Any student of statistics will know how important the topic of regression is. Regression is the process through which a relationship is estimated between two variables. One of these two variables is a dependent variable and the other one independent variable. In a classical approach, these variables are usually denoted by X and Y.
Relationship between these two variables is established by observing how the value of one variable changes with different values of the other variable. Of course, as the terms suggest the dependent variableâ€™s value changes with variation in the independent variable but not vice versa.
There are various methods or approaches when it comes to regression. One of them is called the polynomial regression. It is a linear regression. Here the relationship between two variables X and Y are established based on the nth degree polynomial of the independent variable â€“ generally X.
Although it is linear in nature, it has been used in non-linear relationships as well. It is based on the least squares method, which is a very useful method in data fitting. Polynomial regression has been used in several fields. It has been used to predict the progress of epidemics and also growth rate of tissues, which is a non-linear occurrence.
Types of polynomial regression
Now polynomial regression can be both based on centered and uncentered polynomials. Both are important models but studying both in detail will tell you how centered polynomials have some distinct advantages. Before getting into the advantages of centered models, one must understand the basic uncentered model. For instance, a standard polynomial model is Y = B0 + B1X + B2X2.
Here Y is the dependent variable as one can see. X is independent, and here value of Y depends on the second degree of Xâ€™s value. Now, for higher degree polynomials, obviously, power of X is higher. Therefore, one can understand that higher order equations would mean that both centered and uncentered polynomial homework answers would be complex and large.
Limitations of uncentered models
This was the uncentered model, but there are some limitations to this model. For instance:
- When one has to deal with high values of X, calculating and working with high degree polynomials is a huge headache. It leads to what one calls â€œmath overflow.â€ Large figures appear, and it is a highly difficult to work with such values.
- Even when large values are not involved, there is a case of high covariance and dependency. This leads to occurrence of large errors. In essence, one gets wide and high ranged confidence intervals which may not be useful.
Therefore, one can clearly see that working with a standard uncentered model can bring up problems. But these have been eliminated by working with a centered model, which looks to minimize the values and eliminate errors by utilizing the mean. For more details, you could head over to certain help websites that help you solving questions and getting some centered and uncentered polynomial homework answers.
Centered polynomial model
A centered polynomial model, as mentioned, makes use of the mean value of dependent variable X. Now, its main objective is to minimize the value of X, which not only makes calculations easier but also minimizes errors to a great extent. You will learn this in more detail once you start with centered and uncentered polynomial homework answers.
Here the mean of X is subtracted from each value of the variable. This makes values of dependent variable X smaller. Therefore, a standard centered polynomial model is Y = B0 + B1Xc + B2Xc2. Here Xc = X – , where Â is the mean of X, or the average value of all values of independent variable X.
Advantages of centered polynomial model
One can easily see the advantages of using a centered model over an uncentered model:
- Firstly, by subtracting the mean from values of X or independent variable, there is a huge reduction in values of X. This means that whenever a higher polynomial is involved, calculations automatically become easier than an uncentered model. Not subtracting the mean will produce large figures. Imagine the state of if a 5th-degree polynomial is required. That would mean 5th power of all values of X. On top of that, if there are large values of X, it could be highly distressing to calculate manually.
- As the model is centered, the polynomial model becomes reparameterized. It means that the regression coefficients change and best-fit values are different in this case. Most importantly, covariance is minimum in this case and so are dependencies. As a result, errors too are minimal. This results in smaller confidence intervals, i.e. the ranges of intervals will be much more tight and narrow.
This is why analysts prefer centered polynomial model more than a standard uncentered model. But as a student, you will be expected to know both and come out with both centered and uncentered polynomial homework answers. More such advantages have been explained in these help websites in great detail.
The centering technique
Centering has been used as a technique in not only polynomial models but in various equations and formulae. The basic premise of this technique is to minimize the hassle while calculating large and complex values. As mentioned, it is quite clear that subtracting the mean or rather any constant value from individual variable values does make higher powers easier to determine.
In fact, any constant value can be selected for centering as long as it minimizes the individual values. But generally the mean is selected since it is convenient to have a value that is representative of the whole set and also minimizes the values efficiently.
This part of statistical analysis via polynomial regression is a complex but fundamental portion which cannot be avoided. Work and assignments related to this portion can get highly stressful and therefore it is useful to check out expert websites that assist in finding some centered and uncentered polynomial homework answers.