Title¶

pyGAM with some additions

Description :¶

Download and run in own environment.

This notebook is for the most part exactly the same as Notebook A, with a a few additions. Those additions are clearly marked in the notebook.

CS109B Data Science 2: Advanced Topics in Data Science

Lecture 5.5 - Smoothers and Generalized Additive Models - Model Fitting¶

Harvard University
Spring 2021
Instructors: Mark Glickman, Pavlos Protopapas, and Chris Tanner
Lab Instructor: Eleni Kaxiras

Content: Eleni Kaxiras and Will Claybaugh¶

New material here:¶

• References
• info on B-spline basis functions
• Cleaner formula for csaps

Also cleaner Table of Contents

Table of Contents¶

• 1 - Overview - A Top View of LMs, GLMs, and GAMs to set the stage
• 2 - Splines
• 3 - Generative Additive Models with pyGAM
• 4 - Smooting Splines using csaps
• 5 - Penalized B-splines

1 - Overview¶

image source: Dani Servén Marín (one of the developers of pyGAM)

A - Linear Models¶

First we have the Linear Models which you know from 109a. These models are linear in the coefficients. Very interpretable but suffer from high bias because let's face it, few relationships in life are linear. Simple Linear Regression (defined as a model with one predictor) as well as Multiple Linear Regression (more than one predictors) are examples of LMs. Polynomial Regression extends the linear model by adding terms that are still linear for the coefficients but non-linear when it somes to the predictiors which are now raised in a power or multiplied between them.

$$ \begin{aligned} y = \beta{_0} + \beta{_1}{x_1} & \quad \mbox{(simple linear regression)}\\ y = \beta{_0} + \beta{_1}{x_1} + \beta{_2}{x_2} + \beta{_3}{x_3} & \quad \mbox{(multiple linear regression)}\\ y = \beta{_0} + \beta{_1}{x_1} + \beta{_2}{x_1^2} + \beta{_3}{x_3^3} & \quad \mbox{(polynomial multiple regression)}\\ \end{aligned} $$

• What does it mean for a model to be interpretable?
• Are linear regression models interpretable? Are random forests? What about Neural Networks such as Feed Forward?
• Do we always want interpretability? Describe cases where we do and cases where we do not care.

B - Generalized Linear Models (GLMs)¶

Generalized Linear Models is a term coined in the early 1970s by Nelder and Wedderburn for a class of models that includes both Linear Regression and Logistic Regression. A GLM fits one coefficient per feature (predictor).

C - Generalized Additive Models (GAMs)¶

Hastie and Tidshirani coined the term Generalized Additive Models in 1986 for a class of non-linear extensions to Generalized Linear Models.

$$ \begin{aligned} y = \beta{_0} + f_1\left(x_1\right) + f_2\left(x_2\right) + f_3\left(x_3\right) \\ y = \beta{_0} + f_1\left(x_1\right) + f_2\left(x_2, x_3\right) + f_3\left(x_3\right) & \mbox{(with interaction terms)} \end{aligned} $$

In practice we add splines and regularization via smoothing penalties to our GLMs.

image source: Dani Servén Marín

D - Basis Functions¶

In our models we can use various types of functions as "basis".

• Monomials such as $x^2$, $x^4$ (Polynomial Regression)
• Sigmoid functions (neural networks)
• Fourier functions
• Wavelets
• Regression splines
• Smoothing splines

2 - Piecewise Polynomials a.k.a. Splines¶

Splines are a type of piecewise polynomial interpolant. A spline of degree k is a piecewise polynomial that is continuously differentiable k − 1 times.

Splines are the basis of CAD software and vector graphics including a lot of the fonts used in your computer. The name “spline” comes from a tool used by ship designers to draw smooth curves. Here is the letter $epsilon$ written with splines:

font idea inspired by Chris Rycroft (AM205)

If the degree is 1 then we have a Linear Spline. If it is 3 then we have a Cubic spline. It turns out that cubic splines because they have a continous 2nd derivative (curvature) at the knots are very smooth to the eye. We do not need higher order than that. The Cubic Splines are usually Natural Cubic Splines which means they have the added constrain of the end points' second derivative = 0.

We will use the CubicSpline and the B-Spline as well as the Linear Spline.

scipy.interpolate¶

See all the different splines that scipy.interpolate has to offer: https://docs.scipy.org/doc/scipy/reference/interpolate.html. These routines are based on the FORTRAN library FITPACK written in the '70s.

Let's use the simplest form which is interpolate on a set of points and then find the points between them.

The Linear interpolation does not look very good. Fit a Cubic Spline and plot along the Linear to compare. Feel free to solve and then look at the solution.

• Change the number of knots to 100 and see what happens. What would happen if we run a polynomial model of degree equal to the number of knots (a global one as in polynomial regression, not a spline)?
• What makes a spline 'Natural'?

B-Splines (Curry and Schoenberg(1966) , further study by de Boor, 1978)¶

One way to construct a curve given a set of points is to interpolate the points, that is, to force the curve to pass through the points.

A B-splines (Basis Splines) is defined by a set of control points and a set of basis functions that fit the function between these points. By choosing to have no smoothing factor we force the final B-spline to pass though all the points. If, on the other hand, we set a smothing factor, our function is more of an approximation with the control points as "guidance". The latter produced a smoother curve which is prefferable for drawing software. For more on Splines see: https://en.wikipedia.org/wiki/B-spline) or for a deeper look: De Boor, C. (1978). A practical guide to splines (Applied mathematical sciences (Springer-Verlag New York Inc.) ; v. 27). New York: Springer-Verlag.

We will use scipy.splrep to calulate the coefficients for the B-Spline and draw it.

B-Spline with no smooting¶

What do the tuple values returned by scipy.splrep mean?¶

• The t variable is the array that contains the knots' position in the x axis. The length of this array is, of course, the number of knots.
• The c variable is the array that holds the coefficients for the B-Spline. Its length should be the same as t.

We have number_of_knots - 1 B-spline basis elements to the spline constructed via this method, and they are defined as follows:

$$ \begin{aligned} B_{i, 0}(x) = 1, \textrm{if $ti \le x < t{i+1}$, otherwise $0$,} \ \ B_{i, k}(x) = \frac{x - ti}{t{i+k} - ti} B{i, k-1}(x)

             + \frac{t_{i+k+1} - x}{t_{i+k+1} - t_{i+1}} B_{i+1, k-1}(x)

\end{aligned} $$

• t $\in [t_1, t_2, ..., t_n]$ is the knot vector
• c : are the spline coefficients
• k : is the spline degree

References:¶

• All you wanted to know about B-Splines and were afraid to ask, or how scipy.splrep calculates B-spline representations.
• For more on the basis of B-splines.
• scipy.interpolate.BSpline.basis_element.
• Inside the documentation you will find references to books.

Constructing the B-spline basis elements¶

The degree of the B-spline, k, is inferred from the length of t as len(t)-2. The knot vector is constructed by appending and prepending k+1 elements to internal knots t.

a) The first element is a B-spline of degree $d=0$ with $1$ knot (min number of knots = $d+2$)

$(d=0), B_{i, 0}(x) = 1, \textrm{if $ti \le x < t{i+1}$, otherwise $0$, (0 interior knots, 2 total)}$

b) The second element is a B-spline of degree $d=1$ with $2$ knots

$(d=1), B_{i, 0}(x) = 1, \textrm{if $ti \le x < t{i+1}$, otherwise $0$, (1 interior knot, 3 total)}$

and so on ...

B-Spline with smooting factor s¶

B-Spline with given knots¶

3 - GAMs¶

https://readthedocs.org/projects/pygam/downloads/pdf/latest/

Classification in pyGAM¶

Let's get our (multivariate!) data, the kyphosis dataset, and the LogisticGAM model from pyGAM to do binary classification.

• kyphosis - wherther a particular deformation was present post-operation
• age - patient's age in months
• number - the number of vertebrae involved in the operation
• start - the number of the topmost vertebrae operated on

Outcome dependence on features¶

To help us see how the outcome depends on each feature, pyGAM has the partial_dependence() function.

 pdep, confi = kyph_gam.partial_dependence(term=i, X=XX, width=0.95)

For more on this see the : https://pygam.readthedocs.io/en/latest/api/logisticgam.html

Notice that we did not specify the basis functions in the .fit(). pyGAM figures them out for us by using $s()$ (splines) for numerical variables and $f()$ for categorical features. If this is not what we want we can manually specify the basis functions, as follows:

Regression in pyGAM¶

For regression problems, we can use a linearGAM model. For this part we will use the wages dataset.

https://pygam.readthedocs.io/en/latest/api/lineargam.html

The wages dataset¶

Let's inspect another dataset that is included in pyGAM that notes the wages of people based on their age, year of employment and education.

4 - Smoothing Splines using csaps¶

A smoothing spline is a solution of the problem of minimizing the goodness-of-fit and the smoothing (wiggliness).
$MSE - \lambda\cdot\text{wiggle penalty}$ $=$ $p\sum_{i=1}^N \left(y_i - f(x_i)\right)^2 + (1-p) \int \left(f''(t)\right)^2 dt$,
across all possible functions $f$.

The p smoothing parameter is in the range of [0,1] with 0 giving you: the smoothing spline which is the least-squares straight line fit to the data, and 1: the natural cubic spline interpolant.

5 - Data fitting using pyGAM and Penalized B-Splines¶

When we use a spline in pyGAM we are effectively using a penalized B-Spline with a regularization parameter $\lambda$. E.g.

LogisticGAM(s(0)+s(1, lam=0.5)+s(2)).fit(X,y)

Let's see how this smoothing works in pyGAM. We start by creating some arbitrary data and fitting them with a GAM. The lambda parameter (lam) goes from 0 to infinity. The default for the LinearGAM, is 0.6.

We see that the large $\lambda$ forces a straight line, no flexibility. Let's see now what happens if we make it smaller.

There is some curvature there but still not a good fit. Let's try no penalty. That should have the line fit exactly.

Yes, that is good. Now let's see what happens if we lessen the number of splines. The fit should not be as good.