# Interactive presentation on multi dimensional non parametric regression

2002-Feb-06
version 0.2
Stefan Dirnstorfer

Multi dimensional data sets appear in many areas of financial engineering. They can be modelled mathematically with higher dimensional functions. This page demonstrates the basic concept behind an implementation that handles these functions and interpolates it from given points or known market prices.

## What is multi dimensional regression

• Approximate scattered point clouds
• Smoothing outliers

## Applications

• Volatility surfaces on equity
• 3D surfaces of swaption volatilities (strike, maturity of option, maturity of underlying)
• Spread curves with rating and maturity

# Concept of basis function

In order to express mathematical functions numerically it is a widely adopted concept to store them with a set of basis functions. Those can added and approximate arbitrary functions by linear combination. The numerical algorithm stores and manipulates the linear coefficients.

## Hat functions

A simple set of basis functions is plotted below. These piecewise linear functions are devided into different levels of resolution. In the first row the coursest function spans over the entire domain, while functions on the finest level only have small support. Hat functions with different levels of resolution

## Interpolation example

According to the required accuracy and the available computational resources a function can be approximated by combining basis functions up to certain level of resolution. You can cycle through these levels by clicking on the picture below. In every new level finer basis functions reveal new details of the original mountain. Click to cycle through interpolation steps.

# Multidimensional basisfunctions

The tensor product allows the construction of multi dimensional hat functions. The plot below visualizes tensor product functions generated by a combination of unidirectional hat functions. Each direction x and y is associated with one resolution level. Combinations of levels yields two dimensional basisfunctions with rectangular support. Click to cycle through basis functions.

# Minimization

The minimization problem is described informally by the goodness of interpolation and a measure for the resulting functions smoothness.

Minimize inerpolation error + roughness

Mathematicall this can be written by a regression forumla with a residual measuring roughness. Each point has a weight wi. Low weights yield the interpolation surface to be smooth rather that fitting the points exactly. High weights let the surface be close to that point. An appropriate rougness messure is expressed by the norm of the functions laplace. This yields an algorithms favor for linear functions. # Result

The animation shows an interpolation of a volatility surface. The points represent volatilities computed from market prices. An outlier with far exagerated volatility is added to demonstrate the impact on the interpolation scheme. When the outlier moves into an area where many correct points are located its impact is low. In areas without alternaive information the surface exposes a hump beneath the outlying value. Impact of outliers on the interpolation sceme.