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In many applications, observed signals are contaminated by both random noise and blur. This paper proposes a blind deconvolution procedure for estimating a regression function with possible jumps preserved, by removing both noise and blur when recovering the signals. Our procedure is based on three local linear kernel estimates of the regression function, constructed from observations in a left-side, a right-side, and a two-side neighborhood of a given point, respectively. The estimated function at the given point is then defined by one of the three estimates with the smallest weighted residual sum of squares. To better remove the noise and blur, this estimate can also be updated iteratively. Performance of this procedure is investigated by both simulation and real data examples, from which it can be seen that our procedure performs well in various cases.

Nonparametric regression analysis provides us statistical tools for estimating regression functions from noisy data [

In the literature, there are some existing procedures for estimating regression curves with jumps preserved in cases when only random noise is present in observed data. These procedures include the one-sided kernel estimation methods (e.g., [

In some applications, our observations are both blurred and contaminated by pointwise noise (e.g., signals of groundwater levels in geothermy). It is, therefore, important to remove both noise and blur when estimating the true regression function. In the nonparametric regression literature, we have not seen any discussion about this problem yet. In the context of image processing, which can be regarded as a two-dimensional nonparametric regression problem [

The remaining part of the paper is organized as follows. In next section, our proposed method is discussed in detail. Some comparative results are presented in Section

Suppose that the regression model concerned is

If

(a) Solid lines denote the true regression function, and dotted curve denotes the blurred regression function. (b) Solid lines denote the estimate of the regression function by the proposed method, dotted curve denotes the blurred true regression function, and dashed curve denotes the conventional local linear kernel estimate of the regression function. In both plots, little “

To estimate

Qiu [

Qiu [

Since only part of observations is actually used in

The solid curve denotes

The estimate defined in (

In our procedure, the bandwidth parameter

where

In this section, some simulated examples are presented concerning the numerical performance of our proposed procedure. In all numerical examples presented in this paper, the Epanechnikov kernel function

The function

(a) Solid curve denotes the true regression function

For the proposed procedure, its Mean-Squared Error (MSE) values with several different

MSE values of the estimated regression function. (a)

Next, we compare the proposed procedure (denoted as NEW) with the conventional local linear kernel (LLK) smoothing procedure and the jump-preserving curve estimation (JPCE) procedure by Qiu [

Solid curves denote the estimates by the proposed procedure NEW, dotted curves denote the estimates by LLK, and dashed curves denote the estimates by JPCE.

Tables

Comparison of the MSE values of the three methods in various cases when

Method | |||||||||

New | .0012 | .0051 | .0195 | .0008 | .0022 | .0093 | .0005 | .0010 | 0.0040 |

(.0001) | (.0003) | (.0008) | (.0001) | (.0001) | (.0004) | (.00004) | (.0007) | (.0002) | |

CLLK | .0066 | .0118 | .0275 | .0042 | .0079 | .0203 | .0029 | .0051 | .01218 |

(.0001) | (.0002) | (.0006) | (.0001) | (.0001) | (.0004) | (.00002) | (.0004) | (.0002) | |

JPCF | .0052 | .0106 | .0336 | .0035 | .0056 | .0184 | .0013 | .0024 | .0087 |

(.0003) | (.0004) | (.0009) | (.0002) | (.0002) | (.0005) | (.00007) | (.0007) | (.0002) |

Comparison of the MSE values of the three methods in various cases when

Method | |||||||||

New | .0012 | .0051 | .0195 | .0006 | .0018 | .0088 | .0002 | .0012 | .0038 |

(.0001) | (.0003) | (.0008) | (.00004) | (.0001) | (.0004) | (.00003) | (.00005) | (.0002) | |

CLLK | .0066 | .0118 | .0275 | .00344 | .0066 | .0171 | .0021 | .0041 | .0109 |

(.0001) | (.0002) | (.0006) | (.00003) | (.0001) | (.0004) | (.00002) | (.00004) | (.0002) | |

JPCF | .0052 | .0106 | .0336 | .00222 | .0043 | .0151 | .0017 | .0020 | .0086 |

(.0003) | (.0004) | (.0009) | (.0001) | (.0001) | (.0004) | (.00003) | (.00005) | (.0002) |

In this section, we apply our proposed method to a groundwater level data. Possible jumps in groundwater level arise from changes in subsurface fluid currents, which has become an important predictor of earthquakes. In Figure

Little pluses denote the groundwater level observed by the Seismograph Network Stations of China Earthquake Center during January and May 2007, and solid curve denotes the estimated regression curve by our proposed procedure.

We have presented a blind deconvolution procedure for jump-preserving curve estimation when both noise and blur are present in the observed data. Numerical results show that it performs well in various cases. However, theoretical properties of the proposed method are not available yet, which requires much future research. We believe that the proposed method can be generalized to two-dimensional cases to solve problems such as image deblurring and restoration.

This research was supported in part by an NSF grant. The authors thank the guest editor of this special issue, Professor Ming Li, for help during the paper submission process. They also thank the three anonymous referees for their careful reading of the paper.