com.webcab.ejb.finance.portfolio
Interface Interpolation

All Superinterfaces:

EJBObject, Remote

public interface Interpolation

extends EJBObject

Within this class we offer methods by which the Efficient Frontier can be constructed from a finite set of known points. In particular, we offer methods based around the cubic interpolation (recommended approach) and polynomial interpolation procedures. This Enterprise JavaBean contains several methods for constructing continuous functions from discrete data points. Such methods are generally referred to as Interpolation and Extrapolation methods.

Functionality Offered

With this Enterprise JavaBean we offer the following functionality:

• Polynomial Interpolation
  1. interpolateExtrapolatePolynomial - returns the value and an error estimate of an interpolation point where the interpolation function is a polynomial of a given order.
  2. coefficientsInterpolatingPolynomialStable - this method is similar to the approach given above except that here we have a higher level of stability which we pay for with lower efficiency.
  3. coefficientsInterpolatingPolynomial - this methods evaluates the coefficients of the interpolation polynomial.
• Cubic Spline Interpolation
  1. cubicSplinePointwisePreEvaluation - Evaluates the value of the cubic spline at a given point.
  2. cubicSpline2ndDifferential - Allow the 2nd derivatives of the cubic spline to be evaluated so that they can be consumed within cubicSplinePointwisePreEvaluation
  3. cubicSplinePointwise - Evaluates the cubic spline interpolation constructed from a given set of points. When applying this method you will need to give the values of the derivatives at the end points. A reasonable estimate could be provided by directed evaluation of the slope between either the first or last two known points of the Efficient Frontier.

How to construct the Efficient Frontier using Interpolation

As mentioned above the main aim of these interpolation procedures is to provide the means by which the Efficient Frontier can be constructed from a finite set of points on which it can be evaluated by using the method: calculateEfficientFrontier from Markowitz Enterprise JavaBean. The expected return and risk coordinate components of these points can then be read off by using the methods expectedReturnEfficientFrontier and portfolioRisksEfficientFrontier from Markowitz Enterprise JavaBean.

Once the interpolation points are known we are able to construct the interpolation around these points and then in the case of cubic spline interpolation evaluate the Efficient Frontier at an arbitrary value points by using one of the methods: cubicSplinePointwise, cubicSplinePointwisePreEvaluation.

Method Summary

double[]	coefficientsInterpolatingPolynomial(double[] tabulatedValues, double[] polynomialValues) Evaluates the coefficients of the interpolating polynomial when the tabulation points are known.
double[]	coefficientsInterpolatingPolynomialStable(double[] tabulatedValues, double[] polynomialValues) Evaluates the coefficients of the interpolating polynomial when the tabulation points are known.
double[]	cubicSpline2ndDifferential(double[] tabulationPointsInX, double[] functionValuesAtTabulationPoints, double derivativeInterpolationAt0, double derivativeInterpolationAtn_1) Evaluates the second derivatives of the cubic spline interpolation polynomial at the given functions tabulation points when the first derivative at the boundary (equivalently the end points) is known.
double	cubicSplinePointwise(double[] tabulationPointsInX, double[] functionValuesAtTabulationPoints, double derivativeInterpolationAt0, double derivativeInterpolationAtn_1, double interpolationPoint) Returns the value of the cubic spline interpolation at a given point.
double	cubicSplinePointwisePreEvaluation(double[] tabulationPointsInX, double[] functionValuesAtTabulationPoints, double[] secondDifferential, double interpolationPoint) Returns the cubic spline interpolation of a function at a point.
double[]	interpolateExtrapolatePolynomial(double[] tabulationPointsInX, double[] polynomialValues, double interpolationPoint) This method interpolates (or extrapolates) a given polynomial in one variable.

Methods inherited from interface javax.ejb.EJBObject

getEJBHome, getHandle, getPrimaryKey, isIdentical, remove

Method Detail

interpolateExtrapolatePolynomial

public double[] interpolateExtrapolatePolynomial(double[] tabulationPointsInX,
                                                 double[] polynomialValues,
                                                 double interpolationPoint)
                                          throws InterpolationException,
                                                 InterpolationDemoException,
                                                 RemoteException

This method interpolates (or extrapolates) a given polynomial in one variable. It returns an array of two values the first being the value of the interpolation function evaluated at the interpolation point and the other being the error estimate.

If P(x) is the polynomial of degree `n-1' such that P(xCoordinates[i]) = yCoordinates[i]; i=0,...,n-1; then if the desired point at which the interpolation functions value is required is `interpolationPoint'. Then the returned value will be some double value y, where y = P(interpolationPoint).

The `tabulationPointsInX' array and the `polynomialValues' array should each contain at least one element. If these arrays have different lengths the shorter one will be chosen as reference.

Parameters:

tabulationPointsInX - an array of doubles for which at member we know the corresponding value of the given polynomial. Note, that the corresponding values of the polynomial at these points given by the member of the array yCoordinates which lie in the corresponding position.

polynomialValues - an array of doubles which corresponds to the value of the polynomial at the points tabulationPointsInX. The first point of this array is the value of the given polynomial evaluated at the point tabulationPointsInX[0], th second point of the array is the value of the given polynomial at the point tabulationPointsInX[1].

interpolationPoint - the value of the point at which the interpolation function is evaluated and a corresponding error estimate is given.

Returns:

An array of doubles where the first terms if the value of the interpolation function and the second is an error estimate.

Throws:

InterpolationException - Thrown when the input values do not meet the requirements mentioned above.

InterpolationDemoException

RemoteException

coefficientsInterpolatingPolynomialStable

public double[] coefficientsInterpolatingPolynomialStable(double[] tabulatedValues,
                                                          double[] polynomialValues)
                                                   throws InterpolationException,
                                                          InterpolationDemoException,
                                                          RemoteException

Evaluates the coefficients of the interpolating polynomial when the tabulation points are known. That is, a set of points in the range and the corresponding values in the domain of the interpolation function are known and we will deduce from this information the coefficients of the corresponding interpolation polynomial assuming the degree of this polynomial is one less than the number of tabulation points used.

Further Explanation

If we are given a set of n points on which the interpolation function is known then this method evaluates the n coefficients c_i of the interpolation function c_0 + (c_1 * x) + (c_2 * x * x)+ ....

More explicitly, given a set of tabulation points x = {x[i]: i=0,...,n-1} and y = {y[i]:i=0,...,n-1} which defines a function by y[i] = f(x[i]). This method returns an array of doubles which are the coefficients of the interpolating polynomial where the first term of the array corresponds to the 0th order term, the second term corresponds to the 1st order term (i.e. x's coefficient) and so on.

Comparison with coefficientsInterpolatingPolynomial

The method coefficientsInterpolatingPolynomial differs slightly from this method which is less direct and slower by a power of the number of tabulation points used. However, we have found this approach to be more stable. The essential idea in this approach is that it uses the interpolateExtrapolatePolynomial method with iterative reduction to arrive at the interpolation polynomial.

Remark: If the two arrays have different lengths the shorter one will be used as reference.

Parameters:

tabulatedValues - an array of doubles which represent the values at which the interpolation polynomial is tabulated. That is, is `f' is the interpolating polynomial then we have f(tabulatedValues[i]) = polynomialValues[i]; where `polynomialValues[i]' is the value of the interpolating polynomial at the `tabulatedValues[i]'.

polynomialValues - an array of doubles where the first element `polynomialValues[0]', corresponds to the value of the interpolating polynomial at the first tabulation point `tabulatedValues[0]', and the second values `polynomialValues[1]' corresponds to the value of the interpolating polynomial at the second tabulated point `tabulatedValues[1]', and so on...

Returns:

An array of double where the i-th term is the i-th coefficient within the interpolation function.

Throws:

InterpolationException - Thrown when any of the two parameters are null.

InterpolationDemoException

RemoteException

See Also:

coefficientsInterpolatingPolynomial
, interpolateExtrapolatePolynomial

coefficientsInterpolatingPolynomial

public double[] coefficientsInterpolatingPolynomial(double[] tabulatedValues,
                                                    double[] polynomialValues)
                                             throws InterpolationException,
                                                    InterpolationDemoException,
                                                    RemoteException

Evaluates the coefficients of the interpolating polynomial when the tabulation points are known. That is, a set of points in the range and the corresponding values in the domain of the interpolation function are known and we will deduce from this information the coefficients of the corresponding interpolation polynomial assuming the degree of this polynomial is one less than the number of tabulation points used.

Further Explanation

If we are given a set of n points on which the interpolation function is known then this method evaluates the n coefficients c_i of the interpolation function c_0 + (c_1 * x) + (c_2 * x * x)+ ....

More explicitly, given a set of tabulation points x = {x[i]: i=0,...,n-1} and y = {y[i]:i=0,...,n-1} which defines a function by y[i] = f(x[i]). This method returns an array of doubles which are the coefficients of the interpolating polynomial where the first term of the array corresponds to the 0th order term, the second term corresponds to the 1st order term (i.e. x's coefficient) and so on.

Comparison with coefficientsInterpolatingPolynomialStable

The method coefficientsInterpolatingPolynomialStable differs slightly from this method which is more direct, and faster by a power of the number of tabulation points used. However, we have found this approach to be less stable.

Remark: If the two arrays have different lengths the shorter one will be used as reference.

Parameters:

tabulatedValues - an array of doubles which represent the values at which the interpolation polynomial is tabulated. That is, if `f' is the interpolation polynomial then we have f(tabulatedValues[i]) = polynomialValues[i]; where `polynomialValues[i]' is the value of the interpolation polynomial at the `tabulated values`tabulatedValues[i]'.

polynomialValues - an array of double where the first element `polynomialValues[0]', corresponds to the value of the interpolating polynomial at the first tabulation point `tabulatedValues[0]', and the second values `polynomialValues[1]' corresponds to the value of the interpolating polynomial at the second tabulated point `tabulatedValues[1]', and so on...

Returns:

An array of double where the i-th term is the i-th coefficient within the interpolation function.

Throws:

InterpolationException - Thrown when either of the two parameters is null.

InterpolationDemoException

RemoteException

cubicSpline2ndDifferential

public double[] cubicSpline2ndDifferential(double[] tabulationPointsInX,
                                           double[] functionValuesAtTabulationPoints,
                                           double derivativeInterpolationAt0,
                                           double derivativeInterpolationAtn_1)
                                    throws InterpolationException,
                                           InterpolationDemoException,
                                           RemoteException

Evaluates the second derivatives of the cubic spline interpolation polynomial at the given functions tabulation points when the first derivative at the boundary (equivalently the end points) is known. Knowledge of the second derivative is required in order to uniquely determine the cubic spline.

Description of the parameters

Given arrays tabulationPointsInX[0..n-1] and tabulationPointsInY[0..n-1] containing a tabulated function, i.e. tabulationPointsInY[i] = f(tabulationPointsInY[i]), with tabulationPointsInX[0] < tabulationPointsInX[1] < ... < tabulationPointsInX[n-1], and given values derivativeInterpolationAt0 and derivativeInterpolationAtn_1 for the first derivative of the interpolating function at the points tabulationPointsInX[0] and tabulationPointsInX[n-1], respectively. This method returns an array of length n, that contains the second derivatives of the interpolation function at the tabulation points tabulationPointsInX[i]. If derivativeInterpolationAt0 and/or derivativeInterpolationAtn_1 are equal to 10³⁰ or larger, then the method sets the second derivative at the boundary to be zero.

Remark: If the two arrays have different lengths, the shorter one will be used as reference. The two arrays should be at least 2 elements long.

Parameters:

tabulationPointsInX - an array of doubles which are the values at which the function is tabulated, i.e. functionValuesAtTabulationPoints[i] = f(tabulationPointsInX[i])

functionValuesAtTabulationPoints - an array of doubles which are the values of the function evaluated at the interpolation points, i.e. functionValuesAtTabulationPoints[i] = f(tabulationPointsInX[i])

derivativeInterpolationAt0 - the first derivative of the interpolation function at the point tabulationPointsInX[0]

derivativeInterpolationAtn_1 - the first derivative of the interpolation function at the point tabulatedPointInX[n-1]

Returns:

An array of doubles which are equal to the 2nd derivatives of the interpolation function at the interpolation points.

Throws:

InterpolationException - Thrown when the input values do not meet the requirements mentioned above.

InterpolationDemoException

RemoteException

cubicSplinePointwisePreEvaluation

public double cubicSplinePointwisePreEvaluation(double[] tabulationPointsInX,
                                                double[] functionValuesAtTabulationPoints,
                                                double[] secondDifferential,
                                                double interpolationPoint)
                                         throws InterpolationException,
                                                InterpolationDemoException,
                                                RemoteException

Returns the cubic spline interpolation of a function at a point. We have designed this method so that the 2nd derivatives can be pre-evaluated using the method cubicSpline2ndDifferential.

General Description of Parameters

Given the arrays tabulationPointsInX[0..n-1] and functionValuesAtTabulationPoints[0..n-1], which tabulate a function where the tabulationPointsInX is an array where the elements are monotonically increasing. Moreover, given the array secondDifferenttial[0..n-1], which is the output of the method cubicSpline2ndDifferential, and given a value of the interpolation points interpolationPoint, this method returns the value of the function at interpolationPoint according to the cubic-spline interpolation method.

Description of the Parameters in terms of the Efficient Frontier

With direct regards to the Efficient Frontier the points tabulationPointsInX[0..n-1], will refer to the risk of the portfolios on the Efficient Frontier and be evaluated using portfolioRisksEfficientFrontier of the Markowitz Enterprise JavaBean. The values functionValuesAtTabulationPoints[0..n-1] will refer to the expected returns of the known portfolios on the Efficient Frontier and can be evaluated using expectedReturnEfficientFrontier of the Markowitz Enterprise JavaBean. The 2nd derivatives can be thought of as the rate of change of the increase in the expected return for taken on more risk. That is, the second derivatives express a qualitative property of the Efficient Frontier. The point at which the cubic spline in interpolated namely, interpolationPoint, will refer to the value of the portfolio risk for which the value of corresponding expected return will be returned.

Remarks:

• Care should be taken to ensure that if the method cubicSpline2ndDifferential is used in the evaluation of the 2nd derivatives then all the parameters in both methods are applied in a consistent fashion.
• If the three arrays have different lengths, the shortest one will be used as reference. The three arrays should be at least 2 elements long.

Parameters:

tabulationPointsInX - the array containing the values at which the function is tabulated, i.e. functionValuesAtTabulationPoints[i] = f(tabulationPointsInX[i])

functionValuesAtTabulationPoints - an array of doubles of the values of the function evaluated at the interpolation points, i.e. functionValuesAtTabulationPoints[i] = f(tabulationPointsInX[i])

secondDifferential - an array containing the 2nd differential of the cubic spline at the tabulation points. These 2nd differentials can be evaluated using the method cubicSpline2ndDifferential.

interpolationPoint - the point at which the interpolation function is evaluated and returned. Note that this is the point at which we wished to find the value of the given function which we only knew at the tabulation points

Returns:

The value of the cubic spline interpolation function at a point.

Throws:

InterpolationException - Thrown when the input values do not meet the requirements mentioned above.

InterpolationDemoException

RemoteException

See Also:

cubicSplinePointwise - this is an alternative means by which the value of the cubic spline can be evaluated at a point. Here rather than providing the 2nd differentials we are required to provide the values of the differentials at the end points.
, cubicSpline2ndDifferential - use this method in order to evaluate the 2nd differentials at the tabulation points which most be provided to this method a parameter.

cubicSplinePointwise

public double cubicSplinePointwise(double[] tabulationPointsInX,
                                   double[] functionValuesAtTabulationPoints,
                                   double derivativeInterpolationAt0,
                                   double derivativeInterpolationAtn_1,
                                   double interpolationPoint)
                            throws InterpolationException,
                                   InterpolationDemoException,
                                   RemoteException

Returns the value of the cubic spline interpolation at a given point. In particular, here we apply this method to estimate the value of the Efficient Frontier at a given point when it is only know on a finite set of points.

General Description of Parameters

Given a function which is tabulated at the points tabulationPointsInX[0..n-1] which has elements was are monotonically increasing and takes the values functionValuesAtTabulationPoints[0..n-1] at those points, this methods returns the values of the cubic spline interpolation function evaluated at the given point interpolationPoint.

Description of the Parameters in terms of the Efficient Frontier

With direct regards to the Efficient Frontier the points tabulationPointsInX[0..n-1], will refer to the risk of the portfolios on the Efficient Frontier and be evaluated using portfolioRisksEfficientFrontier of the Markowitz Enterprise JavaBean. The values functionValuesAtTabulationPoints[0..n-1] will refer to the expected returns of the known portfolios on the Efficient Frontier and be evaluated using expectedReturnEfficientFrontier of the Markowitz Enterprise JavaBean. The point at which the cubic spline in interpolated namely, interpolationPoint, will refer to the value of the portfolio risk for which the value of corresponding expected return will be returned.

Notes on Estimating the Derivatives

When applying this method you will need to provide values of the derivative of the interpolation function at the end points. A reasonable estimate with regard to approximating the Efficient Frontier could be provided by directed evaluating the slope between either the first or last two known points of the Efficient Frontier.

Remark: If the two arrays have different lengths, the shorter one will be used as reference. The two arrays should be at least 2 elements long.

Parameters:

tabulationPointsInX - the array containing the values at which the function is tabulated, i.e. functionValuesAtTabulationPoints[i] = f(tabulationPointsInX[i])

functionValuesAtTabulationPoints - an array of double of the values of the function evaluated at the interpolation points, i.e. functionValuesAtTabulationPoints[i] = f(tabulationPointsInX[i]), where f is the function being considered

derivativeInterpolationAt0 - the first derivative of the interpolation function at the point tabulationPointsInX[0]

derivativeInterpolationAtn_1 - the first derivative of the interpolation function at the point tabulatedPointInX[n-1]

interpolationPoint - The value of the `x-coordinate' at which the value of the cubic spline interpolation function is evaluated. Note, that in the case of the Efficient Frontier this will refer to the risk of the portfolio.

Returns:

The value of the cubic spline interpolation function at a point.

Throws:

InterpolationException - Thrown when the input values do not meet the requirements mentioned above.

InterpolationDemoException

RemoteException

See Also:

cubicSplinePointwisePreEvaluation