Writing Homework Help

Discussion 1:Cornilus Bawak

Spline, as implemented in for example generalized additive models or (more limited) in the function in R, allow you to fit flexible models that don’t make the crude assumptions of simple linear models. By fiddling with the degree of smoothness you can find the optimal model. These models are very good for making interpolations and for adjusting for quantitative confounders which have nonlinear effect.

Some of the benefits of using splines are, It is shown that rates of convergence of the prediction error depend on the smoothness of the slope function and on the structure of the predictors. With smoothing splines it could be proven that these rates are optimal in the sense that they are minimax over large classes of possible slope functions and distributions of the predictive curves. They are also valid for identifying causal, nonlinear effects but you might find the causal inference difficult to communicate other than graphically.

Limitations of smoothing spline when use for forecasting/extrapolation the requirement of careful integration of domain knowledge in the model, but sometimes there are other models that make more plausible forecasts. For example if data exhibit cyclical patterns, simple spline regression won’t be able to “discover” it (you will have to specify the cycles explicitly in the model), while there are other approaches that will do the job.

References

https://www.quora.com/What-are-pros-and-cons-of-using-Spline-regression-models

https://www.analyticsvidhya.com/blog/2018/03/introduction-regression-splines-python-codes/

Discussion 2: David

Good afternoon class,

Nonlinearity in data refers to a lack of correlation between the dependent and independent variable. Analysts rely on smoothing splines in order to model nonlinear patterns and fit the data in a more balanced approach to gain further insight on relationships between variables. “In most of the methods in which we fit Non linear Models to data and learn Non linearities is by transforming the data or the variables by applying a Non linear transformation” (Walia, 2018). Smoothing splines certainly provides smoothness and flexibility to a nonlinear model which assists in solving problems. There are several types of splines to choose from which can cause confusion for inexperienced users when selecting the appropriate spline for a model; furthermore, smoothing splines often requires additional effort in estimating parameters.

Reference:

Walia, A. S. (2018, April 04). Cubic and Smoothing Splines in R. Retrieved from https://datascienceplus.com/cubic-and-smoothing-sp…

Discussion 3: Arcelia Rael

Splines are often used to fit nonlinear models and view interactions between our variables (Walia, 2017). One benefit of smoothing splines over cubic splines is that they are more flexible and smoother. Because smoothing splines use the Roughness Penalty, which drives the fitted line’s jaggedness and roughness, they have better control over the fit of the model. Another advantage of smoothing splines is that they “have a Knot for every unique value of (xi)” (Wood, 2017, Smoothing Splines section), so Knots do not have to be accounted for. Additionally, Simpson (2018) writes that smoothing splines focus on balancing the fit to the data and not interloping the data as with cubic splines, which results in a better fit.

The disadvantages of splines include the production of an overly smooth line which may not be representative of the dataset due to the number of parameters (1:1) that must be represented (Simpson, 2018). Additionally, smoothing splines cannot be used with multiple independent variables. Instead, generalized additive models (GAM) can be used to fit the “smoothing function on each predictor independently” (Lander, 2017, p.262).

References

Lander, J. (2017). R for everyone: Advanced analytics and graphics (2nd ed.). Addison-Wesley Professional.

Simpson, G. (2018, January 30). The pros and cons of smoothing spline. Stack

Exchange. https://stats.stackexchange.com/questions/207810/the-pros-and-cons-of-smoothing-spline

Walia, A. S. (2017, June 30). Cubic and smoothing splines in R. DataScience+. https://datascienceplus.com/cubic-and-smoothing-splines-in-r/