the present invention relates to a space - invariant independent component analysis and electronic nose for detection of selective chemicals in an unknown environment , and more specifically , to an approach to analysis of sensor responses to mixtures of unknown chemicals by an electronic nose in an open and changing environment . the following description , taken in conjunction with the referenced drawings , is presented to enable one of ordinary skill in the art to make and use the invention and to incorporate it in the context of particular applications . various modifications , as well as a variety of uses in different applications , will be readily apparent to those skilled in the art , and the general principles , defined herein , may be applied to a wide range of embodiments . thus , the present invention is not intended to be limited to the embodiments presented , but is to be accorded the widest scope consistent with the principles and novel features disclosed herein . furthermore , it should be noted that unless explicitly stated otherwise , the figures included herein are illustrated diagrammatically and without any specific scale , as they are provided as qualitative illustrations of the concept of the present invention . in the following detailed description , numerous specific details are set forth in order to provide a more thorough understanding of the present invention . however , it will be apparent to one skilled in the art that the present invention may be practiced without necessarily being limited to these specific details . in other instances , well - known structures and devices are shown in block diagram form , rather than in detail , in order to avoid obscuring the present invention . the reader &# 39 ; s attention is directed to all papers and documents that are filed concurrently with this specification and are open to public inspection with this specification , and the contents of all such papers and documents are incorporated herein by reference . all the features disclosed in this specification , ( including any accompanying claims , abstract , and drawings ) may be replaced by alternative features serving the same , equivalent or similar purpose , unless expressly stated otherwise . thus , unless expressly stated otherwise , each feature disclosed is one example only of a generic series of equivalent or similar features . furthermore , any element in a claim that does not explicitly state “ means for ” performing a specified function , or “ step for ” performing a specific function , is not to be interpreted as a “ means ” or “ step ” clause as specified in 35 u . s . c . section 112 , paragraph 6 . in particular , the use of “ step of ” or “ act of ” in the claims herein is not intended to invoke the provisions of 35 u . s . c . 112 , paragraph 6 . the present invention provides a space - invariant topology to enable an independent component analysis ( ica ) to solve chemical detection from two unknown mixing chemical sources . the specific system architecture enables the space - invariant ica to achieve a robust independent signal source separation in an unknown open environment . additionally , the system architecture provides an optimal topology for hardware - friendly implementation and an effective architecture for reduced data source requirements . the space - invariant topology along with the techniques of maximum entropy information and natural gradient descent demonstrates that it is effective to separate the two mixed unknown chemical sources with unknown mixing levels to the array of two original sources under insufficient sampled data . from separated sources , they can be identified by projecting them on 11 known chemical sources to find the best match for detection . simulations have shown that 100 % correct detection could be achieved under two cases : a ) under - completed case where the number of input mixtures is larger than number of original chemical sources ; and b ) regular case where the number of input mixtures is the same as the number of original chemical sources . the time - invariant topology approach may face obstacles with an over - complete case , insufficient data and cumbersome topology . four sets of unknown paired mixture sources are collected via an electronic nose , which , in one non - limiting example , is a jpl 16 - enose sensor array ( made at the jet propulsion laboratory , pasadena , calif .). the enose collects the mixture sources in the unknown environment with , at most , 12 samples &# 39 ; data collected . per time - invariant aspect , this appears to be an over - complete case in ica where the number of outputs ( 32 ) is larger than the number of inputs ( 16 ). in one non - limiting example , the approximation of the set up topology of the electronic nose array is shown in fig1 . the collected sensing data x i ( t ) consists of changes in electrical resistance corresponding to sensor response to the unknown mixture of chemical sources s i and their densities ( or concentrations ) at the time t . due to the small separation between the sensors themselves , the input of each sensor resistance is assumed to be uniquely distributed . the sensory data can be modeled as follows : where f i is the unknown non - linear activation , α j is the unknown mixing coefficient of chemical source j , and i is the index of sensor number and n as number of sensor and s j is original source . from equation ( 1 ), using the first order of a taylor expansion , it is rewritten : and s o j is an operating point of the source s j . for each sampling data point in time for the same sensor i , x i fluctuates around its operating point α i and it can be considered as a common bias for x i ( t ) with tε [ t , t + kδt ]. from this argument , equation ( 2 ) can be simplified to : where y is unbiased mixture data , γ is the unknown mixing matrix and s is the chemical source signal . the learning rule based on the maximum entropy algorithm is given by : where g is a non linear function e . g the logistic function or hyperbolic tangent function . the update weight can be calculated as : where w − t is an inverse transport of the nxn weight matrix w , x t is a mixing input vector ( observed vector ), and to simplify equation ( 5 ) using a natural gradient descent technique , the learning rule can be : the most common ica approach is that the number of variables and the number of sources are the same . however , in this study there are two obstacles : 1 ) there are 12 or less samples ( mixing chemical compounds ) from each sensor and the total number of sensor is 16 and they do not have sufficient data set ( at least 16 data samples required ); and 2 ) the number of variables is 16 as number of sensors while the number of compounds in a mixture is 2 and it is considered over complete case . for the time - invariant approach , the data that will flow orthogonal 402 with the time - invariant direction as shown in fig4 and will require 32 outputs ( 16 channels for each chemical compound ). the topology is 16 inputs , 32 outputs and 12 or less sample data which may not be a solvable problem . instead of using a time - invariant approach , the space - invariant approach 404 allows for more data points and enables the square mapping matrix ( the dimension of mixing sources and sensors are the same ). this approach is feasible due to the mathematical model based on equations ( 2 )-( 4 ). the architectures are shown in fig4 . in fig4 , the unbiased input y i ( t ) ( i = 1 − k ) is based on temporal mixture data and the sensory data are spatially invariant . from laboratory set up , a set of single spectra of 11 chemicals using 16 elements is collected in the enose sensor array ; it is averaged and shown in fig5 . a database may also contain the sampled data of air samples from an unknown environment to be used in the calculations . in this embodiment , the number of sensors used is 16 and mixing chemical sources is 2 ; four sets of data will be examined , as shown in table of fig6 . based on the data available provided in fig6 , two experiments were performed : experiment 1 : over - complete case : in this case , a complete sample data ( see column 4 in fig6 ) is used as input to the network and the output size is 2 , as two original sources recovered . experiment 2 : squared case : this is a straight forward case with 2 inputs and 2 outputs when the data was rearranged so that each input is from the same sensor with non - overlapped consecutive sampling times ( t + i * δt and t +( i + 1 )* δt ). in this case , the four subsets of the sampled data in fig6 are studied and all data vectors available are used ( the maximum number of data vectors is 12 ), which is less than the number of sensors ( 16 ). for data set 1 , there are 12 mixtures of chemicals 1 and 7 and the data are shown in fig7 a . fig7 b shows the separated signal sources 1 702 and 7 704 via space - invariant ica of 12 inputs . fig7 c shows the original chemical sources 1 706 and 7 708 by averaging techniques . using the space - invariant ica approach , the recovered signal sources ( chemical 1 and 7 sources ) are shown in fig7 b and the average of the single chemical source 1 and 7 are shown in fig7 c . the spectrum of signal 702 in fig7 b is most closely matched to the signal 706 in fig7 c . similarly the spectrum of signal 704 in fig7 b is most closely matched to the signal 708 in fig7 c . to confirm its performance , the separated sources 1 and 7 were projected by ica technique on the known 11 chemical sources shown in fig5 , the results are provided in the table in fig8 . as can be inferred from fig8 , the single source of chemical 1 has the greatest overlap with the separated source , labeled separated chemical 1 . similarly , single source chemical 7 has the greatest overlap with separated chemical 7 from the mixture shown in fig7 a . data set 2 , a mixture of chemicals 3 and 10 , is plotted in fig9 a below . for the mixture from data set 2 , the performance of space - invariant ica has demonstrated its effective capability to separate the mixture of chemicals 3 and 10 as shown in fig9 b as compared with the original sources in fig9 c . table iv in fig1 clearly indicates that the maximum sensor value in the column for separated chemical 3 was for chemical 3 while the maximum sensor value in the separated chemical 10 column was for chemical 10 . data set 3 , a mixture of chemicals 2 and 13 , is plotted in fig1 a . for the mixture of chemicals 2 and 13 , space - invariant ica has demonstrated its capability to separate the mixture of chemical 2 and 13 as shown in fig1 b as compared with the original sources in fig1 c . fig1 also clearly indicates that the separated chemical with the highest sensor value was chemical 2 among the separated chemical 2 data while chemical 13 captured the highest value among the data for separated chemical 13 . data set 4 , a mixture of chemicals 2 and 5 , is plotted in fig1 a . for mixture from data set 4 shown in fig1 a , space - invariant ica has confirmed its powerful capability to separate the mixture of chemicals 2 and 5 as shown in fig1 b as compared with the original sources in fig1 c . table vi of fig1 shows the correlation between the original and separated source of chemical 2 and 5 . in this experiment , data set 1 ( 16 × 12 ) is paired in columns to obtain the data set 96 × 2 . from this conversion , data values in a single row are the data from the same sensor with consecutive sampling times ( t + i * δt and t +( i + 1 )* δt ); this new data set allows the same number of mixing sources and of original sources . using this new data set , space - invariant ica has produced the results that were validated with the 11 known chemicals . classification was 100 % correct , based on the projection on 11 classes shown in fig5 . to simplify the results , the mean and standard deviation of its projection were tabulated ( separated sources of chemical 1 and chemical 7 ) on each single chemical source , and the results are summarized in the table in fig1 . fig1 is compact information to show that the separation sources from the mixture have successfully identified the original chemical sources . appendix a , incorporated herein by reference , further includes a series of flow chart diagrams providing the order in which data is processed and analyzed in the space - invariant independent component analysis . to separate two ( 2 ) mixing sources from a sixteen ( 16 ) element sensory data array , known as the over - complete case , poses a challenges for mathematical model and network topology . the non - overlapped paire - wise ( i . e . sensor i and sensor i + 1 ) or overlapped pair - wise ( i . e . ( sensor i and sensor i + 1 ) and ( sensor i + 1 and sensor i + 2 ) so on ) may face cumbersome and ineffective techniques . as shown above , the mathematical model has demonstrated space - invariant ica to be an effective topology to overcome insufficient data samples and the over - complete case . moreover , the chemical data itself is fuzzy and inconsistent . the simulation demonstrated that the chemical source separation problem can be solved effectively with complete time sampling data ( k = 12 ) ( under complete case ) and two consecutive sampling data ( k = 2 ). optimal topology may require a model of noise in order to determine the size of the sampling input . moreover , space - invariant ica governed by equation ( 4 ) is only valid when the sampling time is sufficiently small . hence , the sampling time also plays an important role to ensure that the model approach holds . a successful mathematical model to enable the space - invariant ica topology from which infomax and natural gradient descent technique can be applied has been demonstrated , and simulation has confirmed that the modeling is effective and sufficient to perform chemical source separation to enable the smart enose to detect mixtures of chemicals in an open environment .