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Censored Nonparametric Time-Series Analysis with Autoregressive Error Models

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Abstract

This paper focuses on nonparametric regression modeling of time-series observations with data irregularities, such as censoring due to a cutoff value. In general, researchers do not prefer to put up with censored cases in time-series analyses because their results are generally biased. In this paper, we present an imputation algorithm for handling auto-correlated censored data based on a class of autoregressive nonparametric time-series model. The algorithm provides an estimation of the parameters by imputing the censored values with the values from a truncated normal distribution, and it enables unobservable values of the response variable. In this sense, the censored time-series observations are analyzed by nonparametric smoothing techniques instead of the usual parametric methods to reduce modelling bias. Typically, the smoothing methods are updated for estimating the censored time-series observations. We use Monte Carlo simulations based on right-censored data to compare the performances and accuracy of the estimates from the smoothing methods. Finally, the smoothing methods are illustrated using a meteorological time- series and unemployment datasets, where the observations are subject to the detection limit of the recording tool.

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Acknowledgements

We would like to thank the editor, the associate editor, and the anonymous referees for beneficial comments and suggestions.

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  1. Department of Statistics, Faculty of Science, Mugla Sitki Kocman University, 48000, Mugla, Turkey

    Dursun Aydin & Ersin Yilmaz

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  1. Dursun Aydin
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Appendices: Supplemental Technical Materials

Appendices: Supplemental Technical Materials

1.1 Appendix 1: Derivation of Equation (15)

Let \({\varvec{{Z}}}^{(k)}_t\) be the vector \(\varvec{(}Z^{\left( k\right) }_1,\ldots ,Z^{\left( k\right) }_n)^{\top }\). It is easily seen that the residual sum of squares about f can be rewritten as

$$\begin{aligned} \sum ^n_{t=1}{{\left\{ Z^{\left( k\right) }_t-f\left( X_t\right) \right\} }^2{=\left\| {\varvec{\mathrm {Z}}}^{\left( k\right) }-\varvec{\mathrm {f}}\right\| }^2_2}=\left( {\varvec{\mathrm {Z}}}^{\left( k\right) }-\varvec{\mathrm {f}}\right) ^{\top }\left( {\varvec{\mathrm {Z}}}^{\left( k\right) }-\varvec{\mathrm {f}}\right) \end{aligned}$$
(41)

since the vector \(\varvec{\mathrm {f}}\) is definitely the vector of the values \(f\left( X_t\right)\). Determine the penalty term \(\int ^b_a{{\left\{ f^{''}\left( x\right) \right\} }^2dx}\) in (11) as \(\varvec{\mathrm {f}}^{\varvec{{\top }}}\varvec{\mathrm {Kf}}\) from (13) to obtain the penalized residuals sum of squares criterion (14), given by

$$\begin{aligned} PRSS\left( \varvec{\mathrm {f}},\lambda \right) =\left( \varvec{\mathrm {Z}}^{\left( k\right) }-\varvec{\mathrm {f}}\right) ^{\top }\left( \varvec{\mathrm {Z}}^{\left( k\right) }-\varvec{\mathrm {f}}\right) +\lambda {\varvec{\mathrm {f}}}^{\top }\varvec{\mathrm {Kf}} =\varvec{\mathrm {Z}}^{\left( k\right) ^{\top }} \varvec{\mathrm {Z}}^{\left( k\right) }-2{\varvec{\mathrm {Z}}}^{\left( k\right) }\varvec{\mathrm {f}}\varvec{\mathrm {+}}{\varvec{\mathrm {f}}}^{\top }\left( \varvec{\mathrm {I}}+\lambda \varvec{\mathrm {K}}\right) \varvec{\mathrm {f}} \end{aligned}$$
(42)

Taking the derivative with respect to \(\varvec{\mathrm {f}}\) and setting it to zero:

$$\begin{aligned} \frac{\partial PRSS\left( \varvec{\mathrm {f}},\lambda \right) }{\partial \varvec{\mathrm {f}}}=-2{\varvec{\mathrm {Z}}}^{\left( k\right) }\varvec{\mathrm {+}}2\varvec{\mathrm {f}}\left( \varvec{\mathrm {I}}+\lambda \varvec{\mathrm {K}}\right) =0 \end{aligned}$$

Setting Eq. (42) equal to zero and replacing \(\varvec{\mathrm {f}}\) by \({\widehat{\varvec{\mathrm {f}}}}^{\ SS}_{\lambda }\), we obtain

$$\begin{aligned} {\widehat{\varvec{\mathrm {f}}}}^{\ SS}_{\lambda }\left( \varvec{\mathrm {I}}+\lambda \varvec{\mathrm {K}}\right) ={\varvec{\mathrm {Z}}}^{\left( k\right) } \end{aligned}$$
(43)

Hence, by multiplying the term \({\left( \varvec{\mathrm {I}}+\lambda \varvec{\mathrm {K}}\right) }^{-1}\)on the both sides of the equation above, we have the following solution based on smoothing spline for the vector

$$\begin{aligned} {\widehat{\varvec{\mathrm {f}}}}^{\ SS}_{\lambda }={\left( \varvec{\mathrm {I}}+\lambda \varvec{\mathrm {K}}\right) }^{-1}{\varvec{\mathrm {Z}}}^{\left( k\right) } \end{aligned}$$

as expressed in the Eq. (15).

1.2 Appendix 2: Derivation of Equation (25)

Consider the model \({\varvec{\mathrm {Z}}}^{(k)}=\varvec{\mathrm {Xb}}+{\varvec{e}}\) , where \(\varvec{\mathrm {b}}=\left( b_0,b_1,\ldots ,b_q,b_{q+r},r=1,2,\ldots ,m\right)\), with \(b_{q+r}\)the coefficient of the r\(^{th}\) knot. The vector of ordinary least squares residuals can be written as \({\varvec{e}}={\varvec{\mathrm {Z}}}^{(k)}-\) \(\varvec{\mathrm {Xb}}\) and hence

$$\begin{aligned} {\varvec{e}}^{\varvec{\top }}{\varvec{e}}=\left( {\varvec{\mathrm {Z}}}^{\left( k\right) }-\mathrm {\ }\varvec{\mathrm {Xb}}\mathrm {\ }\right) ^{\varvec{\top }}\left( {\varvec{\mathrm {Z}}}^{(k)}-\mathrm {\ }\varvec{\mathrm {Xb}}\mathrm {\ }\right) \end{aligned}$$
(44)

The primary interest is to obtain the estimate \(\widehat{\varvec{\mathrm {b}}}\) that minimizes the least squares residuals defined above. It should be noted that such an unrestricted estimation of the \(b_{q+r}\) leads to a wiggly fit. To overcome this problem, we impose the constraint \(\sum ^m_{r=1}{b^2_{q+r}<C\ }\)(where \(>0\) ) on \(b_{q+r}\) the coefficients. Also, we assume that \({\varvec{\mathrm {S}}}^{PS}_{\lambda }\) is a positive definite and symmetric smoother matrix based on penalized spline, and D is a \(\left( m+2\right) \times \left( m+2\right)\) dimensional diagonal penalty matrix whose first \(\left( q+1\right)\) entries are 0, and the remaining entries are 1. Then, the Eq. (23) expressed in section (11) can be written as

$$\begin{aligned} minimum \ \ \left( {\varvec{\mathrm {Z}}}^{\left( k\right) }-\mathrm {\ }\varvec{\mathrm {Xb}}\mathrm {\ }\right) ^{\varvec{\top }}\left( {\varvec{\mathrm {Z}}}^{(k)}-\mathrm {\ }\varvec{\mathrm {Xb}}\mathrm {\ }\right) \ \ subject \ \ to \ \ \lambda \varvec{\mathrm {b}}^{\top }\varvec{\mathrm {Db}}<C. \end{aligned}$$

By the Lagrange multiplier method, minimizing this optimization problem subject to the constraint is equivalent to the following penalized residuals sum of squares

$$\begin{aligned} PRSS\left( \varvec{\mathrm {b}},\lambda \right) ={\left( {\varvec{\mathrm {Z}}}^{\left( k\right) }-\mathrm {\ }\varvec{\mathrm {Xb}}\mathrm {\ }\right) }^{\varvec{\top }}\left( {\varvec{\mathrm {Z}}}^{\left( k\right) }-\mathrm {\ }\varvec{\mathrm {Xb}}\mathrm {\ }\right) \varvec{+}\lambda {\varvec{\mathrm {b}}}^{\top }\varvec{\mathrm {Db}} ={\varvec{\mathrm {Z}}}^{\left( k\right) }\varvec{\top }{\varvec{\mathrm {Z}}}^{\left( k\right) }\varvec{-}2(\varvec{\mathrm {X}}\varvec{{\top }}{\varvec{\mathrm {Z}}}^{\left( k\right) })\varvec{\mathrm {b}}\varvec{\mathrm {+}}\varvec{\mathrm {b}}\varvec{\mathrm {'(}}\varvec{\mathrm {X}}\varvec{{\top }}\varvec{\mathrm {X}}\varvec{\mathrm {)}}\varvec{\mathrm {b}}\varvec{+}\lambda {\varvec{\mathrm {b}}}^{\top }\varvec{\mathrm {Db}} \end{aligned}$$
(45)

Similar to the ideas in (42), by taking the derivative with respect to \(\varvec{\mathrm {b}}\) in (45) and setting it to zero, we obtain

$$\begin{aligned} \frac{\partial PRSS\left( \varvec{\mathrm {b}},\lambda \right) }{\partial \varvec{\mathrm {b}}}=-2{{\varvec{\mathrm {X}}}^{\varvec{{\top }}}\varvec{\mathrm {Z}}}^{\left( k\right) }\varvec{\mathrm {+}}2\varvec{\mathrm {b}}\left( {\varvec{\mathrm {X}}}^{\varvec{{\top }}}\varvec{\mathrm {X}}\right) +\lambda 2\varvec{\mathrm {bD}}=0 \end{aligned}$$
(46)

Setting (46) equal to zero and replacing \(\varvec{\mathrm {b}}\) by \(\widehat{\varvec{\mathrm {b}}}\), it is easily seen that we obtain the penalized least squares normal equations

$$\begin{aligned} \left( {\varvec{\mathrm {X}}}^{\varvec{{\top }}}\varvec{\mathrm {X}}\varvec{\mathrm {+}} {\uplambda }\varvec{\mathrm {D}}\right) \widehat{\varvec{\mathrm {b}}}\varvec{=}{{\varvec{\mathrm {X}}}^{\varvec{{\top }}}\varvec{\mathrm {Z}}}^{\left( k\right) } \end{aligned}$$
(47)

From (47), the estimated regression coefficients are simply

$$\begin{aligned} \widehat{\varvec{\mathrm {b}}}={\left( {\varvec{\mathrm {X}}}^{\top }\varvec{\mathrm {X}}+\lambda \varvec{\mathrm {D}}\right) }^{-1}{\varvec{\mathrm {X}}}^{\top }{\varvec{\mathrm {Z}}}^{(k)} \end{aligned}$$
(48)

Hence, the fitted values \(\widehat{\varvec{\mathrm {f}}}\) based on RS can be given by

$$\begin{aligned} \widehat{\varvec{\mathrm {f}}}^{\ RS}_{\lambda }\varvec{=}\varvec{\mathrm {X}}\widehat{\varvec{\mathrm {b}}}=\varvec{\mathrm {X}}{\left( {\varvec{\mathrm {X}}}^{\top }\varvec{\mathrm {X}}+\lambda \varvec{\mathrm {D}}\right) }^{-1}{\varvec{\mathrm {X}}}^{\top }{\varvec{\mathrm {Z}}}^{(k)}={\varvec{\mathrm {S}}}^{RS}_{\lambda }{\varvec{\mathrm {Z}}}^{\varvec{(}k\varvec{)}} \end{aligned}$$
(49)

as claimed.

1.3 Appendix 3: Derivation of Equation (28)

Consider the \(RSS\left( \lambda \right)\) defined in Eq. (27). The \(RSS\left( \lambda \right)\) can be rewritten as

$$\begin{aligned} RSS\left( \lambda \right) =\left( {\widehat{\varvec{\mathrm {f}}}}_{\varvec{\lambda }}\varvec{-}{\varvec{\mathrm {Z}}}^{\varvec{(}k\varvec{)}}\right) \varvec{\top }\left( {\widehat{\varvec{\mathrm {f}}}}_{\varvec{\lambda }}\varvec{-}{\varvec{\mathrm {Z}}}^{\varvec{(}k\varvec{)}}\right) \varvec{=}{{\varvec{\mathrm {Z}}}^{\varvec{(}k\varvec{)}}\left( {\varvec{\mathrm {I}}-\varvec{\mathrm {S}}}_{\varvec{\lambda }}\right) }^2{\varvec{\mathrm {Z}}}^{\varvec{(}k\varvec{)}} \end{aligned}$$

It is easily seen that \(RSS\left( \lambda \right)\) can be expressed in a quadratic form. When we take the expected value of this form, we obtain

$$\begin{aligned} E\left[ RSS\left( \lambda \right) \right]= & {} E\left\| {{\varvec{\mathrm {Z}}}^{\left( k\right) }\left( {\varvec{\mathrm {I}}-\varvec{\mathrm {S}}}_{\varvec{\lambda }}\right) }^2{\varvec{\mathrm {Z}}}^{\left( k\right) }\right\| =MSE\left( \lambda \right) \\= & {} {\widehat{\varvec{\mathrm {f}}}}^{\ \varvec{\top }}_{\varvec{\lambda }}{\left( {\varvec{\mathrm {I}}-\varvec{\mathrm {S}}}_{\varvec{\lambda }}\right) }^{2} {\ \widehat{\varvec{\mathrm {f}}}}_{\varvec{\lambda }}\varvec{+}{\sigma }^2tr{\left( {\varvec{\mathrm {I}}-\varvec{\mathrm {S}}}_{\varvec{\lambda }}\right) }^2 \\= & {} {\widehat{\varvec{\mathrm {f}}}}^{\ \varvec{\top }}_{\varvec{\lambda }}{\left( {\varvec{\mathrm {I}}-\varvec{\mathrm {S}}}_{\varvec{\lambda }}\right) }^2{\ \widehat{\varvec{\mathrm {f}}}}_{\varvec{\lambda }}\varvec{+}n{\sigma }^2-2{\sigma }^2tr\left( {\varvec{\mathrm {S}}}_{\varvec{\lambda }}\right) +{\sigma }^2tr\left( {\varvec{\mathrm {S}}}_{\varvec{\lambda }}^{\varvec{\top }}\varvec{\mathrm {S}}_{\varvec{\lambda }}\right) \\= & {} {\widehat{\varvec{\mathrm {f}}}}^{\ \varvec{\top }}_{\varvec{\lambda }}{\left( {\varvec{\mathrm {I}}-\varvec{\mathrm {S}}}_{\varvec{\lambda }}\right) }^2{\ \widehat{\varvec{\mathrm {f}}}}_{\varvec{\lambda }}\varvec{+}{\sigma }^2\left\{ n-2tr\left( {\varvec{\mathrm {S}}}_{\varvec{\lambda }}\right) +tr\left( {\varvec{\mathrm {S}}}_{\varvec{\lambda }}^{\varvec{\top }}\varvec{\mathrm {S}}_{\varvec{\lambda }}\right) \right\} \end{aligned}$$

as defined in the Eq. (28).

1.4 Appendix 4: Proof of the Lemma 4.1

\(SMSE\left( {\widehat{\varvec{\mathrm {f}}}}_{\varvec{\lambda }}\mathrm {\ },\varvec{\mathrm {f}}\right) =E{\left\| {\widehat{\varvec{\mathrm {f}}}}_{\varvec{\lambda }}-\varvec{\mathrm {f}}\right\| }^2\), where as \({\widehat{\varvec{\mathrm {f}}}}_{\varvec{\lambda }}\varvec{=}{\varvec{\mathrm {S}}}_{\varvec{\lambda }}{\varvec{\mathrm {Z}}}^{\varvec{(}k\varvec{)}}\) Then the SMSE matrix can be written as the sum of variance and squared bias

$$\begin{aligned} SMSE\left( {\widehat{\varvec{\mathrm {f}}}}_{\varvec{\lambda }}\mathrm {\ },\varvec{\mathrm {f}}\right) =\sum ^n_{t=1}{{\left\{ E\left( {{\widehat{f}}}_{\lambda }\left( X_t\right) \right) -f\left( X_t\right) \right\} }^2+Var\left( {{\widehat{f}}}_{\lambda }\left( X_t\right) \right) }=\sum ^n_{t=1}{{bias}^2\left( {{\widehat{f}}}_{\lambda }\left( X_t\right) \right) }+\sum ^n_{t=1}{Var\left( {{\widehat{f}}}_{\lambda }\left( X_t\right) \right) } \end{aligned}$$
(50)

As in the usual linear smoother, the bias and Var (i.e, variance) terms in (50) can be written, respectively, as

$$\begin{aligned} \sum ^n_{t=1}{{bias}^2\left( {{\widehat{f}}}_{\lambda }\left( X_t\right) \right) }=\left( E\left( {\widehat{\varvec{\mathrm {f}}}}_{\varvec{\lambda }}\right) -\varvec{\mathrm {f}}\right) '\left( E\left( {\widehat{\varvec{\mathrm {f}}}}_{\varvec{\lambda }}\right) -\varvec{\mathrm {f}}\right) =\left( {E\ (\varvec{\mathrm {S}}}_{\varvec{\lambda }}{\varvec{\mathrm {Z}}}^{\left( k\right) })-\varvec{\mathrm {f}}\right) ^{\varvec{\top }}\left( {E\ (\varvec{\mathrm {S}}}_{\varvec{\lambda }}{\varvec{\mathrm {Z}}}^{\left( k\right) })-\varvec{\mathrm {f}}\right) \varvec{=}{\left\| \left( {\varvec{\mathrm {I}}-\varvec{\mathrm {S}}}_{\varvec{\lambda }}\right) \varvec{\mathrm {f}}\right\| }^2 \end{aligned}$$
(51)

and

$$\begin{aligned} \sum ^n_{t=1}{Var\left( {{\widehat{f}}}_{\lambda }\left( X_t\right) \right) }= & {} tr\left\{ Cov\left( {{\widehat{f}}}_{\lambda }\left( X_t\right) \right) \right\} =tr\left\{ Cov\left( {\widehat{\varvec{\mathrm {f}}}}_{\varvec{\lambda }}\varvec{=}{\varvec{\mathrm {S}}}_{\varvec{\lambda }}{\varvec{\mathrm {Z}}}^{\left( k\right) }\mathrm {\ }\right) \right\} \nonumber \\ {}= & {} tr\left\{ {\varvec{\mathrm {S}}}_{\varvec{\lambda }}\ Cov\left( {\varvec{\mathrm {Z}}}^{\varvec{(}k\varvec{)}}\right) {\varvec{\mathrm {S}}}_{\varvec{\lambda }}^{\varvec{\top }}\right\} \end{aligned}$$
(52)

Assume that \(Cov\left( {\varvec{\mathrm {Z}}}^{\varvec{(}k\varvec{)}}\right) ={\sigma }^2\varvec{\mathrm {I}}\) in (51) and hence, bringing (51) and (52) into (50) yields that

$$\begin{aligned} SMSE\left( {\widehat{\varvec{\mathrm {f}}}}_{\varvec{\lambda }}\mathrm {\ },\varvec{\mathrm {f}}\right) =E{\left\| {\widehat{\varvec{\mathrm {f}}}}_{\varvec{\lambda }}-\varvec{\mathrm {f}}\right\| }^2={\left\| \left( {\varvec{\mathrm {I}}\varvec{\mathrm {-}}\varvec{\mathrm {S}}}_{\lambda }\right) \varvec{\mathrm {f}}\right\| }^2\varvec{+}{{\sigma }^2tr\varvec{\mathrm {(\ }}\varvec{\mathrm {S}}}_{\varvec{\lambda }}{\varvec{\mathrm {S}}}_{\varvec{\lambda }}^{\varvec{\top }}) \end{aligned}$$

This completes the proof of Lemma 4.1.

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Aydin, D., Yilmaz, E. Censored Nonparametric Time-Series Analysis with Autoregressive Error Models. Comput Econ 58, 169–202 (2021). https://doi.org/10.1007/s10614-020-10010-8

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