Electromagnetic interference field prediction method based on multi-scale attention enhancement U-Net
The multi-scale attention-enhanced U-Net method solves the problem of electromagnetic interference field reconstruction under sparse measurements, achieves high-precision electromagnetic interference field prediction, reduces computational costs and infrastructure requirements, and is applicable to electromagnetic interference field prediction in various environments.
Patent Information
- Application Number
- CN202511616275.2
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-11-06
- Publication Date
- 2026-01-23
AI Technical Summary
Existing technologies struggle to accurately reconstruct electromagnetic interference fields from sparse measurements in dynamic environments. In particular, they cannot effectively utilize historical electromagnetic field data and learned patterns in complex and multi-environment environments, resulting in high computational costs and unsuitability for dynamic, large-scale interference management in modern wireless systems.
We employ a multi-scale attention-enhanced U-Net approach, which combines multi-scale feature extraction, source-aware convolution processing, edge preservation, and physical constraint optimization with sparse field measurement data to predict electromagnetic interference fields. This approach integrates specialized attention mechanisms and source-aware processing required for multi-environment electromagnetic field reconstruction.
High-precision electromagnetic interference field reconstruction was achieved under extreme measurement sparsity, reducing the deployment cost of large-scale spectrum management infrastructure and providing a practical solution for dynamic spectrum access and real-time interference mapping.
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Figure CN121396367A_ABST
Abstract
Description
TECHNICAL FIELD
[0001] The application belongs to the technical field of signal processing, and relates to an electromagnetic interference field prediction method based on a multi-scale attention enhanced U-Net. BACKGROUND
[0002] With the explosive growth of wireless communication devices and the spread of electromagnetic interference, accurate interference field prediction becomes crucial for spectrum management and network optimization. Global mobile subscriptions have exceeded 8.8 billion, and IoT devices are expected to reach 75 billion by 2025. This explosive growth in device density creates complex interference patterns that traditional prediction methods cannot fully address. Next-generation wireless systems include 5G / 6G networks, massive MIMO systems, and millimeter-wave communications operating in frequency ranges and deployment densities where interference effects become dominant performance limiting factors.
[0003] Modern wireless networks operate in dense, heterogeneous environments where multiple transmitters, reflecting surfaces, and dynamic obstacles interact to form complex electromagnetic interference patterns that significantly impact system performance. This complexity is particularly evident in special environments that impose unique propagation challenges: urban environments exhibit complex multipath propagation due to high building density and diverse building materials; ocean scenarios present strong reflection effects at the water surface combined with atmospheric ducting phenomena; desert environments introduce severe atmospheric refraction and temperature-dependent propagation variations; and rural environments have vegetation-induced scattering and terrain-dependent shadowing effects.
[0004] Classic electromagnetic field prediction methods mainly rely on numerical solutions to Maxwell's equations through discretization methods. The finite-difference time-domain (FDTD) method remains the gold standard for electromagnetic simulation due to its ability to handle complex geometries and material properties. However, the FDTD method often requires smaller spatial and temporal steps, which means that the time step must be very small when simulating high-speed or high-frequency signals, further increasing computational costs. Alternative methods such as the method of moments (MoM) and the finite element method (FEM) provide computational advantages for specific problem categories, but have similar scalability limitations when applied to large-scale, multi-environment electromagnetic field prediction. Most importantly, these methods cannot leverage historical electromagnetic field data and learned patterns, requiring complete recalculation for each new scenario configuration, making them unsuitable for dynamic, large-scale interference management required by modern wireless systems.
[0005] The pressing need for field reconstruction from sparse measurements stems from the practical impossibility of deploying dense sensor networks over large geographical areas. Economic constraints make a comprehensive measurement infrastructure prohibitively expensive - deploying measurement devices with sufficient spatial density to capture electromagnetic field variations would require sensor spacing in the order of meters to tens of meters, leading to infrastructure costs in the millions of dollars for urban-scale deployments. Logistical constraints in challenging environments such as oceanic regions, desert areas, and access-restricted urban areas make dense sensor deployments extremely difficult. Real-time system requirements necessitate field reconstruction capabilities that can operate with measurement densities as low as 1-5% of the optimal spacing, while maintaining prediction accuracy sufficient for spectrum management applications.
[0006] Physical information neural networks (PINNs) represent a revolutionary approach that directly embeds physical laws into neural network architectures, enabling unprecedented integration of domain knowledge with data-driven learning. However, standard PINN methods lack the architectural complexity required for multi-environment electromagnetic field reconstruction, particularly the specialized attention mechanisms, source-aware processing, and edge-preserving capabilities needed for accurate interference field prediction from sparse measurements.
[0007] Therefore, there is a need for a method for reconstructing electromagnetic interference fields from sparse measurements in dynamic environments, improving prediction accuracy and robustness. SUMMARY
[0008] Therefore, there is a need for a method for reconstructing electromagnetic interference fields from sparse measurements in dynamic environments, improving prediction accuracy and robustness.
[0009] To achieve the above object, the present application provides the following technical scheme: A method for predicting electromagnetic interference fields based on a multi-scale attention-enhanced U-Net, comprising the following steps: S1: Input data preparation: receiving four-channel input data containing sparse field measurement data, sampling mask, obstacle layout, and transmitter location; S2: Multi-scale feature extraction: performing multi-scale feature extraction on the input data through an encoder path, and using a multi-scale attention mechanism to capture electromagnetic propagation characteristics in near-field, mid-field, and far-field; S3: Source-aware convolution processing: using a source-aware convolution layer to dynamically adjust convolution kernel weights according to the location and coupling relationship of electromagnetic sources; S4: Edge-preserving processing: using an edge-preserving module to detect and preserve field discontinuities at obstacle boundaries using a Sobel operator; S5: Decoder reconstruction: up-sampling features to the original resolution through a symmetric decoder path, and using enhanced skip connections to preserve spatial details; S6: Physical constraint optimization: network training using a comprehensive loss function containing reconstruction fidelity loss, spatial gradient consistency loss, unified physical constraint loss, and detail preservation loss; S7: Output complete field distribution: generate complete electromagnetic interference field distribution prediction results.
[0010] Further, in S1, the input data is standardized to 200x200 pixel four-channel data, the first channel is sparse field measurement data, containing field intensity values only at sampling positions; The second channel is a sampling mask, which identifies the measurement positions; The third channel is the obstacle layout, which represents the distribution of obstructions in the environment; The fourth channel is the emission source distribution, which identifies the position and power information of the interference source. Based on Maxwell's electromagnetic theory, the field reconstruction problem can be formulated as a physical constraint optimization:
[0011] The basic electromagnetic constraints derived from Maxwell's equations, S represents the interference source distribution, and O represents the obstacle environment map.
[0012] Further, in S2, the multi-scale attention mechanism extracts features through multiple parallel scale branches, including calculating features on k different scales , including 1x1, 3x3, 5x5, and 7x7 convolution kernels; Calculate physical information attention weights ; Define distance-dependent electromagnetic coupling functions, including near-field coupling, intermediate coupling, far-field coupling, and scattering coupling; Finally, calculate multi-scale feature fusion .
[0013] Further, in S3, the weights of the source-aware convolution layer are dynamically adjusted according to the source position, and the effective coupling function is:
[0014] Including source amplitude factor , propagation attenuation and phase factor and other physical effects.
[0015] Further, in S4, the edge preservation module uses the Sobel operator to calculate the spatial gradient, detects field discontinuity and obstacle boundaries through gradient information, and uses enhancement weights to fuse gradient information with main feature extraction.
[0016] S41: Calculate spatial gradient using Sobel operator:
[0017] The Sobel kernel is defined as:
[0018] S42: detecting field discontinuity and obstacle boundary through gradient information; S43: using enhanced weight Fusing gradient information with main feature extraction.
[0019] Further, in the S5, the decoder adopts a symmetrical structure, and the features are sequentially upsampled through four decoding blocks. Each decoding block uses an upsampling operation to expand the spatial resolution, fuses the features of the corresponding layer of the encoder through enhanced skip connection, and applies dropout at the bottleneck layer to prevent overfitting. The symmetrical structure of the decoder includes: S51: sequentially upsample the features from 12x12x1024 to 200x200x1 through four decoding blocks; S52: each decoding block uses a 2x2 upsampling operation to expand the spatial resolution by 2 times; S53: fuse the features of the corresponding layer of the encoder through enhanced skip connection, and the skip connection integrates a multi-scale attention mechanism; S54: apply dropout (probability 0.15) at the bottleneck layer to prevent overfitting.
[0020] Further, in the S6, the comprehensive loss function includes reconstruction fidelity loss, spatial gradient consistency loss, unified physical constraint loss, and detail preservation loss, wherein the unified physical constraint loss includes source region minimum field strength constraint, obstacle region field decay constraint, and measurement point fidelity constraint. The comprehensive loss function is defined as:
[0021] The weight of each loss component is: ; Reconstruction fidelity loss: ; Spatial gradient consistency loss: ; Unified physical constraint loss: , including source region minimum field strength constraint , obstacle region field decay constraint , and measurement point fidelity constraint ; Detail preservation loss: , , wherein is edge preservation loss, is texture consistency loss.
[0022] Further, the network training of the method adopts AdamW optimizer, cosine annealing learning rate scheduling strategy, batch size of 8 samples, uses gradient accumulation to simulate a larger effective batch size, and adopts gradient clipping and early stopping strategy to prevent overfitting.
[0023] The method is suitable for various propagation environments, including urban environments, rural environments, marine environments and desert environments, and can adapt to different obstacle densities, material properties and meteorological conditions.
[0024] The present application has the advantages of: (1) The present application proposes a physical information deep learning framework, which realizes accurate reconstruction of complete electromagnetic interference field distribution from sparse measurement data (density as low as 3%) by integrating multi-scale attention mechanism, source perception convolution and physical guidance constraint.
[0025] (2) The multi-scale attention mechanism of the present application explicitly integrates the physical principle of electromagnetic field propagation into the network structure, captures the electromagnetic propagation characteristics of near field, local field, intermediate field and far field through parallel multi-scale feature extraction, and for the first time converts implicit source field learning into explicit structure modeling.
[0026] (3) The method of the present application can still maintain high prediction accuracy under extreme measurement sparsity (sampling density 3%), and provides a practical solution for dynamic spectrum access, real-time interference mapping and electromagnetic compatibility analysis, significantly reducing the deployment cost of large-scale spectrum management infrastructure.
[0027] Other advantages, objects and features of the present application will be set forth in part in the following specification, and in part will become apparent to those skilled in the art from the examination of the following specification, or can be learned from practice of the present application. The objects and other advantages of the present application can be realized and attained by the following description. BRIEF DESCRIPTION OF DRAWINGS
[0028] In order to make the purpose, technical scheme and advantages of the present application clearer, the preferred detailed description of the present application will be combined with the drawings as follows, wherein: Figure 1 The implementation step chart of the present application for interference prediction in multiple environments is shown in the figure. Figure 2 The overall architecture chart of MSAE-Net of the present application is shown in the figure. Figure 3 The detailed encoder-decoder structure chart of MSAE-Net of the present application is shown in the figure. DETAILED DESCRIPTION
[0029] The present application is described and explained more fully with reference to the following detailed description. Other advantages of the present application will be realized and appreciated by those skilled in the art, and it will be understood to those skilled in the art that the application can be practiced with modification and alteration, and that the statement and drawings should be construed as illustrative only. Furthermore, it will be understood to those skilled in the art that the following description of exemplary embodiments of the present application are not limiting of the present application, but merely illustrate the principles of the present application. It is therefore contemplated to cover by the present application any and all modifications, combinations, and equivalents that fall within the spirit and scope of the present application.
[0030] The accompanying drawings are included to provide a further understanding of the application, and are incorporated in and constitute a part of this specification. The drawings illustrate embodiments of the present application and, together with the description, serve to explain the principles of the present application. In the drawings:
[0031] The same or similar components in the drawings of the embodiments of the present application correspond to the same or similar components; in the description of the present application, it should be understood that if the terms "upper", "lower", "left", "right", "front", "back" and the like indicate the orientation or positional relationship based on the orientation or positional relationship shown in the drawings, they are only for the convenience of describing the present application and simplifying the description, and do not indicate or imply that the devices or elements referred to must have a particular orientation, be constructed and operated in a particular orientation, therefore the terms describing the positional relationship in the drawings are only for illustrative purposes, and cannot be understood as a limitation of the present application, for those skilled in the art, the specific meaning of the above terms can be understood according to the specific circumstances.
[0032] Figure 1 The embodiment of the method for predicting electromagnetic interference field based on multi-scale attention enhanced U-Net includes the following steps: S1: input data preparation: receiving four-channel input data containing sparse field measurement data, sampling mask, obstacle layout and transmitting source position; S2: multi-scale feature extraction: multi-scale feature extraction of input data is performed through an encoder path, and a multi-scale attention mechanism is used to capture the electromagnetic propagation characteristics of near field, middle field and far field; S3: source perception convolution processing: source perception convolution layer is used to dynamically adjust the convolution kernel weight according to the position and coupling relationship of electromagnetic source; S4: edge preservation processing: the edge preservation module uses Sobel operator to detect and preserve the field discontinuity at the boundary of the obstacle; S5: decoder reconstruction: the features are up-sampled to the original resolution through a symmetrical decoder path, and enhanced skip connection is used to retain spatial details; S6: Physical constraint optimization: network training using a comprehensive loss function containing reconstruction fidelity loss, spatial gradient consistency loss, unified physical constraint loss and detail preservation loss; S7: Output complete field distribution: generate complete electromagnetic interference field distribution prediction results.
[0033] Further, S1 input data is standardized to 200x200 pixel four-channel data, the first channel is sparse field measurement data, containing field intensity values only at sampling positions; the second channel is a sampling mask, identifying measurement positions; the third channel is an obstacle layout, representing the distribution of obstructions in the environment; the fourth channel is a transmission source distribution, identifying the position and power information of the interference source. Based on Maxwell's electromagnetic theory, the field reconstruction problem can be formulated as a physical constraint optimization:
[0034]
[0035] Basic electromagnetic constraints derived for Maxwell's equations, S represents the interference source distribution, and O represents the obstacle environment map.
[0036] Further, Figure 2 The specific structure framework of S2-S7 is as follows: The multi-scale attention mechanism extracts features through multiple parallel scale branches, including calculating features on k different scales , including 1x1, 3x3, 5x5 and 7x7 convolution kernels; Calculate physical information attention weights ; define distance-dependent electromagnetic coupling functions, including near-field coupling, intermediate coupling, far-field coupling and scattering coupling; Finally, calculate the multi-scale feature fusion .
[0037] Further, the weight of the source perception convolution layer in S3 is dynamically adjusted according to the source position, and the effective coupling function is:
[0038] Including source amplitude factor , propagation attenuation and phase factor and other physical effects.
[0039] Further, the edge preservation module in S4 uses the Sobel operator to calculate the spatial gradient, detects field discontinuity and obstacle boundaries through gradient information, and uses enhancement weights to fuse gradient information with main feature extraction.
[0040] S41: Calculate spatial gradient using Sobel operator:
[0041] S42: Detecting field discontinuity and obstacle boundary through gradient information; S43: Using enhanced weight Fusing gradient information with main feature extraction.
[0042] Further, the decoder in S5 adopts a symmetrical structure, sequentially upsamples the features through four decoding blocks, each decoding block uses an upsampling operation to expand the spatial resolution, fuses the features of the corresponding layer of the encoder through enhanced skip connection, and applies dropout at the bottleneck layer to prevent overfitting. The symmetrical structure of the decoder includes: S51: Sequentially upsample the features from 12x12x1024 to 200x200x1 through four decoding blocks; S52: Each decoding block uses a 2x2 upsampling operation to expand the spatial resolution by 2; S53: Fuse the features of the corresponding layer of the encoder through enhanced skip connection, and the skip connection integrates a multi-scale attention mechanism; S54: Apply dropout (probability 0.15) at the bottleneck layer to prevent overfitting.
[0043] Further, the comprehensive loss function in S6 includes reconstruction fidelity loss, spatial gradient consistency loss, unified physical constraint loss, and detail preservation loss, wherein the unified physical constraint loss includes source region minimum field strength constraint, obstacle region field attenuation constraint, and measurement point fidelity constraint. The comprehensive loss function is defined as:
[0044] The weight of each loss component is: ; Reconstruction fidelity loss: ; Spatial gradient consistency loss: ; Unified physical constraint loss: , including source region minimum field strength constraint , obstacle region field attenuation constraint , and measurement point fidelity constraint ; Detail preservation loss: , , wherein is edge preservation loss, is texture consistency loss.
[0045] Further, the network training of the method adopts an AdamW optimizer, a cosine annealing learning rate scheduling strategy, a batch size of 8 samples, uses gradient accumulation to simulate a larger effective batch size, and adopts gradient clipping and early stopping strategies to prevent overfitting.
[0046] Figure 2 and Figure 3 The specific steps are as follows: 1) Input layer design: The network input layer receives four-channel data with a spatial dimension of 200x200 pixels: the first channel: sparse field measurement data , containing field intensity values only at sampling locations; the second channel: sampling mask identifies measurement locations; the third channel: obstacle layout O, representing the distribution of obstructions in the environment; the fourth channel: emitter distribution S, identifying the location and power information of the interference source.
[0047] All input data is normalized before being fed into the network, standardizing the numerical range to the [0, 1] interval.
[0048] 2) Encoder path implementation: The encoder consists of four levels, each level containing: E-Block 1: input 200x200x4, output 200x200x64, using 3x3 convolution + BN + ReLU; E-Block 2: input 100x100x64, output 100x100x128, 2x2 max pooling followed by 3x3 convolution; E-Block 3: input 50x50x128, output 50x50x256, same pooling and convolution operation; E-Block 4: input 25x25x256, output 25x25x512, continue downsampling and feature extraction.
[0049] Before the encoder starts, the source position information is extracted through a source-aware convolution layer and integrated into the encoder. The source-aware convolution uses two 1x1 convolution layers followed by a sigmoid activation, with 32 output channels. The extracted source features are connected to the main feature map after the first encoding block.
[0050] 3) Bottle neck layer design: The bottle neck layer is located between the encoder and the decoder, with a feature map dimension of 12x12x512. This layer contains: two consecutive 3x3 convolution layers, expanding the channel number to 1024; a Dropout layer (probability 0.15) to prevent overfitting; batch normalization and ReLU activation function.
[0051] 4) Multi-scale attention mechanism: At each level of the encoder, the multi-scale attention module processes the features in parallel: Branch 1: 1x1 convolution, capturing point source characteristics; Branch 2: 3x3 convolution, capturing local field characteristics; Branch 3: 5x5 convolution, capturing intermediate field characteristics; Branch 4: 7x7 convolution, capturing far-field and scattering characteristics.
[0052] The output of each branch is fused by weighting with physical information attention weights:
[0053] where the electromagnetic coupling function According to the distance r and the wave number k, :
[0054] 5) Edge preservation module integrated in each convolution block of the encoder and decoder: Main path: standard 3x3 convolution + BN + ReLU; Edge path: gradient information is extracted using the Sobel operator, including:
[0055] Fusion: edge features are integrated into the main features by a weighting factor; λ.
[0056] 6) Decoder path implementation: The decoder uses a symmetric upsampling structure: D-Block 1: input 12x12x1024, 2x2 upsampling to 25x25x512; D-Block 2: input 25x25x512, upsampling to 50x50x256; D-Block 3: input 50x50x256, upsampling to 100x100x128; D-Block 4: input 100x100x128, upsampling to 200x200x64.
[0057] Each decoding block contains: 2x2 transpose convolution for upsampling; skip connection with the corresponding layer of the encoder; two 3x3 convolutions + BN + ReLU.
[0058] 7) Detail fusion layer: Collect auxiliary outputs Aux1-Aux4 from the four decoding levels, adjust all auxiliary outputs to the same number of channels through 1x1 convolution, use learned weights for weighted fusion, and finally output a single-channel prediction result through 1x1 convolution.
[0059] 8) Output layer: The final output layer uses a 1x1 convolution to map the 64-channel feature map to a single channel with a spatial dimension of 200x200, representing the normalized predicted interference field intensity distribution. The Sigmoid activation function is used to ensure that the output value is in the range [0, 1].
[0060] Finally, it is to be explained that the above embodiments are only used to illustrate the technical solutions of the present application but not to limit the present application. Although the present application is described in detail with reference to the preferred embodiments, it should be understood by those skilled in the art that the technical solutions of the present application can be modified or equivalently replaced without departing from the purpose and scope of the technical solutions, which should be covered in the scope of claims of the present application.
Claims
1. A method for predicting electromagnetic interference fields based on multi-scale attention-enhanced U-Net, characterized in that: The method includes the following steps: S1: Input data preparation: Receive four-channel input data including sparse field measurement data, sampling mask, obstacle layout and emission source location; S2: Multi-scale feature extraction: Multi-scale feature extraction is performed on the input data through the encoder path, and a multi-scale attention mechanism is used to capture the electromagnetic propagation characteristics of the near field, mid field and far field; S3: Source-aware convolution processing: The source-aware convolutional layer dynamically adjusts the kernel weights based on the location and coupling relationship of the electromagnetic source; S4: Edge Preservation Processing: The Sobel operator is used in the edge preservation module to detect and preserve field discontinuities at obstacle boundaries; S5: Decoder Reconstruction: Features are upsampled to the original resolution via a symmetrical decoder path, and spatial details are preserved by utilizing enhanced skip connections; S6: Physical constraint optimization: The network is trained using a comprehensive loss function that includes reconstruction fidelity loss, spatial gradient consistency loss, unified physical constraint loss, and detail preservation loss; S7: Output complete field distribution: Generates complete electromagnetic interference field distribution prediction results.
2. The electromagnetic interference field prediction method based on multi-scale attention-enhanced U-Net according to claim 1, characterized in that: In step S1, the input data is standardized to 200×200 pixel four-channel data, including: First channel: Sparse field measurement data Field intensity values are included only at the sampling location; Second channel: Sampling mask identifies the measurement location; Third passage: Obstacle layout O, representing the distribution of obstructions in the environment; Fourth channel: Transmitter distribution S, indicating the location and power information of the interference source.
3. The electromagnetic interference field prediction method based on multi-scale attention-enhanced U-Net according to claim 1, characterized in that: In S2, the multi-scale attention mechanism extracts features through multiple parallel scale branches. The specific steps are as follows: S21: Calculate features at k different scales The scale includes 1×1, 3×3, 5×5, and 7×7 convolutional kernels; S22: Calculate the physical information attention weights. ,in: S23: Define the distance-dependent electromagnetic coupling function: For wave number, S24: Calculate multi-scale feature fusion, where the function is of order zero. .
4. The electromagnetic interference field prediction method based on multi-scale attention-enhanced U-Net according to claim 1, characterized in that: In S3, the weights of the source-aware convolutional layer are dynamically adjusted according to the source location, specifically defined as follows: Source-aware weights The calculation is as follows: Effective coupling function It includes multiple physical effects: For the source amplitude factor, Indicates propagation attenuation. This is the phase factor.
5. The electromagnetic interference field prediction method based on multi-scale attention-enhanced U-Net according to claim 1, characterized in that: In step S4, the specific implementation steps of the edge preservation module are as follows: S41: Calculate the spatial gradient using the Sobel operator: The Sobel kernel is defined as: S42: Detect field discontinuities and obstacle boundaries using gradient information; S43: Use augmented weights. The gradient information is fused with the main feature extraction.
6. The electromagnetic interference field prediction method based on multi-scale attention-enhanced U-Net according to claim 1, characterized in that: In S5, the decoder adopts a symmetrical structure, including: S51: Upsampling the features from 12×12×1024 to 200×200×1 sequentially through four decoding blocks; S52: Each decoding block uses a 2×2 upsampling operation to double the spatial resolution; S53: Fusing the features of the corresponding layer of the encoder through enhanced skip connections, which integrate a multi-scale attention mechanism; S54: Applying dropout to the bottleneck layer to prevent overfitting, with a dropout probability of 0.
15.
7. The electromagnetic interference field prediction method based on multi-scale attention-enhanced U-Net according to claim 1, characterized in that: In S6, the comprehensive loss function is defined as: The weights of each loss component are: ; Reconstruction fidelity loss: ; Spatial gradient consistency loss: ; Unified physical constraint loss: Including minimum field strength constraints in the source region Field attenuation constraints in obstacle regions and measurement point fidelity constraints ; Loss of detail: , ,in To preserve loss at the edge, For texture consistency loss.
8. The electromagnetic interference field prediction method based on multi-scale attention-enhanced U-Net according to claim 7, characterized in that: The specific definition of the physical constraint loss is: Source constraints: Ensure minimum field strength near the electromagnetic source ; Obstacle constraints: Field attenuation in the forced shading area ; Consistency constraints: To maintain the fidelity of the observed data points.