Probability Density Function Curves Research Articles

A new empirical distribution for the design wave heights under the impact of typhoons

In the South China Sea, where storm surges are frequent, accurate calculation of the design wave heights is crucial to the coastal structures and subsequent disaster prevention and relief. The data sampling methods have developed from annual extreme value sampling to over threshold value sampling to use the precious observed wave heights more effectively. However, the problems still exist, resulting in "sharp peak" and "thick tail" for the over threshold data. And traditional design wave heights estimation models cannot fully fit the tail of the data. The strong typhoon also makes the variation patterns of associated wave height factors highly uncertain. To address these issues, this study proposes a Poisson—Gumbel-Pareto distribution model based on the cumulative distribution functions and the combined extreme value distribution theory. The new model not only takes the impacts of typhoon frequency on wave heights into consideration when the wave heights are over the threshold, it can better represent the peak of the data, and also makes the tail of the probability density function curves gradually descend, that is, the tail data can be fully utilized and fitted using this new method. Taking the measured wave and typhoon data from the Naozhou Ocean Observatory in the South China Sea from 1990 to 2016, the design wave heights in the western Guangdong waters have been calculated on the level of once-in-many-years, the design wave heights for 100-year and 200-year are 9.51m and 10.11m respectively. The analysis shows that the new empirical distribution improves the previous extreme value models' disadvantages, with reasonable and stable projections in the high return periods, which is especially favorable when dealing with extreme sea conditions.

Application of Maximum Entropy Method in Error Statistic of Vertical Launching Device

The installation accuracy of the vertical launcher will have varying degrees of influence on the initial accuracy of the test missile launching. Therefore, it is of great practical significance to study the assembly error characteristics of the launcher. This article will perform ANSYS simulation of the vertical launcher based on the marine force environment, use the maximum entropy method to perform statistical analysis on the extracted simulation error data, and calculate and solve the probability density function and distribution curve of the error data. The analysis results provide guidance for the next step in the accuracy control research of the launcher.

Investigation on the PAPR performance of odd-bit QAM constellations for DFT spread OFDM systems

<p>Adaptive quadrature amplitude modulation (QAM) is a crucial scheme that enables the modern communication systems to overcome the adverse effects of channel fluctuations and maintain an acceptable spectral efficiency. In order to enhance adaptive modulation even further, adoption of odd-bit QAM constellations alongside even constellations had been suggested to improve the transmission efficiency of adaptive QAM modulation. Hence, odd-bit QAM had been extensively studied, analyzed, and tested by many researchers for various patterns, sizes, and communication systems in terms of bit error rate (BER) and peak to average power ratio (PAPR). However, the PAPR performance of odd-bit QAM constellation with single carrier transmission systems adopted in the uplink of the 4G long term evolution (LTE) standards caught almost no research interest. In this paper, the PAPR performance of both cross and rectangular odd-bit QAM constellations are investigated for DFT-S-OFDM systems. Complementary cumulative distribution functions (CCDFs) and probability density functions (PDFs) curves for PAPR are also obtained. Finally, an equation for PAPR PDF is empirically derived for odd-bit cross QAM based DFT-S-OFDM. The results show that cross odd-bit QAM outperforms the rectangular odd-bit QAM in terms of PAPR by 1.02 dB for 8-QAM and 1.3 dB for 32-QAM. This proves that cross odd-bit QAM is a better choice in terms of PAPR for DFT-S-OFDM systems. </p>

Optimization of Distributed Solar Photovoltaic Power Generation in Day-ahead Electricity Market Incorporating Irradiance Uncertainty

This paper proposes a simple and practical approach to model the uncertainty of solar irradiance and determines the optimized day-ahead (DA) schedule of electricity market. The problem formulation incorporates the power output of distributed solar photovoltaic generator (DSPVG) and forecasted load demands with a specified level of certainty. The proposed approach determines the certainty levels of the random variables (solar irradiance and forecasted load demand) from their probability density function curves. In this process of optimization, the energy storage system (ESS) has also been modeled based on the fact that the energy stored during low locational marginal price (LMP) periods and dispatched during high LMP periods would strengthen the economy of DA schedule. The objective of the formulated non-linear optimization problem is to maximize the social welfare of market participants, which incorporates the assured generation outputs of DSPVG, subject to real and reactive power balance and transmission capability constraints of the system and charging/dis-charging and energy storage constraints of ESS. The simulation has been performed on the Indian utility 62-bus system. The results are presented with a large number of cases to demonstrate the effectiveness of the proposed approach for the efficient, economic and reliable operation of DA electricity markets.

Stochastic transition behaviors in a generalized Van der Pol oscillator with fractional damping under Gaussian white noise

The stochastic P-bifurcation behavior of bi-stability in a generalized Van der Pol oscillator with a fractional damping under multiplicative Gaussian white noise excitation is investigated. Firstly, using the principle of minimal mean square error, the non-linear stiffness terms can be equivalent to a linear stiffness which is a function of the system amplitude, and the original system is simplified to an equivalent integer order Van der Pol system. Secondly, the system amplitude?s stationary probability density function is obtained by stochastic averaging. Then according to the singularity theory, the critical parametric conditions for the sys-tem amplitude?s stochastic P-bifurcation are found. Finally, the types of the system?s stationary probability density function curves of amplitude are qualitatively analyzed by choosing the corresponding parameters in each area divided by the transition set curves. The consistency between the analytical results and the numerical results obtained from Monte-Carlo simulation verifies the theoretical analysis in this paper and the method used in this paper can directly guide the design of the fractional order controller to adjust the response of the system.

Stochastic bifurcation analysis of a bistable Duffing oscillator with fractional damping under multiplicative noise excitation

The stochastic P-bifurcation behavior of bi-stability in a Duffing oscillator with fractional damping under multiplicative noise excitation is investigated. Firstly, in order to consider the influence of Duffing term, the non-linear stiffness can be equivalent to a linear stiffness which is a function of the system amplitude, and then, using the principle of minimal mean square error, the fractional derivative term can be equivalent to a linear combination of damping and restoring forces, thus, the original system is simplified to an equivalent integer order Duffing system. Secondly, the system amplitude?s stationary probability density function is obtained by stochastic averaging, and then according to the singularity theory, the critical parametric conditions for the system amplitude?s stochastic P-bifurcation are found. Finally, the types of the system?s stationary probability density function curves of amplitude are qualitatively analyzed by choosing the corresponding parameters in each area divided by the transition set curves. The consistency between the analytical results and the numerical results obtained from Monte-Carlo simulation verifies the theoretical analysis, and the method used in this paper can directly guide the design of the fractional order controller to adjust the behaviors of the system.

Gamma distribution model of diffusion MRI for the differentiation of primary central nerve system lymphomas and glioblastomas.

The preoperative imaging-based differentiation of primary central nervous system lymphomas (PCNSLs) and glioblastomas (GBs) is of high importance since the therapeutic strategies differ substantially between these tumors. In this study, we investigate whether the gamma distribution (GD) model is useful in this differentiation of PNCSLs and GBs. Twenty-seven patients with PCNSLs and 57 patients with GBs were imaged with diffusion-weighted imaging using 13 b-values ranging from 0 to 1000 sec/mm2. The shape parameter (κ) and scale parameter (θ) were obtained with the GD model. Fractions of three different areas under the probability density function curve (f1, f2, f3) were defined as follows: f1, diffusion coefficient (D) <1.0×10-3 mm2/sec; f2, D >1.0×10-3 and <3.0×10-3 mm2/sec; f3, D >3.0 × 10-3 mm2/sec. The GD model-derived parameters were compared between PCNSLs and GBs. Receiver operating characteristic (ROC) curve analyses were performed to assess diagnostic performance. The correlations with intravoxel incoherent motion (IVIM)-derived parameters were evaluated. The PCNSL group's κ (2.26 ± 1.00) was significantly smaller than the GB group's (3.62 ± 2.01, p = 0.0004). The PCNSL group's f1 (0.542 ± 0.107) was significantly larger than the GB group's (0.348 ± 0.132, p<0.0001). The PCNSL group's f2 (0.372 ± 0.098) was significantly smaller than the GB group's (0.508 ± 0.127, p<0.0001). The PCNSL group's f3 (0.086 ± 0.043) was significantly smaller than the GB group's (0.144 ± 0.062, p<0.0001). The combination of κ, f1, and f3 showed excellent diagnostic performance (area under the curve, 0.909). The f1 had an almost perfect inverse correlation with D. The f2 and f3 had very strong positive correlations with D and f, respectively. The GD model is useful for the differentiation of GBs and PCNSLs.

Probability Density Forecasting of Wind Speed Based on Quantile Regression and Kernel Density Estimation

Based on quantile regression (QR) and kernel density estimation (KDE), a framework for probability density forecasting of short-term wind speed is proposed in this study. The empirical mode decomposition (EMD) technique is implemented to reduce the noise of raw wind speed series. Both linear QR (LQR) and nonlinear QR (NQR, including quantile regression neural network (QRNN), quantile regression random forest (QRRF), and quantile regression support vector machine (QRSVM)) models are, respectively, utilized to study the de-noised wind speed series. An ensemble of conditional quantiles is obtained and then used for point and interval predictions of wind speed accordingly. After various experiments and comparisons on the real wind speed data at four wind observation stations of China, it is found that the EMD-LQR-KDE and EMD-QRNN-KDE generally have the best performance and robustness in both point and interval predictions. By taking conditional quantiles obtained by the EMD-QRNN-KDE model as the input, probability density functions (PDFs) of wind speed at different times are obtained by the KDE method, whose bandwidth is optimally determined according to the normal reference criterion. It is found that most actual wind speeds lie near the peak of predicted PDF curves, indicating that the probabilistic density prediction by EMD-QRNN-KDE is believable. Compared with the PDF curves of the 90% confidence level, the PDF curves of the 80% confidence level usually have narrower wind speed ranges and higher peak values. The PDF curves also vary with time. At some times, they might be biased, bimodal, or even multi-modal distributions. Based on the EMD-QRNN-KDE model, one can not only obtain the specific PDF curves of future wind speeds, but also understand the dynamic variation of density distributions with time. Compared with the traditional point and interval prediction models, the proposed QR-KDE models could acquire more information about the randomness and uncertainty of the actual wind speed, and thus provide more powerful support for the decision-making work.

Reliability analysis of laterally loaded piles in marine environments considering the degradation of flexural rigidity

In this paper, the governing equation of chloride diffusion in reinforced concrete square piles is presented. The deterioration process of a pile caused by chloride-induced reinforcement corrosion is analyzed. The time-varying degradation coefficient of the flexural rigidity of piles induced by reinforcement corrosion is obtained. The time-varying failure probability and reliability index under the displacement failure criterion are calculated by using the Monte Carlo simulation method. The sensitivity of the coefficient of variation (COV) and mean value of the significant parameters for the time-varying probability density function (PDF) curves is analyzed. The results indicate that the degradation coefficient of the flexural rigidity and the reliability index decrease with increasing service time, while the failure probability increases with increasing service time. The peak value of the PDF curve of the pile head displacement decreases as the COV of the lateral load, soil elastic modulus and mean value of the lateral load and bending moment increase. The displacement of the pile head corresponding to the peak value increases substantially with increasing lateral load and bending moment, and decreases significantly with increasing soil elastic modulus and concrete elastic modulus.

Study on the Capacity Fading Effect of Low-Rate Charging on Lithium-Ion Batteries in Low-Temperature Environment

By taking a cylindrical LiFePO4 power battery as the research object, the cycle performance test was conducted under different charging current aging paths in a preset low-temperature environment and combined with EIS results to analyze the dynamic characteristics of the battery during the aging process, using the PDF (Probability Density Function) curve to analyze the change of battery energy storage characteristics, and analyze the aging mechanism of the power battery by analyzing the change in the lithium precipitation energy difference. The experimental results showed that under a low-temperature environment, the effect of increasing the charge rate is mainly reflected in slowing down the phase transformation reaction. From the analysis of lithium precipitation of the battery, it can be seen that the main mechanism of the aging of the battery is the loss of active lithium under the conditions of low-rate cycling at sub-zero temperature. The products from the side reaction between the lithium plating and the electrolyte build up on the SEI (Solid Electrolyte Interphase) film, which significantly increases the battery impedance late in the cycle. The work in this paper complements the mechanistic studies of lithium-ion batteries under different aging paths and is also useful for capacity estimation models and research on battery health.

Probabilistic analysis of strip footings on spatially variable soils with linearly increasing shear strength

Abstract This paper presents a probabilistic approach to obtain bearing capacities of strip footings resting on spatially variable marine clay deposits of increasing undrained shear strength with depth. A spatially variable feature is generated with a non-stationary random field where the ratio of the mean and standard deviation of undrained shear strength is maintained. The sparse polynomial chaos expansion is adopted to improve the calculation efficiency of random numerical simulation. Similar to the deterministic case, for a given degree of soil heterogeneity κ, the dimensionless vertical capacity is independent of the change in the slope of undrained shear strength in the stochastic case. The probability density function (PDF) and cumulative distribution function (CDF) curves of normalised bearing capacity for varying κ are constructed and the effect of soil parameters, such as κ, coefficient of variations and autocorrelation distances, on the probabilistic bearing capacity is discussed. Results show a remarkable improvement in efficiency in computation with a minimal number of realisations. Parametric studies obtained demonstrate that κ has the lowest effect on the normalised bearing capacity among the above parameters. In addition, for an anisotropic autocorrelation distance of δh/δv = 5, the greatest reduction in the random vertical capacity vu,p at a low p occurs when the dimensionless vertical autocorrelation distance δv/B = 0.5, which offers a guide in the design of offshore shallow foundations.

541 Anchoring the CLASI-A, a clinical outcome assessment (COA), to the patients’ perspective of their disease

The CLASI-A is a validated tool for physicians to quantify skin disease activity in cutaneous lupus and has been successfully used in several phase 1-3 clinical trials. Here, we evaluated the CLASI-A based on recent guidance from the FDA (USA) to focus on “meaningful within-patient change (i.e., improvement or deterioration from the patients’ perspective) in the concepts assessed by COAs.” We performed a prospective study on 250 patients with cutaneous lupus (1-25 clinic visits each). At nearly every visit, we recorded both a CLASI-A score by the physician (0-70; 70 is worst, 0 is best) and a visual analogue scale by the patient (ptVAS) to indicate his or her perception of cutaneous disease activity (0-10; 0 is worst, 10 is best), i.e., a patient global impression of severity (PGIS). At the initial visit, CLASI-A and ptVAS strongly correlated (Spearman’s rho=-0.226, p=0.0003). To assess within-patient change for each patient with >1 visit (n=229), we selected the pairs of visits with the largest and the smallest differences in ptVAS scores (range of Δs = -10 to +10, in a bell-shaped distribution, two data points per patient). We grouped these data points into five categories according to ΔptVAS: [-10,-6] (markedly worse from the patient’s perspective), [-5,-2] (moderately worse), [-1,1] (minimal or no change), [2,5] (moderately better), and [6,10] (markedly better). Changes in CLASI-A scores between the same two visits differed amongst the five ΔptVAS categories (p<0.0001 by ANOVA on ranks). Empirical cumulative distribution function (eCDF) curves and probability density function (PDF) curves for the changes in CLASI scores showed progressive improvements in CLASI-A as the patients’ perception improved across the five ΔptVAS categories. Moderate worsening or improvement from the patients’ perspectives corresponded to median changes in the CLASI-A score of +16.7% and -32.1%. Our data indicate that the CLASI-A can be anchored to the ptVAS, to assess meaningful within-patient changes in cutaneous disease from the patients’ perspective.

SINR and Rate Distributions for Downlink Cellular Networks

Signal-to-Interference-plus-Noise Ratio (SINR) measures the quality of the radio link between a base station and a User Equipment (UE). This quality indicator is important for the operator as it is used to measure and to optimize network coverage and the maximum user bitrate (which can be deduced from the SINR using the Shannon formula). In this paper, we study the SINR distribution for the downlink assuming that the received signals from serving and interfering cells undergo a path-loss and log-normal shadowing fading. We show that SINR can be approximated by a log-normal random variable and we give analytical expressions of characteristics parameters of its distribution. We then derive the average SINR expression. Also, we give a closed-expression of the rate probability density function and the outage probability and we obtain closed expressions of the average rate and lower and upper bounds of the average rate. For validation, we first compare theoretical results to Monte-Carlo network simulations and show that all the theoretical curves approximate well those obtained by simulations. Then using real measurements obtained by a measurement campaign, we validate the SINR distribution assumption. Based on a test of normality, we find that the recorded SINR samples are normally distributed in the logarithmic domain. We also show that the theoretical probability density function curves and the histogram of the measured samples of SINR are close.

Probabilistic assessment of steel buildings installed with passive control devices under multi-hazard scenario of earthquake and wind

The multi-hazard effects on multi-story steel buildings equipped with energy dissipative passive vibration response control devices are investigated for their performance under earthquake- and wind-induced forces. The passive response control devices include steel bracings, viscous, and viscoelastic dampers. The buildings without and with the passive control devices are modeled as multi-degree of freedom (M-DOF) systems, with the seismic masses lumped at each floor level. The governing differential equations of motion for the uncontrolled and controlled buildings are solved by using Newmark’s time integration approach. The dynamic response quantities are compared in terms of statistical distribution, which include the statistics of top floor acceleration, inter-storey drift, column base shear, and top floor displacement, for the uncontrolled and controlled buildings subjected to earthquake- and wind-induced forces. Conditional probabilities of failure are determined by overlapping the probability density function curves and conducting fragility analysis through cumulative distribution function curves to assess the effectiveness of the control devices against the natural hazards. It is observed that the design of the passive control devices employing additional damping may account for the performance of the structures under the multi-hazard scenario. Moreover, the variability in either of the demands affects the failure probability of the passively controlled structures extensively. Therefore, the probability of failure for a structure experiencing high-amplitude seismic hazard and a maximum wind speed within the critical frequency band during the design life varies largely for different damper schemes, which calls upon careful selection for design of a structure considering such multi-hazard scenario.

Two-Dimensional Discrepancies in Fracture Geometric Factors and Connectivity Between Field-Collected and Stochastically Modeled DFNs: A Case Study of Sluice Foundation Rock Mass in Datengxia, China

This study takes sluice foundation rock mass in Datengxia Hydropower Station, China as an example to examine two-dimensional (2D) discrepancies in fracture geometric factors and connectivity between field-collected and stochastically modeled discrete fracture networks (DFNs). We discover that the trace lengths of field-collected and corresponding modeled DFNs diverge, especially with relatively large lengths. A new variable called minimum spacing sequence (MSS) is proposed, which lists minimum spacing between each fracture midpoint and all the other fracture midpoints. The probability density function curve of MSS shows that the fracture locations do not follow a homogeneous (Poisson) model. The following is performed to examine whether the differences will result in noticeable DFN application errors. The 2D fracture connectivity, which is calculated by depth-first search algorithm, is applied to quantify the discrepancies between field-collected and statistically modeled 2D DFNs. Results show that the statistically modeled DFNs have small clustered fracture path numbers and ratios but with considerably large maximum and average lengths for paths (or for paths longer than certain thresholds) owing to the concentration disadvantage and connection advantage of scattered fractures. We comprehensively compare different 2D DFNs (including field-collected DFN, totally modeled DFNs, DFNs with field fracture size, and DFNs with field fracture locations) and conclude that generating statistically modeled DFNs with identical connectivity features is extremely difficult. Mechanical means that consider connections among fractures are recommended for DFN applications.

PIH57 THE BLEEDING AND PELVIC DISCOMFORT SCALE: MEASURING PATIENT-REPORTED SYMPTOMS IN UTERINE FIBROIDS

The symptom severity scale of the Uterine Fibroid Symptom and Quality of Life (UFS-QOL) instrument contains 8 items and is multi-factorial. The objectives were to determine the subscale (factor) consisting of items relevant to most women with heavy menstrual bleeding (HMB) associated with uterine fibroids (UF), assess its psychometric properties, and determine its responder threshold. Analyses were based on pooled, blinded data from the first one third of patients with HMB enrolled in two Phase 3 studies of relugolix in uterine fibroids (N=254). Factor analyses were done on the UFS-QOL symptom severity scale. The subscale consisting of symptoms experienced by most patients was chosen for further analysis and its psychometric properties were evaluated. The meaningful change threshold was derived by anchor-based methods with change in PGA of symptom severity as an anchor. Within and between anchor group changes in the subscale scores from Baseline to Week 24 were evaluated using pair t-test analysis of variance. Cumulative distribution function and probability density function curves were used as supportive information. Factor analyses revealed that the UFS-QOL symptom severity scale consisted of 3 factors. Factor 1 had three items representing common symptoms of HMB in UF; this factor was named the Bleeding and Pelvic Discomfort (BPD) scale. The BPD scale had no ceiling effects; all response options were used by patients. The items of the BPD scale were found to work cohesively to inform the total score and to adequately distinguish between patients of different severities. Descriptive statistics supported the construct validity and responsiveness. A 20-point change on the BPD scale score (range from 0 - 100) is recommended as the responder threshold. The BPD scale is a short patient-reported outcome measure relevant to patients with HMB due to UF that can be used to assess their symptom burden.

Performance Assessment of Non-Gaussian Control Systems Based on Mixture Correntropy

The performance assessment of any control system plays a key role in industrial control systems. To meet the real-time requirements of modern control systems, a quick and accurate evaluation of the performance of a system is necessary. In this paper, a performance assessment method of a non-Gaussian control system based on mixture correntropy is proposed for non-Gaussian stochastic systems. Mixture correntropy can solve the problem of minimum entropy translation invariance. When the expected output of a system is unavailable, mixture correntropy combined with the estimation of distribution algorithm (EDA) is used for system identification and noise distribution estimation so as to calculate the benchmark of entropy-based performance assessment. When the expected output of a system is available, the mixture correntropy is directly used as the index to evaluate the performance of the system. To improve the real-time aspect of the performance assessment, an improved EDA is presented to obtain the evaluation index more quickly. For both Gaussian and non-Gaussian systems, the mixture correntropy and the improved identification algorithm are used for system performance assessment, and the results are compared with the minimum entropy index and the probability density function (PDF) curve coincident area index. The comparisons verify the rationality and effectiveness of the correntropy index and the rapidity of the improved EDA algorithm.

Stochastic P-bifurcation in a bistable Van der Pol oscillator with fractional time-delay feedback under Gaussian white noise excitation

The stochastic P-bifurcation behavior of a bistable Van der Pol system with fractional time-delay feedback under Gaussian white noise excitation is studied. Firstly, based on the minimal mean square error principle, the fractional derivative term is found to be equivalent to the linear combination of damping force and restoring force, and the original system is further simplified to an equivalent integer order system. Secondly, the stationary Probability Density Function (PDF) of system amplitude is obtained by stochastic averaging, and the critical parametric conditions for stochastic P-bifurcation of system amplitude are determined according to the singularity theory. Finally, the types of stationary PDF curves of system amplitude are qualitatively analyzed by choosing the corresponding parameters in each area divided by the transition set curves. The consistency between the analytical solutions and Monte Carlo simulation results verifies the theoretical analysis in this paper.

Comparative Study on Probabilistic Forecasts of Heavy Rainfall in Mountainous Areas of the Wujiang River Basin in China Based on TIGGE Data

Based on the ensemble precipitation forecast data in the summers of 2014–2018 from the Observing System Research and Predictability Experiment (THORPEX) Interactive Grand Global Ensemble (TIGGE), a comparative study of two multi-model ensemble methods, the Bayesian model average (BMA) and the logistic regression (LR), was conducted. Meanwhile, forecasts of heavy precipitation from the two models over the Wujiang River Basin in China for the summer of 2018 were compared to verify their performances. The training period sensitivity test results show that a training period of 2 years was the best for BMA probability forecast model. Compared with the BMA method, the LR model required more statistical samples and its optimal length of the training period was 5 years. According to the Brier score (BS), for precipitation events exceeding 10 mm with lead times of 1–7 days, the BMA outperformed the LR and the raw ensemble prediction system forecasts (RAW) except for forecasts with a lead time of 1 day. Furthermore, for heavy rainfall events exceeding 25 and 50 mm, the RAW and the BMA performed much the same in terms of prediction. The reliability diagram of the two multi-model ensembles (i.e., BMA and LR) was more reliable than the RAW for heavy and moderate rainfall forecasts, and the BMA model had the best performance. The BMA probabilistic forecast can produce a highly concentrated probability density function (PDF) curve and can also provide deterministic forecasts through analyzing percentile forecast results. With regard to the heavy rainfall forecast in mountainous areas, it is recommended to refer to the forecast with a larger percentile between the 75th and 90th percentiles. Nevertheless, extreme events with low probability forecasts may occur and cannot be ignored.

Stochastic P-bifurcation in a generalized Van der Pol oscillator with fractional delayed feedback excited by combined Gaussian white noise excitations

This paper investigates the stochastic P-bifurcation behavior of bistability in a generalized Van der Pol system with fractional time-delay feedback under additive and multiplicative Gaussian white noise excitations. First, using the minimal mean square error principle, the fractional derivative is found to be equivalent to a linear combination of damping and restoring forces, and the original system is transformed into an equivalent integer-order system. Second, the stationary probability density function of system amplitude is obtained by stochastic averaging, and according to singularity theory, the critical parameters for stochastic P-bifurcation of the system are found. Finally, the different types of stationary probability density function curves of the system amplitude are qualitatively analyzed by choosing the corresponding parameters in each region divided by the transition set curves. The consistency between the analytical solutions and the Monte Carlo simulation results verifies the theoretical analysis in this paper. The method used in this paper can directly guide the design of the fractional-order controller to adjust the response of the system.