representation used by the model.
• Learning algorithms used to estimate the coefficients in the model.
• Rules of thumb to consider when preparing data for use with linear regression.

Try out linear regression and get comfortable with it.

Do you have any questions about linear regression or about this post?
Leave a comment and ask, I will do my best to answer.

from https://machinelearningmastery.com/linear-regression-for-machine-learning/#:~:text=Linear%20regression%20is%20a%20linear,the%20input%20variables%20(x).