Answer: In theory, the delta width values of x, y are approaching zero.Therefore, the interpolated f(x,y) at given input x1, y1 is approaching thevalue of the given input f1. Note that in practical, numerical calculation,the delta width values would be small but finite and the interpolated f(x,y)at x1, y1 can be made within 1% of the given input f1, by adjusting thevalue of delta width accordingly. Please check this property out onour web site (RDIC).
Answer: Our method starts with an identity integral (which is analogous to Reproducing Kernel Hilbert Space -RKHS method) and its derivation depends on Monte-Carlo method. The error analysis of our method can be produced analytically. The estimate on the convergence of our method has shown that the convergence depends on two terms: (a) bias term; (b) random error term. The bias term depends on the dimension as 
where M' is the number of sample data points used in the interpolation and n is the dimension, and the random error term is inversely proportional to the square root of M' and independent of the dimension. Both terms are independent of the smoothness of the function to interpolate. Please download our supplementary material in Microsoft WORD file, or in PDF file, regarding "accuracy and convergence" issues.
Answer: Our method starts with (Eq. A) and (Eq. A-1) on the Introduction page. (Eq. A) uses the concept of Dirac delta function and (Eq. A-1) is the "computable" variant of (Eq. A). The derivation of interpolant formula is established via Monte-Carlo integration. The form of the interpolant found at RDIC indeed looks the same as "a Nadaraya-Watson estimator with a modified Cauchy kernel". However, the error analysis of our interpolants is different from the kernel regression method. More on this issue, please read the supplementary material (See the previous question and answer). In addition, the width (bandwidth) selection in our method has followed a different algebraic process.
Answer: In 2-D and 3-D space, Dirac-Monte Carlo method can easily produce interpolants for polar, spherical and cylindrical coordinate systems. These coordinates are non-Cartesian. The major advantages of having these interpolants are: (a) to deal with interpolation applications which are better treated by non-Cartesian coordinates (These interpolants shall provide efficientsolutions for global weather study, mining industry, earthgravity and magnetic field survey, as well as geometric surfacereconstruction.); (b) to deal with non-convex domain (For example, between two concentric circles or two concentric spherical surfaces, L-shape corridor, etc. Currently, there are no interpolants/estimators thatcan do this correctly.).
Answer: In nonparametric kernel regression literature, Cauchy kernel has beenshunned by researchers because mean and variance, or higher moments in thiscase, are not defined for the unbounded domain. But the long tail of Cauchykernel is a desired, needed property to give long-range influence andsensitivity in high dimension space for applications with bounded domainsupport. We shall employ Gaussian distribution to approximate Dirac delta function for later study.
Answer: Our interpolant provides a simple nonparametric kernel smoothing treatment which does notdepend on "distance" but on the "coordinate separation". The user needs toprovide 2 width values in order to do interpolation in 2-D. Though themethod posted on the site deals with Cartesian coordinates, it isstraightforward to be extended to polar coordinates if desired.
Answer: Our method does not follow the standard approach of deriving nonparametric kernel regression estimator. In addition, It does not use Mean-Square-Error (MSE) to obtain the optimized bandwidth values. Our approach is to find out the first-cut values of bandwidth (analytically simple, download at: http://www.fanginc.com/texas2.doc). Then we fine tune the width value (generally reducing the first-cut value) so that the reproduced function values at input locations of global maximum and minimum are within a few percent (about 5% roughly) of the prescribed, input values. Overly reducing the width value to match the prescribed values, say less than 1%, is not recommended because it will cause large variation and less smoothness of the interpolated function values. This fine-tuning part can be automated, if needed. Our program on the web site is "rudimentary" and for demo only, and the entire interpolation center will be upgraded in the very near future.
Answer: The answer is "yes". If you agree that there is a functionwhich depends on "time" within an interval, say, between t1 and t2, then itis ok to have function at a future time t3 to give "weighted contribution "to the interpolated value at a past time t4. Of course, t3 and t4 are bothwithin the interval defined between t1 and t2. One reason for us to say so is that if you make a Taylor's series expansion of the function at t3, andif it is convergent, it is ok mathematically to use the function expansionfor time value greater than t3 or smaller than t3. In a way of speaking,"causality" does not play a role here. Currently, we are planning to use interpolant to do communication signal reconstruction in the timedomain.
Answer: Please download the answer.
Answer: We do not have pseudo-code available for our program. If you tell us your need regarding the dimension you need and the domain values of your variables, we may be able to provide a software demo version for you. In the near future, our software shall be available in the commercial market.
Answer: Please note that we only used 12 randomly located input data for the interpolation. The errorassociated with the interpolant is documented in supplementary material. The more input data are used, the better the final answer willbe.
Answer: Please visit our web site at: http://www.fanginc.com/main.htm to findout technical information on our interpolation algorithm. Contact us foradditional data if needed. Our algorithm can easily accommodate 5 variables. We shall provide a Java version of our program at a later time.
Answer: Please read the supplementary material to find out how to calculate the optimized delta width for your applications.
However, there is one characteristics of our method which should be pointed out. As can be seen in our interpolant, it is basically a ratio of "numerator" over "denominator", and both are in the form of summation of rational functions. If the position to be interpolated has one coordinate value very close or equal to the same coordinate value of any given input locations and if delta width is a small value, then the interpolated function value will have a direct impact by the given input function value. In other words, for example if the input location (x=a, y=b) for 2D case has the input function value F0, then any interpolated locations within or close to x=a ± 
or y=b ± (That is, a narrow but long "cross" centered at x=a, y=b in the problem domain.) will yield values strongly influenced by F0 value. In this situation, the user should use a larger delta width value, and it will remove this "artifact" feature in the interpolation process. Generally speaking, the artifact will not show up by use of the optimized delta width value found in our method. As a final remark, it is emphasized that the user needs to be aware of this long-range dependence effect.
Answer: The delta width value found in our analysis has been optimized for the entire domain. However, It can be further fine-tuned if there is any specific goal to study the neighborhoods of sample input locations. Read the supplementary material for finding more info about delta width values in multi-dimension space as well as non-square problem domains.
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