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Article

Power-Type Functions of Prediction Error of Sea Level Time Series

1
Ocean College, Zhejiang University, Hangzhou 310058, China
2
School of Information Science and Technology, East China Normal University, Shanghai 200241, China
*
Author to whom correspondence should be addressed.
Entropy 2015, 17(7), 4809-4837; https://doi.org/10.3390/e17074809
Submission received: 27 April 2015 / Revised: 21 June 2015 / Accepted: 3 July 2015 / Published: 9 July 2015
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory I)

Abstract

:
This paper gives the quantitative relationship between prediction error and given past sample size in our research of sea level time series. The present result exhibits that the prediction error of sea level time series in terms of given past sample size follows decayed power functions, providing a quantitative guideline for the quality control of sea level prediction.

1. Introduction

Time series prediction gains interests of people in various fields of sciences and engineering, see for example [120], simply mentioning a few. Prediction of sea level time series (sea level for short) is desired in geosciences and ocean engineering [2123], for many purposes such as understanding fluctuations of sea level rise, its dynamics and so forth [2431]. Various methods and technologies have been reported for time series prediction, such as linear predictors [3241], nonlinear predictions [4246], and those based on artificial intelligence, including fuzzy systems [4751] and artificial neural networks (ANN) [5270].
This paper presents our study in the aspect of sea level time series prediction using the real sea-level data recorded by the National Data Buoy Center (NDBC) at the time scale of hour. The data are publicly accessible via http://www.ndbc.noaa.gov/historical_data.shtml. The predictor we utilized in this research is an ANN predictor of the back propagation (BP) type, which is available in the toolbox of the software Matlab. Two highlights of this paper are as follows.
  • We established the quantitative relationship between the past sample size, used for the predicting 40 points ahead of sea level time series as a case, and the prediction error characterized in the form of mean square at measurement stations of sea level time series we investigated.
  • We obtained the analytic expression of that relationship in the form of power function, providing a guideline for the quality control in sea level prediction.
The rest of the paper is organized as follows. The research background, including the problem statement, is explained in Section 2. The research method is discussed in Section 3. Results are given in Section 4, which is followed by conclusions.

2. Research Background

2.1. Data

NDBC is an abbreviation, standing for a part of the US National Weather Service (NWS). It provides data for academic research, including sea level (see http://www.ndbc.noaa.gov/historical_data.shtml). We use the data recorded at the stations named LKWF1, SMKF1, LONF1, SAUF1, SPGF1, and VENF1, respectively. In terms of the names of the measurement stations, LKWF1 implies the station at Lake Worth, FL, USA ( http://www.ndbc.noaa.gov/station_page.php?station=LKWF1); SMKF1 is the station at Sombrero Key, FL, USA ( http://www.ndbc.noaa.gov/station_page.php?station=SMKF1); the station LONF1 is the one at Long Key, FL, USA ( http://www.ndbc.noaa.gov/station_page.php?station=LONF1); the station SAUF1 is at St. Augustine, FL, USA ( http://www.ndbc.noaa.gov/station_page.php?station=SAUF1); SPGF1 is at Settlement Point, GBI, Bahamas ( http://www.ndbc.noaa.gov/station_page.php?station=SPGF1); and VENF1 is at Venice, FL, USA ( http://www.ndbc.noaa.gov/station_page.php?station=VENF1). They are located in Florida, on the east coast of the Gulf of Mexico. The data are under the directory of Water Level, which is publicly accessible ( http://www.ndbc.noaa.gov/historical_data.shtml#wlevel), referring Gilhousen [71] as an instance of research using the data by NDBC.
Six real traces of sea level time series described in Table 1 are selected in our research, where x_lkwf1_1999(t) denoted the sea level time series in 1999 at the station LKWF1, x_ven1_2003(t) the sea level time series in 2003 at the station VENF1, and so forth. Our selection of the above data is quite arbitrary in this research, simply for the purpose of study of prediction error of sea level without any conflict of interests regarding other manuscripts.

2.2. Problem Statement

Let n > 0 be the past sample size. Denote by m ≥ 0 the step number of prediction. Then, random variables x(i)s for i < n are the past records. Rewrite the n past records by x(i − 1), x(i − 2), …, x(in). Following Kolmogorov [72], with the selection of proper real coefficient as, one may obtain the linear combination of the n past records expressed by
L = s = 1 n a s x ( i s ) .
The above L may be used to predict a future value expressed by x(i + m). Denote by e2 the mean square error (MSE) given by
e 2 e 2 ( n , m ) = E [ x ( i + m ) L ) ] 2 ,
where E is the mean operator. Kolmogorov stated that the e2(n, m) does not increase as n increase (Kolmogorov [72]), which yields a consequence we express as Corollary 1.
Corollary 1. For a given m, the following holds
e 2 ( n 2 , m ) e 2 ( n 1 , m ) if n 2 > n 1 .
Note that the above is a general expression in quality with respect to e2(n, m). For a given time series x(i), however, it does not tell us, quantitatively, how large e2(n, m) is with the past sample size n and the x(m) to be predicted. For that reason, we consider the solution to the problem described below, aiming at establishing a quantitative relationship between the past sample size n and the given error provided that the step number of prediction m is predetermined.
Problem 1. Let x(i) be a sea level time series. For a fixed m and given error e1, what is the past sample size, denoted by N1, required such that
e 2 ( n , m ) | n = N 1 = e 2 ( N 1 , m ) = e 1 2 ?
Note 1. The solution to the above problem is to find the function denoted by e1(N1, m). An alternative solution is N1(e1, m), which is in fact the inverse of e1(N1, m) with respect to N1. □

2.3. Research Thoughts

We divide the task mentioned in Note 1 into two subtasks as follows.
  • First, we find the e1(n, m) expressed only by an empirical curve based on processing real data of sea level.
  • Then, by fitting the data, we may find the analytic expression of e1(n, m) as well as e1(N1, m).

2.4. BP-ANN Predictor

Artificial neural networks (ANNs) are computational non-linear models that mimic the structures and functions of animals’ nervous systems. ANNs, composed of interconnected neurons, are used to “learn” the complex relationship between input data and output data. Back propagation (BP) neural networks, one of the most important models in ANNs [73], are widely used in the fields of economy, ecology, meteorology, hydrology, medicine, etc., see, for example, [7477].
A BP neural network (BPNN), composed of one input layer, one or more middle layers called hidden layers and one output layer, can be seen as a non-linear input-output map, in which any two of these adjacent layers are interconnected directly while neutrons in each layer are independent.
As previously mentioned, the current value of a time series is based on the previous observed values. In this paper, every 21st sample is obtained by its previously successive 20 samples. Thus, the three-layer BP neural network is comprised of 20 inputs and 1 output (see Section 4.5). The n samples are divided into two sets to train and test the neural network predictor. Specifically, the first (n = 20) groups of samples are training data and the last 20 data are used to test the predictor. Every time the sample size changes, the number of samples for training is changed, while the numbers of neurons in input and output layers are constant. For example, if the sample size is 200, the inputs of the BPNN predictor are 20-dimensional vectors, containing the samples from the 1st to the 20th, from the 2nd to the 21st, and so forth, up to the last vector which contains the testing samples from the 180th to the 199th to train the network. Output data are the 21st to the 200th of the one-dimensional original data, corresponding to the 180 vectors in the input layer. For the testing sample, the 201st value can be obtained from its former data, referring to the samples from the 181st to the 200th. The 202nd value is the output of the predictor, whose inputs are the data from the 182nd to the 200th and the 201st value, which is a predicted value. The rest predicted values can be generated by this means. In this paper, annual data from the 5001st to the 5040th recorded by each station are predicted. The predicting process is shown in Figure 1.
Once the sizes of input and output layer are fixed, the optimal number of hidden layer and the number of nodes in hidden layer becomes a major challenge, see, for example, [78,79]. Any continuous and bounded real-valued function, based on Kolmogorov theorem [72,80], can be expressed as a three-layer back propagation neural network, in which there are (2k + 1) nodes for the middle layer, where k is the number of the input nodes. Thus, in this paper, one single hidden layer with 41 nodes is used for the BPNN model.
The transfer function, called activation function, represents the relationship between input and output of each node. Activation functions are required to be monotonic increasing and differentiable [81]. Sigmoid function, as a continuous and bounded function, is commonly used in input and hidden layers [82,83], which is expressed by
f ( u ) = 1 1 + e u ,
where u stands for the input of a single neuron. For the input layer, u is the input vector, which is the normalized sea level time series x(t), while for the hidden layer, u is the weighted sum of outputs of the previous layer subtract the threshold value of the layer. Linear transfer function is used for the neurons in output layer, so that the outputs of the entire network can take any values.
BP neural networks are trained by adjusting the weights between two layers and threshold values based on the gradient steepest descent method, that is, if the errors between actual outputs and expected ones are not satisfied, the errors will propagated backward to adjust weights and threshold values repeatedly until the performance of the system such as mean square error (MSE) is minimized.
The available data are always divided into two sets, namely training and testing datasets. In order to obtain precise results and to reduce running time, it is necessary to normalize the available datasets to a range between zero and one [84], before training process with the formula below:
x ¯ = x x min x max x min ,
where xmax and xmin are, respectively, the maximum and minimum values of the sample data x(t). The final forecasting results are obtained by the following inverse normalized formula:
x = x ¯ ( x max x min ) + x min .

3. Experimental Methods

In this research, Neural Network (NN) Toolbox of the software MATLAB R2012b is used to realize the neural network predicting model. MATLAB is a multi-paradigm environment for numerical computing developed by MathWorks, Natick, USA. NN Toolbox, based on neural network theory, provides functions to design, train, visualize and simulate all kinds of networks such as linear network, BPNN, radial basis function NN, regression NN and so on [85,86]. Three functions: newff, train and sim are used in the learning procedure in the MATLAB commands [87].
As previously mentioned, the original dataset is divided into training and testing sets. In order to reduce prediction error, the data values are normalized between 0 and 1 via the function mapminmax. Function newff, which is used to generate and initialize a BP neural network, defines sizes and activation function of every layer and type of training algorithm. In this research, we use the function logsig between the input and hidden layers. The outputs of logsig are limited over the interval from 0 to 1 while its inputs can be taken with any value from negative infinity to positive infinity [88,89]. The transfer function purelin is used between the hidden and output layers so that the outputs of the network could be arbitrary values. After creating a BP neural network, functions train and sim are used to train and simulate the net.
Without loss of generality, data used as samples are recorded at the six aforementioned stations, a total of approximately 8700 values annually in every experiment. For each station, the first set of samples is sea level from the 4901st to the 5000th of the original data, 100 values in total. In this paper, the step number of prediction is 40, that is, values from the 5001st to 5040th are predicted. Compared with the actual data, we can get the mean squared prediction error from the Equation (2). The second set of sample data are from the 4801st to the 5000th, 200 data totally. Then original data from the 4701st to the 5000th are chosen as the third set and so on, until the 15th set of sea level data from the 3501th to the 5000th. Fiften mean square errors, which are 15 points generated by 15 sets of samples, constitute a curve envelope, which can be fitted by specific functions. In this way, we can get the quantitative relationship between sample size and prediction error.

4. Results

4.1. Prediction Results

We selected the data recorded at the station LKWF1 in 1999 for demonstration. The original sea level series contains 8641 points of data, some of these are shown in Figure 2. When we used the past sample records of 100 data points from the 4901st point to the 5000th to predict the next 40 data points, the prediction error, MSE, is 0.38664. The original data of 40 points and their predictions are indicated in Figure 3. Figure 4 shows the prediction results when the past sample size is increased to be 200, that is, the past sample records contain the data from the 4801st point to the 5000th, to predict the next 40 data points. By eye, one may see that the prediction error in Figure 4 is smaller than that in Figure 3. When increasing the past sample size to 300, which includes the data from the 4701st point to the 5000th, to predict the next 40 data points, we obtained the results shown in Figure 5, which clearly exhibits that the prediction error in this case is smaller than that described in Figure 4. In Figures 617, the demonstrated prediction results based on the past sample records are from 400 to 1500; that is, using the past sample records from the 4601st–3401st point up to the 5000th, to predict the next 40 data points. Those figures again exhibit that the prediction error is decreased as the past sample size increases, which is consistent with the description of Kolmogorov in [72].
The values of prediction errors are summarized in Table 2. The relationship between the past sample size and prediction error is demonstrated by Figure 18, of which the generation process is shown in Figure 19.
Note 2. FromFigure 18, as well as Figure 20, we see that MSE decreases rapidly when the past sample size increases from 100 to 300. Then, prediction errors appear to decay slowly when the past sample size increases. Table 3 summarizes the results.
Prediction results at the stations LONF1, SAUF1, SPGF1 and VENF1 are summarized (see Tables 47). The corresponding figures are also as follows (see Figures 2124). Ignoring decay rates of the six stations, all the figures indicate that the prediction errors taper off on the whole as sample sizes increase, and curves decay fast at first but slowly in the end. When the sample size rises to a certain extent, the prediction accuracy cannot be obviously improved. That is to say, if the range of prediction errors has been determined, there is no need to take very large sample size. In the next section, we will fit the curve to explore the precise relationship between prediction errors and sample sizes.

4.2. Prediction Error Curve Fitting

Data fitting is used to select the proper type of curve to fit the observed data, and analyze the relationship between these two variables based on the curve fitting equation. In this paper, power function satisfactorily fits the prediction error curve according to the shape characteristic of the curve. Figures 2530 show the prediction errors’ fitting results of the six aforementioned stations. The corresponding power functions are also listed in Table 8.
Both figures and table indicate that prediction errors at different stations have the same function form in power law. Power functions fit prediction error curves satisfactorily. The quantitative functions may help us determine the range of sample size in the case that prediction accuracy is ensured, which improves utilization of data.
The above experimental results with respect to prediction errors are expressed as
f ( n ) = A n a , A > 0 , a > 0.
According to the physical meaning of past sample size, we have n > 0. Thus, by denoting H(n) the Heaviside unit step function, we rewrite Equation (8) as
f ( n ) = A n a H ( n ) , A > 0 , a > 0.
The above expression may serve as a solution to the problem we raised in Section 2.
Note 2. The prediction error f(n) provides us with a guideline to control the prediction error in terms of a given value of past sample size.
Note 3. Our present result about f(n) may serve as a specific case for the quantitative description of prediction error for that f(n) → 0 for n → ∞, which was qualitatively described by Kolmogorov, see Corollary 1.

4.3. Discussions

Sea level time series has the property of long-range dependence (LRD), as can be seen from Ercan et al. [90], Barbosa et al. [91], Beretta et al. [92], and Li et al. [93]. Denote by y(t) a time series in general. Its autocorrelation function (ACF) is denote by ryy(τ) = E[y(t)y(t + τ)]. By LRD (Beran [94]), we mean
0 r y y ( τ ) d τ = .
A typical case of ryy(τ), which satisfies the above, is a decayed power function expressed as
r y y ( τ ) ~ c τ b ( 0 < b < 1 ) ,
where b = 2 − 2H, H ∈ (0.5, 1) is the Hurst parameter. Time series with LRD, in addition to sea level time series, have wide applications in various fields, ranging from physics to computer science, see e.g., [95124], just mentioning a few with the preference by Mandelbrot [125] and references therein, where H plays a key role. At the moment, however, we have not got enough knowledge to establish the relationship between H and the past sample size with respect to prediction errors of sea level time series. Our future work will work at it so as to understand deeply the issue of prediction errors of sea level time series. Another future work of ours is to investigate whether power-type functions of prediction error might appear as a law in the aspect of prediction of time series with LRD in general besides sea level time series.
As previously mentioned in the Introduction, there are two categories of predictors, namely, linear predictors and nonlinear ones [223,3270,7389], which may be considered for a specific type of time series, such as sea level. Since sea level is of LRD [9093], which is nonlinear, and since, ANN is a nonlinear predictor, it was consequently used in this research.
In the future, we shall use other types of predictors, for instance, autoregressive (AR) type, to study the relationship between prediction error and past sample size of sea level time series, and compare it with the one presented in this research.
Note that the present relationship between prediction error and past sample size may be a law specifically for sea level time series. In general, we never imply that it may fit with other types of time series, such as ocean surface waves or network traffic. Nevertheless, it might yet be a reference for them in this regard.
Finally, we note that this research is in the domain of statistics. Therefore, though the data used in this research (see Table 1) are from 1999 to 2003, we statistically infer that the present law is available for hourly recorded sea level in the past and future.

6. Conclusions

We have established the relationship between prediction error and past sample size for the prediction of sea level time series based on real data measured at six stations on the east coast of the Gulf of Mexico in Florida. The closed form of that relationship, in decayed power functions, has been obtained. It may yet be useful, on the one hand, for one to control prediction error according to a given past sample size or, on the other hand, for us to require the size of past sample according to a predetermined prediction error.
Sea water levels at six stations on the east coast of the Gulf of Mexico, Florida are predicted. A type of artificial neural network, back propagation (BP), is used with the Neural Network Toolbox in MATLAB. Prediction errors with different values are generated according to changes in sample sizes.

Acknowledgments

This paper was partly supported by the National Natural Science Foundation of China under the Project Grant Nos. 61272402 and 61070214. We appreciate the National Data Buoy Center that provides its measured data of sea level, which makes our research possible.

Conflicts of Interest

The authors declare that there is no conflict of interests regarding the publication of this paper.

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Figure 1. Process of predicting sea level time series from the 5001st value to the 5040th value.
Figure 1. Process of predicting sea level time series from the 5001st value to the 5040th value.
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Figure 2. Sea level series at the Station LKWF1 in 1999.
Figure 2. Sea level series at the Station LKWF1 in 1999.
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Figure 3. Prediction results with the sample size 100 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.
Figure 3. Prediction results with the sample size 100 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.
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Figure 4. Prediction results with the sample size 200 at the Station LKWF1in 1999. Solid line: predicted values, dashed line: original values.
Figure 4. Prediction results with the sample size 200 at the Station LKWF1in 1999. Solid line: predicted values, dashed line: original values.
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Figure 5. Prediction results with the sample size 300 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.
Figure 5. Prediction results with the sample size 300 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.
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Figure 6. Prediction results with the sample size 400 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.
Figure 6. Prediction results with the sample size 400 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.
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Figure 7. Prediction results with the sample size 500 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.
Figure 7. Prediction results with the sample size 500 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.
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Figure 8. Prediction results with the sample size 600 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.
Figure 8. Prediction results with the sample size 600 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.
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Figure 9. Prediction results with the sample size 700 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.
Figure 9. Prediction results with the sample size 700 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.
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Figure 10. Prediction results with the sample size 800 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.
Figure 10. Prediction results with the sample size 800 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.
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Figure 11. Prediction results with the sample size 900 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.
Figure 11. Prediction results with the sample size 900 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.
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Figure 12. Prediction results with the sample size 1000 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.
Figure 12. Prediction results with the sample size 1000 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.
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Figure 13. Prediction results with the sample size 1100 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.
Figure 13. Prediction results with the sample size 1100 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.
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Figure 14. Prediction results with the sample size 1200 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.
Figure 14. Prediction results with the sample size 1200 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.
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Figure 15. Prediction results with the sample size 1300 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.
Figure 15. Prediction results with the sample size 1300 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.
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Figure 16. Prediction results with the sample size 1400 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.
Figure 16. Prediction results with the sample size 1400 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.
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Figure 17. Prediction results with the sample size 1500 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.
Figure 17. Prediction results with the sample size 1500 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.
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Figure 18. Relationship between the prediction error and the past sample size at the Station LKWF1 in 1999.
Figure 18. Relationship between the prediction error and the past sample size at the Station LKWF1 in 1999.
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Figure 19. Generation process of relationship between prediction errors and past sample sizes.
Figure 19. Generation process of relationship between prediction errors and past sample sizes.
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Figure 20. Relationship between the prediction error and the past sample size at the Station SMKF1 in 2003.
Figure 20. Relationship between the prediction error and the past sample size at the Station SMKF1 in 2003.
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Figure 21. Relationship between the prediction error and the past sample size at the Station LONF1 in 2003.
Figure 21. Relationship between the prediction error and the past sample size at the Station LONF1 in 2003.
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Figure 22. Relationship between the prediction error and the past sample size at the Station SAUF1 in 2001.
Figure 22. Relationship between the prediction error and the past sample size at the Station SAUF1 in 2001.
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Figure 23. Relationship between the prediction error and the past sample size at the Station SPGF1 in 1996.
Figure 23. Relationship between the prediction error and the past sample size at the Station SPGF1 in 1996.
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Figure 24. Relationship between the prediction error and the past sample size at the Station VENF1 in 2003.
Figure 24. Relationship between the prediction error and the past sample size at the Station VENF1 in 2003.
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Figure 25. Curve fitting of the Station LKWF1 in 1999. Solid line: original curve, dashed line: fitting curve.
Figure 25. Curve fitting of the Station LKWF1 in 1999. Solid line: original curve, dashed line: fitting curve.
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Figure 26. Curve fitting of the Station SMKF1 in 2003. Solid line: original curve, dashed line: fitting curve.
Figure 26. Curve fitting of the Station SMKF1 in 2003. Solid line: original curve, dashed line: fitting curve.
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Figure 27. Curve fitting of the Station LONF1 in 2003. Solid line: original curve, dashed line: fitting curve.
Figure 27. Curve fitting of the Station LONF1 in 2003. Solid line: original curve, dashed line: fitting curve.
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Figure 28. Curve fitting of the Station SAUF1 in 2001. Solid line: original curve, dashed line: fitting curve.
Figure 28. Curve fitting of the Station SAUF1 in 2001. Solid line: original curve, dashed line: fitting curve.
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Figure 29. Curve fitting of the Station SPGF1 in 1996. Solid line: original curve, dashed line: fitting curve.
Figure 29. Curve fitting of the Station SPGF1 in 1996. Solid line: original curve, dashed line: fitting curve.
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Figure 30. Curve fitting of the Station VENF1 in 2003. Solid line: original curve, dashed line: fitting curve.
Figure 30. Curve fitting of the Station VENF1 in 2003. Solid line: original curve, dashed line: fitting curve.
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Table 1. Measured data of six traces of sea level time series.
Table 1. Measured data of six traces of sea level time series.
Series NameRecord Date and TimeData SizeMeasurement Station
x_lkwf1_1999(t)0:00, 1 Jan.–23:00, 31 Dec. 19998760LKWF1
x_smkf1_2003(t)0:00, 1 Jan.–23:00, 31 Dec. 20035851SMKF1
x_lonf1_2003(t)0:00, 1 Jan.–23:00, 31 Dec. 20038697LONF1
x_sauf1_2001(t)0:00, 1 Jan.–21:00, 31 Dec. 20018758SAUF1
x_spgf1_1996(t)0:00, 1 Jan.–23:00, 15 Dec. 19968616SPGF1
x_ven1_2003(t)0:00, 1 Jan.–23:00, 31 Dec. 20038760VENF1
Table 2. Prediction errors investigated using x_lkwf1_1999(t) in the Station LKWF1.
Table 2. Prediction errors investigated using x_lkwf1_1999(t) in the Station LKWF1.
Past sample size: nNumber of data to be predicted: mPast samples usedPredicted dataMSE: e2(n, m)
100404901st–5000th/4:00, 26 July–7:00, 30 July8:00, 30 Jul.–23:00, 31 Jul.0.38664
200404801st–5000th/23:00, 21 Jul.–7:00, 30 Jul.8:00, 30 Jul.–23:00, 31 Jul.0.17422
300404701st–5000th/19:00, 17 Jul.–7:00, 30 Jul.8:00, 30 Jul.–23:00, 31 Jul.0.09026
400404601st–5000th/15:00, 13 Jul.–7:00, 30 Jul.8:00, 30 Jul.–23:00, 31 Jul.0.09519
500404501st–5000th/10:00, 9 Jul.–7:00, 30 Jul.8:00, 30 Jul.–23:00, 31 Jul.0.06699
600404401st–5000th/5:00, 5 Jul.–7:00, 30 Jul.8:00, 30 Jul.–23:00, 31 Jul.0.04418
700404301st–5000th/1:00, 1 Jul.–7:00, 30 Jul.8:00, 30 Jul.–23:00, 31 Jul.0.04385
800404201st–5000th/21:00, 26 Jun.–7:00, 30 Jul.8:00, 30 Jul.–23:00, 31 Jul.0.03291
900404101st–5000th/17:00, 22 Jun.–7:00, 30 Jul.8:00, 30 Jul.–23:00, 31 Jul.0.02826
1000404001st–5000th/13:00, 18 Jun.–7:00, 30 Jul.8:00, 30 Jul.–23:00, 31 Jul.0.02601
1100403901st–5000th/9:00, 14 Jun.–7:00, 30 Jul.8:00, 30 Jul.–23:00, 31 Jul.0.02185
1200403801st–5000th/5:00, 10 Jun.–7:00, 30 Jul.8:00, 30 Jul.–23:00, 31 Jul.0.02012
1300403701st–5000th/20:00, 5 Jun.–7:00, 30 Jul.8:00, 30 Jul.–23:00, 31 Jul.0.02028
1400403601st–5000th/16:00, 1 Jun.–7:00, 30 Jul.8:00, 30 Jul.–23:00, 31 Jul.0.02313
1500403501st–5000th/12:00, 28 May–7:00, 30 Jul.8:00, 30 Jul.–23:00, 31 Jul.0.01762
Table 3. Prediction errors investigated using x_smkf1_2003(t) in the Station SMKF1.
Table 3. Prediction errors investigated using x_smkf1_2003(t) in the Station SMKF1.
Past sample size: nNumber of data to be predicted: mPast samples usedPredicted dataMSE: e2(n, m)
100404901st–5000th/13:00, 24 Jul.–16:00, 28 Jul.17:00, 28 Jul.–8:00, 30 Jul.0.37867
200404801st–5000th/9:00, 20 Jul.–16:00, 28 Jul.17:00, 28 Jul.–8:00, 30 Jul.0.16937
300404701st–5000th/5:00, 16 Jul.–16:00, 28 Jul.17:00, 28 Jul.–8:00, 30 Jul.0.11727
400404601st–5000th/0:00, 12 Jul.–16:00, 28 Jul.17:00, 28 Jul.–8:00, 30 Jul.0.05214
500404501st–5000th/20:00, 7 Jul.–16:00, 28 Jul.17:00, 28 Jul.–8:00, 30 Jul.0.02879
600404401st–5000th/16:00, 3 Jul.–7:00, 30 Jul.17:00, 28 Jul.–8:00, 30 Jul.0.03622
700404301st–5000th/12:00, 29 Jun.–16:00, 28 Jul.17:00, 28 Jul.–8:00, 30 Jul.0.02651
800404201st–5000th/8:00, 25 Jun.–16:00, 28 Jul.17:00, 28 Jul.–8:00, 30 Jul.0.02235
900404101st–5000th/4:00, 21 Jun.–16:00, 28 Jul.17:00, 28 Jul.–8:00, 30 Jul.0.02442
1000404001st–5000th/0:00, 17 Jun.–16:00, 28 Jul.17:00, 28 Jul.–8:00, 30 Jul.0.02077
1100403901st–5000th/20:00, 12 Jun.–16:00, 28 Jul.17:00, 28 Jul.–8:00, 30 Jul.0.02426
1200403801st– 5000th/16:00, 8 Jun.–16:00, 28 Jul.17:00, 28 Jul.–8:00, 30 Jul.0.02265
1300403701st–5000th/11:00, 4 Jun.–16:00, 28 Jul.17:00, 28 Jul.–8:00, 30 Jul.0.02037
1400403601st–5000th/7:00, 31 May–16:00, 28 Jul.17:00, 28 Jul.–8:00, 30 Jul.0.02085
1500403501st–5000th/3:00, 27 May–16:00, 28 Jul.17:00, 28 Jul.–8:00, 30 Jul.0.01981
Table 4. Prediction errors investigated using x_lonf1_2003(t) in the Station LONF1.
Table 4. Prediction errors investigated using x_lonf1_2003(t) in the Station LONF1.
Past sample size: nNumber of data to be predicted: mPast samples usedPredicted dataMSE: e2(n, m)
100404901st–5000th/18:00, 28 Jul.–21:00, 1 Aug.22:00, 1 Aug.–13:00, 3 Aug.0.15316
200404801st–5000th/14:00, 24 Jul.–21:00, 1 Aug.22:00, 1 Aug.–13:00, 3 Aug.0.13100
300404701st–5000th/10:00, 20 Jul.–21:00, 1 Aug.22:00, 1 Aug.–13:00, 3 Aug.0.12900
400404601st–5000th/5:00, 16 Jul.–21:00, 1 Aug.22:00, 1 Aug.–13:00, 3 Aug.0.11930
500404501st–5000th/1:00, 12 Jul.–21:00, 1 Aug.22:00, 1 Aug.–13:00, 3 Aug.0.10686
600404401st–5000th/21:00, 7 Jul.–21:00, 1 Aug.22:00, 1 Aug.–13:00, 3 Aug.0.10962
700404301st–5000th/17:00, 3 Jul.–21:00, 1 Aug.22:00, 1 Aug.–13:00, 3 Aug.0.10348
800404201st–5000th/13:00, 29 Jun.–21:00, 1 Aug.22:00, 1 Aug.–13:00, 3 Aug.0.10434
900404101st–5000th/4:00, 21 Jun.–21:00, 1 Aug.22:00, 1 Aug.–13:00, 3 Aug.0.10594
1000404001st–5000th/9:00, 25 Jun.–21:00, 1 Aug.22:00, 1 Aug.–13:00, 3 Aug.0.10692
1100403901st–5000th/23:00, 16 Jun.–21:00, 1 Aug.22:00, 1 Aug.–13:00, 3 Aug.0.10334
1200403801st–5000th/19:00, 12 Jun.–21:00, 1 Aug.22:00, 1 Aug.–13:00, 3 Aug.0.09628
1300403701st–5000th/15:00, 8 Jun.–21:00, 1 Aug.22:00, 1 Aug.–13:00, 3 Aug.0.09980
1400403601st–5000th/11:00, 4 Jun.–21:00, 1 Aug.22:00, 1 Aug.–13:00, 3 Aug.0.10160
1500403501st–5000th/16:00, 28 May–21:00, 1 Aug.22:00, 1 Aug.–13:00, 3 Aug.0.09752
Table 5. Prediction errors investigated using x_sauf1_2001(t) in the Station SAUF1.
Table 5. Prediction errors investigated using x_sauf1_2001(t) in the Station SAUF1.
Past sample size: nNumber of data to be predicted: mPast samples usedPredicted dataMSE: e2(n, m)
100404901st–5000th/21:00, 28 Jul.–8:00, 2 Aug.9:00, 2 Aug.–0:00, 4 Aug.3.80104
200404801st–5000th/16:00, 24 Jul.–8:00, 2 Aug.9:00, 2 Aug.–0:00, 4 Aug.2.56308
300404701st–5000th/12:00, 20 Jul.–8:00, 2 Aug.9:00, 2 Aug.–0:00, 4 Aug.1.45882
400404601st–5000th/3:00, 16 Jul.–8:00, 2 Aug.9:00, 2 Aug.–0:00, 4 Aug.1.28306
500404501st–5000th/21:00, 11 Jul.–8:00, 2 Aug.9:00, 2 Aug.–0:00, 4 Aug.0.96792
600404401st–5000th/17:00, 7 Jul.–8:00, 2 Aug.9:00, 2 Aug.–0:00, 4 Aug.0.95034
700404301st–5000th/11:00, 3 Jul.–8:00, 2 Aug.9:00, 2 Aug.–0:00, 4 Aug.0.89086
800404201st–5000th/7:00, 29 Jun.–8:00, 2 Aug.9:00, 2 Aug.–0:00, 4 Aug.0.63288
900404101st–5000th/3:00, 25 Jun.–8:00, 2 Aug.9:00, 2 Aug.–0:00, 4 Aug.0.61248
1000404001st–5000th/21:00, 20 Jun.–8:00, 2 Aug.9:00, 2 Aug.–0:00, 4 Aug.0.48420
1100403901st–5000th/16:00, 16 Jun.–8:00, 2 Aug.9:00, 2 Aug.–0:00, 4 Aug.0.41888
1200403801st–5000th/12:00, 12 Jun.–8:00, 2 Aug.9:00, 2 Aug.–0:00, 4 Aug.0.33480
1300403701st–5000th/7:00, 8 Jun.–8:00, 2 Aug.9:00, 2 Aug.–0:00, 4 Aug.0.33480
1400403601st–5000th/3:00, 4 Jun.–8:00, 2 Aug.9:00, 2 Aug.–0:00, 4 Aug.0.33488
1500403501st – 5000th/23:00, 30 May – 8:00, 2 Aug.9:00, 2 Aug.–0:00, 4 Aug.0.32614
Table 6. Prediction errors investigated using x_spgf1_1996(t) in the Station SPGF1.
Table 6. Prediction errors investigated using x_spgf1_1996(t) in the Station SPGF1.
Past sample size: nNumber of data to be predicted: mPast samples usedPredicted dataMSE: e2(n, m)
100404901st–5000th/3:00, 21 Jul.–7:00, 25 Jul.8:00, 25 Jul.–23:00, 26 Jul.0.38766
200404801st–5000th/23:00, 16 Jul.–7:00, 25 Jul.8:00, 25 Jul.–23:00, 26 Jul.0.18390
300404701st–5000th/14:00, 12 Jul.–7:00, 25 Jul.8:00, 25 Jul.–23:00, 26 Jul.0.15660
400404601st–5000th/8:00, 8 Jul.–7:00, 25 Jul.8:00, 25 Jul.–23:00, 26 Jul.0.14834
500404501st–5000th/2:00, 4 Jul.–7:00, 25 Jul.8:00, 25 Jul.–23:00, 26 Jul.0.13988
600404401st–5000th/21:00, 4 Jul.–7:00, 25 Jul.8:00, 25 Jul.–23:00, 26 Jul.0.11374
700404301st–5000th/14:00, 29 Jun.–7:00, 25 Jul.8:00, 25 Jul.–23:00, 26 Jul.0.10464
800404201st–5000th/10:00, 25 Jun.–7:00, 25 Jul.8:00, 25 Jul.–23:00, 26 Jul.0.08100
900404101st–5000th/6:00, 21 Jun.–7:00, 25 Jul.8:00, 25 Jul.–23:00, 26 Jul.0.07846
1000404001st–5000th/1:00, 17 Jun.–7:00, 25 Jul.8:00, 25 Jul.–23:00, 26 Jul.0.07160
1100403901st–5000th/20:00, 12 Jun.–7:00, 25 Jul.8:00, 25 Jul.–23:00, 26 Jul.0.06674
1200403801st–5000th/16:00, 8 Jun.–7:00, 25 Jul.8:00, 25 Jul.–23:00, 26 Jul.0.06592
1300403701st–5000th/11:00, 4 Jun.–7:00, 25 Jul.8:00, 25 Jul.–23:00, 26 Jul.0.06926
1400403601st–5000th/7:00, 31 May–7:00, 25 Jul.8:00, 25 Jul.–23:00, 26 Jul.0.06532
1500403501st–5000th/2:00, 27 May–7:00, 25 Jul.8:00, 25 Jul.–23:00, 26 Jul.0.06512
Table 7. Prediction errors investigated using x_venf1_2003(t) in the Station VENF1.
Table 7. Prediction errors investigated using x_venf1_2003(t) in the Station VENF1.
Past sample size: nNumber of data to be predicted: mPast samples usedPredicted dataMSE: e2(n, m)
100404901st–5000th/20:00, 24 Jul.–23:00 28 Jul.0:00, 29 Jul.–15:00, 30 Jul.0.44356
200404801st–5000th/16:00, 20 Jul.–23:00, 28 Jul.0:00, 29 Jul.–15:00, 30 Jul.0.12978
300404701st–5000th/12:00, 16 Jul.–23:00, 28 Jul.0:00, 29 Jul.–15:00, 30 Jul.0.08738
400404601st–5000th/8:00, 12 Jul.–23:00, 28 Jul.0:00, 29 Jul.–15:00, 30 Jul.0.04698
500404501st–5000th/23:00, 7 Jul.–23:00, 28 Jul.0:00, 29 Jul.–15:00, 30 Jul.0.03906
600404401st–5000th/19:00, 3 Jul.–23:00, 28 Jul.0:00, 29 Jul.–15:00, 30 Jul.0.03502
700404301st–5000th/15:00, 29 Jun.–23:00, 28 Jul.0:00, 29 Jul.–15:00, 30 Jul.0.02538
800404201st–5000th/10:00, 25 Jun.–23:00, 28 Jul.0:00, 29 Jul.–15:00, 30 Jul.0.02060
900404101st–5000th/6:00, 21 Jun.–23:00, 28 Jul.0:00, 29 Jul.–15:00, 30 Jul.0.01870
1000404001st–5000th/2:00, 17 Jun.–23:00, 28 Jul.0:00, 29 Jul.–15:00, 30 Jul.0.01878
1100403901st–5000th/22:00, 12 Jun.–23:00, 28 Jul.0:00, 29 Jul.–15:00, 30 Jul.0.01894
1200403801st–5000th/18:00, 8 Jun.–23:00, 28 Jul.0:00, 29 Jul.–15:00, 30 Jul.0.01672
1300403701st–5000th/14:00, 4 Jun.–23:00, 28 Jul.0:00, 29 Jul.–15:00, 30 Jul.0.01594
1400403601st–5000th/10:00, 31 May–23:00, 28 Jul.0:00, 29 Jul.–15:00, 30 Jul.0.01716
1500403501st–5000th/5:00, 27 May–23:00, 28 Jul.0:00, 29 Jul.–15:00, 30 Jul.0.01888
Table 8. Curve fitting results of the six stations. f(n) = e2(n, m)|m = 40, n is the past sample size.
Table 8. Curve fitting results of the six stations. f(n) = e2(n, m)|m = 40, n is the past sample size.
Station namePower functions
LKWF1 f ( n ) = 78.680 n 1.1550
SMKF1 f ( n ) = 121.600 n 1.252
LONF1 f ( n ) = 0.3194 n 0.1645
SAUF1 f ( n ) = 195.50 n 0.8476
SPGF1 f ( n ) = 9.0370 n 0.6948
VENF1 f ( n ) = 562.50 n 1.5400

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Li, M.; Li, Y.; Leng, J. Power-Type Functions of Prediction Error of Sea Level Time Series. Entropy 2015, 17, 4809-4837. https://doi.org/10.3390/e17074809

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Li M, Li Y, Leng J. Power-Type Functions of Prediction Error of Sea Level Time Series. Entropy. 2015; 17(7):4809-4837. https://doi.org/10.3390/e17074809

Chicago/Turabian Style

Li, Ming, Yuanchun Li, and Jianxing Leng. 2015. "Power-Type Functions of Prediction Error of Sea Level Time Series" Entropy 17, no. 7: 4809-4837. https://doi.org/10.3390/e17074809

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