Projection pursuit regression

In statistics, projection pursuit regression (PPR) is a statistical model developed by Jerome H. Friedman and Werner Stuetzle which is an extension of additive models. This model adapts the additive models in that it first projects the data matrix of explanatory variables in the optimal direction before applying smoothing functions to these explanatory variables.

Model overview

The model consists of linear combinations of non-linear transformations of linear combinations of explanatory variables. The basic model takes the form

y_{i}=\beta _{0}+\sum _{j=1}^{r}f_{j}(\beta _{j}^{\mathrm {T} }x_{i})+\varepsilon ,

where x_i is a 1 × p row of the design matrix containing the explanatory variables for example i, y_i is a 1 × 1 prediction, {β_j} is a collection of r vectors (each a unit vector of length p) which contain the unknown parameters, {f_j} is a collection of r initially unknown smooth functions that map from ℝ → ℝ, and r is a hyperparameter. Good values for r can be determined through cross-validation or a forward stage-wise strategy which stops when the model fit cannot be significantly improved. As r approaches infinity and with an appropriate set of functions {f_j}, the PPR model is a universal estimator, as it can approximate any continuous function in ℝ^p.

Model estimation

For a given set of data $\{(y_{i},x_{i})\}_{i=1}^{n}$ , the goal is to minimize the error function

\min _{f_{j},\beta _{j}}S=\sum _{i=1}^{n}\left[y_{i}-\sum _{j=1}^{r}f_{j}(\beta _{j}^{\mathrm {T} }x_{i})\right]^{2}

over the functions $f_{j}$ and vectors $\beta _{j}$ . Because no method exists for solving over all variables at once, this problem has to be broken down. First, consider each $(f_{j},\beta _{j})$ pair individually: Let all other parameters be fixed, and find a "residual", the variance of the output not accounted for by those other parameters, given by

r_{i}=y_{i}-\sum _{l\neq j}f_{l}(\beta _{l}^{\mathrm {T} }x_{i})

The task of minimizing the error function now reduces to solving

\min _{f_{j},\beta _{j}}S'=\sum _{i=1}^{n}\left[r_{i}-(\beta _{j}^{\mathrm {T} }x_{i})\right]^{2}

for each j in turn. Typically new $(f_{j},\beta _{j})$ pairs are added to the model in a forward stage-wise fashion.

Aside: Previously-fitted pairs can be readjusted after new fit-pairs are determined by an algorithm known as backfitting, which entails reconsidering a previous pair, recalculating the residual given how other pairs have changed, refitting to account for that new information, and then cycling through all fit-pairs this way until parameters converge. This process typically results in a model that performs better with fewer fit-pairs, though it takes longer to train, and it is usually possible to achieve the same performance by skipping backfitting and simply adding more fits to the model (increasing r).

Solving the simplified error function to determine an $(f_{j},\beta _{j})$ pair can be done with alternating optimization, where first a random $\beta _{j}$ is used to project $X$ in to 1D space, and then the optimal $f_{j}$ is found to describe the relationship of that projection versus the residual via your favorite regression method. Then if $f_{j}$ is held constant, the optimal updated weights $\beta _{j}$ can be found by solving a weighted least squares problem. To derive this weighted least squares form, first Taylor expand $f_{j}(\beta _{j}^{T}x_{i})\approx f_{j}(\beta _{j,old}^{T}x_{i})+{\dot {f_{j}}}(\beta _{j,old}^{T}x_{i})(\beta _{j}^{T}x_{i}-\beta _{j,old}^{T}x_{i})$ , then plug it back in to the simplified error function $S'$ and do some algebraic manipulation to put it in the form

S'=\sum _{i=1}^{n}{\dot {f_{j}}}(\beta _{j,old}^{T}x_{i})^{2}\left[\left(\beta _{j,old}^{T}x_{i}+{\frac {r_{i}-f_{j}(\beta _{j,old}^{T}x_{i})}{{\dot {f_{j}}}(\beta _{j,old}^{T}x_{i})}}\right)-\beta _{j}^{T}x_{i}\right]^{2}

{\underset {\beta _{j}}{\operatorname {arg\,min} }}\left\|{\hat {r}}-X\beta _{j}\right\|_{W}^{2}=(X^{\mathrm {T} }WX)^{-1}X^{\mathrm {T} }W{\hat {r}}

After estimating the smoothing functions $f_{j}$ , one generally uses the Gauss–Newton iterated convergence technique to solve for $\beta _{j}$ ; provided that the functions $f_{j}$ are twice differentiable.

It has been shown that the convergence rate, the bias and the variance are affected by the estimation of $\beta _{j}$ and $f_{j}$ . It has also been shown that $\beta _{j}$ converges at an order of $n^{\frac {1}{2}}$ , while $\beta _{j}$ converges at a slightly worse order.

Discussion

The PPR model takes the form of a basic additive model but with the additional β_j component, so each f_j fits a scatter plot of $\beta _{j}^{\mathrm {T} }X^{\mathrm {T} }$ vs the residual (unexplained variance) during training rather than using the raw inputs themselves. This constrains the problem of finding each f_j to low dimension, making it solvable with common least squares or spline fitting methods and sidestepping the curse of dimensionality at this stage. Because f_j is taken of a projection of X, the result looks like a "ridge" in high dimension, so {f_j} are often called "ridge functions". The directions β_j are chosen to optimize the fit of their corresponding ridge functions.

Note that because PPR attempts to fit projections of the data, it can be difficult to interpret the fitted model as a whole, because each input variable has been accounted for in a complex and multifaceted way. This can make the model more useful for prediction than for understanding the data, though visualizing individual ridge functions and considering which projections the model is discovering can yield some insight.

Advantages of PPR estimation

It uses univariate regression functions instead of their multivariate form, thus effectively dealing with the curse of dimensionality
Univariate regression allows for simple and efficient estimation
Relative to generalized additive models, PPR can estimate a much richer class of functions
Unlike local averaging methods (such as k-nearest neighbors), PPR can ignore variables with low explanatory power.

Disadvantages of PPR estimation

PPR requires examining an M-dimensional parameter space in order to estimate $\beta _{j}$ .
One must select the smoothing parameter for $f_{j}$ .
The model is often difficult to interpret

Extensions of PPR

Alternate smoothers, such as the radial function, harmonic function and additive function, have been suggested and their performances vary depending on the data sets used.
Alternate optimization criteria have been used as well, such as standard absolute deviations and mean absolute deviations.
Ordinary least squares can be used to simplify calculations as often the data does not have strong non-linearities.
Sliced Inverse Regression (SIR) has been used to choose the direction vectors for PPR.
Generalized PPR combines regular PPR with iteratively reweighted least squares (IRLS) and a link function to estimate binary data.

PPR vs neural networks (NN)

Both projection pursuit regression and neural networks models project the input vector onto a one-dimensional hyperplane and then applies a nonlinear transformation of the input variables that are then added in a linear fashion. Thus both follow the same steps to overcome the curse of dimensionality. The main difference is that the functions $f_{j}$ being fitted in PPR can be different for each combination of input variables and are estimated one at a time and then updated with the weights, whereas in NN these are all specified upfront and estimated simultaneously.

Thus, PPR estimation is more straightforward than NN and the transformations of variables in PPR are data driven whereas in NN, these transformations are fixed.

References

Friedman, J.H. and Stuetzle, W. (1981) Projection Pursuit Regression. Journal of the American Statistical Association, 76, 817–823.
Hand, D., Mannila, H. and Smyth, P, (2001) Principles of Data Mining. MIT Press. ISBN 0-262-08290-X
Hall, P. (1988) Estimating the direction in which a data set is the most interesting, Probab. Theory Related Fields, 80, 51–77.
Hastie, T. J., Tibshirani, R. J. and Friedman, J.H. (2009). The Elements of Statistical Learning: Data Mining, Inference and Prediction. Springer. ISBN 978-0-387-84857-0
Klinke, S. and Grassmann, J. (2000) ‘Projection Pursuit Regression’ in Smoothing and Regression: Approaches, Computation and Application. Ed. Schimek, M.G.. Wiley Interscience.
Lingjarde, O. C. and Liestol, K. (1998) Generalized Projection Pursuit Regression. SIAM Journal of Scientific Computing, 20, 844-857.