Recursive Research Articles

Using the Non-Parametric Classifier CART to Model Wood Density

To identify the stand attributes that best explain the variability in wood density, Pinus radiata plantations located in the Chilean coastal sector were studied and modeled. The study area corresponded to stands located in sedimentary soil between the zones of Constitucion and Cobquecura. Within each sampling sector, individual tree variables were recorded and the most relevant stand parameters were estimated. Fifty trees were sampled in each sector, obtaining from each one six wood discs from different stem heights. Each disc was weighed in green and then dried to anhydrous weight, and its basic density was calculated. The profile identification to classify basic density according to stand characteristics was performed through regression trees, a technique based in the use of predictor variables to partition the database using recursive algorithms in regions with similar responses. The objective of the regression tree method is to obtain highly homogenous groups (branches), which are identified using pruning techniques that successively eliminate the branches that least contribute to the classification of the variable of interest. The results found that the stand attributes that contributed significantly to basic density classification were the basal area, the number of trees per hectare, and the mean height.

Non-linear modelling of elastic hysteretic damping in the time domain

Elastic hysteretic damping is defined as the dissipation of energy at a rate that is weakly dependent on frequency of vibration. In this article, we propose that the elastic hysteretic damping can be achieved by a simple modification to the viscous damping model. The proposed modification is based on computing an instantaneous correction factor that recursively depends on the state variables of the system. This correction factor is related to the rate by which the velocity changes with respect to the displacement. The new model compares quite favourably with the other existing solutions in the time-domain and differences between the solutions become evident for higher damping ratios. It is found that the new model predicts consistently the weak variation in the loss factor as a function of frequency. In addition to its simple mathematical formulation, the proposed model is superior to the existing solutions in that it does not require knowledge of the past history of motion neither the knowledge of the excitation frequency and is extensible to any type of loading. Various aspects pertaining to the linearity of the proposed approach are finally discussed.

Experimental Parameter Identification for Moving Base Nonholonomic Robot Consisting Primary Joints

Main aim of this paper is to study dynamic modeling and parameter identification of a nonholonomic moving based robot with chain of serial manipulators with primary joints. One of the main scenarios for analyzing mechanical systems including the no holonomic constraints will be carried out by using Lagrangian formulation and its associated “Lagrange multipliers”. Most research on movable manipulators are limited to robots with only rotating joints. The dynamic equations of robots with rotational and prismatic movement is a very important subject with many applications. The combination of this system with a movable base will bring capability of operating in a wider area than a fixed base manipulator. The moving ability added to the agility of the manipulator, is requirement of many application such as; space explores, rescue operations, operation in hazardous locations, agriculture, and so on. Eliminating these variables from the obtained equations is a time-consuming and cumbersome task. To prevent computing the Lagrange multipliers associated with the nonholonomic constraints; Gibbs-Appell formulation shall be implemented. To automatically derive the motion equations and upgrade the computational efficiency, a recursive algorithm has been derived in the simulation of the system. In the concept of this algorithm, all the mathematical scheme is carried out by only 3 × 3 and 3 × 1 matrices. In the last section, computational modeling for a chain of the manipulator with 3 links and primary joints in each arm is performed to show the aptitude of the proposed scheme in implementation of the motion equations and parametric analysis for complex systems at this level. One of the main purposes of this study is to develop rover technologies for interplanetary explorations.

Prediction of pork supply via the calculation of pig population based on population prediction model

Based on the perspective of pig population system prediction, and accorded to principle of pig months transfer, this paper refers to the modeling principle and method of discrete population quantity prediction model. Then the prediction model of pork supply is derived and established: Firstly, the recursive model of pig population system and estimation model of pork supply was established. Then this study estimated the sum of monthly mortality and culling rate of breeding sows. Furthermore, the method for new left gilts in each month and estimation of breeding sows at each month of age was established. Last, this research established the estimation method model of the initial state of pig population. On this basis, an example calculation is made to predict the monthly pork supply in Heilongjiang Province from January 2016 to March 2018 in the future. The results showed that the prediction model of pork supply based on the prediction of pig population system is an effective perspective to study the forecast of pork supply. In the prediction stage, the prediction accuracy of the number of slaughtered fattened hogs was 96.36% and 97.54%, and the prediction accuracy of pork supply was 98.08% and 93.82%.This study not only lay a theoretical foundation for further study on the balance between pork supply and demand, but also helps to guide pork producers and governments at all levels to make relevant production decisions and plans. Keywords: pig population, breeding sow, pig production, pork supply prediction, population prediction model DOI: 10.25165/j.ijabe.20201302.5303 Citation: Zhang F, Wang F L. Prediction of pork supply via the calculation of pig population based on population prediction model. Int J Agric & Biol Eng, 2020; 13(2): 208–217.

MR T1WI based radiomics and machine learning model for predicting the histopathological grades of soft tissue sarcomas

Objective To explore the value of T1WI based optimal radiomics and machine learning model in predicting histological grades of soft tissue sarcoma. Methods The preoperative MR T1WI data of 113 patients with soft tissue sarcoma in Affiliated Hospital of Qingdao University from May 2009 to November 2018 was analyzed retrospectively. The patients were divided into training set (n=80) and validation set (n=33) using randomly stratified sampling mothed. According to the French Federation Nationale des Centres de Lutte Contre le Cancer (FNCLCC) system, the soft tissue sarcomas were divided into 3 pathological levels (grade Ⅰ-Ⅲ). Grade Ⅰ was defined as low grade, grade Ⅱ and Ⅲ were defined as high grade. In the training set, there were 18 cases with low-grade lesions, 62 cases with high-grade lesions. In the validation set, there were 7 cases with low-grade lesions and 26 cases with high-grade lesions. After a normalizationapproach applied on the image, the radiomics features were extracted in the regions of interest using A.K software. Based on different feature selection methods [with or without recursive feature elimination (RFE)], machine learning algorithm [random forest (RF) or support vector machine (SVM)] and sampling technology [without subsampling, with the synthetic minority oversampling technique (SMOTE) or with random oversampling examples], a total of 12 models were built and each machine-learning combination model was trained using leave-one-out cross validation. The receiver operating characteristic (ROC) curves were used to evaluate the efficacy of the model in predicting the pathological grade of soft tissue sarcoma. Results Among the 12 different machine learning models, the optimal classification model for the prediction of soft tissue sarcoma pathological grade was a combination of RF, RFE and SMOTE, with an area under the curve of 0.909 (95% confidence interval, 0.808-1.000) in the validation set, and the accuracy, sensitivity, and specificity were 84.85%, 86.21%, and 75.00%, respectively. Conclusion The radiomics based machine learning model can be used as an attractive application approach for predicting histological grades of soft tissue sarcoma. Key words: Magnetic resonance imaging; Soft tissue neoplasms; Sarcoma; Radiomics; Artificial intelligence

Assessment of Thailand Socio-Economic Impact towards Greenhouse Gas Mitigation Actions in 2030 Using a Computable General Equilibrium Model

The threat of climate change has urged increased global awareness and apprehension concerning interactions between society, the economy and the environment. Thailand has made a pledge to the United Nations in the form of NDC (National Determined Contributions) committing that the Kingdom will reduce its GHG emissions within 2030 by 20 % through its own efforts and by 25 % with international support on technology transfer and capacity building compared to the emissions in BaU. To support the GHG emissions reduction target, the country has adopted several mitigation plans including its own sustainable development plan, energy efficiency plan, and renewable energy plan, as well as a sustainable transport plan and others. This study aims to elaborate on the country’s socio-economic impact as it strives to achieve the NDC mitigation target by 2030 using a variety of sustainability scenarios. The research approach used a recursive dynamic computable general equilibrium model of Thailand. The result of employing the model showed that introducing GHG mitigation actions in industrial energy efficiency improvement could help increase GDP by 1.9 % in 2030 compared to BaU. However, it would also increase the GHG price by 2,517 THB/t CO2-eq due to the increase of energy usage from enhanced devices and machinery efficiency. Cryogenic carbon capture technology could minimize GDP loss by 0.4 % while drastically reducing GHG price by 3,978 THB/t CO2-eq compared to a scenario without mitigation action. Combined energy efficiency improvement and cryogenic carbon capture actions show the best result since it potentially increases GDP by 2.1 % compared to BaU while also reducing GHG price by 2,747 THB/t CO2-eq compared to a scenario without mitigation action in 2030. Primary energy supplies in this study have also been shifted from high carbon content to clean energy in all mitigation action scenarios.

Seismic response of a highway bridge in case of vehicle-bridge dynamic interaction

The vehicle-bridge interaction (VBI) analysis might be cumbersome and computationally expensive in bridge engineering due to the necessity of solving large number of coupled system of equations. However, VBI analysis can provide valuable insights into the dynamic behavior of highway bridges under specific loading conditions. Hence, this paper presents a numerical study on the dynamic behavior of a conventional highway bridge under strong near-field and far-field earthquake motions considering the VBI effects. A recursive substructuring method, which enables solving bridge and vehicle equations of motion separately and suitable to be adapted to general purpose finite element softwares, was used. A thorough analysis that provides valuable information about the effect of various traffic conditions, vehicle velocity, road roughness and effect of soil conditions under far-field and near-field strong earthquake motions has been presented. A real-life concrete highway bridge was chosen for numerical demonstrations. In addition, sprung mass models of vehicles consist of conventional truck and car models were created using physical and dynamic properties adopted from literature. Various scenarios, of which the results may help to highlight the different aspects of the dynamic response of concrete highway bridges under strong earthquakes, have been considered.

Constrained Recursive Input Estimation of Blending and Mixing Systems

Blending, mixing processes are often supported by advanced process control systems to maximise margins from available component and heat streams. Since these model-based solutions require accurate and reliable data, in weakly instrumented processes, the unknown inlet concentrations and temperatures should be estimated based on the measured outflows. This work presents a method for the reliable estimation of multiple input variables of process units. The key idea is that the input estimation problem is formulated as a constrained recursive estimation task. The applicability of the method is illustrated based on a benchmark model of a blending system. The performance of the method is compared to the moving window and Kalman Filter based solutions. The results show the superior performance of the proposed method and confirm that the apriori knowledge-based constraints improve the robustness of the estimates.

Towards Emotion-Sensitive Conversational User Interfaces in Healthcare Applications.

Perception of emotions and adequate responses are key factors of a successful conversational agent. However, determining emotions in a healthcare setting depends on multiple factors such as context and medical condition. Given the increase of interest in conversational agents integrated in mobile health applications, our objective in this work is to introduce a concept for analyzing emotions and sentiments expressed by a person in a mobile health application with a conversational user interface. The approach bases upon bot technology (Synthetic intelligence markup language) and deep learning for emotion analysis. More specifically, expressions referring to sentiments or emotions are classified along seven categories and three stages of strengths using treebank annotation and recursive neural networks. The classification result is used by the chatbot for selecting an appropriate response. In this way, the concerns of a user can be better addressed. We describe three use cases where the approach could be integrated to make the chatbot emotion-sensitive.

Development of Mathematical Model Based on Artificial Neural Network to Predict Density in Polymerization Process of Styrene

In the chemical industry is important to control the process in order to guarantee the quality and repeatability of the final product. Using sensors in the industrial plant allows a large volume of data to be captured regarding the process. These data can be used for modelling to better understanding and predict the properties of the product in the process. In this work, two types of Artificial Neural Networks (ANN) and the hybrid model Adaptive Neuro Fuzzy Inference System (ANFIS) were used to predict the density of polystyrene along the styrene polymerization process. The dataset used was extracted from the batch of polymerization reactions performed in open-loop, manual control and closed-loop and monitored in each 5 seconds. The Feedforward and Elman ANN has coefficient of correlation (R) equal 94.2%. However, the best topology obtained to Feedforward ANN presents 2 hidden layers and error index RMSE (Root Mean Squared Error) equal to 2.69x10-2. The Elman ANN presents only 1 hidden layer and RMSE of 3.39x10-2. The ANFIS model, in turn, presented R equal to 91% and RMSE of 0.2123. Therefore, ANFIS model did not prove to be the most adequate for the prediction of the polystyrene density in the studied process.Polymerization process pose significant challenges to the industrial community as it is difficult to control with high nonlinearity behaviour and fast dynamic response. The monitoring and control of polymer processes guarantee to the final product the qualities required by the market. Muhammad and Aziz (2017), for instance, studied the production of low density polyethylene (LDPE) and presented a review of the control strategies developed for the LDPE process. The strategies presented were developed in tubular and autoclave reactors and highlights the importance of nonlinear control in polymerization process.The use of sensors to monitor production allows a large volume of process data to be collected. Therefore, it is necessary to construct mathematical models of prediction to interpret and correlate significant patterns, indispensable to assist in the management of decisions and risk analysis.The development of phenomenological models for polymerization is complex and requires deep knowledge about the processes involved in each step. Modelling using artificial intelligence is a strategy that can provide valuable information about the process and allows the construction of an intelligent model capable of predicting process response based on parameters provided. ANN and ANFIS are artificial intelligence tools that can be used to build predictive models. In supervised learning, the data are presented to the network and its main objective is to provide a model that correctly correlates the pairs inputs - outputs of the problems. The use of ANN and ANFIS to predict the density of the polystyrene produced in the process becomes attractive since it is a type of non-linear modelling.Jumari and Mohd-Yusof (2017) presents models to measure melt flow index (MFI) in industrial polypropylene loop reactors using first principle (FP) model and ANN model. The authors state that the prediction of the ANN model is more accurate compare to the MFI calculated by the FP model. Furthermore, the CPU time recorded that ANN model is much faster than FP model.This work aims to develop direct and recursive ANN and ANFIS models from a set of experimental data from a controlled styrene polymerization plant capable of predicting the density of the product satisfactorily.

Towards a digital twin realization of the blade system design study wind turbine blade

This paper describes the application of a novel virtual prototyping methodology to wind turbine blade design. Numeric modelling data and experimental data about turbine blade geometry and structural/dynamical behaviour are combined to obtain an affordable digital twin model useful in reducing the undesirable uncertainties during the entire turbine lifecycle. Moreover, this model can be used to track and predict blade structural changes, due for example to structural damage, and to assess its remaining life. A new interactive and recursive process is proposed. It includes CAD geometry generation and finite element analyses, combined with experimental data gathered from the structural testing of a new generation wind turbine blade. The goal of the research is to show how the unique features of a complex wind turbine blade are considered in the virtual model updating process, fully exploiting the computational capabilities available to the designer in modern engineering. A composite Sandia National Laboratories Blade System Design Study (BSDS) turbine blade is used to exemplify the proposed process. Static, modal and fatigue experimental testing are conducted at Clarkson University Blade Test Facility. A digital model was created and updated to conform to all the information available from experimental testing. When an updated virtual digital model is available the performance of the blade during operation can be assessed with higher confidence.

Application of multiple support vector machine recursive feature elimination model in cancer feature gene selection

Objective To analyze the cancergene expression profile data using multi-support vector machine recursive feature elimination algorithm (MSVM-RFE) and calculate the genetic ranking score to obtain the optimal feature gene subset. Methods Gene expression profiles of bladder cancer, breast cancer, colon cancer and lung cancer were downloaded from GEO (Gene Expression Omnibus) database. The differentially expressed genes were obtained by differential expression analysis. The differential gene expressions were sequenced by MSVM-RFE algorithm and the average test errors of each gene subset were calculated. Then the optimal gene subsetsof four kinds of cancer were obtained according to the minimum average test errors. Based on the datasets of four kinds of cancer characteristic genes before and after screening, linear SVM classifiers were constructed and the classification efficiencies of the optimal feature gene subsets were verified. Results Using the optimal feature gene subsetobtained by MSVM-RFE algorithm, the classification accuracy was improved from (96.77±1.28)% to (99.85±0.46)% for the bladder cancer data, improved from (83.77±4.93)% to (88.30±3.85)% for the breast cancer data, and improved from (72.69±2.41)% to (90.21±3.31)% for the lung cancer data.Besides, theoptimal feature gene subsetkept the classification accuracy of colon cancer classifierat a high level (>99.5%). Conclusions The feature gene extraction based on MSVM-RFE algorithm can improve the classification efficiency of cancer. Key words: Gene expression profile; Recursive feature elimination; Support vector machine; Feature gene

Recursive algorithm for NoGo based on value evaluation

Recursive algorithm for NoGo based on value evaluation

Multivariable Recursively Generated Weighted Shifts and the 2-variable Subnormal Completion Problem

Multivariable Recursively Generated Weighted Shifts and the 2-variable Subnormal Completion Problem

Real-time diagnosis of micro-short circuit for Li-ion batteries utilizing low-pass filters

The fast diagnosis of micro-short circuit cells is crucial for the safety of battery packs. Based on the difference between the “median cell” and other cells in a battery pack, we propose a method that can identify micro-short circuit cells under dynamic conditions in real time. We model cell differences and analyze the model from the perspective of its low frequency variation characteristics. We find that approximate open-circuit voltage differences can be obtained when terminal voltage differences are passed through low-pass filters. Then approximate electric quantity differences can be obtained by utilizing the open-circuit voltage differences and the data smoothing function of low-pass filters. For onboard applications of diagnosis method, the recursive least square is adopted to estimate micro-short circuit currents and resistances utilizing the change of electric quantity differences. We verify and analyze the feasibility of the diagnosis method by using simulation data when the cells in a battery pack have temperature, state of charge, capacity, and internal resistance inconsistency, respectively. Finally, the effectiveness of the diagnosis method is further verified by the triggering experiments of micro-short circuits for real battery packs.

The establishment of a decision tree model for the individualized treatment of spinal metastases based on RPA

Objective To evaluate the prognostic factors of spinal metastases by recursive partitioning analysis (RPA) and establish a decision tree model that can guide clinicians to select individualized treatment. Methods Between January 2011 and December 2015, three institutional databases were searched to identify 169 patients with metastatic spinal tumors underwent surgery. The ratio of male and female was 1.48∶1 (102 males and 67 females). The average age was 59.2±11.1 years. One-hundred eighteen cases of patients were randomly selected as training samples and the remaining 51 cases were verified samples. Preoperative factors were collected and analyzed by RPA methods, including primary tumor, KPS score, Frankel grade, gender, age, visceral metastasis, bone metastasis, spinal metastasis, blood glucose, blood pressure, surgery site, symptoms, surgery interval, serum albumin level and other risk factors. Pearson's Chi-square test was performed to judge the segmentation point of the decision tree model. The decision tree model is built using the Chi-square automatic interaction detector (CHIAD) algorithm and sensitivity and specificity was automatically calculated. Results The median postoperative survival time was 12.6±1.2 months [95%CI(10.1, 15.0)]. 70% were randomly selected as the experimental group (118 cases), and 30% were the verification group (51 cases). The sensitivity was 96.9%. The specificity was 89.8%. The Kappa coefficient was 0.874 in the experimental group. The sensitivity was 95.4%. The specificity was 90.8%. The Kappa coefficient was 0.810 in the test group. The prognostic factors (weight from high to low) based on RPA were Frankel grade (F=8.132, P=0.005), the primary tumor and the KPS score (Equal, F=9.871, P=0.000 and F=11.945, P=0.003), serum albumin and movement time (Equal, F=7.566, P=0.018 and F=9.966, P=0.008). The decision tree model consists of 7 types. Survival time was 51 months, 18 months, 13 months, 8 months, 4 months, 5 months, 9 months in the class I~VII, respectively. According to the difference of RPA survival time, the operation was classified as 3 grade. Total spinal resection was regarded as grade 1, including class I and class II, with an average survival time of over 18 months. Limited operation was regarded as grade 2, including class III, V and VII with survival time in 6-18 months. The conservative treatment was regarded as grade 3, including class Ⅳ and Ⅵ with the survival time was less than 6 months. Conclusion The decision tree model based on RPA for predicting the survival time of spinal metastases can not only identify the prognostic factors, but also classify and grade various prognostic factors; the decision tree model is simple and can guide clinicians to choose the best surgical plan by predicting the survival time. Key words: Methods; Decision trees; Multicenter study; Individualized medicine

Clinical study of intensity modulated radiation therapy for whole-brain radiotherapy with simultaneous integrated boost for brain metastases

Objective To evaluate the efficacy and safety of intensity modulated radiation therapy (IMRT) for whole-brain radiotherapy with simultaneous integrated boost for brain metastases. Methods From January 2015 to January 2017, 120 patients with brain metastases undergoing hospital radiotherapy at the First People′s Hospital and the Second People′s Hospital of Yulin City of Guangxi Zhuang Autonomous Region were randomly divided into three dimensional conformal radiation therapy (3DCRT) group (n=60) and IMRT group (n=60) by envelope method. The curative effects of the two groups and the changes of patient′s mental states before and after radiotherapy were compared. Results The median survival times of patients in 3DCRT group was 11.5 months, and the 6 months and 1 year survival times rates were 83.3% and 35.0% respectively. The median survival times of patients in IMRT group was 12.9 months, and the 6 months and 1 year survival rates were 86.7% and 38.3% respectively. There was no significant difference in prognosis between the two groups (χ2=0.143, P=0.705). There were 8 cases with complete remission, 38 cases with partial remission, 11 cases with stable disease, 3 cases with progressive disease in 3DCRT group, and there were 9, 40, 9, 2 cases in IMRT group respectively. There was no statistically significant difference in short-term efficacy between the two groups (Z=-0.641, P=0.520). Univariate analysis results showed that recursive partitioning analysis (RPA) classification (χ2=53.484, P<0.001), extracranial lesions control (χ2=13.392, P<0.001), whether the midline offset (χ2=4.427, P=0.035) were prognostic factors. Multivariate analysis results showed that RPA classification (HR=2.631, 95%CI: 1.884-3.673, P<0.001), extracranial lesions control (HR=1.697, 95%CI: 1.034-2.786, P=0.037), whether the midline offset (HR=1.787, 95%CI: 1.118-2.856, P=0.015) were the independent prognostic factors for the overall survival. According to the results of activities of mini-mental state examination, the scores of 3DCRT group before and after radiotherapy were 27.12±2.42 and 26.08±2.42 respectively, with a significant difference (t=2.723, P=0.009); and the scores of IMRT group before and after radiotherapy were 26.57±3.09 and 27.20±2.46 respectively, with no significant difference (t=-1.610, P=0.113). Conclusion IMRT for whole-brain radiotherapy with simultaneous integrated boost is an effective treatment for multiple brain metastases, which can relieve symptoms, prolong survival time and improve quality of life. Key words: Neoplasm metastasis; Radiotherapy, intensity-modulated; Carcinoma; Cranial irradiation

Prognostic role of neutrophil-to-lymphocyte ratio in advanced pancreatic ductal adenocarcinoma: impact of baseline fluctuation and changes during chemotherapy.

Inflammation has been implicated in carcinogenesis and progression of pancreatic cancer. The neutrophil-to-lymphocyte ratio is an index of systemic inflammation. We examined the prognostic role of the neutrophil-to-lymphocyte ratio at baseline and the significance of intrapersonal variability of the ratio before and during chemotherapy. Advanced pancreatic adenocarcinoma patients who had received chemotherapy were included. Baseline clinical and biochemical parameters, including the neutrophil-to-lymphocyte ratio, were extracted and analyzed. The neutrophil-to-lymphocyte ratio threshold was determined via recursive partitioning and assessed at diagnosis, prior to chemotherapy and during treatment. Overall survival was estimated via the Kaplan-Meier method and compared between groups with the logrank test. Between 2005 and 2011, 85 patients with locally advanced (n = 38) and metastatic disease were identified: 68% with a neutrophil-to-lymphocyte ratio >3 had shorter median overall survival than patients with a neutrophil-to-lymphocyte ratio <3 (3.4 vs 9.4 months, P = 0.001). Pretreatment, 35% of repeat neutrophil-to-lymphocyte ratios crossed the threshold of 3. A persistently elevated neutrophil-to-lymphocyte ratio >3 suggested a worse overall survival than in patients with a decreasing, increasing or persistently low neutrophil-to-lymphocyte ratio (1.9 vs 8.2, 12.3 and 11.7 months, respectively, P <0.001). Twenty-three percent of patients had a >50% decrease in neutrophil-to-lymphocyte ratio following 4 weeks of chemotherapy, with a trend towards improvement in overall survival (12.5 vs 5.0 mo, P = 0.068). The baseline neutrophil-to-lymphocyte ratio is a validated marker for a poor prognosis. Multiple assessments of the pre-treatment neutrophil-to-lymphocyte ratio might be required. Reduction in the neutrophil-to-lymphocyte ratio during chemotherapy may be associated with improved survival.

Visualisation and ‘diagnostic classifiers’ reveal how recurrent and recursive neural networks process hierarchical structure

We investigate how neural networks can learn and process languages with hierarchical, compositional semantics. To this end, we define the artifical task of processing nested arithmetic expressions,...

SCALING EXPONENT OF COMPACT POLYMER CONFORMATIONS IN NON HOMOGENEOUS MEDIA

We studied compact conformations of a ring polymer adsorbed on the non homogeneous (e. g. porous) substrates. Substrates are represented by the generalization of modified rectangular (MR) lattice – a hierarchically constructed family of fractal lattices embedded in 2d space and parameterized with an integer p >1. Analyzing the exact set of recursive relations for arbitrary value of p, we established an asymptotic form of a number of conformations. As a correction to the leading exponential factor we obtained the stretched exponential factor with the exponent s = 1/2 on each member of the fractal family. Although it is believed that the critical exponent s on fractal lattices is determined not only by the fractal dimension of the underlying lattice but also by other lattice parameters, here we found that s had the same value on different fractals with the same fractal dimension (d f = 2).