CN109977354B - MASH hydrology time series trend analysis smoothing parameter selection method based on significant change rate - Google Patents

MASH hydrology time series trend analysis smoothing parameter selection method based on significant change rate Download PDF

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CN109977354B
CN109977354B CN201811504903.8A CN201811504903A CN109977354B CN 109977354 B CN109977354 B CN 109977354B CN 201811504903 A CN201811504903 A CN 201811504903A CN 109977354 B CN109977354 B CN 109977354B
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朱迪
梅亚东
贲月
吴贞晖
陈俊鸿
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Abstract

The invention provides a MASH smoothing parameter selection method based on a significant change rate, which is characterized by comprising the following steps: step 1, calculating a time series MASH matrix: performing moving average on time series data in two dimensions of the year and the year, and constructing a MASH matrix; step 2, prewhitening the MASH matrix, and removing autocorrelation of data; step 3, defining a statistic S, constructing a statistic Z related to S and the time sequence length n, carrying out M-K trend test on each row of data of each MASH matrix after the pre-whitening treatment, and recording the number of significant change days passing the significance test; step 4, calculating a significant rate of change, and drawing a significant rate of change distribution map; step 5, after calculating the change value of the significant change rate, counting the number of the minimum change values and the corresponding Y value and w value, and drawing the points in a significant change rate distribution graph; and 6, determining proper smoothing parameters Y and w.

Description

MASH hydrology time series trend analysis smoothing parameter selection method based on significant change rate
Technical Field
The invention belongs to the field of hydrologic time series trend analysis, and particularly relates to a MASH hydrologic time series trend analysis smoothing parameter selection method based on a significant change rate.
Technical Field
Time series trend analysis methods can be mainly divided into two categories, namely exploratory data analysis methods and mathematical statistics methods. The exploratory data analysis method is mainly used for processing the original data in the modes of drawing, tabulating, calculating characteristic quantity and the like, so that the characteristics and the rules of the original data are known. The mathematical statistical method utilizes statistical test to quantify the trend characteristics of research time series by constructing statistical quantity, and comprises the following steps: M-K test method, linear regression method, Spumann correlation coefficient method, etc. The two methods supplement each other and complement each other. Wherein, the former is biased to qualitative analysis and is more visual; the latter biased towards quantitative analysis.
In recent years, as people pay more attention to the influence of factors such as climate change and human activities on hydrological variables, a trend analysis method of time series is widely applied to the aspects of precipitation, air temperature, runoff and the like. As far as the present is concerned, most time series trend analysis methods can effectively analyze data annual trend change characteristics, but the analysis methods for time series annual and annual trend change conditions are not many. Among them, the moving average over shifting horizon (MASH) is an exploratory data analysis method that can simultaneously analyze the time series annual and annual trend changes. However, when the MASH method is applied, the smooth parameter selection is empirical and lacks certain basis. Therefore, the MASH smoothing parameter selection method is provided, has important significance for perfecting and developing the MASH method and enriching the time sequence trend method, and can provide technical support for hydrologic time sequence trend analysis.
Disclosure of Invention
The present invention has been made to solve the above-mentioned problems, and an object of the present invention is to provide a MASH hydrological time series trend analysis smoothing parameter selection method based on a significant change rate, which can determine MASH method smoothing parameters and use them for time series trend analysis. In order to achieve the purpose, the invention adopts the following scheme:
the invention provides a method for selecting smoothing parameters of MASH hydrological time series trend analysis based on a significant change rate, which is characterized by comprising the following steps of:
step 1, calculating a time sequence MASH matrix
The time series is a long-term data of years with days. Constructing a MASH matrix as shown by the following formula by performing a running average on the time series data in two dimensions, year and year:
Figure GDA0002564099480000021
in the formula, NhRepresenting the number of time levels of MASH moving averages, the moving average at day t of the h level in the MASH matrix is calculated as follows:
ut,h＝meany∈[h,h+Y-1][meand∈[t-w,t+w](xd,y)]，
in the formula, xd,yRepresenting the observed value of the day runoff on the day d of the y year of the original runoff time series; each list of MASH matricesShow a horizontal smoothing result (a column of 365 data, corresponding to 365 days a year); y and w are parameters of MASH moving average, representing the size of the sliding window; y represents the interplanetary moving average time interval; 2w +1 represents the annual moving average time interval; n is a radical ofh＝Ny-Y+1，NyThe length of the original time series is shown, which shows that the sequence length of the original time series is reduced after MASH smoothing.
Step 2, pre-whitening MASH matrix
Since the raw time series data is subjected to MASH smoothing, there is an autocorrelation. Therefore, the MASH matrix calculated in step 1 is processed by a pre-whitening method to remove the autocorrelation of the data, and the pre-whitening processing formula is as follows:
X′t＝Xt-r1Xt-1，
in the formula, r1Representing the first order autocorrelation coefficient, XtRepresents original sequence, X'tRepresenting the sequence after decorrelation.
Step 3, calculating the number of significant change days by using M-K trend test
Based on the pre-whitened sequence processed in the step 2, calculating the days exceeding the significance level in the sequence by using an M-K trend test method, and specifically comprising the following sub-steps of:
step 3-1. define a statistic S as shown in the following equation:
Figure GDA0002564099480000022
in the formula, sgn is a sign function and represents when (x)j-xk) When the value is less than, equal to or greater than zero, respectively taking-1, 0 or 1; x is the number ofjAnd xkRespectively representing the jth and kth values in the pre-whitened time series;
step 3-2. construct statistic Z as follows:
Figure GDA0002564099480000023
wherein n represents a time-series length; such asIf Z is greater than zero, it indicates that the sequence has an increasing trend, if Z is less than zero, it indicates that the sequence has a decreasing trend, and given a confidence level of α, the corresponding test value in the standard normal distribution table is U1-0.5αWhen the absolute value of Z is greater than or equal to U1-0.5αThe time is shown to pass the significance test with the confidence degree of 1- α, namely, the sequence has the trend of significant increase or decrease;
and 3, assuming that the MASH smoothing parameter Y is 1,2, 3. cndot. M and w is 1,2, 3. cndot. N, the MASH smoothing parameter has M × N MASH matrixes, and each MASH matrix has 365 × N MASH matrixeshA piece of data; performing M-K trend test on each row of data of each MASH matrix after pre-whitening treatment to obtain a statistic Zi,j,kThe value (i ═ 1,2, · · M, j ═ 1,2, · · N, k · 1,2, · · 365), Zi,j,kAnd (3) representing the trend test value of M-K on the K-th line when Y is equal to i and w is equal to j. Defining days of significant Change Di,j(i ═ 1,2,3 · M, j ═ 1,2,3 · N), the formula is shown below:
Figure GDA0002564099480000031
wherein i ═ 1,2, 3. cndot. M, j ═ 1,2, 3. cndot. N; i Zi,j,kI represents the trend check value Z of M-K on the K-th line when Y is equal to i and w is equal to ji,j,kAbsolute value of (d); sign is defined as follows:
Figure GDA0002564099480000032
step 4. calculating the significant rate of change
Defining the significant rate of change θ is shown as follows:
Figure GDA0002564099480000033
significant rate of change θi,jReflecting that when Y is i years and w is j days, the total number of N × M corresponding to the ratio of the total days of significant change in MASH matrix year to the total days of one year form a theta matrix with N rows and M columnsi,jPoints are plotted on the Y-axisIn the standard, a significant change rate distribution graph is drawn on a graph with w as a vertical coordinate;
step 5, calculating the change value of the significant change rate
Change value change (theta) of significant rate of changei,j) Calculated as follows:
Figure GDA0002564099480000034
wherein the content of the first and second substances,
Figure GDA0002564099480000035
and
Figure GDA0002564099480000036
respectively representing the difference values of two adjacent columns and two rows of the theta matrix;
after the change value of the significant change rate is calculated, counting the number of the minimum change values and the corresponding Y value and w value, and plotting the minimum change values in a significant change rate distribution graph;
step 6, determining proper smoothing parameters Y and w
And selecting the Y and w values corresponding to the minimum change value as MASH smoothing parameters based on the significant change rate distribution graph under different Y and w.
The MASH hydrological time series trend analysis smoothing parameter selection method based on the significant change rate provided by the invention can also have the following characteristics: in step 1, when the annual smoothing is performed, when the days in the beginning and the end of the year are less than 2w +1, the boundary smoothing phenomenon can be processed by adopting a symmetrical extension method. For example, when w is 2, the data 354d and 365d at the end of the year may be copied before the data 1d, the data 1d and 2d may be copied after the data 365d, and the original sequence with the length of 365d is extended to 369d, so that the sequence length after smoothing in the year is still 365 d.
The method for selecting smoothing parameters of MASH hydrological time series trend analysis based on the significant change rate can also be characterized in that in the step 3-2, the confidence level α is given to be 0.05, and the corresponding test value in the standard normal distribution table is U1-0.5α1.96, when the absolute value of Z is largeAt or equal to 1.96, this indicates a significance test with a 95% confidence pass.
The MASH hydrological time series trend analysis smoothing parameter selection method based on the significant change rate provided by the invention can also have the following characteristics: in step 6, when there are a plurality of Y and w values corresponding to the minimum variation value, the influence of Y and w on the smoothing effect should be considered; the larger or smaller the Y value is, the trend change characteristics among time series years are not easily reflected; the larger or smaller w is, the change of the time series in the year is not easy to reflect, so that a scheme with moderate values of Y and w is selected as a smooth parameter.
Action and Effect of the invention
The method for determining the MASH hydrological time series trend analysis smoothing parameters, provided by the invention, is based on the MASH matrix, calculates the obvious change rate by using the pre-whitening treatment and the M-K trend inspection method, and selects the MASH smoothing parameters through the change value of the obvious change rate, so that the implementation is convenient, and the result is visual. The method provides a concept and a calculation method of the obvious change rate for the first time, is an important innovation in the technical field, has an important significance for the development and the perfection of the MASH method, and has a very good popularization and use value.
Drawings
Fig. 1 is a significant change rate distribution graph obtained by plotting daily runoff data of a certain hydrological station according to an embodiment of the present invention;
FIG. 2 is a MASH smoothing graph based on different smoothing parameters plotted with daily runoff data of a certain hydrological station according to an embodiment of the present invention; fig. 2(a) is a MASH smoothing chart with smoothing parameters Y being 11 and w being 18; fig. 2(b) is a MASH smoothing map with smoothing parameters Y-14 and w-26; fig. 2(c) is a MASH smoothing map with smoothing parameters Y-18 and w-14; fig. 2(d) is a MASH smoothing map with smoothing parameters Y of 30 and w of 5.
Detailed Description
The selection method of the smoothing parameters of the MASH hydrological time series trend analysis based on the significant change rate is explained in detail in the following with the attached drawings.
< example >
In this embodiment, taking daily runoff data of a certain hydrological station from 1950 to 2016 as an example for description, the method for selecting smoothing parameters for MASH hydrological time series trend analysis based on significant change rate provided by this embodiment includes the following steps:
step 1, calculating a time sequence MASH matrix
Taking runoff data of a hydrological station from 1950 to 2016 as time series data, and performing sliding average on the time series data in two dimensions of the year and the year to construct a MASH matrix shown as the following formula:
Figure GDA0002564099480000051
in the formula, NhRepresenting the number of MASH running averages, the running average at day t of the h level in the MASH matrix is calculated as follows:
ut,h＝meany∈[h,h+Y-1][meand∈[t-w,t+w](xd,y)]，
in the formula, xd,yRepresenting the observed value of the day runoff on the day d of the y year of the original runoff time series; each column of the MASH matrix represents one level of smoothing results; y and w are parameters of MASH moving average, representing the size of the sliding window; y represents the interplanetary moving average time interval; 2w +1 represents the annual moving average time interval; n is a radical ofh＝Ny-Y+1，NyThe length of the original time series is shown, which shows that the sequence length of the original time series is reduced after MASH smoothing. During the annual smoothing, the days in the beginning of the year and the days in the end of the year are less than 2w +1, and the marginal smoothing phenomenon can be processed by adopting a symmetrical extension method. For example, when w is 2, the data 354d and 365d at the end of the year may be copied before the data 1d, the data 1d and 2d may be copied after the data 365d, and the original sequence with the length of 365d is extended to 369d, so that the sequence length after smoothing in the year is still 365 d.
Step 2, pre-whitening MASH matrix
Processing the MASH matrix obtained in the step 1 by adopting a pre-whitening method to remove autocorrelation of data, wherein the formula of the pre-whitening processing is as follows:
X′t＝Xt-r1Xt-1，
in the formula, r1Representing the first order autocorrelation coefficient, XtRepresents original sequence, X'tRepresenting the sequence after decorrelation.
Step 3, calculating the number of significant change days by using M-K trend test
The day beyond significance level in the sequence was calculated using M-K trend test. Specifically, first, a statistic S is defined as shown by:
Figure GDA0002564099480000061
in the formula, sgn is a sign function and represents when (x)j-xk) When the value is less than, equal to or greater than zero, respectively taking-1, 0 or 1; x is the number ofjAnd xkRespectively representing the j and k values in the pre-whitened time series.
Secondly, when the statistic S is greater than, equal to, or less than zero, the statistic Z is constructed as follows:
Figure GDA0002564099480000062
in the formula, n represents a time-series length. If Z is greater than zero, it indicates that the sequence has an increasing trend; if Z is less than zero, it indicates that the sequence has a decreasing trend. When the absolute value of Z is greater than or equal to 1.96, it indicates a pass confidence of 95% significance test, i.e.: there is a tendency for the sequence to increase or decrease significantly.
In the calculation of the embodiment, assuming that the MASH smoothing parameter Y is 1,2,3 · 30, and w is 1,2,3 · 30, there are 30 × 30 MASH matrices, each of which has 365 × NhAnd (4) data. And (3) performing pre-whitening treatment in the step (2) on each row of data of each MASH matrix, and performing M-K trend test to obtain a statistic Zi,j,kThe value (i ═ 1,2, · 30, j ═ 1,2, · 30, k · 1,2, · · 365), Zi,j,kAnd (3) representing the trend test value of M-K on the K-th line when Y is equal to i and w is equal to j. Defining days of significant Change Di,j(i ═ 1,2,3 · 30, j · 1,2,3 · 30), whose formula is shown below:
Figure GDA0002564099480000063
wherein, | Zi,j,kI represents Zi,j,kAbsolute value of (d); sign is defined as follows:
Figure GDA0002564099480000064
step 4, calculating the significant change rate
Defining the significant rate of change θ is shown as follows:
Figure GDA0002564099480000071
significant rate of change θi,jReflecting that when Y is i years and w is j days, the total number of the days in which the significant change occurs in the MASH matrix year is 30 × 30 corresponding to the ratio of the total days in one year, and theta matrix with 30 rows and 30 columns is formedi,jThe points are plotted on a graph with Y as the abscissa and w as the ordinate, and a significant rate of change distribution graph is plotted.
Step 5, calculating the change value of the significant change rate
Significant rate of change value change (θ)i,j) Calculated as follows:
Figure GDA0002564099480000072
wherein the content of the first and second substances,
Figure GDA0002564099480000073
and
Figure GDA0002564099480000074
respectively representing the difference values of two adjacent columns and two adjacent rows of the theta matrix. After calculating the change value of the significant change rate, counting the number of the minimum change values and the corresponding Y value and w value, and plotting the minimum change values in the significant change rate scoreIn the layout, see FIG. 1.
Step 6, determining proper smoothing parameters Y and w
According to the significant change rate distribution diagram, as shown in fig. 2, it can be seen that the rate of significant change of the flow rate ranges from 0.05 to 0.5, and the decrease of w shows an increasing trend with the increase of Y. The significant rate of change minimum exists at 4, shown as a black dot in fig. 2, indicating that there are 4 smoothing parameter options available for reference, Y on the abscissa and w on the ordinate, which are (11, 18), (14, 26), (18, 14) and (30, 5), respectively. Thus, alternative slip smoothing parameters are Y-11, 14, 18 and 30, and w-18, 26, 14 and 5, respectively. When determining MASH moving average parameters, a scheme with moderate values of Y and w should be selected from 4 groups of smoothing parameters. For convenience of explanation, the present embodiment draws a MASH smoothing map of daily runoff under 4 schemes, which is shown in fig. 2(a), (b), (c), and (d), and the smoothing parameter values of each map are illustrated in the accompanying drawings. As can be seen from fig. 2, in fig. 2(a), because the smoothing parameter Y is the smallest, the number of the smoothed curves is the largest, the distribution of the curves is disordered, and the change of the time series annual trend is not easily reflected; in fig. 2(d), because the smoothing parameter Y is the largest, the number of smoothed curves is small, the curves are distributed concentratedly, the annual difference is smaller, and the annual trend change of the time series is not easily reflected; in fig. 2(b), the smoothing parameter w is the maximum, and the smoothed curve has small annual variation difference, so that the annual variation of the time series is not easily reflected. The values of Y and w in fig. 2(c) are moderate in the 4 schemes, and the plotted MASH smoothing curve graph can better reflect the trend characteristics of time series within a year and between years, so that the embodiment selects Y-18 and w-14 as MASH smoothing parameters.
The invention is mainly used in the field of time series trend analysis, and is an improvement of an MASH smoothing method in a time series trend analysis method. The concept of the remarkable change rate theta provided by the invention for the first time has the following significance: the greater the significant change rate theta, the greater the number of days in the sequence that show a significant upward or downward trend, and the greater the variability; the sliding parameters Y and w take different values, similar or equal obvious change rate theta values can occur, and the fact that the curve difference is not large after MASH smoothing is shown in a certain value range of Y and w is shown, so that the values of Y and w are more suitable for MASH smoothing. In view of this, the invention calculates the change condition of the change value of the significant change rate, and takes the Y and w values corresponding to the minimum position of the change value of the significant change rate as MASH smoothing parameters, so that the parameter selection is more based, and the popularization and the use of the MASH method are facilitated.
The above embodiments are merely illustrative of the technical solutions of the present invention. The selection method of the smoothing parameters of MASH hydrological time series trend analysis based on the significant change rate is not limited to the contents described in the above embodiments, but is subject to the scope defined by the claims. Any modification or supplement or equivalent replacement made by a person skilled in the art on the basis of this embodiment is within the scope of the invention as claimed in the claims.

Claims (2)

1. The MASH hydrological time series trend analysis smoothing parameter selection method based on the significant change rate is characterized by comprising the following steps of:
step 1, calculating a time sequence MASH matrix
Performing a moving average on the time series data in two dimensions of the year and the year, and constructing a MASH matrix shown as the following formula:
Figure FDA0002587097390000011
in the formula, NhRepresenting the number of time levels of MASH moving averages, the moving average at day t of the h level in the MASH matrix is calculated as follows:
ut,h＝meany∈[h,h+Y-1][meand∈[t-w,t+w](xd,y)]，
in the formula, xd,yRepresenting the observed value of the day runoff on the day d of the y year of the original runoff time series; each column of the MASH matrix represents one level of smoothing results; y and w are parameters of MASH moving average, representing the size of the sliding window; y represents the interplanetary moving average time interval; 2w +1 represents the annual running averageTime interval; n is a radical ofh＝Ny-Y+1，NyRepresenting the original time series length;
step 2, pre-whitening MASH matrix
And (3) processing the MASH matrix obtained by the calculation in the step 1 by adopting a pre-whitening method, removing autocorrelation of data, wherein a pre-whitening processing formula is as follows:
X′t＝Xt-r1Xt-1，
in the formula, r1Representing the first order autocorrelation coefficient, XtRepresents original sequence, X'tRepresenting the sequence after the autocorrelation is removed;
step 3, calculating the number of significant change days by using M-K trend test
Calculating the days exceeding the significance level in the sequence by using an M-K trend test method based on the pre-whitened sequence processed in the step 2, and comprising the following sub-steps of:
step 3-1. define a statistic S as shown in the following equation:
Figure FDA0002587097390000012
in the formula, sgn is a sign function and represents when (x)j-xk) When the value is less than, equal to or greater than zero, respectively taking-1, 0 or 1; x is the number ofjAnd xkRespectively representing the jth and kth values in the pre-whitened time series;
step 3-2. construct statistic Z as follows:
Figure FDA0002587097390000021
where n represents the time series length and given a confidence level of α, the corresponding test value is U in a standard normal distribution table1-0.5αWhen the absolute value of Z is greater than or equal to U1-0.5αWhen, it represents a significance test with confidence 1- α;
and 3, assuming that the MASH smoothing parameter Y is 1,2, 3. cndot. M and w is 1,2, 3. cndot. N, the MASH smoothing parameter has M × N MASH matrixes, and each MASH matrix has 365 × N MASH matrixeshA piece of data; performing M-K trend test on each row of data of each MASH matrix after the pre-whitening treatment, and recording the number of significant change days D passing the significance testi,j：
Figure FDA0002587097390000022
Wherein i ═ 1,2, 3. cndot. M, j ═ 1,2, 3. cndot. N; i Zi,j,kI represents the trend check value Z of M-K on the K-th line when Y is equal to i and w is equal to ji,j,kAbsolute value of (d); sign is defined as follows:
Figure FDA0002587097390000023
step 4. calculating the significant rate of change
Defining the significant rate of change θ is shown as follows:
Figure FDA0002587097390000024
significant rate of change θi,jReflecting that when Y is i years and w is j days, the total number of N × M corresponding to the ratio of the total days of significant change in MASH matrix year to the total days of one year form a theta matrix with N rows and M columnsi,jPoints are plotted on a graph with Y as an abscissa and w as an ordinate, and a significant change rate distribution graph is plotted;
step 5, calculating the change value of the significant change rate
Change value change (theta) of significant rate of changei,j) Calculated as follows:
Figure FDA0002587097390000025
wherein the content of the first and second substances,
Figure FDA0002587097390000026
Figure FDA0002587097390000027
and
Figure FDA0002587097390000028
respectively representing the difference values of two adjacent columns and two rows of the theta matrix;
after the change value of the significant change rate is calculated, counting the number of the minimum change values and the corresponding Y value and w value, and plotting the minimum change values in a significant change rate distribution graph;
step 6, determining proper smoothing parameters Y and w
And selecting the Y and w values corresponding to the minimum change value as MASH smoothing parameters based on the significant change rate distribution graph under different Y and w.
2. The significant rate of change-based MASH hydrologic time series trend analysis smoothing parameter selection method of claim 1, characterized in that:
where, in step 3-2, given a confidence level α of 0.05, the corresponding test value in the standard normal distribution table is U1-0.5αWhen the absolute value of Z is greater than or equal to 1.96, it indicates a significance test with a 95% confidence pass.
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