Wetland forest and grass ecosystem restoration power quantitative evaluation method under drought stress

Abstract

The invention provides a wetland forest and grass ecosystem restoration strength evaluation method under drought stress. The method comprises the steps of data collection, daily-scale ecosystem service function index ESS construction, ecological drought identification and restoration elasticity curve construction. Wherein the time resolution of the index is accurate to the day, the capture of short-time drought below the monthly scale is enhanced, and the accurate description of the dynamic process of ecological system restoration under the drought stress is realized; combined identification is carried out on the drought events in combination with the standardized ecological water deficit, and the accuracy of identification of the drought events causing significant differences of ecological system service functions is improved; a quantile regression model is utilized to quantify the ecological system recovery capability of different vegetation types, and data characteristics of different recovery time distributions of the ecological system under the same drought intensity are considered, so that the defect that a traditional mean value regression model cannot fully describe drought complex characteristics is overcome; and a theoretical support is provided for sustainable development of an ecological system under an extreme climate condition.

Description

Technical Field

[0001] The present invention belongs to the technical fields of ecological restoration and ecological assessment, and particularly relates to a method for quantitatively evaluating the restoration power of a wetland forest and grass ecosystem under drought stress. Background Art

[0002] Ecological drought refers to an abnormal water cycle event that occurs when the available water resources in a region cannot meet the ecological water demand. Such an abnormal water cycle phenomenon can cause consequences such as tree death, vegetation degradation, and biodiversity loss, thereby damaging the stability of the ecosystem and leading to abnormal ecosystem service functions. Ecological resilience refers to the ability of an ecosystem to recover from a disturbed state affected by drought to a balanced state, and it is an important means for the resource environment to maintain sustainable development under extreme climate conditions. In recent years, under the influence of extreme climate conditions, the stress of drought on the ecosystem has become increasingly serious. Conducting research on ecological resilience is crucial for maintaining the sustainable development of the ecosystem under extreme climate conditions.

[0003] Current research on the resilience of ecosystems mostly focuses on constructing an index evaluation system and using methods such as linear weighting and fuzzy analytic hierarchy process to integrate various factor indicators subjectively selected to generate a resilience evaluation index. Although the above research can fully evaluate the state of the ecosystem after being disturbed, its correlation with the disturbance factors is not close enough. Especially in the face of complex climate phenomena such as drought, different levels of drought will cause different degrees of disturbance to the ecosystem. Considering the dynamic change process of ecosystem resilience with drought intensity can provide more valuable theoretical support for drought prevention and post-disaster governance.

[0004] Existing research on the resilience of ecosystems under drought disturbances mainly focuses on phenotypic characteristics of vegetation, such as annual aboveground net primary productivity (ANPP), enhanced vegetation index (EVI), and normalized difference vegetation index (NDVI). However, this research perspective ignores the impact of drought stress on the overall ecosystem service functions and fails to fully reveal the complex characteristics of drought. Dodd et al. (2023) comprehensively evaluated ecosystem functions and their changes under extreme climates by selecting 38 indicators covering aboveground to underground levels. Similarly, Au et al. (2023) used forest carbon pools and carbon fluxes as indicators of ecosystem service functions and analyzed the impact of drought on the resilience of forest ecosystems using carbon models. Lu and Yan (2023) focused on the carbon cycle as the core of their research, evaluated the overall ecosystem service value through indicators such as GPP, NE, and RE, and explored the relationship between these indicators and vapor pressure deficit, revealing the response mechanism of ecosystems to drought. Although these studies provide important support for theoretical development, focusing on a single ecosystem function cannot comprehensively reflect the impact of drought on ecosystem services. In addition, not all drought events will lead to abnormal ecosystem service values, and traditional hydro-meteorological indices (such as SPI and SPEI) often cannot accurately identify the causes of significant abnormalities in ecosystem services, causing certain interference to the research on the response of ecosystems under drought stress. Therefore, developing a method that can accurately identify ecological drought and strengthening the causal relationship between drought and ecosystem abnormalities are of great significance for drought research.

[0005] Under current climate conditions, the frequency of short-duration drought events (usually lasting only 2 to 4 weeks) is increasing, having a non-negligible impact on plant growth and carbon cycling within a short duration, which is a key factor in evaluating ecosystem resilience. Most existing indicators for monitoring ecosystem functions or identifying drought events are based on annual or monthly scales and cannot capture the rapid occurrence and dynamic changes of drought. Therefore, improving the temporal resolution of assessments is crucial for a deeper understanding of the resilience of ecosystems under drought stress.

[0006] Most of the current methods for quantifying the recovery ability of ecosystems under drought disturbances adopt mean regression analysis. However, the recovery of ecosystems is a complex and dynamic process, which is jointly affected by multiple factors such as biodiversity, extreme climate, and human activities. Even under the same drought intensity, the recovery time of ecosystems often shows significant differences. The mean regression method simplifies different recovery times to an average value, ignoring the data distribution characteristics reflected by different recovery times under the same drought intensity, which will lead to biases in the assessment of the ecosystem recovery ability and cannot fully demonstrate its complexity. In contrast, the quantile regression model retains the complete distribution information of the response variable by fitting the recovery times at different percentile points under the same drought intensity, reveals the heterogeneity of ecosystems under the same drought stress, effectively makes up for the deficiencies of mean regression, and more comprehensively captures the characteristics of the data. Summary of the Invention

[0007] Aiming at the problems existing in the prior art, the purpose of the present invention is to provide a method for quantitatively evaluating the recovery force of wetland forest and grass ecosystems under drought stress. The method comprehensively evaluates the ecosystem service function from the perspectives of supply function, regulation function, etc. by establishing a daily-scale ecosystem service function comprehensive index ESS, combining the water production value, carbon storage value, and habitat quality value of the ecosystem, and proposes an ecological drought joint identification method combined with the standardized ecological water deficit index (SEWDI). The Bayesian nonparametric quantile regression model is used to construct the ecosystem recovery elasticity curve to quantitatively evaluate the recovery elasticity of forest and grassland ecosystems.

[0008] The purpose of the present invention is achieved by the following technical solutions:

[0009] The present invention provides a method for quantitatively evaluating the recovery force of wetland forest and grass ecosystems under drought stress, including the following steps:

[0010] Step 1, data collection:

[0011] Collect topographic data, meteorological data, soil data, and vegetation data in the study area within a certain period, and perform data cleaning and preprocessing.

[0012] Step 2, construction of the daily-scale ecosystem service function index ESS:

[0013] S21, based on the various data collected in Step 1, use the InVEST model to evaluate the ecosystem service function at the annual scale in the study area, specifically including calculating the annual water production, carbon storage, and habitat quality in the study area;

[0014] S22. Downscale and standardize the annual-scale ecosystem service function assessment results obtained in S21 in combination with daily-scale meteorological data, and further construct a daily-scale ecosystem service function index ESS through linear weighting.

[0015] Step 3. Ecological drought identification:

[0016] Use the daily-scale ecosystem service function index ESS constructed in Step 2 and the daily-scale standardized ecological water deficit index SEWDI to identify anomalies in the ecological drought in the study area, and obtain ESS anomaly events and SEWDI anomaly events respectively.

[0017] Jointly identify the ESS anomaly events and SEWDI anomaly events to determine the ecological drought events that cause significant anomalies in the ecosystem service value.

[0018] Step 4. Construction of recovery resilience curve:

[0019] Calculate the drought intensity and drought recovery time of each identified abnormal ecological drought event in Step 3, and use the Bayesian nonparametric quantile regression model to construct the ecosystem recovery resilience curves of different vegetation types, so as to realize the quantitative evaluation of the ecosystem resilience in the study area.

[0020] The beneficial effects of the present invention compared with the prior art are as follows:

[0021] In the method for quantitatively evaluating the recovery resilience of wetland forest and grass ecosystems under drought stress of the present invention, the ecological recovery resilience is comprehensively evaluated from the perspective of ecosystem service functions by combining the water production value, carbon storage value and habitat quality value of the ecosystem; the time resolution of indexes such as ESS is accurate to days, enhancing the capture of short-term droughts below the monthly scale and realizing the accurate characterization of the dynamic process of ecosystem recovery under drought stress; combined with the standardized ecological water deficit to jointly identify drought events, improving the accuracy of identifying drought events that cause significant differences in ecosystem service functions; selecting the quantile regression model instead of the mean regression to more fully describe the complex characteristics of drought and realizing the quantification of the recovery resilience of different vegetation ecosystems, providing important reference significance for the sustainable development of ecosystems under extreme climate conditions. Description of the drawings

[0022] The present invention will be further described below with reference to the drawings and embodiments:

[0023] Figure 1 is a schematic flow chart of the method for quantitatively evaluating the recovery resilience of wetland forest and grass ecosystems under drought stress;

[0024] Figure 2 is the geographical location and terrain distribution of Baiyangdian, the application scenario of the embodiment.

[0025] Figure 3 Distribution map of ecosystem services in the Baiyangdian application scenario of the embodiment; among them, (a) is the water yield distribution, (b) is the habitat quality distribution, (c) is the carbon storage distribution, and (d) is the ESS index distribution;

[0026] Figure 4 Flow chart for identifying abnormal ecosystem service values using the ESS index in the embodiment;

[0027] Figure 5 A typical ecological drought event identified in the embodiment;

[0028] Figure 6 Ecological system recovery resilience curve constructed in the embodiment; among them, from (a) to (f) correspond to low-coverage grassland, medium-coverage grassland, high-coverage grassland, sparse forest land, shrub forest, and forest land in sequence. Detailed implementation manners

[0029] To make the objectives, technical solutions, and advantages of the present invention clearer, the following will further describe the disclosed implementation manners of the present invention in detail with reference to the accompanying drawings. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0030] Embodiment 1

[0031] As Figure 1 shown, this embodiment takes the Baiyangdian Basin as an example to provide a quantitative assessment method for the restoration force of wetland forest and grass ecosystems under drought stress, constructs the ecological system resilience restoration curves of forest land and grassland in the Baiyangdian Basin, and conducts instance simulations.

[0032] Baiyangdian (latitude 38°44′ - 38°59′, longitude 115°45′ - 116°06′) is the largest freshwater wetland in the North China Plain and plays a crucial role in providing ecosystem service functions such as water resource regulation, carbon storage, water quality purification, and protection of biodiversity.

[0033] Figure 2 Shows the geographical location and topographic distribution of Baiyangdian. Baiyangdian is located at the junction of Baoding and Cangzhou, near the confluence of the alluvial fans of the Yongding River and the Hutuo River, adjacent to the Taihang Mountains, with a total area of 366 square kilometers. As part of the Daqing River in the Haihe River Basin, Baiyangdian has a warm temperate, semi-humid continental monsoon climate. The region is dry and windy in spring and cold in winter, with an average annual temperature of 12.2°C and an average annual precipitation of 529.7 millimeters. Its vast water area and rich aquatic biodiversity are crucial for protecting the ecological environment, supporting biodiversity, promoting local economic development, and creating ecological value.

[0034] The method includes the following steps:

[0035] Step 1, data collection:

[0036] Collect topographic data, meteorological data, soil data, and vegetation data in the Baiyangdian Basin during the period from 1982 to 2019, and perform data cleaning and preprocessing.

[0037] Step 2, construction of the daily-scale ecosystem service function index ESS:

[0038] S21, based on the various data collected in Step 1, use the InVEST model to evaluate the annual-scale ecosystem service functions of the study area. Specifically: use the InVEST model to calculate the annual water yield, carbon storage, and habitat quality in the Baiyangdian Basin from 1982 to 2019; use the daily-scale precipitation, normalized difference vegetation index NDVI, and leaf area index LAI as the distribution standards for the annual water yield, carbon storage, and habitat quality respectively, and calculate the daily water yield, carbon storage, and habitat quality in the Baiyangdian from 1982 to 2019.

[0039] S22, combine the daily-scale meteorological data to downscale and standardize the annual-scale ecosystem service function evaluation results obtained in S21, and further perform linear weighting to construct the daily-scale ecosystem service function index ESS. Specifically:

[0040] Perform Z-score standardization on the daily-scale water yield, carbon storage, and habitat quality sequences from 1982 to 2019 respectively to eliminate the dimensional differences among the three.

[0041] Classify the water yield into ecosystem supply services, carbon storage into regulating services, and habitat quality into supporting services. Calculate the weights of the water production value, carbon storage value, and habitat quality value according to the ecosystem service value equivalent factors of the three. Perform linear weighting on the above standardized daily-scale dimensionless water yield, carbon storage, and habitat quality to obtain the daily-scale ecosystem service function index ESS.

[0042] Table 1 shows the corresponding weights of the water yield, carbon storage, and habitat quality after conversion according to the ecosystem service value equivalent factors.

[0043] Table 1

[0044]

[0045] Figure 3 Show the spatial distributions of the average daily-scale water yield, carbon storage, habitat quality, and ESS index in the Baiyangdian Basin from 1982 to 2019.

[0046] Step 3, identification of ecological drought:

[0047] Using the daily-scale ecosystem service function index ESS constructed in step 2 and the daily-scale standardized ecological water deficit index SEWDI to identify anomalies in the ecological drought of the study area, obtaining ESS anomaly events and SEWDI anomaly events respectively.

[0048] As Figure 4 shown, the specific method for identifying anomalies using ESS includes: extracting the cumulative values of the same period of each year from 1982 to 2019 to form a time series, normalizing it and calculating the standard deviation SD, and using -0.5SD as the measurement standard for the ESS of the selected date. If the ESS of a certain day is lower than the corresponding -0.5SD of that day, it is considered that the ecosystem service function has an anomaly. If the ESS of a certain day is higher than -0.5SD and does not fall below -0.5SD within the next 30 days, it is considered that the ecosystem service function has returned to normal.

[0049] Using SEWDI to identify anomalies, specifically: using the three-threshold run theory with 0.5, -0.5, and -1.5 as thresholds, and using the standardized ecological water deficit SEWDI index to identify the ecological drought events in Baiyangdian from 1982 to 2019.

[0050] Jointly identify the ESS anomaly events and SEWDI anomaly events, and make a joint judgment on the ESS anomaly events and SEWDI anomaly events. If at least half of the time during an abnormal period identified by ESS is also identified as abnormal by SEWDI, then retain this abnormal period as the ecological drought event that causes the ecosystem to be abnormal.

[0051] Figure 5 This is a typical drought event identified in this embodiment. Among them, SEWDI began to show anomalies on March 19, 1995 and returned to the normal level on May 18, 1995. After the drought event continued to develop for some time, the ecosystem service function began to be affected, that is, there was a certain time lag between the ESS anomaly and the SEWDI anomaly. ESS began to show anomalies on April 24, 1995 and returned to normal on May 29, 1995. During this period, SEWDI was below the normal level for more than half of the time (from April 24, 1995 to May 18, 1995). Therefore, this ecological drought event is retained, and the entire abnormal period of ESS (from April 24, 1995 to May 29, 1995) is used as the recovery period of this drought.

[0052] Step 4, construction of the resilience curve:

[0053] Calculate the drought intensity and drought recovery time of each abnormal ecological drought event identified in calculation step 3, and use the Bayesian nonparametric quantile regression model to construct the ecosystem recovery resilience curve of different vegetation types, so as to realize the quantitative evaluation of the ecosystem resilience in the study area.

[0054] Specifically: Take the total time experienced from the start of the anomaly of ESS to its return to normal as the drought recovery time, take the absolute value of the cumulative value of SEWDI during the drought recovery time as the drought intensity, and use Bayesian non-linear quantile regression to fit the two on different vegetation cover types.

[0055] Use the cubic polynomial of the B-spline function as the basis function for model fitting. To improve the accuracy and efficiency of quantile estimation, replace the original general test function with the ELF (xtended log-f) loss function.

[0056] The expression is:

[0057]

[0058] In the formula, x is the drought intensity, y is the response variable, that is, the drought recovery time, μ τ (x) is the conditional distribution of the response variable at the τ quantile, τ takes 0.1, 0.5, 0.9, b n (x) is the spline basis function for constructing the drought recovery time distribution, E is the expected value of the response variable y under the condition of the given covariate x, μ is all possible quantile values, K is the basis function coefficient, n is the loss function for quantile estimation, where z is the difference between the predicted value and the observed value, λ is the smoothing parameter, and σ is the scale parameter, all of which affect the model complexity and overfitting risk. Based on the ELF loss function, introduce the Bayesian algorithm to update and calibrate the model, and solve the spline basis function coefficient. The specific process is as follows:

[0059] (1) Use the integrated KL divergence (IKL) as the calibration loss function, and use the Brent algorithm to calculate 1 / σ0 when the loss function is minimized as the baseline learning rate.

[0060] (1) Take the integrated KL divergence (IKL) as the calibration loss function, and use the Brent algorithm to calculate 1 / σ0 when the loss function is minimized as the baseline learning rate.

[0061]

[0062] In the formula, is the posterior variance obtained based on the covariance matrix of the gradient of the ELF loss function, is the posterior variance obtained based on the model predicted value, is a positive parameter, here take 1 / 2; Q is the number of samples, and q is the index value of the traversed samples.

[0063] (2) Calculate the optimal scaling loss bandwidth based on the principle of minimizing the asymptotic mean square error (MSE) of the conditional variance and the ELF loss function, and determine the smoothing parameter λ and the learning rate σ(x) related to the independent variable accordingly.

[0064]

[0065] σ(x) = λσ0

[0066] Wherein, is the optimal scaling loss bandwidth, f z is the probability density function of the error term, f′ z is the first derivative of f z F z is the cumulative distribution function of the error term, F z -1 (τ) is the τ - quantile of the error term, κ(x) is the conditional variance, d is the dimension of the basis function coefficient K n The dimension of, Q is the number of samples, 1 / σ0 is the baseline learning rate.

[0067] (3) Use Laplace approximation to calculate the ELF marginal likelihood function and perform numerical optimization to select the smoothing parameter γ.

[0068]

[0069] Wherein, γ, λ are smoothing parameters, σ(x) is the learning rate, G{γ, σ(x), λ} is the optimal solution of the regression coefficient obtained by internal iteration, is the bias function of the ELF density function, is the formula The minimum point of, is the saturated log - likelihood function, X T WX is the weighted outer product matrix of the design matrix X, S λ is the precision matrix of the prior distribution, M p is for S λ The dimension of the null space.

[0070] (4) Adopt the penalized iterative reweighted least squares (PIRLS) algorithm to optimize the objective function, and the maximum a posteriori estimate value of the spline basis function coefficient can be obtained, thus determining the relationship between drought intensity and recovery time.

[0071]

[0072] Wherein, γ, λ are smoothing parameters, σ(x) is the learning rate, K is the basis function coefficient, is the i - th bias component of the likelihood function based on the ELF density, P is the number used to construct the precision matrix, Sp is a positive definite matrix and is used to construct the precision matrix. is the deviation of the likelihood function.

[0073] In summary, the resilience characteristics of the forest and grass ecosystem in Baiyangdian are as follows:

[0074] Figure 6 are the ecosystem resilience curves of different vegetations in Baiyangdian. From (a) to (f), they correspond to low-coverage grassland, medium-coverage grassland, high-coverage grassland, sparse forest land, shrub forest, and forest land in sequence. The horizontal axis is the independent variable, i.e., the drought intensity, and the vertical axis is the response variable, i.e., the drought recovery time. Three quantiles, 0.1, 0.5, and 0.9, are selected to fit the drought recovery resilience. This resilience curve reflects the time required for the system to return to the normal state under a certain ecological drought intensity. Different quantiles reflect the recovery ability of the ecosystem facing drought in different states. The area enclosed by the 0.9 quantile resilience curve and the horizontal axis can be regarded as the resilience value of this ecosystem facing drought. The smaller the area, the stronger the recovery ability.

[0075] The resilience curves of all quantiles show that as the drought intensity increases, the recovery time increases. However, under extreme drought conditions, the 0.9 quantile curves of some vegetation types show anomalies. The 0.9 quantile curves of low-coverage grassland, medium-coverage grassland, and sparse forest land show a stable or even decreasing trend. This indicates that once the drought intensity exceeds a certain threshold, the recovery ability of these ecosystems will remain stable or increase slightly. If the drought intensity continues to rise, the ecosystems of these vegetation types will be damaged and unable to recover to the state before the disturbance. In contrast, the resilience of high-coverage grassland, shrub forest, and forest land decreases significantly under extreme drought conditions. As the drought intensity increases, the recovery ability of these ecosystems decreases significantly. For the Baiyangdian Basin, under medium and low intensity droughts, the recovery resilience of forest land is greater than that of grassland (forest land > high-coverage grassland > shrub forest > medium-coverage grassland > sparse forest land >), but with the continuous increase of the drought degree, the ecosystem recovery resilience of forest land and grassland shows an opposite trend. Under high intensity, the recovery resilience of grassland is greater than that of forest land.

[0076] Finally, it should be noted that the above is only used to illustrate the technical solution of the present invention and not to limit it. Although the present invention has been described in detail with reference to the preferred arrangement, those of ordinary skill in the art should understand that the technical solution of the present invention (such as the application of various formulas, the sequence of steps, etc.) can be modified or equivalently replaced without departing from the spirit and scope of the technical solution of the present invention.

Claims

1. A method for quantitatively assessing the resilience of wetland forest and grassland ecosystems under drought stress, characterized in that: The method comprises the following steps: Step 1, data collection: Collect terrain data, meteorological data, soil data and vegetation data within a certain period of time in the study area, and clean and pre-process the data; Step 2: Construction of daily-scale ecosystem service function index ESS: S21, based on the data collected in step 1, use the InVEST model to evaluate the ecosystem service function of the study area on an annual scale, including calculating the annual water yield, carbon storage and habitat quality in the study area; S22, downscale and standardize the annual ecosystem service function assessment results obtained in S21 by combining daily meteorological data, and further construct the daily ecosystem service function index ESS through linear weighting; Step 3, ecological drought identification: The daily-scale ecosystem service function index ESS constructed in step 2 and the daily-scale standardized ecological water deficit index SEWDI were used to identify the abnormal ecological drought in the study area, and the abnormal events of ESS and SEWDI were obtained respectively; Jointly identify ESS abnormal events and SEWDI abnormal events to determine ecological drought events that cause significant abnormalities in ecosystem service value; Step 4: Restore elasticity curve construction: The drought intensity and drought recovery time of each abnormal ecological drought event identified in step 3 were calculated, and the Bayesian nonparametric quantile regression model was used to construct the ecosystem recovery elasticity curves of different vegetation types, thereby achieving a quantitative assessment of the ecosystem resilience in the study area.

2. The method according to claim 1, characterized in that In step 2 S22, daily rainfall, normalized difference vegetation index and leaf area index are used as downscaling basis for annual water production, carbon storage and habitat quality respectively.

3. The method according to claim 1, characterized in that In step 2, S22, Z-score standardization is used to eliminate dimensional differences.

4. The method according to claim 1, characterized in that: The specific method of using ESS to identify anomalies in step 3 includes: A specific date was selected, and the cumulative ESS value of the 30 days before that day was calculated with a time scale of 30 days; the 30-day cumulative value of all years corresponding to that day was taken as the time series, which was normalized and the standard deviation SD was calculated, and -0.5 times the SD was used as the criterion for judging whether the daily ESS was abnormal; if the daily ESS was lower than the corresponding -0.5SD of the day, it was considered that the ecosystem service function was abnormal; if the ESS was higher than -0.5SD and did not fall below -0.5SD in the next 30 days, it was considered that the ecosystem service function had returned to normal.

5. The method according to claim 1, characterized in that In step 3, SEWDI is used for anomaly identification, specifically: The SEWDI index is used to identify abnormal phenomena using the three-threshold run theory with -0.5, -1.5, and 0.5 as standards.

6. The method according to any one of claims 1, 4 or 5, characterized in that: In step 3, ESS abnormal events and SEWDI abnormal events are jointly identified, specifically: The ESS abnormal events and SEWDI abnormal events are jointly judged. If at least half of the time in an abnormal period identified by ESS is also identified as abnormal by SEWDI, the abnormal period is retained as an ecological drought event that causes abnormal ecosystems.

7. The method according to claim 1, characterized in that In step 4, the drought intensity is the cumulative value of the absolute value of SEWDI during the drought recovery time, and the drought recovery time is the total time from the beginning of the abnormality of ESS to the recovery of ESS in step 3.

8. The method according to claim 1, characterized in that In step 4, the generalized additive model with ELF as the loss function is used to fit the drought recovery time and drought intensity using the Bayesian algorithm to construct the ecosystem recovery elasticity curve, which is expressed as: In the formula, x is the drought intensity, y is the drought recovery time, μ τ (x) is the conditional distribution of the response variable under the τ quantile, τ is 0.1, 0.5, 0.9, b n (x) is the spline basis function for constructing drought recovery time distribution, K n is the basis function coefficient, E is the condition under which the covariate x is given, Regarding the expected value of the response variable y, μ is all possible quantile values, is the loss function for quantile estimation, where z is the difference between the predicted value and the observed value, λ is the smoothing parameter, and σ is the scale parameter.

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