measuring the strangeness of strange attractors we study the correlation exponent v introduced recently as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise the exponent v is closely related to the fractal dimension and the information dimension but its computation is considerably easier its usefulness in characterizing experimental data which stem from very high dimensional systems is stressed algorithms for extracting v from the time series of a single variable are proposed the relations between the various measures of strange attractors and between them and the lyapunov exponents are discussed it is shown that the conjecture of kaplan and yorke for the dimension gives an upper bound for v various examples of finite and infinite dimensional systems are treated both numerically and analytically

application of the karhunen loeve procedure for the characterization of human faces the use of natural symmetries mirror images in a well defined family of patterns human faces is discussed within the framework of the karhunen loeve expansion this results in an extension of the data and imposes even and odd symmetry on the eigenfunctions of the covariance matrix without increasing the complexity of the calculation the resulting approximation of faces projected from outside of the data set onto this optimal basis is improved on average

graphical representations and cluster algorithms i discrete spin systems abstract graphical representations similar to the fk representation are developed for a variety of spin systems in several cases it is established that these representations have fkg monotonicity properties which enables characterization theorems for the uniqueness phase and the low temperature phase of the spin system certain systems with intermediate phases and or first order transitions are also described in terms of the percolation properties of the representations in all cases these representations lead in a natural fashion to swendsen wang type algorithms hence at least in the above mentioned instances these algorithms realize the program described by kandel and domany phys rev b 43 1991 8539 8548 all of the algorithms are shown to satisfy a li sokal bound which at least for systems with a divergent specific heat implies critical slowing down however the representations also give rise to invaded cluster algorithms which may allow for the rapid simulation of some of these systems at their transition points

critical frontier of the potts and percolation models on triangular type and kagome type lattices i closed form expressions we consider the potts model and the related bond site and mixed site bond percolation problems on triangular type and kagome type lattices and derive closed form expressions for the critical frontier for triangular type lattices the critical frontier is known usually derived from a duality consideration in conjunction with the assumption of a unique transition our analysis however is rigorous and based on an established result without the need of a uniqueness assumption thus firmly establishing all derived results for kagome type lattices the exact critical frontier is not known we derive a closed form expression for the potts critical frontier by making use of a homogeneity assumption the closed form expression is unique and we apply it to a host of problems including site bond and mixed site bond percolations on various lattices it yields exact thresholds for site percolation on kagome martini and other lattices and is highly accurate numerically in other applications when compared to numerical determination

suboptimal minimum cluster volume cover based method for measuring fractal dimension a new method for calculating fractal dimension is developed in this paper the method is based on the box dimension concept however it involves direct estimation of a suboptimal covering of the data set of interest by finding a suboptimal cover this method is better able to estimate the required number of covering elements for a given cover size than is the standard box counting algorithm moreover any decrease in the error of the covering element count directly increases the accuracy of the fractal dimension estimation in general our method represents a mathematical dual to the standard box counting algorithm by not solving for the number of boxes used to cover a data set given the size of the box instead the method chooses the number of covering elements and then proceeds to find the placement of smallest hyperellipsoids that fully covers the data set this method involves a variant of the fuzzy c means clustering algorithm as well as the use of the minimum cluster volume clustering algorithm a variety of fractal dimension estimators using this suboptimal covering method are discussed finally these methods are compared to the standard box counting algorithm and wavelet decomposition methods for calculating fractal dimension by using one dimensional cantor dust sets and a set of standard brownian random fractal images

discontinuity of the magnetization in one dimensional 1 x y 2 ising and potts models results from percolation theory are used to study phase transitions in one dimensional ising and q state potts models with couplings of the asymptotic form j sub x y approx const absolute value x y sup 2 for translation invariant systems with well defined lim sub x infinity x sup 2 j sub x j sup possibly o or infinity we establish 1 there is no long range order at inverse temperatures beta with beta j sup less than or equal to1 2 if beta j sup q then by sufficiently increasing j sub 1 the spontaneous magnetization m is made positive 3 in models with 0 j sup infinity the magnetization is discontinuous at the transition point as originally predicted by thouless and obeys m beta sub c less than or equal to 1 beta sub c j sup sup 1 2 4 for ising q 2 models with j sup infinity it is noted that the correlation function decays as beta approx c beta absolute value x y sup 2 whenever beta beta sub c points 1 3 are deduced from previous percolation results by utilizing the fortuin kasteleyn representation which also yields other results of independent interest relating potts models with different values of q

on local intrinsic dimension estimation and its applications in this paper we present multiple novel applications for local intrinsic dimension estimation there has been much work done on estimating the global dimension of a data set typically for the purposes of dimensionality reduction we show that by estimating dimension locally we are able to extend the uses of dimension estimation to many applications which are not possible with global dimension estimation additionally we show that local dimension estimation can be used to obtain a better global dimension estimate alleviating the negative bias that is common to all known dimension estimation algorithms we illustrate local dimension estimation s uses towards additional applications such as learning on statistical manifolds network anomaly detection clustering and image segmentation

intrinsic dimensionality estimation with optimally topology preserving maps a new method for analyzing the intrinsic dimensionality id of low dimensional manifolds in high dimensional feature spaces is presented compared to a previous approach by fukunaga and olsen 1971 the method has only linear instead of cubic time complexity with respect to the dimensionality of the input space moreover it is less sensitive to noise than the former approach experiments include id estimation of synthetic data for comparison and illustration as well as id estimation of an image sequence

nonlinear projection trick in kernel methods an alternative to the kernel trick in kernel methods such as kernel principal component analysis pca and support vector machines the so called kernel trick is used to avoid direct calculations in a high virtually infinite dimensional kernel space in this brief based on the fact that the effective dimensionality of a kernel space is less than the number of training samples we propose an alternative to the kernel trick that explicitly maps the input data into a reduced dimensional kernel space this is easily obtained by the eigenvalue decomposition of the kernel matrix the proposed method is named as the nonlinear projection trick in contrast to the kernel trick with this technique the applicability of the kernel methods is widened to arbitrary algorithms that do not use the dot product the equivalence between the kernel trick and the nonlinear projection trick is shown for several conventional kernel methods in addition we extend pca l1 which uses l1 norm instead of l2 norm or dot product into a kernel version and show the effectiveness of the proposed approach

numerical material representation using proper orthogonal decomposition and diffuse approximation from numerical point of view analysis and optimization in computational material engineering require efficient approaches for microstructure representation this paper develops an approach to establish an image based interpolation model in order to efficiently parameterize microstructures of a representative volume element rve based on proper orthogonal decomposition pod reduction of density maps snapshots when the parameters of the rve snapshot are known a priori the geometry and topology of individual phases of a parameterized snapshot is given by a series of response surfaces of the projection coefficients in terms of these parameters otherwise a set of pseudo parameters corresponding to the detected dimensionality of the data set are taken from learning the manifolds of the projection coefficients we showcase the approach and its potential applications by considering a set of two phase composite snapshots the choice of the number of retained modes is made after considering both the image reconstruction errors as well as the convergence of the effective material constitutive behavior obtained by numerical homogenization

on the critical percolation probabilities we prove that the critical probabilities of site percolation on the square lattice satisfy the relation p c p 1 furthermore we prove the continuity of the function percolation probability

simple exponential family pca principal component analysis pca is a widely used model for dimensionality reduction in this paper we address the problem of determining the intrinsic dimensionality of a general type data population by selecting the number of principal components for a generalized pca model in particular we propose a generalized bayesian pca model which deals with general type data by employing exponential family distributions model selection is realized by empirical bayesian inference of the model we name the model as simple exponential family pca sepca since it embraces both the principal of using a simple model for data representation and the practice of using a simplified computational procedure for the inference our analysis shows that the empirical bayesian inference in sepca formally realizes an intuitive criterion for pca model selection a preserved principal component must sufficiently correlate to data variance that is uncorrelated to the other principal components experiments on synthetic and real data sets demonstrate effectiveness of sepca and exemplify its characteristics for model selection

topological localization via signals of opportunity we consider problems of localization disambiguation and mapping in a domain filled with signals of opportunity generated by transmitters one or more static or mobile receivers utilize these signals and from them characterize the domain localize disambiguate etc the tools we develop are topological in nature and rely on interpreting the problem as one of embedding the domain into a sufficiently high dimensional space of signals via a signal profile function varying kinds of signal processing toa tdoa doa etc and discretization are addressed finally we describe experiments that demonstrate the feasibility of these ideas in practice

on the uniqueness of the infinite occupied cluster in dependent two dimensional site percolation we consider dependent site percolation on the two dimensional square lattice the underlying probability measure being invariant and ergodic under each of the translations and invariant under axis reflections if this measure satisfies the fkg condition and if percolation occurs then we show that the infinite occupied cluster is unique with probability 1 and that all vacant star clusters are finite

can the fractal dimension of images be measured abstract fractal dimension is a popular parameter for explaining certain phenomena and for describing natural textures the problem of estimating the fractal dimension of a profile or an image is more difficult and devious than theory suggests this paper studies the accuracy and robustness of two common estimators of fractal dimension box counting and the variation method using two types of data brownian and takagi poor results are demonstrated from applying theory directly called naive estimation data is then interpreted in the most optimistic way possible by matching the estimator to the known fractal dimension experiments quantify the effects of resolution or fineness of sampling and quantization or rounding of sampled values increasing resolution enhances the estimators when true dimension d is large but may possibly due to quantization effect degrade estimators when d is small quantization simply causes shifts in estimates the results suggest that one should not place much reliance in the absolute value of a fractal estimate but that the estimates do vary monotonically with d and might be useful descriptors in tasks such as image segmentation and description

fundamental limitations for estimating dimensions and lyapunov exponents in dynamical systems we show that values of the correlation dimension estimated over a decade from the grassberger procaccia algorithm cannot exceed the value 2 log10n if n is the number of points in the time series when this bound is saturated it is thus not legitimate to conclude that low dimensional dynamics is present the estimation of lyapunov exponents is also discussed

curvilinear component analysis a self organizing neural network for nonlinear mapping of data sets we present a new strategy called curvilinear component analysis cca for dimensionality reduction and representation of multidimensional data sets the principle of cca is a self organized neural network performing two tasks vector quantization vq of the submanifold in the data set input space and nonlinear projection p of these quantizing vectors toward an output space providing a revealing unfolding of the submanifold after learning the network has the ability to continuously map any new point from one space into another forward mapping of new points in the input space or backward mapping of an arbitrary position in the output space

a new approach to dimensionality reduction theory and algorithms this paper applies whitney s embedding theorem to the data reduction problem and introduces a new approach motivated in part by the constructive proof of the theorem the notion of a good projection is introduced which involves picking projections of the high dimensional system that are optimized such that they are easy to invert the basic theory of the approach is outlined and algorithms for finding the projections are presented and applied to several test cases a method for constructing the inverse projection is detailed and its properties including a new measure of complexity are discussed finally well known methods of data reduction are compared with our approach within the context of whitney s theorem

a new estimator of intrinsic dimension based on the multipoint morisita index the size of datasets has been increasing rapidly both in terms of number of variables and number of events as a result the empty space phenomenon and the curse of dimensionality complicate the extraction of useful information but in general data lie on non linear manifolds of much lower dimension than that of the spaces in which they are embedded in many pattern recognition tasks learning these manifolds is a key issue and it requires the knowledge of their true intrinsic dimension this paper introduces a new estimator of intrinsic dimension based on the multipoint morisita index it is applied to both synthetic and real datasets of varying complexities and comparisons with other existing estimators are carried out the proposed estimator turns out to be fairly robust to sample size and noise unaffected by edge effects able to handle large datasets and computationally efficient highlightsa new estimator of intrinsic dimension is suggested comparisons with commonly used estimators of intrinsic dimension are conducted the suggested estimator turns out to be robust to sample size and noise the suggested estimator is able to handle large datasets the suggested estimator is computationally efficient

nonlinear component analysis as a kernel eigenvalue problem a new method for performing a nonlinear form of principal component analysis is proposed by the use of integral operator kernel functions one can efficiently compute principal components in high dimensional feature spaces related to input space by some nonlinear map for instance the space of all possible five pixel products in 16 16 images we give the derivation of the method and present experimental results on polynomial feature extraction for pattern recognition

riemannian manifold learning recently manifold learning has been widely exploited in pattern recognition data analysis and machine learning this paper presents a novel framework called riemannian manifold learning rml based on the assumption that the input high dimensional data lie on an intrinsically low dimensional riemannian manifold the main idea is to formulate the dimensionality reduction problem as a classical problem in riemannian geometry that is how to construct coordinate charts for a given riemannian manifold we implement the riemannian normal coordinate chart which has been the most widely used in riemannian geometry for a set of unorganized data points first two input parameters the neighborhood size k and the intrinsic dimension d are estimated based on an efficient simplicial reconstruction of the underlying manifold then the normal coordinates are computed to map the input high dimensional data into a low dimensional space experiments on synthetic data as well as real world images demonstrate that our algorithm can learn intrinsic geometric structures of the data preserve radial geodesic distances and yield regular embeddings

dimensionality of object representations in monkey inferotemporal cortex we have calculated the intrinsic dimensionality of visual object representations in anterior inferotemporal ait cortex based on responses of a large sample of cells stimulated with photographs of diverse objects because dimensionality was dependent on data set size we determined asymptotic dimensionality as both the number of neurons and number of stimulus image approached infinity our final dimensionality estimate was 93 sd 11 indicating that there is basis set of approximately 100 independent features that characterize the dimensions of neural object space we believe this is the first estimate of the dimensionality of neural visual representations based on single cell neurophysiological data the dimensionality of ait object representations was much lower than the dimensionality of the stimuli we suggest that there may be a gradual reduction in the dimensionality of object representations in neural populations going from retina to inferotemporal cortex as receptive fields become increasingly complex

danco an intrinsic dimensionality estimator exploiting angle and norm concentration abstract in the past decade the development of automatic intrinsic dimensionality estimators has gained considerable attention due to its relevance in several application fields however most of the proposed solutions prove to be not robust on noisy datasets and provide unreliable results when the intrinsic dimensionality of the input dataset is high and the manifold where the points are assumed to lie is nonlinearly embedded in a higher dimensional space in this paper we propose a novel intrinsic dimensionality estimator danco and its faster variant fastdanco which exploit the information conveyed both by the normalized nearest neighbor distances and by the angles computed on couples of neighboring points the effectiveness and robustness of the proposed algorithms are assessed by experiments on synthetic and real datasets by the comparative evaluation with state of the art methodologies and by significance tests

intrinsic dimension estimation of manifolds by incising balls dimensionality reduction is a very important tool in data mining intrinsic dimension of data sets is a key parameter for dimensionality reduction however finding the correct intrinsic dimension is a challenging task in this paper a new intrinsic dimension estimation method is presented the estimator is derived by finding the exponential relationship between the radius of an incising ball and the number of samples included in the ball the method is compared with the previous dimension estimation methods experiments have been conducted on synthetic and high dimensional image data sets and on data sets of the santa fe time series competition and the results show that the new method is accurate and robust

intrinsic dimension estimation the paper reviews state of the art of the methods of intrinsic dimension id estimation the paper defines the properties that an ideal id estimator should have the paper reviews under the above mentioned framework the major id estimation methods underlining their advances and the open problems dimensionality reduction methods are preprocessing techniques used for coping with high dimensionality they have the aim of projecting the original data set of dimensionality n without information loss onto a lower m dimensional submanifold since the value of m is unknown techniques that allow knowing in advance the value of m called intrinsic dimension id are quite useful the aim of the paper is to review state of the art of the methods of id estimation underlining the recent advances and the open problems

on the unicity of discontinuous transitions in the two dimensional potts and ashkin teller models we investigate the uniqueness of the discontinuous phase transition of the q state potts model and the symmetric q s s cubic model generalized ashkin teller model on the integer square lattice with q 1 for the ferromagnetic cases when there is a discontinuous phase transition we show that the discontinuity can only occur at a point of self duality

convergence of numerical box counting and correlation integral multifractal analysis techniques abstract a systematic study of the rate of convergence for a numerical box counting and a numerical correlation integral algorithm for determining the generalized fractal dimension d q are described the algorithms are applied to euclidean point sets koch constructions and a symmetric chaotic mapping the results provide a basis for estimating the size of a fractal subset needed for measurement of the generalized dimension d q in particular the number of points n 5 required to assure 5 convergence of the algorithms is given within a factor of 4 by log 10 n 5 2 54 d q 0 11 for the fractal sets studied here approximately 25 times as many points are needed for 1 convergence approximately 0 1 times as many points are needed for 25 convergence the box based correlation integral algorithm employed in the present studies which is well suited to the analysis of large data sets is also described

potts models and random cluster processes with many body interactions known differential inequalities for certain ferromagnetic potts models with pair interactions may be extended to potts models with many body interactions as a major application of such differential inequalities we obtain necessary and sufficient conditions on the set of interactions of such a potts model in order that its critical point be astrictly monotonic function of the strengths of interactions the method yields some ancillary information concerning the equality of certain critical exponents for potts models this amounts to a small amount of rigorous universality these results are achieved in the context of a fortuin kasteleyn representation of potts models with many body interactions for such a potts model the corresponding random cluster process is a random hypergraph

density and uniqueness in percolation two results on site percolation on thed dimensional lattice d 1 arbitrary are presented in the first theorem we show that for stationary underlying probability measures each infinite cluster has a well defined density with probability one the second theorem states that if in addition the probability measure satisfies the finite energy condition of newman and schulman then there can be at most one infinite cluster with probability one the simple arguments extend to a broad class of finite dimensional models including bond percolation and regular lattices

on the uniqueness of the infinite cluster in the percolation model we simplify the recent proof by aizenman kesten and newman of the uniqueness of the infinite open cluster in the percolation model our new proof is more suitable for generalization in the direction of percolation type processes with dependent site variables

multi class pairwise linear dimensionality reduction using heteroscedastic schemes linear dimensionality reduction ldr techniques have been increasingly important in pattern recognition pr due to the fact that they permit a relatively simple mapping of the problem onto a lower dimensional subspace leading to simple and computationally efficient classification strategies although the field has been well developed for the two class problem the corresponding issues encountered when dealing with multiple classes are far from trivial in this paper we argue that as opposed to the traditional ldr multi class schemes if we are dealing with multiple classes it is not expedient to treat it as a multi class problem per se rather we shall show that it is better to treat it as an ensemble of chernoff based two class reductions onto different subspaces whence the overall solution is achieved by resorting to either voting weighting or to a decision tree strategy the experimental results obtained on benchmark datasets demonstrate that the proposed methods are not only efficient but that they also yield accuracies comparable to that obtained by the optimal bayes classifier

integral opponent colors features for computing visual target distinctness abstract this paper presents a computational target distinctness model to predict human visual search performance for a set of color images visual distinctness metrics can be used to compare and rank target detectability and to quantify background or scene complexity the subjective ranking induced by the psychophysically determined visual target distinctness for 64 human observers is adopted as the reference order the ranking produced by the proposed measure and several computational models are compared with the subjective rank order

estimating the intrinsic dimension of data with a fractal based method in this paper the problem of estimating the intrinsic dimension of a data set is investigated a fractal based approach using the grassberger procaccia algorithm is proposed since the grassberger procaccia algorithm 1983 performs badly on sets of high dimensionality an empirical procedure that improves the original algorithm has been developed the procedure has been tested on data sets of known dimensionality and on time series of santa fe competition

random cluster models on the triangular lattice we study percolation and the random cluster model on the triangular lattice with 3 body interactions starting with percolation we generalize the star triangle transformation we introduce a new parameter the 3 body term and identify configurations on the triangles solely by their connectivity in this new setup necessary and sufficient conditions are found for positive correlations and this is used to establish regions of percolation and non percolation next we apply this set of ideas to the q 1 random cluster model we derive duality relations for the suitable random cluster measures prove necessary and sufficient conditions for them to have positive correlations and finally prove some rigorous theorems concerning phase transitions

an evaluation of intrinsic dimensionality estimators the intrinsic dimensionality of a data set may be useful for understanding the properties of classifiers applied to it and thereby for the selection of an optimal classifier in this paper the authors compare the algorithms for two estimators of the intrinsic dimensionality of a given data set and extend their capabilities one algorithm is based on the local eigenvalues of the covariance matrix in several small regions in the feature space the other estimates the intrinsic dimensionality from the distribution of the distances from an arbitrary data vector to a selection of its neighbors the characteristics of the two estimators are investigated and the results are compared it is found that both can be applied successfully but that they might fail in certain cases the estimators are compared and illustrated using data generated from chromosome banding profiles

data dimensionality estimation methods a survey in this paper data dimensionality estimation methods are reviewed the estimation of the dimensionality of a data set is a classical problem of pattern recognition there are some good reviews algorithms for clustering data prentice hall englewood cliffs nj 1988 in literature but they do not include more recent developments based on fractal techniques and neural autoassociators the aim of this paper is to provide an up to date survey of the dimensionality estimation methods of a data set paying special attention to the fractal based methods

limitations of nonlinear pca as performed with generic neural networks kramer s 1991 nonlinear principal components analysis nlpca neural networks are feedforward autoassociative networks with five layers the third layer has fewer nodes than the input or output layers this paper proposes a geometric interpretation for kramer s method by showing that nlpca fits a lower dimensional curve or surface through the training data the first three layers project observations onto the curve or surface giving scores the last three layers define the curve or surface the first three layers are a continuous function which we show has several implications nlpca projections are suboptimal producing larger approximation error nlpca is unable to model curves and surfaces that intersect themselves and nlpca cannot parameterize curves with parameterizations having discontinuous jumps we establish results on the identification of score values and discuss their implications on interpreting score values we discuss the relationship between nlpca and principal curves and surfaces another nonlinear feature extraction method

time series analysis the fractals of interest in this paper are sets of states attainable by a system whose dynamical behaviour is governed by nonlinear evolution equations experimentally such sets normally associated with chaotic dynamics can occur in chemical reaction systems fluid flows electronic oscillators driven neurons the list has been growing rapidly over the past 10 15 years experimental studies of these systems generally produce time series of measurements that have no obvious fractal properties classical signal processing methods provide little help however recently techniques have been developed that are capable of extracting geometrical information from time series data this paper will review some of these methods and their application to the study of the geometry of invariant sets underlying the dynamics and bifurcations of some experimental and model systems

testing for nonlinearity in time series the method of surrogate data we describe a statistical approach for identifying nonlinearity in time series the method first specifies some linear process as a null hypothesis then generates surrogate data sets which are consistent with this null hypothesis and finally computes a discriminating statistic for the original and for each of the surrogate data sets if the value computed for the original data is significantly different than the ensemble of values computed for the surrogate data then the null hypothesis is rejected and nonlinearity is detected we discuss various null hypotheses and discriminating statistics the method is demonstrated for numerical data generated by known chaotic systems and applied to a number of experimental time series which arise in the measurement of superfluids brain waves and sunspots we evaluate the statistical significance of the evidence for nonlinear structure in each case and illustrate aspects of the data which this approach identifies

critical point of a triangular potts model with two and three site interactions the q state potts model on the triangular lattice with nearest neighbour interactions and three site interactions in half of the triangular faces is considered the exact duality relation is re examined from the point of view of determining its critical point using the continuity and uniqueness arguments the authors determine the exact critical point in the ferromagnetic model it is argued that a transition exists in an antiferromagnetic model only for q 3 a conjecture is then made on the phase diagram for the q 3 isotropic model these results are used to determine the exact criticality of a dilute potts model on the honeycomb lattice

a bond percolation critical probability determination based on the star triangle transformation the bond percolation critical probability of a planar graph with square and triangular faces obtained by inserting a diagonal in every other face of the square lattice is the root of 1 p 6p2 6p3 p5 0 in 0 1 which is approximately 0 404 518 the proof uses the star triangle transformation to determine the parameter value for which the percolative behaviour of the lattice and its dual lattice are identical

graphical representations and cluster algorithms ii we continue the study initiated in part i of graphical representations and cluster algorithms for various models in or related to statistical mechanics for certain models e g the blume emery griffths model and various generalizations we develop fortuin kasteleyn type representations which lead immediately to swendsen wang type algorithms for other models e g the random cluster model that are defined by a graphical representation we develop cluster algorithms without reference to an underlying spin system in all cases phase transitions are related to percolation or incipient percolation in the graphical representation which via the ic algorithm allows for the rapid simulation of these systems at the transition point pertinent examples include the continuum widom rowlinson model the restricted 1 step solid on solid model and the xy model

potts and percolation models on bowtie lattices we give the exact critical frontier of the potts model on bowtie lattices for the case of q 1 the critical frontier yields the thresholds of bond percolation on these lattices which are exactly consistent with the results given by ziff et al j phys a 39 15083 2006 for the q 2 potts model on a bowtie a lattice the critical point is in agreement with that of the ising model on this lattice which has been exactly solved furthermore we do extensive monte carlo simulations of the potts model on a bowtie a lattice with noninteger q our numerical results which are accurate up to seven significant digits are consistent with the theoretical predictions we also simulate the site percolation on a bowtie a lattice and the threshold is s c 0 5479148 7 in the simulations of bond percolation and site percolation we find that the shape dependent properties of the percolation model on a bowtie a lattice are somewhat different from those of an isotropic lattice which may be caused by the anisotropy of the lattice