CN112215299B - Block bootstrap method for hydrological space data mean value estimation - Google Patents
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Abstract
The invention provides a block bootstrapping method for estimating a mean value of hydro-meteorological space data, which comprises the following steps: s1: acquiring information of N hydrological space data points from an observation area to form a sample set X= { X 1 ,x 2 ,...,x N -a }; s2: b times of block bootstrap sampling are carried out on the samples in the sample set to obtain a B group resampling sample set; s3: respectively carrying out mean value estimation on the B group resampled sample sets to obtain an estimated mean value sequenceThe invention provides a block bootstrapping method for estimating the mean value of hydro-meteorological space data, which ensures that hydro-meteorological space data points in a certain range are extracted as a whole by block bootstrapping sampling, avoids destroying the dependent structure of the data, and solves the problem that the prior bootstrapping method is used for processing the space-related and non-independent samples with the same distribution, such as hydro-meteorological variables, can fail.
Description
Technical Field
The invention relates to the technical field of hydrological weather, in particular to a block bootstrap method for estimating the mean value of space data of the hydrological weather.
Background
The hydro-meteorological variables (such as rainfall, air temperature, evaporation and radiation and the like) have complex space variability and non-stationarity, so the difficulty of carrying out statistical inference on the surface average value of the regional hydro-meteorological variables and quantitatively describing the uncertainty caused by the space correlation is relatively large. Currently, bootstrap (Bootstrap) analog mean sampling statistical inference is generally adopted. The bootstrap method is based on the original data, and the sample capacity is expanded through resampling, so that the empirical distribution of the sample mean value sequence is obtained. However, bootstrap requires independent co-distribution of samples, which can be ineffective for spatially dependent, non-independent co-distributed samples such as hydrographic variables.
In the prior art, for example, in the chinese patent published on the 2019, the method for determining the optimal resolution of the spatial interpolation of the hydro-meteorological elements, with the publication number CN106340018B, the effective data contained in the obtained spatial interpolation result image is maximized by selecting the optimal resolution, which has more reference value, but the problem that the hydro-meteorological variables are invalid when the spatially related and non-independently co-distributed samples are processed by using the bootstrap method is not solved.
Disclosure of Invention
The invention provides a block bootstrap method for estimating the mean value of the space data of the hydrological weather, which aims to overcome the technical defect that the prior bootstrap method is used for processing the space-related and non-independent samples with the same distribution, such as the hydrological weather variable, can fail.
In order to solve the technical problems, the technical scheme of the invention is as follows:
a block bootstrapping method for estimating a mean value of hydro-meteorological space data comprises the following steps:
s1: acquiring information of N hydrological space data points from an observation area to form a sample set X= { X 1 ,x 2 ,...,x N -a }; wherein sample x i Information for the ith hydrographic space data point, i=1, 2, …, N;
s2: b times of block bootstrap sampling are carried out on the samples in the sample set to obtain a B group resampling sample set;
s3: respectively carrying out mean value estimation on the B group resampled sample sets to obtain an estimated mean value sequenceWherein (1)>B=1, 2, …, B, which is the estimated mean of the B-th set of resampled samples.
In the scheme, the characteristic that the dependence exists in the hydrological space data is considered, the hydrological space data in a certain range is ensured to be extracted as a whole by the block bootstrapping sampling, and the dependence structure of the data is prevented from being damaged, so that the bootstrapping method is prevented from being invalid.
Preferably, in step S1, the acquired information of the hydro-meteorological space data point is coordinates of the hydro-meteorological space data point.
Preferably, the coordinates of the hydrological space data points are latitude and longitude coordinates.
Preferably, in step S2, the block bootstrapping sample comprises the steps of:
s2.1: setting a search radius r;
s2.2: randomly selecting one sample from the sample set, taking the coordinates of the selected sample as the center, and taking the sample falling in the searching range and the selected sample into a regenerated sample set;
s2.3: judging whether the number of samples in the regenerated sample set exceeds N;
if yes, selecting the first N samples in the regenerated sample set to form a resampled sample setCompleting one-time block bootstrap sampling; wherein (1)>For regenerating the kth sample in the sample set, k=1, 2, …, N;
if not, returning to the step S2.2.
Preferably, in step S2.2, a sample is randomly selected from the sample set according to the principle of equal probability per sample.
Preferably, the probability of each sample being selected is
Preferably, in step S2.2, the distance d between the sample i and the selected sample j is calculated by ij Comparing with the searching radius r to judge whether the sample i falls in the searching range of the sample j;
if the distance d ij If the search radius r is smaller than or equal to the search radius r, the sample i falls in the search range of the sample j;
otherwise, sample i does not fall within the search range of sample j.
Preferably, the distance d between samples i and j is calculated by Euclidean method ij The calculation formula is as follows:
wherein u is i Longitude coordinate of sample i, v i Latitude coordinates of the sample i; u (u) j Is the longitude coordinate of sample j, v j Is the latitude coordinate of sample j.
Preferably, in step S3, a calculation formula for performing mean value estimation on the resampled sample set is:
wherein t is * The estimated mean value of the resampled sample set,to regenerate the kth sample in the sample set, k=1, 2, …, N.
Preferably, in step S3, after obtaining the estimated mean value sequence, interval estimation is further performed on the mean value of the sample set according to the estimated mean value sequence.
Compared with the prior art, the technical scheme of the invention has the beneficial effects that:
the invention provides a block bootstrapping method for estimating the mean value of hydro-meteorological space data, which considers the characteristic that the hydro-meteorological space data has dependency inside, and ensures that hydro-meteorological space data points in a certain range are extracted as a whole by block bootstrapping sampling, thereby avoiding destroying the dependency structure of the data and avoiding the failure of the bootstrapping method.
Drawings
FIG. 1 is a flow chart showing the steps performed in the embodiment 1 of the present invention;
FIG. 2 is a flowchart illustrating steps of block bootstrapping samples according to embodiment 1 of the present invention;
FIG. 3 is a spatial distribution diagram of average precipitation over four seasons for example 2 of the present invention;
FIG. 4 is a graph of an estimated australian/New Zealand plane mean precipitation interval obtained by a conventional bootstrap method for example 2 of the present invention;
FIG. 5 is a graph of the average precipitation interval of southeast America face obtained by the conventional bootstrap method of example 2 of the present invention;
FIG. 6 is a graph of the estimated Australian/New Zealand surface mean precipitation interval obtained by the block bootstrapping method of the one type of hydrological spatial data mean estimation of example 2 of the present invention;
fig. 7 is a graph of the estimated southeast face average precipitation interval of south america obtained by the block bootstrapping method of the present invention.
Detailed Description
The drawings are for illustrative purposes only and are not to be construed as limiting the present patent;
for the purpose of better illustrating the embodiments, certain elements of the drawings may be omitted, enlarged or reduced and do not represent the actual product dimensions;
it will be appreciated by those skilled in the art that certain well-known structures in the drawings and descriptions thereof may be omitted.
The technical scheme of the invention is further described below with reference to the accompanying drawings and examples.
Example 1
As shown in fig. 1, a block bootstrapping method for estimating a mean value of hydro-meteorological space data includes the following steps:
s1: acquiring information of N hydrological space data points from an observation area to form a sample set X= { X 1 ,x 2 ,...,x N -a }; wherein sample x i Information for the ith hydrographic space data point, i=1, 2, …, N;
s2: b times of block bootstrap sampling are carried out on the samples in the sample set to obtain a B group resampling sample set;
s3: respectively carrying out mean value estimation on the B group resampled sample sets to obtain an estimated mean value sequenceWherein (1)>B=1, 2, …, B, which is the estimated mean of the B-th set of resampled samples.
In the specific implementation process, the characteristic that the inside of the hydrological space data has the dependency is considered, and the hydrological space data points in a certain range are ensured to be extracted as a whole block by the block bootstrap sampling, so that the dependency structure of the data is prevented from being damaged; for resampled samples based on radius range search, independent co-distributed bootstrap methods may be applied for mean estimation.
More specifically, in step S1, the acquired information of the hydro-meteorological space data point is coordinates of the hydro-meteorological space data point.
More specifically, the coordinates of the hydrological space data points are longitude and latitude coordinates.
More specifically, as shown in fig. 2, in step S2, the block bootstrap sampling includes the steps of:
s2.1: setting a search radius r;
s2.2: randomly selecting one sample from the sample set, taking the coordinates of the selected sample as the center, and taking the sample falling in the searching range and the selected sample into a regenerated sample set;
s2.3: judging whether the number of samples in the regenerated sample set exceeds N;
if yes, selecting the first N samples in the regenerated sample set to form a resampled sample setCompleting one-time block bootstrap sampling; wherein (1)>For regenerating the kth sample in the sample set, k=1, 2, …, N;
if not, returning to the step S2.2.
In the implementation process, each block bootstrap sampling obtains a group of resampling sample set, and B groups of resampling sample sets are correspondingly obtained by repeating the block bootstrap sampling for B times.
More specifically, in step S2.2, a sample is randomly selected from the sample set according to the principle of equal probability per sample.
More specifically, the probability that each sample is selected is
More specifically, in step S2.2, the distance d between the sample i and the selected sample j is calculated by ij Comparing with the searching radius r to judge whether the sample i falls in the searching range of the sample j;
if the distance d ij If the search radius r is smaller than or equal to the search radius r, the sample i falls in the search range of the sample j;
otherwise, sample i does not fall within the search range of sample j.
More specifically, the distance d between samples i and j is calculated by Euclidean method ij The calculation formula is as follows:
wherein u is i Longitude coordinate of sample i, v i Latitude coordinates of the sample i; u (u) j Is the longitude coordinate of sample j, v j Is the latitude coordinate of sample j.
More specifically, in step S3, the calculation formula for performing the mean value estimation on the resampled sample set is:
wherein t is * The estimated mean value of the resampled sample set,to regenerate the kth sample in the sample set, k=1,2,…,N。
More specifically, in step S3, after obtaining the estimated mean value sequence, interval estimation is further performed on the mean value of the sample set according to the estimated mean value sequence.
Example 2
This example 2 was conducted based on the method of example 1, taking the global seasonal grid precipitation data of the united states Climate Prediction Center (CPC) 1982-2010 as an example, calculating average precipitation over years for different seasons precipitation 1982-2010, and calculating the face average for average precipitation over years for different partitions worldwide.
Figure 3 shows the spatial distribution of average precipitation over 1982-2010 in northern hemisphere winter, spring, summer and autumn (denoted DJF, MAM, JJA and SON respectively). It can be seen that the seasonal precipitation distribution exhibits a pronounced spatial correlation and seasonal nature. During DJF, global rain belts are mainly distributed in the southern hemisphere, wherein the regional rainfall is large in the northern Australia, philippines, indonesia, the central south America, the central Africa and the like. From DJF to MAM and JJA, the rain belt gradually moves north. During JJA, the rainfall is high in the east Asia zone, the south Asia zone, the middle of the America zone and the middle of Africa zone. A fifth inter-government climate change specialized committee (Intergovernmental Panel on Climate Change) was introduced to evaluate reporting reference areas and compare the annual average seasonal rainfall characteristics of different areas. The reference region locations are identified in fig. 3 by black boxes, each region being distinguished by a different data mark.
For each sub-zone, the zone-surface average precipitation is calculated as follows
Wherein P is i The average precipitation is observed for many years for grid i, N being the total number of grids in the area.
Firstly, a common bootstrap method is adopted to estimate the average rainfall distribution interval of the regional surface. For each region, extracting N data points, calculating an average value, repeating 1000 times to obtain a group of region face average precipitation sequences, and forming empirical distribution for face average precipitation confidence interval estimation. Fig. 4 shows an average precipitation interval estimate for the australian/new zealand land precipitation surface based on the normal bootstrap method, the corresponding region being identified as region 26 in fig. 3. In the figure, the red origin represents the surface average value of the average precipitation observed for many years, the dark blue interval represents the estimated 50% confidence interval, and the light blue interval represents the estimated 80% confidence interval. It can be seen that there is a strong seasonal nature of australia/new zealand precipitation, and that during DJF the seasonal average precipitation exceeds 250mm, whereas MAM, JJA and SON precipitation are overall lower. Furthermore, the precipitation intervals are narrow for both four seasons, especially for both JJA and SON seasons. As can be seen by comparing fig. 3, the precipitation of one zone in australia/new zealand is spatially distributed evenly and overall lower in both season JJA and SON. Similarly, fig. 5 shows an interval estimate of average precipitation for four seasonal planes in the southeast region of south america, which corresponds to region 10 in fig. 3. The overall seasonal precipitation is weaker in the southeast region of south america compared to the australia/new zealand region, with four season observables differing by 50mm. Overall, DJF and MAM precipitation was more abundant, and JJA and SON precipitation was lower.
Then, the block bootstrap method for the average value estimation of the hydro-meteorological space data provided by the invention is adopted to carry out interval estimation on the surface average precipitation. The radius of the preset search is 5 degrees, and resampling is carried out by a method of searching the accumulated sample points of the adjacent grids by the radius. Fig. 6 shows an australian/new zealand regional plane average precipitation interval estimation based on a block bootstrapping method of hydrological space data mean estimation provided by the invention. It is evident that the interval range of four season estimates is larger than the normal bootstrap method. In particular DJF, the 80% area of the surface average precipitation exceeds 100mm, and compared with the spatial distribution diagram of FIG. 3, during DJF, the North Australia presents obvious rain drops, the rainfall decreases from north to south, and the rainfall presents strong spatial dependency characteristics. The common bootstrap method resamples based on all data points, and the strong spatial dependence leads to estimation failure, so that the distribution range existing when the ground average precipitation exists can not be estimated. Fig. 7 shows a block bootstrap method sampling an estimated area average precipitation interval based on a block bootstrap method of the hydrological space data mean estimation provided by the invention in the southeast region of south america. Compared to fig. 5, the overall precipitation interval increases for four seasons, especially JJA, with 80% intervals exceeding 100mm.
The experimental result shows that the block bootstrapping method for the mean value estimation of the hydrographic space data provided by the invention can effectively consider the characteristics of multi-dimensionality, irregularity and autocorrelation of the space data.
It is to be understood that the above examples of the present invention are provided by way of illustration only and not by way of limitation of the embodiments of the present invention. Other variations or modifications of the above teachings will be apparent to those of ordinary skill in the art. It is not necessary here nor is it exhaustive of all embodiments. Any modification, equivalent replacement, improvement, etc. which come within the spirit and principles of the invention are desired to be protected by the following claims.
Claims (7)
1. The blocky bootstrap method for the hydrological space data mean value estimation is characterized by comprising the following steps of:
s1: acquiring information of N hydrological space data points from an observation area to form a sample set X= { X 1 ,x 2 ,...,x N -a }; wherein sample x i Information for the ith hydrographic space data point, i=1, 2, …, N; the acquired information of the hydro-meteorological space data points is coordinates of the hydro-meteorological space data points;
s2: b times of block bootstrap sampling are carried out on the samples in the sample set to obtain a B group resampling sample set; wherein the bulk bootstrap sampling comprises the steps of:
s2.1: setting a search radius r;
s2.2: randomly selecting one sample from the sample set, taking the coordinates of the selected sample as the center, and taking the sample falling in the searching range and the selected sample into a regenerated sample set; wherein by the distance d between sample i and the selected sample j ij Comparing with the searching radius r to judge whether the sample i falls in the searching range of the sample j; if it isDistance d ij If the search radius r is smaller than or equal to the search radius r, the sample i falls in the search range of the sample j; otherwise, sample i does not fall within the search range of sample j;
s2.3: judging whether the number of samples in the regenerated sample set exceeds N; if yes, selecting the first N samples in the regenerated sample set to form a resampled sample setCompleting one-time block bootstrap sampling; wherein (1)>For regenerating the kth sample in the sample set, k=1, 2, …, N; if not, returning to the step S2.2;
s3: respectively carrying out mean value estimation on the B group resampled sample sets to obtain an estimated mean value sequenceWherein (1)>B=1, 2, …, B, which is the estimated mean of the B-th set of resampled samples.
2. The blocky bootstrap method of the mean value estimation of the hydro-meteorological space data according to claim 1, wherein coordinates of the hydro-meteorological space data points are longitude and latitude coordinates.
3. The blocky bootstrap method of the mean value estimation of the hydrographic spatial data according to claim 1, characterized in that in step S2.2, a sample is randomly selected from the sample set according to the principle of equal probability of each sample.
4. A block bootstrapping method of hydrological spatial data mean estimation according to claim 3, wherein the probability of each sample being selected is
5. The blocky bootstrap method of hydrographic spatial data mean estimation as defined in claim 4, characterized in that a distance d between samples i and j is calculated by euclidean method ij The calculation formula is as follows:
wherein u is i Longitude coordinate of sample i, v i Latitude coordinates of the sample i; u (u) j Is the longitude coordinate of sample j, v j Is the latitude coordinate of sample j.
6. The blocky bootstrap method of the mean value estimation of the hydro-meteorological space data according to claim 1, wherein in step S3, a calculation formula for performing the mean value estimation on the resampled sample set is:
wherein t is * The estimated mean value of the resampled sample set,to regenerate the kth sample in the sample set, k=1, 2, …, N.
7. The blocky bootstrap method of the mean value estimation of the hydro-meteorological space data according to claim 1, wherein in step S3, after obtaining the estimated mean value sequence, the method further comprises performing interval estimation on the mean value of the sample set according to the estimated mean value sequence.
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