Moving beyond linearity Part VI Moving Beyond Linearity As of Dec 4, 2019 Some of the figures in this presentation are taken from ”An Introduction to Statistical Learning, with applications in R” (Springer, 2013) with permission from the authors: G. James, D. Witten, T. Hastie and R. Tibshirani Seppo Pynn¨ onen Applied Multivariate Statistical Analysis
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Moving beyond linearity
Part VI
Moving Beyond Linearity
As of Dec 4, 2019Some of the figures in this presentation are taken from ”An Introduction to Statistical
Learning, with applications in R” (Springer, 2013) with permission from the authors:
G. James, D. Witten, T. Hastie and R. Tibshirani
Seppo Pynnonen Applied Multivariate Statistical Analysis
Moving beyond linearity
1 Moving beyond linearity
Polynomial regressions
Step functions
Basis functions
Regression splines
Smoothing splines
Local regressions
Generalized additive models
Seppo Pynnonen Applied Multivariate Statistical Analysis
Moving beyond linearity
Linear models may be too rough approximations of the underlyingrelationships.
However, because the true (non-linear) model is unknown, severalapproaches have been developed to approximate the underlyingtrue relationship.
Polynomial regression.
Step functions.
Regression splines.
Smoothing splines.
Local regression.
Generalized additive models.
Seppo Pynnonen Applied Multivariate Statistical Analysis
Moving beyond linearity
Polynomial regressions
1 Moving beyond linearity
Polynomial regressions
Step functions
Basis functions
Regression splines
Smoothing splines
Local regressions
Generalized additive models
Seppo Pynnonen Applied Multivariate Statistical Analysis
Moving beyond linearity
Polynomial regressions
Polynomial regression fits instead of the simple regression
y = β0 + β1x + ε (1)
polynomial function
y = β0 + β1x + β2x2 + · · ·+ βdx
d + ε. (2)
The motivation is to increase flexibility of the fitted model tocapture possible non-linearities.
Usually the degree of the polynomial is at most three or four.
Polynomial regression can be applied in logit and probit models aswell.
Seppo Pynnonen Applied Multivariate Statistical Analysis
Moving beyond linearity
Polynomial regressions
Example 1
Fit a fourth order polynomial regression in the Wage data by regressing wage onthe fourth order polynomial of age. All coefficient are statistically significant atthe 5 % level (the fourth order is on the border line).
> (fit <- lm(wage ~ poly(age, 4), data = Wage)) # polynomial model of order 4
> lines(x = age.range[1]:age.range[2], y = wage.pred$fit, col = "steel blue") # fitted line
> ## 95% lower bound of the fitted line
> lines(x = age.range[1]:age.range[2], y = wage.pred$fit - 2 * wage.pred$se.fit, col = "red") #
> ## 95% upper bound
> lines(x = age.range[1]:age.range[2], y = wage.pred$fit + 2 * wage.pred$se.fit, col = "red") #
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20 30 40 50 60 70 80
5010
015
020
025
030
0
age
wag
e
Seppo Pynnonen Applied Multivariate Statistical Analysis
Moving beyond linearity
Step functions
1 Moving beyond linearity
Polynomial regressions
Step functions
Basis functions
Regression splines
Smoothing splines
Local regressions
Generalized additive models
Seppo Pynnonen Applied Multivariate Statistical Analysis
Moving beyond linearity
Step functions
Polynomial functions impose global structure on x .
Step functions impose local structures by converting a continuousvariable into an ordered categorical variable
Co(x) = I (x < c1),C1(x) = I (c1 ≤ x < c2),C3(x) = I (c2 ≤ x < c3),
...CK−1(x) = I (cK−1 ≤ x < cK ),CK (x) = I (cK ≤ x),
(3)
where I (·) is and indicator function that returns 1 if the conditionis true, and zero otherwise.
Seppo Pynnonen Applied Multivariate Statistical Analysis
Moving beyond linearity
Step functions
Because C0(x) + C1(x) + · · ·+ CK (x) = 1, C0(x) becomes thereference class, and the dependent variable y is regressed on Cj(x),j = 1, . . . ,K , resulting to (a dummy variable regression)
where bj(x) are fixed known functions, called basis functions.
Other popular basis functions are based on Fourier series, wavelets,and on different spline functions.
Seppo Pynnonen Applied Multivariate Statistical Analysis
Moving beyond linearity
Regression splines
1 Moving beyond linearity
Polynomial regressions
Step functions
Basis functions
Regression splines
Smoothing splines
Local regressions
Generalized additive models
Seppo Pynnonen Applied Multivariate Statistical Analysis
Moving beyond linearity
Regression splines
Spline basis representation
The basis model in (5) can be used to define a piecewise degree dpolynomial under the constraint that it and possibly its d − 1 derivativesare continuous.
Typically, again d is fairly low, like 3 (cubic).
There are many different ways to represent polynomial splines.
The most direct ways to represent for example the cubic spline with Kknots, where knots define points at which polynomial coefficients canchange, is to start off with a basis for a cubic polynomial, x , x2, x3, andadd one truncated power basis function per knot.
A truncated power basis function is defined as
h(x , ξ) = (x − ξ)3+ =
{(x − ξ)3 if x > ξ
0 otherwise,(6)
where ξ is the knot.
Seppo Pynnonen Applied Multivariate Statistical Analysis
Moving beyond linearity
Regression splines
The regression is of the form
y = β0+β1x+β2x2+β3x
3+β4h(x , ξ1)+· · ·+βK+3h(x , ξK )+ε (7)
where ξj are the K knots.
Therefore the regression amounts to estimating K + 4 regressioncoefficients and thus uses K + 4 degrees of freedom.
Because splines can have high variance at the outer range ot thepredictors (very small and very large values), additional boundaryconstraints are often imposed.
A natural spline is a regression spline, where the function isrequired to be linear at the boundary (region where x is smallerthan the smallest knot, or larger than the largest knot).
Seppo Pynnonen Applied Multivariate Statistical Analysis
Moving beyond linearity
Regression splines
The number of places of the knots can be chosen for example bycross-validation or just choosing the places and number such thatthe curve looks nice.
Fewer knots can be selected in places were the function seems tochange slower and more knots in places where the function appearschange faster.
Compared to polynomial regression, splines can produce moreflexibility with less estimated coefficients.
Seppo Pynnonen Applied Multivariate Statistical Analysis
Moving beyond linearity
Regression splines
Example 3
Splines in R can be fitted utilizing splines package which is part of the basepackage and is therefore readily available.
Function bs() generates basis functions for splines with the specified set of knots(see help(bs), which shows that the default is a cubic spline).
Fitting in the Work data set wage on spline function of age with three knots atξ1 = 25, ξ2 = 40, and ξ3 = 60.
> attr(ns(Wage$age, knots = c(25, 40, 60)), which = "knots")
[1] 25 40 60
> attr(ns(Wage$age, df = 4), which = "knots")
25% 50% 75%
33.75 42.00 51.00
Finally, by degree parameter in bs() and ns() functions the can be defined the
degree of the polynomial (default is 3, i.e., cubic).
Seppo Pynnonen Applied Multivariate Statistical Analysis
Moving beyond linearity
Regression splines
Choosing the number and location of knots
Number and location of the knots?
Number of knots define the flexibility of the spline–more knots,more flexibility.
Place more knots in places where the function varies mostrapidly.
Fewer knots where the function seems more stable.
In practice, however, often the knots are placed uniformly. In Rthis can be done by defining df as seen in the above example.
Cross-validation can be used to select the degrees of freedom.
Seppo Pynnonen Applied Multivariate Statistical Analysis
Moving beyond linearity
Smoothing splines
1 Moving beyond linearity
Polynomial regressions
Step functions
Basis functions
Regression splines
Smoothing splines
Local regressions
Generalized additive models
Seppo Pynnonen Applied Multivariate Statistical Analysis
Moving beyond linearity
Smoothing splines
Fitting some ’smooth’ curve g(x) is another way to fit regression lines.
This amounts to minimizing RSS =∑n
i=1(yi − g(xi ))2.
In order to make g(x) smooth, i.e., that it is not too jigged (the extremeis that it goes through all data points making RSS zero), an approach is tofind the function g that minimizes
n∑i=1
(yi − g(xi ))2 + λ
∫g ′′(t)2dt, (8)
where λ is a nonnegative tuning parameter and g ′′(t) is the secondderivative of g .
The function g that minimizes (8) is known as a smoothing spline.
Similar to lasso, equation (8) takes the “loss + penalty” formulation,where the sum is the loss and the integral part is the penalty term thatpenalizes the variability in g .
Seppo Pynnonen Applied Multivariate Statistical Analysis
Moving beyond linearity
Smoothing splines
It can be sown that g(x) that minimizes (8) is a piecewise cubicpolynomial with knots at the unique values of x1, . . . , xn and it islinear outside the extreme knots.
Thus, g(x) that minimizes (8) is a natural cubic spline with knotsx1, . . . , xn.
However, g(x) is not the same natural cubic spline that one wouldget if one applied the basis function approach.
The smoothness of the spline is controlled by the penalty term,i.e., by the magnitude of λ.
Seppo Pynnonen Applied Multivariate Statistical Analysis
Moving beyond linearity
Smoothing splines
Choosing the smoothing parameter λ
The tuning parameters controls the roughness of the smoothingspline, λ = 0 results to a solution of g that goes through everyobservation xi , and as λ→∞, results to a linear function such thatthe second derivative g ′′ is zero, i.e., perfectly smooth function.
In the former case the effective degrees of freedom is n and in thelater 2 (a linear function goes through 2 points).
Degrees of freedom refer to number of free parameters in themodel.
Smoothing spline has n parameters and hence n nominal degrees offreedom.
Seppo Pynnonen Applied Multivariate Statistical Analysis
Moving beyond linearity
Smoothing splines
Choosing λ
However, these n parameters are heavily constrained or shrunkdown by the penalty term and thereby reducing the flexibility ofthe smoothing spline.
The effective degrees of freedom is denoted as dfλ which is ameasure of flexibility of the smoothing spline.
High dfλ indicates high flexibility (lower bias but higher variance)while low dfλ implies low flexibility (higher bias but lower variance).
For a given λ the smoothing spline is defined in terms of an n × nmatrix Sλ, such that
gλ = Sλy , (9)
where g is the solution of (8) for a given λ, i.e., it is a n-vector offitted values of the smoothing spline at the training pointsx1, . . . , xn.
Seppo Pynnonen Applied Multivariate Statistical Analysis
Moving beyond linearity
Smoothing splines
Choosing λ
The effective degrees of freedom is defined as the trace of the matrix Sλ, i.e.,
dfλ = tr(Sλ) =n∑
i=1
{Sλ}ii , (10)
where (Sλ)ii is the ith diagonal element of Sλ.
The problem is to find optimal λ.
This is done by cross-validation, like LOOCV.
It turns out that LOOCV can be computed very efficiently (need only a singlefit), using
RSScv (λ) =n∑
i=1
(yi − g(−i)λ (xi ))2 =
n∑i=1
(yi − gλ(xi )
1− {Sλ}ii
)2
, (11)
where g(−i)λ (xi ) denotes the fitted value estimated without the ith observation
(xi , yi ).
Seppo Pynnonen Applied Multivariate Statistical Analysis
Moving beyond linearity
Smoothing splines
Example 4
Continuing with the orangeWage data, fitting smoothing splines can beperformed by smooth.spline() function in the splines package.
> fit.smooth3$lambda # lambda produced by ’generalized’ CV
[1] 0.0348627
Seppo Pynnonen Applied Multivariate Statistical Analysis
Moving beyond linearity
Smoothing splines
Smoothing spline fits with fixed and CV selected df values.
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20 30 40 50 60 70 80
5010
015
020
025
030
0
Smoothing Spline
Age
Wag
eFixed 16 dfLOOCV 6.8 df
Seppo Pynnonen Applied Multivariate Statistical Analysis
Moving beyond linearity
Local regressions
1 Moving beyond linearity
Polynomial regressions
Step functions
Basis functions
Regression splines
Smoothing splines
Local regressions
Generalized additive models
Seppo Pynnonen Applied Multivariate Statistical Analysis
Moving beyond linearity
Local regressions
Local regression gives another approach for fitting flexiblenon-linear functions.
The general idea is to fit weighted least squares in the neighboringpoints around a given point x0.
The (relative) number of points around which the (weighted) leastsquares is applied is defined by the span parameter s.
The weighting function Ki0 is often referred to as the weightfunction.
Seppo Pynnonen Applied Multivariate Statistical Analysis
Moving beyond linearity
Local regressions
The graph below illustrates the idea of local regressions for asimulated data, the yellow areas refers to the weighting function,the blue line is the underlying true function f (x), the orange is thefitted local regression estimate f (x), and the red lines refer to thelocal weighted regressions around the particular x value.
0.0 0.2 0.4 0.6 0.8 1.0
−1
.0−
0.5
0.0
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1.0
1.5
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OOO
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0.0 0.2 0.4 0.6 0.8 1.0
−1
.0−
0.5
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1.0
1.5
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OOO
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Local Regression
Source: James et al. (2013) Fig 7.9
Seppo Pynnonen Applied Multivariate Statistical Analysis
Seppo Pynnonen Applied Multivariate Statistical Analysis
Moving beyond linearity
Generalized additive models
1 Moving beyond linearity
Polynomial regressions
Step functions
Basis functions
Regression splines
Smoothing splines
Local regressions
Generalized additive models
Seppo Pynnonen Applied Multivariate Statistical Analysis
Moving beyond linearity
Generalized additive models
Generalizes additive models (GAMSs) deal with multiple regressionwith several predictors, x1, . . . , xp by allowing nonlinear functionsof each predictor, while maintaining additivity.
The multiple linear regression model
y = β0 + β1x1 + · · ·+ βpxp + ε (12)
can be extended by replacing the linear componets, βjxj , with a(smooth) nonlinear function fj(xj), such that
It is notable that if fjs are selected as natural splines, fitting themodel reduces just to an OLS fit with natural splines of xjs asexplanatory variables.
Seppo Pynnonen Applied Multivariate Statistical Analysis