# Evaluation of Interpolants in Their Ability to Fit Seismometric Time Series

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## Abstract

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## 1. Introduction

## 2. Background of the Data

## 3. Spline Smoothing Methods

#### 3.1. Nearest-Neighbor Interpolation

#### 3.2. Bilinear Interpolation

#### 3.2.1. Algorithm

#### 3.2.2. Unit Square

#### 3.2.3. Nonlinear

#### 3.3. Bicubic Interpolation

#### 3.4. Shape-Preserving (PCHIP)

#### Interpolation on a Single Interval

**Unit interval**$(0,1)$**:**On the unit interval $(0,1)$, consider the starting point ${p}_{0}$ at $t=0$ and ending point ${p}_{1}$ at $t=1$ with the starting tangent ${m}_{0}$ at $t=0$ and the ending tangent ${m}_{1}$ at $t=1$, the polynomial can be defined by,$$p\left(t\right)=(2{t}^{3}-3{t}^{2}+1){p}_{0}+({t}^{3}-2{t}^{2}+t){m}_{0}+(-2{t}^{3}+3{t}^{2}){p}_{1}+({t}^{3}-{t}^{2}){m}_{1}$$**Interpolation on an arbitrary interval:**Interpolating x in an arbitrary interval (${x}_{k},{x}_{k+1}$) is done by mapping the latter to $[0,1]$ through an affine (degree 1) change of variable. The formula is$$p\left(x\right)={h}_{00}\left(t\right){p}_{k}+{h}_{10}\left(t\right)({x}_{k+1}-{x}_{k}){m}_{k}+{h}_{01}\left(t\right){p}_{k+1}+{h}_{11}\left(t\right)({x}_{k+1}-{x}_{k}){m}_{k+1}.$$**Uniqueness:**The formulae specified above provides the unique third-degree polynomial path between the two points with the given tangents.

## 4. Statistical Analysis of Variability

#### 4.1. Loess Smoothing

**Loess curve**, particularly when the smoothed value is obtained by a weighted quadratic least squares regression over the span of values of the y-axis scattergram criterion variable. Similarly, the same process is called

**Lowess curve**when each smoothed value is given by weighted linear least squares regression over the span. Although some literature consider

**Lowess**and

**Loess**as synonymous, some key features of the local regression models are:

**Definition of a Lowess/Loess model:**Lowess/Loess, originally proposed by Cleveland and further improved by Cleveland and Devlin [9], specifically denoted a method that is also known as locally weighted polynomial regression. At each point in the data set a low-degree polynomial is fitted to a subset of the data, with explanatory variable values near the point whose response is being estimated. Weighted least square method is implemented in order to fit the polynomial where more weightage is given to points near the point whose response is being estimated and less weightage to points further away. The value of the regression function at the point is then evaluated by calculating the local polynomial by using the explanatory variable values for that data point. One needs to compute the regression function values for each of the n data points in order to complete the Lowess/Loess process. Many details in this method, such as degree of the polynomial model and weights, are flexible.**Local subsets of data:**The subset of data used for each weighted least square fit in Lowess/Loess is decided by a nearest neighbors algorithm. One can predetermine the specific input to the process called the “bandwidth" or “smoothing parameter", it determines how much of the data is used to fit each local polynomial according to its needs. The smoothing parameter α, takes values between $\frac{(\lambda +1)}{n}$ and 1, with λ denoting the degree of the local polynomial, the value of α is the proportion of data used in each fit. The subset of data used in each weighted least squares fit comprises the $n\alpha $ points (rounded to the next larger integer) whose explanatory variable values are closest to the point at which the response is being evaluated. The smoothing parameter α is named because it controls the flexibility of the Lowess/Loess regression function. Large values of α produce the smoothest functions that wiggle the least in response to fluctuations in the data. The smaller α is, the closer the regression function will conform to the data, but using a very small value for the smoothing parameter is not desirable since the regression function will eventually start to capture the random error in the data. For the majority of the Lowess/Loess applications, α values are chosen in the $[0.25,05]$ interval.**Degree of local polynomials:**First and second degree polynomial are used to fit local polynomials to each subset of data. That means, either a locally linear or quadratic function are most useful; using a zero polynomial turns Lowess/Loess into a weighted moving average. Such a simple model might work well for some situations, and may approximate the underlying function well enough. Higher-degree polynomial methods perform very good in theory although this method doesn't agree with the spirit of Lowess/Loess method: Lowess/Loess is based on the idea that any function can be approximated in a small neighborhood by a low-degree polynomial and simple models can be easily fitted to data. High-degree polynomials would tend to overfit the data in each subset and are numerically unstable, making precise calculation almost impossible.**Weight function:**The weight function assigns more weight to the data points nearest the point of evaluation and less weight to the data points that are further away. The idea behind this method is that the points near each other in the explanatory variable space are more likely to be associated to each other in a simple way than the points that are located further away. Following this argument, points that are likely to follow the local model best influence the local parameter estimate the most. Points that are less likely to actually conform to the local model have less influence on the local model parameter estimation. The Lowess/Loess methods use the traditional tri-cubed weight function. However, any other weight function that satisfies the properties that are presented in [10] could also be taken into consideration. The process of calculating the weight for a specific point in any localized subset of data is done by evaluating the weight function at the distance between the point and the point of estimation, after scaling the distance so that the maximum absolute distance over all possible points in the subset of data is exactly one.**Advantages and Disadvantages of Lowess/Loess:**As mentioned above, the biggest advantage of the Lowess/Loess methods over many other methods is the fact that they do not require the specification of a function to fit a model over the global sample data. Instead, an analyst has to provide a smoothing parameter value and the degree of the local polynomial. Moreover, the flexibility of this process makes it very appropriate for modeling complex processes for which no theoretical model exists. The simplicity for executing the methods, make these processes very popular among the modern era regression methods that fit the general framework of least squares regression, but having a complex deterministic structure. Although they are less obvious than some of the other methods related to linear least squares regression, Lowess/Loess also enjoy most of the benefits generally shared by the other methods, the most important of those is the theory for computing uncertainties for prediction, estimation and calibration. Many other tests and processes used for validation of least square models can also be extended to Lowess/Loess.The major drawback of Lowess/Loess is the inefficient use of data compared to other least square methods. Typically it requires fairly large, densely sampled data sets in order to create good models, the reason behind is that the Lowess/Loess methods rely on the local data structure when performing the local fitting, thus proving less complex data analysis in exchange of increased computational cost. The Lowess/Loess methods do not produce a regression function that is represented by a mathematical formula, what may be a disadvantage: It can make it very difficult to transfer the results of the numerical analysis to other researchers, in order to transfer the regression function to others, they would need the data set and the code for Lowess/Loess calculations. In non-linear regression, on the other hand, it is only necessary to write down a functional form in order to provide estimates of the unknown parameters and the estimated uncertainty. In particular, the simple form of Lowess/Loess can not be applied to mechanistic modeling where the fitted parameters specify particular physical properties of the system. It is worth mentioning the computation cost associated with this procedure, although this should not be a problem in the modern computing environment, unless the data sets being used are very large. Lowess/Loess also have a tendency to be affected by the outliers in the data set, like any other least square methods. There is an iterative robust version of Lowess/Loess (see [10]) that can be applied to reduce sensitivity to outliers, but if there exist too many extreme outliers, this robust version also fails to produce the desired results.

#### 4.2. Smoothing Spline

## 5. Results and Discussions

#### 5.1. Mathematical Models Applied to the Time Series

#### 5.2. Statistical Analysis of Local Variation in Time Series

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Basu, K.; Mariani, M.C.; Serpa, L.; Sinha, R.
Evaluation of Interpolants in Their Ability to Fit Seismometric Time Series. *Mathematics* **2015**, *3*, 666-689.
https://doi.org/10.3390/math3030666

**AMA Style**

Basu K, Mariani MC, Serpa L, Sinha R.
Evaluation of Interpolants in Their Ability to Fit Seismometric Time Series. *Mathematics*. 2015; 3(3):666-689.
https://doi.org/10.3390/math3030666

**Chicago/Turabian Style**

Basu, Kanadpriya, Maria C. Mariani, Laura Serpa, and Ritwik Sinha.
2015. "Evaluation of Interpolants in Their Ability to Fit Seismometric Time Series" *Mathematics* 3, no. 3: 666-689.
https://doi.org/10.3390/math3030666