Tuning the Multivariate Poisson Mixture Model for Clustering Supermarket Shoppers Tom Brijs a Dimitris Karlis b Gilbert Swinnen a Koen Vanhoof a Geert Wets a
a Department of Economics
b Department of Statistics
Limburg University, Universitaire Campus Athens University of Economics
B-3590 Diepenbeek, BELGIUM 76 Patision, Str., 10434 Athens, GREECE ++3211 268757 ++3010 8203503
tom.brijs@luc.ac.be karlis@aueb.gr
ABSTRACT
This paper describes a multivariate Poisson mixture model for clustering supermarket shoppers based on their purchase frequency in a set of product categories. The multivariate nature of the model accounts for cross-selling effects that may exist between the purchases made in different product categories. However, because of computational difficulties, most multivariate approaches limit the covariance structure of the model by including just one common interaction term. Although this reduces the number of parameters significantly, it is often too simplistic in practice since typically multiple interactions exist on different levels. This paper proposes a theoretically correct variance/covariance structure of the multivariate Poisson model, based on domain knowledge or preliminary statistical analysis of significant purchase interaction effects in the data in order to produce the most parsimonious clustering model. As a result, the model does not contain more parameters than necessary whilst still accounting for the existing covariance in the data. From a practical point of view, the model can be used by retail category managers to devise customized merchandising strategies as illustrated in the text.
Categories and Subject Descriptors
[Methods and Algorithms]
Keywords
Mixture models, clustering, EM algorithm, multivariate Poisson. 1. INTRODUCTION
Today’s competition forces consumer goods manufacturers and retailers to differentiate from their competitors by specializing and by offering goods/services that are tailored towards one or more subgroups or segments of the market. The retailer in the fast moving consumer goods (FMCG) sector is, however, highly limited in his ability to segment the market and to focus on the most promising segments, since the typical attraction area of the retail store is too small to afford neglecting a subgroup within the store’s attraction area [7]. Nevertheless, given the huge amount of transactional and loyalty card data being collected by retail stores today, this leads to the intriguing question whether customer segmentation may provide a viable alternative to discover homogeneous customer segments for in-store segmentation, based on their shopping behavior. Indeed, scanner data reflect the frequency of purchases of products or product categories within the retail store and, as a result, they are extremely useful for modeling consumer purchase behavior. From a practical point of view, the discovery of different user groups will help the retail category manager to optimize his marketing mix towards different customer subgroups.
In this context, the use of mixture models (also called model based clustering) has recently gained increased attention as a statistically grounded approach to clustering [16, 18, 20]. More specifically, the multivariate Poisson mixture model will be introduced in this paper and it will be shown that the fully-parameterized model can be greatly simplified by preliminary statistical analysis of the existing purchase interactions in the transactional data, which will enable to remove free parameters from the variance/covariance matrix.
Cadez et al. [5] also used a mixture framework to model sales transaction data. However, their approach is different from ours in a number of respects. Firstly, they assume individual transactions to arise from a mixture of multinomials, whereas in our approach we model a customer’s transaction history (i.e. the aggregation of the counts over the collection of transactions per individual) by means of a multivariate Poisson mixture model. The use of the Poisson distribution in this context is motivated in section 2. Secondly, the focus of their contribution is not really on finding groups of customers in the data, but rather on making predictions and profiling each individual customer’s behavior. This is in contrast with our approach where the objective is to provide a methodology to discover groups of customers in the data having similar purchase rates in a number of product categories. Furthermore, our approach (being multivariate) explicitly accounts for interdependency effects between products which will lead to additional marketing insights as discussed in section 6 of this paper.
Another approach was taken by Ordonez et al. [19]. They used a mixture of Normal distributions to fit a sparse data set of binary vectors corresponding to the raw market baskets in a sales transaction database. Neither do they take correlations between product purchases into account by assuming diagonal covariance matrices.
Other examples include the use of the univariate Poisson mixture model [9] to find groups of customers with similar purchase rates of a particular candy product. The model identified light users (from 0 to 1.08 packs per week), heavy users (from 3.21 to 7.53 packs per week) and extremely heavy users (from 11.1 to 13.6 packs per week) of candy.
An example outside the marketing area includes the bivariate Poisson mixture model by Li et al. [15] for modeling outcomes of manufacturing processes producing numerous defect-free products. The mixture model was used to detect specific process equipment problems and to reduce multiple types of defects simultaneously.
To summarize, our model differs with existing approaches in basically two respects: 1) the use of the Poisson distribution instead of using Normal or multinomial distribution, 2) multivariate character of the model whereas most models treat the joint distribution as the product of the marginal univariate distributions.
This paper is structured as follows. Firstly, section 2 introduces the concept of model based clustering. Secondly, section 3 introduces the general formulation of the multivariate Poisson distribution. The reason is that the model proposed in this paper is a methodological enhancement to the more general multivariate Poisson model. Then, in section 4, model based clustering by using the multivariate Poisson mixture model is discussed together with its limitations. Subsequently, section 5 contains the core methodological contribution of this paper, i.e. the simplification of the general multivariate Poisson mixture model towards a more parsimonious formulation with restricted variance/covariance structure. The idea is implemented by using a concrete retail data example, which clearly illustrates the suggested simplification. Section 6 reports the results of the model and proposes a number of interesting strategic merchandising issues based on those results. Finally, section 7 is reserved for conclusions and topics for future research.
2. MODEL BASED CLUSTERING Historically, cluster analysis has developed mainly through ad hoc methods based on empirical arguments. The last decade, however, there is an increased interest in model-based methodologies, which allow for clustering procedures based on statistical arguments and methodologies. The majority of such procedures are based on the multivariate normal distribution, see [3, 16] and others. The central idea of such models is the use of finite mixtures of multivariate normal distributions.
In general, in model-based clustering, the observed data are assumed to arise from a number of apriori unknown segments that are mixed in unknown proportions. The objective is then to ‘unmix’ the observations and to estimate the parameters of the underlying density distributions within each segment. The idea is that observations (in our case supermarket shoppers) belonging to the same class are similar with respect to the observed variables in the sense that their observed values are assumed to come from the same density distributions, whose parameters are unknown quantities to be estimated. The density distribution is used to estimate the probability of the observed values of the segmentation variable(s), conditional on knowing the mixture component from which those values were drawn. The population of interest thus consists of k subpopulations and the density (or probability function) of the q-dimensional observation y from the j-th subpopulation is f (y | θj)for some unknown vector of parameters θj . The interest lies on finding the values of the non-observable vector ? = (φ1 ,φ2 , …, φn ) which contains the cluster labels for each observation (1, …, n) and φi= j if the i-th observation belongs to the j-th subpopulation. Since, we do not observe the cluster labels, the unconditional density of the vector y is a mixture density of the form
)
|
(
)
(
1
j
i
k
j
j
i
y
f
p
y
fθ
?
=
=
where 0 < p j< 1, and 1
1
=
?
=
k
j
j
p are the mixing proportions. Note
that the mixing proportion is the probability that a randomly selected observation belongs to the j-th cluster.
This is the classical mixture model (see [4, 18]). The purpose of model-based clustering is to estimate the parameters
)
,...,
,
,...,
(1
1
1k
k
p
pθ
θ
?
. Following the maximum likelihood (ML) estimation approach, this involves maximizing the loglikelihood
÷
÷
?
?
?
?
è
?
=?
?
=
=
)
|
(
ln
)
,
;
(
1
1
j
i
k
j
j
n
i
y
f
p
p
y
Lθ
θ
which is not easy since there is often not a closed-form solution for calculating these parameters. Fortunately, due to the finite mixture representation, an expectation-maximization (EM) algorithm is applicable.
The majority of model-based clustering is based on the multivariate normal distribution and hence it is based on the assumption of continuous data. If the data are not continuous, one can circumvent the problem by transforming the data to continuous data by using appropriate techniques, e.g. like correspondence analysis for categorical data. Such approaches, however, have serious limitations because useful information can be lost during transformation. Moreover, the normality assumption for each component may not be useful, especially for the case of count data with zeros, or when the normal approximation of the discrete underlined distribution is poor. For this reason, we will base our clustering on the multivariate Poisson distribution to allow for modeling the discrete nature of our data.
3. MULTIVARIATE POISSON DISTRIBUTION
Consider a vector X = (X1,X2 , …, X m) where X i‘s are independent and each follows a Poisson Po(λi), i = 1, …, m distribution. Usually, multivariate Poisson distributions (MP) are defined with Poisson marginals by multiplying the vector X with a matrix A of zeros and ones. Suppose that the matrix A has dimensions q x m, then the vector of random variables Y, defined
as Y=A X, follows a multivariate Poisson distribution. The marginal distributions are simple Poisson distributions due to the properties of the Poisson distribution. In practice, the matrix A is structured so as to depict the covariances between the variables Y i and Y j. Such a structure assumes that A has the form
[]m A A A (2)
1
=A
where A i is a matrix of dimensions ÷÷
?
?
??è?×i q q where the columns of the matrix are all the combinations containing exactly i ones and q -i zeros. For instance, for q =3, we need
úúú?ùêêê?é=1000100011A úúú?ùêêê?é=1101010112A ú
úú
?ù
êêê?é=1113A
This construction of the matrix A has been used to define the multivariate Poisson distribution in its general form [11]. An implication of the above definition is that a multivariate Poisson distribution can be defined via a multivariate reduction technique. Suppose that q = 3 and slightly changing the notation to help the exposition, if one starts with independent Poisson random variables X i , with means λi , i ∈S, with S = {1, 2, 3, 12, 13, 23, 123}, then the following representation is obtained for the Y i ’s:
Y 1 = X 1 + X 12 + X 13 + X 123 Y 2 = X 2 + X 12 + X 23 + X 123 Y 3 = X 3 + X 13 + X 23 + X 123
where the form of the matrix A is now:
ú
úú?ù
êêê
?é=111010010110101101001A
and X = (X 1 , X 2 , X 3 , X 12 , X 13 , X 23 , X 123).
Interesting is the fact that X 12 implies a covariance term between Y 1 and Y 2 whilst the term X 123 implies a 3-fold covariance term. Furthermore, it is important to recognize that the λi ‘s have a similar interpretation. Indeed, it can be seen that the mean vector and the covariance matrix of the vector Y is given as:
E (Y ) = A M and Var (Y ) = A ΣA T
where
M = E (X ) = (λ1 , λ2 , …, λm )
and Σ is the variance/covariance matrix of X and is given as:
Σ = Var (X ) = diag (λ1 , λ2 , …, λm )
This brings out the idea to create multivariate distributions with
chosen covariances, i.e. not to include all the possible covariance terms but only to select covariance terms that are useful. Indeed, using all the m -fold covariance terms imposes too much structure while complicating the whole procedure without adding any further insight into the data. For this reason, after a preliminary examination, one may identify interesting covariance terms that may be included into the model and to exclude the others. This corresponds to fix the value of the Poisson parameter, i.e. the corresponding λ’s.
The multivariate Poisson (MP) distribution in its general form, i.e. with all the m -fold covariance terms included, is computationally quite complicated as it involves multiple summations. Perhaps, this difficulty in the calculation of the probability mass function has been the major obstacle in the use of the multivariate Poisson distribution in its most general form. Kano and Kawamura [12] described recursive schemes to reduce the computational burden, but the calculation remains computationally demanding for large dimensions.
The next section introduces the construction, the estimation and the limitations of the general multivariate Poisson mixture model.
4. MODEL BASED CLUSTERING USING
MULTIVARIATE POISSON
Let θ denote the vector of parameters of the general multivariate Poisson distribution. Then it holds that θ = (λ1 , λ2 , …, λm ). We denote the multivariate Poisson distribution with parameter vector θ as MP(θ) and its probability mass function by f (y | θ ).
Consider the case where there are k clusters in the data and each cluster has parameter vector θj , j = 1,…k and an observation Y i conditional on the j -th cluster follows a MP(θj ) distribution. Then unconditionally, each Y i follows a k -component multivariate Poisson mixture. To proceed, one has to estimate using ML the parameters of the distribution. This task can be difficult, especially for the general case of a multivariate Poisson distribution with full covariances. In fact, estimation of the parameters can be carried out using the EM algorithm for finite mixtures. The main problem is that one has to maximize the likelihood of several multivariate Poisson distributions, which is extremely cumbersome. In a recent paper, Karlis [13] described the multivariate Poisson distribution. An extended version of this EM algorithm will be used later in section 5.2.
Another important feature is the following. Even if we start with independent Poisson distributions, i.e. without assuming any covariance term, the finite mixture of such distributions will lead to non-zero covariances among the variables. The covariance is induced by the mixing distribution, i.e. the probability distribution with positive probability p j at the simplex θj . This validates the above comment about the lack of need for imposing so much structure. If there is some covariance at the simple multivariate Poisson distribution, then the unconditional covariance can be decomposed in two parts, one due to the intrinsic covariance from the multivariate Poisson distribution and one from the mixing distribution.
5. RESTRICTED COVARIANCE MODEL 5.1 The model
The main contribution of this paper lies within the insight that the variance/covariance structure of the multivariate Poisson mixture model can be greatly simplified by examining the interaction effects between the variables of interest. Indeed, the statistical technique of log-linear analysis [2] is very well suited to assess the statistical significance of all k -fold associations between categorical variables in a multi-way contingency table [10]. The log-linear model is one of the specialized cases of generalized linear models for Poisson-distributed data.
More specifically, for the retailing example under study, this means that non-significant purchase interaction effects (i.e. cross-selling or substitution effects) between products or product categories can be used to cancel out free parameters in the variance/covariance structure of the multivariate Poisson model. In fact, the data used in this study contains the purchase history of 155 households over a period of 26 weeks in 4 product categories, i.e. cake mix (C), cake frosting (F), fabric detergent (D) and fabric softener (S). Log-linear analysis of the frequencies of co-occurrence showed that there was a strong purchase relationship between the 2-fold interactions cake mix and cake frosting, and between fabric detergent and fabric softener, but other k -fold combinations (e.g. 3-fold interaction between cake mix, cake frosting and fabric detergent) did not show to be statistically significant. The significance of the two-fold interactions is also illustrated by scatter matrix shown in figure 1.
These scatter plots show that there exists a strong positive relation between cake mix and cake frosting, and fabric softener and fabric detergent, but not between other combinations of these products. This is further validated by calculating the sample Pearson correlation coefficients for the two-way product combinations. Only two correlations are significantly bigger than zero, i.e., r (mix, frosting) = 0.66, and r (detergent, softener) = 0.48, where r (A,B) denotes the Pearson correlation between the variables A and B.
Therefore, we make use of the latent variables X = (X C , X F , X D , X S , X CF , X DS ) i.e. we use only two covariance terms and, for example, the term X DS is the covariance term between detergent and softener. The interpretation of the parameters λij are similar. The form of the matrix A is now:
ú
ú
ú
ú
ú
?ù
êêêê
ê?
é=101000100100010010010001A
and the vector of parameters is now θ = (λC , λF , λD , λS , λCF ,
λDS ). Thus we have
Y C = X C + X CF Y F = X F + X CF Y D = X D + X DS Y S = X S + X DS
Our definition of the model, in fact, assumes that the conditional probability function is the product of two bivariate Poisson distributions [14], one bivariate Poisson for cake mix and cake frosting, and another bivariate Poisson for fabric detergent and fabric softener. In general, we denote the probability mass
the bivariate Poisson (BP) distribution as 1 , λ2 , λ12), where λ1, λ2, λ12 are the parameters and i
y y i y y C i i y i y y e y e ÷
÷?
?
??è?÷÷????è?÷÷????è?=
?
=??2112),min(0
212211221!!
!),,212
211λλλλλλλλ ...
probability function of an observation D ,Y S ) is given as )
,,;,(),,;,()
|,,,(DS S D S D CF F C F C S D F C λλλy y BP λλλy y BP θy y y y P
k -components model by
?==k
j j S D F C j θy y y y P p y P 1
)|,,,()(
As previously mentioned, the model assumes covariance between all the variables; the covariance is imposed by the mixing distribution. In addition, variables Y C and Y F and Y D and Y S have increased covariance due to their intrinsic covariance induced by our model.
The major issue of the above-defined model is how one can estimate the parameters of the model. For a model with k components the number of parameters equals 7k -1. The likelihood function is quite complicated for direct maximization. Therefore, an EM type of algorithm is used. The algorithm is described in the next section.
5.2 Estimation: The EM Algorithm
The EM algorithm is a popular algorithm for ML estimation in statistics (see, e.g. Dempster et al [8] and McLachlan and Krishnan [17] ). It is applicable to problems with missing values Suppose that we observe data Y obs and that there are unobservable/missing data Y mis , that are perhaps missing values or even non-observable latent variables that we would like to be able to observe. The idea is to augment the observed and the unobserved data, taking the complete data Y com = (Y obs , Y mis ). The key idea is to iterate between two steps, the first step, the E-step, estimates the missing data using the observed data and the current values of the parameters, whilst the second step, the M-step, maximizes the complete data likelihood.
In our case, consider the multivariate reduction proposed above.
The observed data are the q -dimensional vectors Y i = (Y Ci , Y Fi ,
Y Di , Y Si ) whereas the unobservable data are the vectors X i = (X Ci ,
X Fi , X Di , X Si , X CFi , X DSi ) that correspond to the original variables
that led to the multivariate Poisson model plus the vectors Z i = (Z 1i , Z 2i , …, Z ki ) that correspond to the component
memberships with Z ji = 1 if the i -th observation belongs to the j -th
component, and 0 otherwise. Thus, the complete data are the
vectors (Y i , X i , Z i ). If we denote with φ the vector of parameters, then at the E-step of the algorithm, one has to calculate the conditional expectations E (X ji | Y i , φ) for i=1,…n, },,,,,{DS CF S D F C j ∈and E (Z ji | Y i , φ) for i=1,…n, j=1,…k . Note that, given the conditional expectations of the E-step, updating of the parameters is straightforward, as it is simply ML estimation of Poisson parameters. More formally, the procedure can be described as follows:
E-step : Using the current values of parameters calculate
)
()|()|(i j i j ji ij y P y P p data Z E w θ=
=, i=1,…n, j=1,…k
,
)
()
|()|()|(
),,;,()
,min(0
1i y y r CFj Fj Fi Cj Ci k
j DSj Sj Dj Si Di j CFi y P r Po r y Po r y rPo y y BP p x F C ??==??×
=λλλλλλ ,
)
()
|()|()|(
),,;,()
,min(0
1i y y r DSj Sj Si Dj Di k
j CFj Fj Cj Fi Ci j DSi y P r Po r y Po r y rPo y y BP p x S D ??==??×
=λλλλλλ ,
, , ,Dsi Si Si DSi Di Di CFi Fi Fi CFi Ci Ci x y x x y x x y x x y x ?=?=?=?=
M-step : Update the parameters
n
w p n
i ij
j ?==
1
, j =1,…k
??===
n
i ij
n
i ji
ij ij w x w 1
1λ, j =1,…k, },,,,,{DS CF S D F C i ∈
If some convergence criterion is satisfied, stop iterating, otherwise
go back to the E-step.
The similarities with the standard EM algorithm for finite mixtures is obvious. The quantities w ij at the termination of the algorithm are the posterior probabilities that the i -th observation belongs to the j -th cluster and thus they can be used to assign observations to the cluster with higher posterior probability.
Scalability: An important feature of our model is that we may use frequency tables to simplify the calculations. However, the description of the EM algorithm above is given without using frequencies. Indeed, the discrete nature of the data allows for using frequency tables instead of raw data. As a result, the sample size is not at all important for the computing time, since the original data are collapsed to frequency tables. Consequently, our clustering model is scalable to very large databases. In fact, even
with a very large database, the clustering is done without any additional effort. This is of particular interest in real applications, where usually the amount of observations is large. In general, in order to examine the scalability of our algorithm, two issues should be taken into account, the dimensions of the problem and the covariance structure being considered. In fact, it is well known that the speed of the EM algorithm depends on the ‘missing’ information. One could measure the missing information as the ratio of the observed information to the missing information. Our specification of the model has only two latent variables. Adding more latent variables lead to more ‘missing’ information and thus adds more computing time.
The same is true as far as the number of dimensions is concerned. More dimensions lead to more latent variables. If the structure is not more complicated, the algorithm will perform relatively the same, but if the structure is more complicated, then we expect more effort. This is a strong indication that the structure imposed before the fit of the model must remain in moderate levels. However, for a category manager, who is usually responsible for a limited number of product categories, this will pose no practical problems.
Finally, it is worth mentioning that if the model is expanded to a larger number of product categories, the discovery of significant purchase interactions to include in the variance-covariance structure involves the log-linear analysis of a larger collection of contingency tables. However, the cell frequencies in each contingency table can be calculated quite easily from the frequent itemsets in association rule mining [1]. Indeed, it can be shown that the cells in a multi-way contingency table, containing a particular combination of items, can be calculated from their corresponding frequent itemsets. Thus, the specification of the variance-covariance structure for larger model specifications is not really more complicated than for smaller models.
6. EMPIRICAL STUDY
The EM algorithm was implemented sequentially for 1 to 16 components (k = 1,...,16) because the loglikelihood stopped increasing after k >16 (see figure 2). A well-documented drawback of the EM is its dependence on the initial values. In order to overcome this problem, 10 different sets of starting values were chosen at random. In fact, the mixing proportions (p ) were uniform random numbers and rescaled so as to sum at 1, while the λ’s were generated from a uniform distribution on the range of the data points. For each set of starting values, the algorithm was run for 100 iterations without caring about any convergence criterion. Then, from the solution with the largest likelihood, EM iterations were continued until a rather strict convergence criterion was satisfied, i.e. until the relative change of the loglikelihood between two successive iterations was smaller than 10-12. This procedure was repeated 10 times for each value of k . As expected, problems with multiple maxima occurred for large values of k , while for smaller values of k the algorithm usually terminated at the same solution with small perturbations.
There is a wealth of procedures for selecting the number of clusters (or equivalently the number of components) in a mixture [18]. The majority of them have been proposed for the case of clustering via multivariate normal mixtures. In this paper, we
based our selection on the Akaike Information Criterion (AIC) given as:
AIC = -2L k +2 d k
where L k is the value of the maximized loglikelihood for a model with k components and d k is the number of parameters for this model. In our case d k = 7k - 1. Other criteria could have also been used.
number of components for the restricted MVP mixture model
Figure 2 shows that the AIC criterion selects 6 components as the optimal number of components. The depicted values are rescaled so as to be comparable to the loglikelihood. However, it is interesting to note that the AIC for the 5 component solution is extremely close to the 6 component solution and thus for reasons
of parsimony, the 5 component solution could be selected. Furthermore, as we will describe in the sequel, the differences between the 5 and 6 components solutions are minor, hence, it seems plausible to choose the smaller model with 5 components. Figure 3 shows the values of the mixing proportions for the entire range of models used (values of k ranging from 2 to 16). It is apparent from the graph that usually the additional component corresponds to a split of an existing component in two parts, perhaps with some minor modification for the rest components, especially if they have estimates close to the component split. This illustrates the stability of the model and the existence of two larger components, which together cover almost 80% of all observations (see also table 1 and 2).
It is also quite interesting to see that the solution with 5 and 6 components differ slightly. This is extremely interesting from the retailer point of view for which the existence of a limited number of clusters is very important. Indeed, if a large number of clusters would exist, it is impossible for the retailer to manage all segments separately, i.e. it would neither be cost-effective, nor practical to set-up different merchandising strategies for each (small) segment.
to 16 components
Figure 4. Bubble plots for selected pairs of parameters.
Figure 4 on the previous page shows selected bubble-plots for pairs of the parameters. In fact, each graph depicts the joint
mixing distribution for the selected pair. The plots depict both the
5 and the
6 cluster solution. The circles represent the 6-cluster solution and the squares the 5 cluster solution. The size of the circle/square reflects the mixing proportion, the larger the size the larger the mixing proportion. It is clear from the graph that the
two solutions differ only slightly and that the 6 cluster solution
just splits up one of the existing clusters as indicated by the arrows on figure 4.
Table 1 and table 2 contain the parameter estimates for the model with 5 components and 6 components respectively. One can see
that all the components of the 5-cluster solution still exist in the 6-cluster solution, but an additional component appeared (number 2
in the 6-cluster solution) that takes observations from the old components 1 and 3 of the 5-cluster solution. In both solutions, there are 2 clusters of large size that are very similar, indicating
the existence of two relatively stable clusters, which together account for almost 80% of all the customers.
parameters cluster λC λF λCF λD λS λDS p
1 0.207 0.295 1.507 8.431 4.639 0.000 0.088
2 0.427 0.279 1.09
3 1.347 0.031 1.955 0.575
3 0.908 0.441 0.555 0.000 1.030 0.977 0.216
4 2.000 0.792 4.292 0.000 0.524 1.187 0.062
5 4.668 0.000 1.223 3.16
6 1.161 0.702 0.059 Table 1: Estimated parameters for the 5-components model
parameters cluster λC λF λCF λD λS λDS p
1 0.205 0.171 1.523 6.116 0.000 2.061 0.066
2 0.356 0.000 2.06
3 8.698 10.33 0.000 0.019
3 0.42
4 0.311 1.061 1.27
5 0.02
6 2.083 0.578
4 0.897 0.42
5 0.587 0.000 1.047 0.972 0.215
5 1.975 0.77
6 4.28
7 0.000 0.521 1.192 0.062
6 4.684 0.000 1.219 3.040 1.085 0.782 0.059 Table 2: Estimated parameters for the 6-components model
The rule for allocating observation to clusters is the higher posterior probability rule. An observation is allocated to the cluster for which the posterior probability, as measured by w ij, is larger. Note that w ij are readily available after the termination of
the EM algorithm, as they constitute the E-step of the algorithm. Careful examination of the results about the cluster that each customer belongs to reveals that there are only 10 customers that changed cluster between the two solutions, 8 of them switched to
the new cluster.
In order to assess the quality of clustering we calculated the entropy criterion [18] as:
)
/1
ln(
)
ln(
1
)
(11
k
n
k
I
n
i
k
j
ij
ij w
w
??
==
?
=
with the convention that w ij ln(w ij) = 0 if w ij=0. In the case of a perfect classification, for each i there is only one w ij=1 and all the rest are 0. This implies a value for the criterion equal to 1.
On the contrary, for the worst case clustering the value of the criterion is 0. Thus, values near 1 show a good clustering. For our data, we found that I(5)=0.81 and I(6)=0.78, indicating a very good separation between the clusters.
Figure 5: Pairwise clusters
Figure 5 depicts the clusters for the 5-clusters solution. All the pairs of variables are included and the different clusters are indicated by different symbols. Note that pair wise customers with the same values for some pair belong to different clusters due to their values for the rest variables. However some clusters can be identified by particular pairs, like the cluster indicated by ‘*’ for the pair softener and detergent.
An interesting feature of the results in table 2 is the interpretation
of the zero values. If the zero value corresponds to a covariance parameters (i.e. λCF , λDS) then this implies that the two variables are not correlated at all for this component, i.e. the purchase rate
of a product is independent from the purchase rate of the other product. The interpretation of a zero value for the other lambdas is
a little more complicated. A zero value leads to high correlation between the two variables, because the value for this product is usually very similar to the values for the other product.
cluster Cakemix Frosting Detergent Softener Obs.
1 1.667 1.750 9.250 4.833 12
2 1.505 1.419 3.290 2.022 93
3 1.618 0.971 0.912 2.000 34
4 7.12
5 5.625 1.000 1.500 8
5 6.250 1.125 4.125 1.875 8 Overall
mean
2.077 1.548
3.155 2.200 155 Table 3: Cluster centers for the 5-component mixture model
In order to interpret the cluster differences with regard to the original data, table 3 contains the cluster centers for the 5-components solution. The last row contains the sample centers. Looking at the two major clusters (cluster 2 and 3) in table 1, it can be observed that they have a rather different profile. Especially with regard to fabric detergent and fabric softener, both clusters show indeed a rather different behavior.
Cluster 2 shows a very low average purchase rate of fabric softener (λS = 0.031) but a rather high covariance between fabric detergent and fabric softener (λDS = 1.955). This means that, for this cluster, the purchases of fabric softener are largely due to cross-selling with fabric detergent.
This is shown in table 3: people in cluster 2 have an average
purchase rate of fabric softener of 2.022, which is largely due to the covariance with fabric detergent (λDS = 1.955). Consequently, sales of fabric softener in cluster 2 are almost non-existent and if they occur, they have occurred as a result of cross-selling with fabric detergent.
In contrast, cluster 3 shows a somewhat opposite profile. Cluster 3 shows no purchases of fabric detergent at all (λD = 0.000), but again a relatively strong covariance with fabric softener (λDS = 0.977). This means that, for this cluster, the purchases of fabric detergent are exclusively due to cross-selling with fabric softener. This is again shown in table 3: people in cluster 3 have an average purchase rate of fabric detergent of 0.912, which is rather low compared to the total sample average, but this purchase rate is exclusively due to the covariance with fabric softener (λDS = 0.977). Consequently, it can be concluded that sales of fabric
detergent on its own in cluster 3 are non-existing and if they
occur, they have occurred exclusively as a result of cross-selling with fabric softener. These are important findings since they have interesting
implications for marketing decision making, e.g. for targeted
merchandising strategies. For instance, if it is known that in
cluster 3 detergent is purchased only if one also purchases
softener, then softener could be positioned as a loss-leader brand
in this segment. It means that by reducing the price of a leader
brand like of softener (even making a loss on softener) customers
will not only buy more softener, but they will also buy more
detergent, which is highly dependent on softener (as shown in our
analysis). This way, an overall positive profit can be achieved
because the high margin detergent purchase will compensate for
the low margin softener purchase. As such, the softener acts as a
decoy, i.e., the softener brand is positioned in an attractive way in
order to encourage people to purchase detergent too.
Moreover, it is a common practice in retailing to allocate special
display space (e.g. at the end-of-aisle) to some products in the
store. This is done to stimulate the sales of the product on display. For instance, by putting softener on special display, more people will be encouraged to purchase softener. But, since people who buy softener also buy detergent, some of them in this segment will actually purchase detergent too. As a result, by putting softener on special display, sales of both products are boosted. Similarly, coupon programs can be designed to encourage
consumers to increase the number of categories considered on a
shopping trip. In this respect, knowledge about correlated category usage patterns enables category managers to implement
cross-category marketing strategies. For instance, Catalina Marketing [6] sells point-of-purchase electronic couponing systems that can be implemented to print coupons for a particular category, based on the purchases made in other categories. For example, cluster 2 consumers in table 1 could be given detergent coupons, not only to stimulate sales of detergent, but to stimulate sales of softener too, given that the sales of softener are dependent on detergent sales.
Another interesting merchandising strategy would be to put highly interdependent products closer together in the store to please the shopping behavior of so-called ‘run-shoppers’ who typically don’t want to waste time by looking for items in the store.
Finally, cluster 1 in the 5-component solution shows another interesting, yet different profile compared to the two bigger clusters discussed before. Customers in this cluster purchase large quantities of fabric detergent (λD = 8.431) and fabric softener (λS = 4.639), however, the purchases are not correlated at all (λDS = 0.000). This means that although customers in cluster 1 purchase high amounts of detergent and softener, their values are
not the result of cross-selling between both products. Consequently, whatever promotional campaign (like price reduction or special display) on one of both products (say detergent), it will not influence the sales of the other product (softener). An explanation could be that customers in cluster 1 are deal-prone customers, and thus will only purchase a product when it is on promotion or when it is out of stock at home. 7. CONCLUSIONS AND FUTURE RESEARCH 7.1 Conclusions In this paper, a multivariate Poisson mixture model was
introduced to cluster supermarket shoppers based on their purchase rates in a number of predefined product categories. However, instead of using the classical definition of the multivariate Poisson distribution, i.e. with a fully-saturated variance/covariance matrix, we showed that the number of free parameters can be reduced significantly by preliminary examination of the interdependencies between the product category purchases. Knowledge about these interactions can be obtained in different ways. For instance, via a data mining approach where product interdependencies can be discovered by means of association rule analysis, or via statistical analysis of the multi-way contingency table containing the frequencies of product category purchase co-occurrences. The result is a much more parsimonious version of the multivariate Poisson mixture model that is easier and faster to estimate whilst it still accounts for the
existing covariances in the underlying data. Furthermore, an EM
algorithm was presented to estimate the parameters of the model.
The model was tested on a real supermarket dataset including the purchases of 155 households over a period of 26 weeks in 4 product categories. The results of the model indicated that two big clusters, accounting for almost 80% of the observations, could be found with a distinct purchasing profile in terms of the purchase rates and purchase interactions between the product
categories considered. Moreover, a number of concrete merchandising strategies were proposed for the discovered clusters based on this purchasing profile.
7.2 Limitations
The model, however, also has some limitations.
First of all, neither pricing, nor promotional elasticity data were available to this study. As a result, the suggested merchandising strategies should be adopted with care. Indeed, the price and promotion sensitivity of consumers is not known so that results of merchandising strategies, based on knowledge about purchase rates, cannot be predicted with great certainty.
Secondly, the segment specific purchase rates are treated as static parameters in the model, whereas in practice, they can probably change over time. This could result in customers switching from one cluster to another, or entire clusters to change profile over time, i.e. moving from one position to another. This dynamic aspect has not been accounted for in the study.
Finally, although it was shown in this paper how to significantly reduce the complexity of the multivariate Poisson mixture model, and how the model scales towards more observations (e.g. more customers or a longer transaction history), the scalability towards including more product categories requires more empirical study. 7.3 Future Research
Future research topics include the evaluation of Markov Chain Monte Carlo (MCMC) methods to estimate the parameters for larger versions of the multivariate Poisson mixture model. Moreover, EM algorithms for the general multivariate Poisson mixture model discussed in section 4 are of interest. Lastly, issues related to our model will be addressed too. Such an example is the construction of efficient recursive schemes for the evaluation of the probability mass function of a multivariate Poisson model. Such schemes can speed up the estimation of our model considerably.
8. ACKNOWLEDGEMENT
The authors want to thank Prof. Puneet Manchanda from the Graduate School of Business at the University of Chicago for his data support.
9. REFERENCES
[1] Agrawal, R., Mannila, H., Srikant, R., Toivonen, H., and
Verkamo, A. (1996), Fast discovery of association rules, in: Fayyad, U.M., Piatetsky-Shapiro, G., Smyth, P., and Uthurusamy, R. (eds.), Advances in Knowledge Discovery and Data Mining: AAAI Press, pp. 307-328.
[2] Agresti, A., An introduction to categorical data analysis,
Wiley Series in Probability and Statistics, 1996.
[3] Banfield, J.D., and Raftery, A.E., Model-based Gaussian and
non-Gaussian clustering, in: Biometrics, Vol. 49, pp. 803-821.
[4] B?hning, D., Computer assisted analysis of mixtures &
applications in meta-analysis, disease mapping & others, CRC press, 1999. [5] Cadez, I.V., Smyth, P., and Mannila, H. Probabilistic
modeling of transaction data with applications to profiling, visualization, and prediction, in: Proceedings of the Seventh ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, San Francisco (CA), USA, pp.
37-46, 2001.
[6] Catalina Marketing, Catalina CheckOut Coupon ?, Catalina
Marketing Website, https://www.doczj.com/doc/935810046.html,.
[7] Corstjens, M.L., and Corstjens, J., Store Wars: The battle for
mindspace and shelfspace. Wiley, 1995.
[8] Dempster, A.P., Laird, N.M., and Rubin, D., Maximum
likelihood from incomplete data via the EM algorithm.
Journal of the Royal Statistical Society, B 39, 1-38, 1977. [9] Dillon, W.R., and Kumar, A., Latent structure and other
mixture models in marketing: an integrative survey and overview, in: Advanced Methods in Marketing Research, Richard P. Bagozzi (ed.), Cambridge, MA: Blackwell, pp.
295-351, 1994.
[10] DuMouchel, W., Pregibon, D., Empirical bayes screening for
multi-item associations, in: Proceedings of the Seventh ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, San Francisco (CA), USA, pp. 67-76, 2001. [11] Johnson, N., Kotz, S., and Balakrishnan, N., Discrete
multivariate distributions. Wiley, 1997.
[12] Kano, K., and Kawamura, K., On recurrence relations for the
probability function of multivariate generalized Poisson distribution, in: Communications in Statistics – Theory and Methods, Vol. 20, pp. 165-178, 1991.
[13] Karlis, D. An EM algorithm for multivariate Poisson
distribution and related models. (submitted), 2001
[14] Kocherlakota, S., and Kocherlakota, K., Bivariate discrete
distributions, New York: Marcel Dekker, 1992.
[15] Li, C., Lu, J., Park, J., Kim, K., Brinkley P.A., and Peterson,
J.P., Multivariate zero-inflated poisson models and their applications, in: Technometrics, Vol. 41(1), pp. 29-38, 1999. [16] McLachlan, G.J., and Basford, K.E., Mixture models:
inference and applications to clustering, New York: Marcel Dekker, 1988.
[17] McLachlan, G.J. and T. Krishnan, The EM algorithm and
extensions. Wiley, New York, 1997.
[18] McLachlan, G., and Peel, D., Finite mixture models, Wiley
series in probability and statistics, 2000.
[19] Ordonez, C., Omiecinski, E., and Ezquerra, N., A fast
algorithm to cluster high dimensional basket data, in : Proceedings of the IEEE International Conference on Data Mining, San Jose (CA), USA, pp. 633-636, 2001.
[20] Wedel, M., and Kamakura, W.A., Market segmentation:
conceptual and methodological foundations, Kluwer: Dordrecht, 1999.
目录 1命名原因 2分布特点 3关系 4应用场景 5应用示例 6推导 7形式与性质 命名原因 泊松分布实例
泊松分布（Poisson distribution），台译卜瓦松分布（法语：loi de Poisson，英语：Poisson distribution，译名有泊松分布、普阿松分布、卜瓦松分布、布瓦松分布、布阿松分布、波以松分布、卜氏分配等），是一种统计与概率学里常见到的离散机率分布（discrete probability distribution）。泊松分布是以18～19 世纪的法国数学家西莫恩·德尼·泊松（Siméon-Denis Poisson）命名的，他在1838年时发表。这个分布在更早些时候由贝努里家族的一个人描述过。 分布特点 泊松分布的概率函数为： 泊松分布的参数λ是单位时间(或单位面积)内随机事件的平均发生次数。泊松分布适合于描述单位时间内随机事件发生的次数。 泊松分布的期望和方差均为特征函数为 关系 泊松分布与二项分布 泊松分布 当二项分布的n很大而p很小时，泊松分布可作为二项分布的近似，其中λ为np。通常当n≧20,p≦时，就可以用泊松公式近似得计算。 事实上，泊松分布正是由二项分布推导而来的，具体推导过程参见本词条相关部分。 应用场景 在实际事例中，当一个随机事件，例如某电话交换台收到的呼叫、来到某公共汽车站的乘客、某放射性物质发射出的粒子、显微镜下某区域中的白血球等等，以固定的平均瞬时速率λ（或称密度）随机且独立地出现时，那么这个事件在单位时间（面积或体积）内出现的次数或个数就近似地服从泊松分布P(λ)。因此,泊松分布在管理科学、运筹学以及自然科学的某些问题中都占有重要的地位（在早期学界认为人类行为是服从泊松分布，2005年在nature上发表的文章揭示了人类行为具有高度非均匀性）。 应用示例
图像工程作业1：梯度域图像融合(Gradient-Domain Fusion) O VERVIEW This project explores gradient-domain processing, a simple technique with a broad set of applications including blending, tone-mapping, and non-photorealistic rendering. For the core project, we will focus on "Poisson blending"; tone-mapping and NPR can be investigated as bells and whistles. The primary goal of this assignment is to seamlessly blend an object or texture from a source image into a target image. The simplest method would be to just copy and paste the pixels from one image directly into the other. Unfortunately, this will create very noticeable seams, even if the backgrounds are well-matched. How can we get rid of these seams without doing too much perceptual damage to the source region? The insight is that people often care much more about the gradient of an image than the overall intensity. So we can set up the problem as finding values for the target pixels that maximally preserve the gradient of the source region without changing any of the background pixels. Note that we are making a deliberate decision here to ignore the overall intensity! So a green hat could turn red, but it will still look like a hat. We can formulate our objective as a least squares problem. Given the pixel intensities of the source image "s" and of the target image "t", we want to solve for new intensity values "v" within the source region "S":
多聚焦图像融合方法综述 摘要：本文概括了多聚焦图像融合的一些基本概念和相关知识。然后从空域和频域两方面将多聚焦图像融合方法分为两大块，并对这两块所包含的方法进行了简单介绍并对其中小波变换化法进行了详细地阐述。最后提出了一些图像融合方法的评价方法。 关键词：多聚焦图像融合；空域；频域；小波变换法；评价方法 1、引言 按数据融合的处理体系，数据融合可分为：信号级融合、像素级融合、特征级融合和符号级融合。图像融合是数据融合的一个重要分支，是20世纪70年代后期提出的概念。该技术综合了传感器、图像处理、信号处理、计算机和人工智能等现代高新技术。它在遥感图像处理、目标识别、医学、现代航天航空、机器人视觉等方面具有广阔的应用前景。 Pohl和Genderen将图像融合定义为：“图像融合是通过一种特定的方法将两幅或多幅图像合成一幅新图像”，其主要思想是采用一定的方法，把工作于不同波长范围、具有不同成像机理的各种成像传感器对同一场景成像的多幅图像信息合成一幅新的图像。 作为图像融合研究重要内容之一的多聚焦图像融合，是指把用同一个成像设备对某一场景通过改变焦距而得到的两幅或多幅图像中清晰的部分组合成一幅新的图像，便于人们观察或计算机处理。图像融合的方法大体可以分为像素级、特征级、决策级3中，其中，像素级的图像融合精度较高，能够提供其他融合方法所不具备的细节信息，多聚焦融合采用了像素级融合方法，它主要分为空域和频域两大块，即： （1）在空域中，主要是基于图像清晰部分的提取，有梯度差分法，分块法等，其优点是速度快、方法简单，不过融合精确度相对较低，边缘吃力粗糙； （2）在频域中，具有代表性的是分辨方法，其中有拉普拉斯金字塔算法、小波变换法等，多分辨率融合精度比较高，对位置信息的把握较好，不过算法比较复杂，处理速度比较慢。 2、空域中的图像融合 把图像f(x,y)看成一个二维函数，对其进行处理，它包含的算法有逻辑滤波器法、加权平均法、数学形态法、图像代数法、模拟退火法等。 2.1 逻辑滤波器法 最直观的融合方法是两个像素的值进行逻辑运算,如:两个像素的值均大于特定的门限值,
泊松分布及其在实际中的应用 摘要：本文从泊松分布的定义和基本性质出发，举例讨论了泊松分布在实际中的重要应用。 关键字：泊松分布；应用；运筹学；分子生物学；核衰变 泊松分布是法国数学家泊松于1837年引入的，是概率论中的几大重要分布之一。作为一种常见的离散型随机变量的分布，其在实际中有着非常广泛的应用。 1泊松分布的定义及基本知识 1.1定义： （1）若随机变量X 的分布列为 ）， ?=>= =-,2,1,0(0,! )(k k e k X P k λλλ 则称X 服从参数为λ的泊松分布，并用记号X~P(λ)表示。 （2）泊松流： 随机质点流：随机现象中源源不断出现的随机质点构成的序列。 若质点流具有平稳性、无后效性、普通性, 则称该质点流为泊松事件流(泊松流)。 例如某电话交换台收到的电话呼叫数; 到某机场降落的飞机数; 一个售货员接待的顾客数等这些事件都可以看作泊松流。 1.2有关泊松分布的一些性质 （1）满足分布列的两个性质：P(X=k)≥0(k=0,1,2,…）， 且有 1! ! )(0 =?====-∞ =-∞=∞ =-∑∑∑ λλλ λ λλe e k e k e k X P k k k o k k . （2）若随机变量X 服从参数为λ的泊松分布，则X 的期望和方差分别为：E （X)=λ; D(X)=λ. （3）以n ，p 为参数的二项分布，当n →∞，p →0时，使得np=λ保持为正常数，则 λλ--→ -e k p p C k k n k k n ! ) 1(对于k=0,1,2，…一致成立。 由如上定理的条件λ=np 知，当n 很大时，p 很小时，有下面的近似公式 λλ--→ -=e k p p C k P k k n k k n n ! ) 1()( 2泊松分布的应用 对于试验成功概率很小而试验次数很多的随机过程, 都可以很自然的应用于泊松分布的理论。在泊松分布中的概率表达式只含一个参数λ，减少了对参数的确定与修改工作量, 模型构建比较简单, 具有很重要的实际意义。 以下具体举例说明泊松分布在实际中的重要应用。 （1）泊松分布在经济生活中的应用： 泊松分布是经济生活中的一种非常重要的分布形式，尤其是经常被运用在运筹学研究中的一个分布模型。如物料订单的规划，道路交通信号灯的设计，生产计划的安排，海港发
摘要：介绍了遥感影像三种常用的图像融合方式。进行实验，对一幅具有高分辨率的SPOT全色黑白图像与一幅具有多光谱信息的SPOT图像进行融合处理，生成一幅既有高分辨率又有多光谱信息的图像，简要分析比较三种图像融合方式的各自特点，择出本次实验的最佳融合方式。 关键字：遥感影像；图像融合；主成分变换；乘积变换；比值变换；ERDAS IMAGINE 1. 引言 由于技术条件的限制和工作原理的不同，任何来自单一传感器的信息都只能反映目标的某一个或几个方面的特征，而不能反应出全部特征。因此，与单源遥感影像数据相比，多源遥感影像数据既具有重要的互补性，也存在冗余性。为了能更准确地识别目标，必须把各具特色的多源遥感数据相互结合起来，利用融合技术，针对性地去除无用信息，消除冗余，大幅度减少数据处理量，提高数据处理效率；同时，必须将海量多源数据中的有用信息集中起来，融合在一起，从多源数据中提取比单源数据更丰富、更可靠、更有用的信息，进行各种信息特征的互补，发挥各自的优势，充分发挥遥感技术的作用。[1] 在多源遥感图像融合中，针对同一对象不同的融合方法可以得到不同的融合结果，即可以得到不同的融合图像。高空间分辨率遥感影像和高光谱遥感影像的融合旨在生成具有高空间分辨率和高光谱分辨率特性的遥感影像，融合方法的选择取决于融合影像的应用，但迄今还没有普适的融合算法能够满足所有的应用目的，这也意味着融合影像质量评价应该与具体应用相联系。[2] 此次融合操作实验是用三种不同的融合方式（主成分变换融合，乘积变换融合，比值变换融合），对一幅具有高分辨率的SPOT全色黑白图像与一幅具有多
光谱信息的SPOT图像进行融合处理，生成一幅既有高分辨率又有多光谱信息的图像。 2. 源文件 1 、 imagerycolor.tif ，SPOT图像，分辨率10米，有红、绿、两个红外共四个波段。 2 、imagery-5m.tif ，SPOT图像，分辨率5米。 3. 软件选择 在常用的四种遥感图像处理软件中，PCI适合用于影像制图，ENVI在针对像元处理的信息提取中功能最强大，ER Mapper对于处理高分辨率影像效果较好，而ERDAS IMAGINE的数据融合效果最好。[3] ERDAS IMAGINE是美国Leica公司开发的遥感图像处理系统。它以其先进的图像处理技术，友好、灵活的用户界面和操作方式，面向广阔应用领域的产品模块，服务于不同层次用户的模型开发工具以及高度的RS/GIS（遥感图像处理和地理信息系统）集成功能，为遥感及相关应用领域的用户提供了内容丰富而功能强大的图像处理工具。 2012年5月1日，鹰图发布最新版本的ERDAS IMAGINE，所有ERDAS 2011软件用户都可以从官方网站上下载最新版本 ERDAS IMAGINE 11.0.5. 新版本包括之前2011服务包的一些改变。相比之前的版本，新版本增加了更多ERDAS IMAGINE和GeoMedia之间的在线联接、提供了更为丰富的图像和GIS产品。用户使用一个单一的产品，就可以轻易地把两个产品结合起来构建一个更大、更清
很多人在上概率论这门课的时候就没搞明白过泊松分布到底是怎么回事，至少我就是如此。如果我们学习的意义是为了通过考试，那么我们大可停留在“只会做题”的阶段，因为试卷上不会出现“请发表一下你对泊松公式的看法”这样的题目，因为那样一来卷子就变得不容易批改，大部分考试都会出一些客观题，比如到底是泊松分布还是肉松分布。而如果我们学习的目的是为了理解一样东西，那么我们就有必要停下来去思考一下诸如“为什么要有泊松分布？”、“泊松分布的物理意义是什么？”这样的“哲学”问题。 如果我们要向一个石器时代的人解释什么是电话，我们一定会说：“电话是一种机器，两个距离很远的人可以通过它进行交谈”，而不会说：“电话在1876 年由贝尔发明，一台电话由几个部分构成……”（泊松分布在1876 年由泊松提出，泊松分布的公式是……）所以我们问的第一个问题应该是“泊松分布能拿来干嘛？” 泊松分布最常见的一个应用就是，它作为了排队论的一个输入。比如在一段时间t（比如 1 个小时）内来到食堂就餐的学生数量肯定不会是一个常数（比如一直是200 人），而应该符合某种随机规律：假如在 1 个小时内来200 个学生的概率是10%，来180 个学生的概率是20%……一般认为，这种随机规律服从的就是泊松分布。 这当然只是形象化的理解什么是泊松分布，若要公式化定义，那就是：若随机变量X 只取非负整数值0,1,2,..., 且其概率分布服 从则随机变量X 的分布称为泊松分布，记作P(λ)。这个分布是S.-D.泊松研究二项分布的渐近公式时提出来的。泊松分布P (λ)中只有一个参数λ ，它既是泊松分布的均值，也是泊松分布的方差。生活中，当一个随机事件，例如来到某公共汽车站的乘客、某放射性物质发射出的粒子、显微镜下某区域中的白血球等等，以固定的平均瞬时速率λ(或称密度)随机且独立地出现时，那么这个事件在单位时间（面积或体积）内出现的次数或个数就近似地服从
泊松分布的概念及表和查表方法 Poisson分布，是一种统计与概率学里常见到的离散概率分布，由法国数学家西莫恩·德 目录 1命名原因 2分布特点 3关系 4应用场景 5应用示例 6推导 7形式与性质
命名原因 泊松分布实例 泊松分布（Poisson distribution），台译卜瓦松分布（法语：loi de Poisson，英语：Poisson distribution，译名有泊松分布、普阿松分布、卜瓦松分布、布瓦松分布、布阿松分布、波以松分布、卜氏分配等），是一种统计与概率学里常见到的离散机率分布（discrete probability distribution）。泊松分布是以18～19 世纪的法国数学家西莫恩·德尼·泊松（Siméon-Denis Poisson）命名的，他在1838年时发表。这个分布在更早些时候由贝努里家族的一个人描述过。 分布特点 泊松分布的概率函数为： 泊松分布的参数λ是单位时间(或单位面积)内随机事件的平均发生次数。泊松分布适合于描述单位时间内随机事件发生的次数。 泊松分布的期望和方差均为特征函数为 关系 泊松分布与二项分布 泊松分布 当二项分布的n很大而p很小时，泊松分布可作为二项分布的近似，其中λ为np。通常当n≧20,p≦0.05时，就可以用泊松公式近似得计算。 事实上，泊松分布正是由二项分布推导而来的，具体推导过程参见本词条相关部分。应用场景
在实际事例中，当一个随机事件，例如某电话交换台收到的呼叫、来到某公共汽车站的乘客、某放射性物质发射出的粒子、显微镜下某区域中的白血球等等，以固定的平均瞬时速率λ（或称密度）随机且独立地出现时，那么这个事件在单位时间（面积或体积）内出现的次数或个数就近似地服从泊松分布P(λ)。因此,泊松分布在管理科学、运筹学以及自然科学的某些问题中都占有重要的地位（在早期学界认为人类行为是服从泊松分布，2005年在nature上发表的文章揭示了人类行为具有高度非均匀性）。 应用示例 泊松分布适合于描述单位时间（或空间）内随机事件发生的次数。如某一服务设施在一定时间内到达的人数，电话交换机接到呼叫的次数，汽车站台的候客人数，机器出现的故障数，自然灾害发生的次数，一块产品上的缺陷数，显微镜下单位分区内的细菌分布数等等。 观察事物平均发生m次的条件下，实际发生x次的概率P（x）可用下式表示： 例如采用0.05J/㎡紫外线照射大肠杆菌时，每个基因组（～4×106核苷酸对）平均产生3个嘧啶二体。实际上每个基因组二体的分布是服从泊松分布的，将取如下形式： …… 是未产生二体的菌的存在概率，实际上其值的5%与采用0.05J/㎡照射时的大肠杆菌uvrA-株，recA-株（除去既不能修复又不能重组修复的二重突变）的生存率是一致的。由于该菌株每个基因组有一个二体就是致死量，因此就意味着全部死亡的概率。 推导 泊松分布是最重要的离散分布之一，它多出现在当X表示在一定的时间或空间内出现的事件个数这种场合。在一定时间内某交通路口所发生的事故个数，是一个典型的例子。泊松分布的产生机制可以通过如下例子来解释。
最优化大作业 Poisson图像编辑 实验背景 图像合成是图像处理的一个基本问题，其通过将源图像中一个物体或者一个区域嵌入到目标图像生成一个新的图像。在对图像进行合成的过程中，为了使合成后的图像更自然，合成边界应当保持无缝。但如果原图像和目标图像有着明显不同的纹理特征，则直接合成后的图像会存在明显的边界。针对此问题，Prez等[1]提出了一种利用构造Poisson方程求解像素最优值的方法，在保留了源图像梯度信息的同时，可以很好的融合源图像与目标图像的背景。该方法根据用户指定的边界条件求解一个Poisson方程，实现了梯度域上的连续，从而达到边界处的无缝融合。 实验原理 Poisson图像编辑的主要思想是，根据原图像的梯度信息以及目标图像的边界信息，利用插值的方法重新构建出和成区域内的图像像素。为表述方便，在图1中标出了各个字母所代表 的区域。其中，g代表了原图像中被合成的部分，V是g的梯度场，S是合并后的图像，Ω是合并后目标图像中被覆盖的区域，?Ω是其边界。设合并后图像在Ω内的像素值由f表示，在Ω外的像素值由*f表示。 图1各区域代表字母示意 注意到图像合并的要求是使合并后的图像看上去尽量的平滑，没有明显的边界。所以，Ω内的梯度值应当尽可能的小。因此，f的值可以由(1)得出
2 * min ..f f s t f f Ω ?Ω ?Ω ?=?? (1) 根据Euler-Lagrange 公式，f 最小值应该满足(2)式 * ..f s t f f ?Ω ?Ω ?== (2) 其中?代表Laplacian 算子。然而，(2)式所求得的解中并没有包含源图像的相关信息。为使合并后的图像中Ω内的图像尽量接近于g ，可以利用g 的梯度场V 作为求解(1)式时的引导场，即将(1)式改写为如下形式 2 *min ..f f V s t f f Ω ?Ω ?Ω ?-=?? (3) 其最优解应满足如(4)的Poisson 方程： * ..f divV s t f f ?Ω ?Ω ?== (4) 对于一副图像，其像素点的值并不是连续函数，而是离散的数值。因此，需要建立(4)式的离散形式。对于S 中的每一个像素点p ，定义p N 代表图像中p 4临域点的集合，定义 ,p q <>代表p 与p N 内一点q 的点对。那么Ω的边界?Ω可以由(5)式表示 {\:0}p p S N ?Ω=∈Ω?Ω≠ (5) 这样，可以改写(3)为： ,0 * min () ..,p q pq f p q p p f f v s t f f p Ω <>?Ω≠--=∈?Ω ∑ (6) 同样，根据最优性条件，最优解满足(7)式： *p p p p p q q pq q N q N q N N f f f v ∈?Ω ∈??Ω ∈- = + ∑ ∑ ∑ (7) 这是一个线性方程，解之即可得出Ω内的像素值。如果像素较多时，系数矩阵规模较大， 求解时可以采用迭代的方法。Gauss-Seidel 迭代是一种简单易行的迭代法，应此被选作求解(7)式的算法。
第九章spss的回归分析 1、利用习题二第4题的数据，任意选择两门课程成绩作为解释变量和被解释变量，利用SPSS 提供的绘制散点图功能进行一元线性回归分析。请绘制全部样本以及不同性别下两门课程成绩的散点图，并在图上绘制三条回归直线，其中，第一条针对全体样本，第二和第三条分别针对男生样本和女生样本，并对各回归直线的拟和效果进行评价。 选择fore和phy两门成绩做散点图 步骤：图形→旧对话框→散点图→简单散点图→定义→将phy导入X轴、将fore导入Y 轴，将sex导入设置标记→确定 图标剪辑器内点击元素菜单→选择总计拟合线→选择线性→确定→再次选择元素菜单→点击子组拟合线→选择线性→确定 分析：如上图所示，通过散点图，被解释变量y与fore有一定的线性相关关系。 2、线性回归分析与相关性回归分析的关系是怎样的？ 线性回归分析是相关性回归分析的一种，研究的是一个变量的增加或减少会不会引起另一个变量的增加或者减少。
3、为什么需要对线性回归方程进行统计检验？一般需要对哪些方面进行检验？ 线性回归方程能够较好地反映被解释变量和解释变量之间的统计关系的前提是被解释变量和解释变量之间确实存在显著的线性关系。 回归方程的显著性检验正是要检验被解释变量和解释变量之间的线性关系是否显著，用线性模型来描述他们之间的关系是否恰当。一般包括回归系数的检验，残差分析等。 4、SPSS多元线性回归分析中提供了哪几种解释变量筛选策略？ 包括向前筛选策略、向后筛选策略和逐步筛选策略。 5、先收集到若干年粮食总产量以及播种面积、使用化肥量、农业劳动人数等数据，请利用建立多元线性回归方程，分析影响粮食总产量的主要因素。数据文件名为“粮食总产量.sav”。 步骤：分析→回归→线性→粮食总产量导入因变量、其余变量导入自变量→确定 结果如图： Variables Entered/Removed b Model Variables Entered Variables Removed Method 1 农业劳动者人数(百万人), 总播种面积(万公顷), 风灾 面积比例(%), 粮食播种面 积(万公顷), 施用化肥量 (kg/公顷), 年份a . Enter a. All requested variables entered. b. Dependent Variable: 粮食总产量(y万吨) ANOVA b Model Sum of Squares df Mean Square F Sig. 1 Regression 2.025E9 6 3.375E8 414.944 .000a Residual 2.278E7 28 813478.405 Total 2.048E9 34 a. Predictors: (Constant), 农业劳动者人数(百万人), 总播种面积(万公顷), 风灾面积比例(%), 粮食播种面积(万公顷), 施用化肥量(kg/公顷), 年份 b. Dependent Variable: 粮食总产量(y万吨) Coefficients a Model Unstandardized Coefficients Standardized Coefficients t Sig. B Std. Error Beta
泊松过程与泊松分布的基本知识泊松过程是随机过程的一个经典模型，是一种累积随机事件的发生次数的独立增量过程。也就是说，每次事件的发生是相互独立的。那么泊松分布和泊松过程又什么关系呢？可以说泊松分布是描述稀有事件的统计规律，即可以描述一段时间内发生某个次数的概率。而泊松过程呢，就适合刻画“稀有事件流”的概率特性。 比较：泊松分布 泊松过程的主要公式： 其实没多少不一样对不对？不一样的是泊松过程是一个可以查看在时间t内发生次数的概率，这个t是可变的。泊松分布则是给定了时间。 泊松过程的关键在于，它的到达间隔序列Tn，即每两次发生的时间是服从的独立同指数分布的。如果每次发生的间隔时间不服从指数分布，那么这个随机过程就会更一般化，我们成为是更新过程，这也是随机过程的推广。 泊松过程分为齐次泊松过程和非齐次泊松过程，齐次的意思很简单，就是说过程并不依赖于初始时刻，强度函数是一个常数，从上面的公式也看得出来。而非齐次则是变成了,这意味着什么呢？这以为着随着与时间的改变，强度是会改变的，改变服从强度函数，说了这
么久，强度究竟是个什么概念？强度的意思就是泊松过程的该事件发生的频率，或者说快慢，泊松分布中我们知道期望就是,实际含义就是，在一段时间内，发生的次数平均水平是次。 复合泊松过程：泊松过程我们已经知道，用描述一段时间累积发生的次数，但是如果每次发生带来的后果都是不一样的，我们怎么描述这个过程呢？比如，火车站到达的乘客是服从泊松过程的，但是每个乘客携带有不同重量的行李，我们如何刻画在[0,t]时间内行李总重量呢，这个过程就是复合泊松过程。复合泊松过程的均值函数和方差函数一般可以用全期望和全方差公式进行计算，因为简单泊松过程的期望很容易求。 更新过程： 上文已经说到，更新过程作为泊松过程的推广，更具有一般性，那么在讨论更新过程时，我们更多地讨来更新函数，更新函数是更新过程的均值函数m(t)=E[N(t)]，怎么理解呢，就是说需要用t时刻的累积计数的期望特性来表达更新过程。有一条定理： 这个定理是可以证明的，Fn（t）是分布函数，就是说：在t时刻，更新函数值就是在这个时刻，n取遍所有值的分布之和。 那么是否可以这样理解，更新过程和泊松过程的区别就是更新间隔序列不同，那么如果已知了更新间隔序列的概率密度函数，就可以求解该过程的更新函数了，详细的推导就不写了。扔结论出来：对间隔序列概率密度函数做拉氏变换得到Lf(s),然后求 Lm(s)=Lf(s)/s(1-Lf(s)),再对Lm(s)进行逆变换，就得到了m(t),这就是更新函数。
基于Gram-Schmidt的图像融合方法概述 摘要遥感图像融合的目的是综合来自不同空间分辨率和光谱分辨率的遥感信息，生成一幅具有新空间特征和波谱特征的合成图像。它具有重要的意义和广泛的应用前景。而由于采用的算法或变换方法的不同，融合方法有多种。在众多的融合方法相互比较的过程中，我们发现Gram-Schmidt具有较高的图像保真效果，是一种高效的图像融合方法。由于该算法在遥感图像融合中的应用尚处于起步阶段，对于Gram-Schmidt光谱锐化高保真的影像融合算法的了解尚不全面。对此，对Gram-Schmidt的原理、方法、优势等做了较为详尽的介绍。 关键词遥感融合保真Gram-Schmidt 概述 1 引言 对于光学系统的遥感影像，其空间分辨率和光谱分辨率一直存在着不可避免矛盾。在一定的信噪比的情况下，光谱分辨率的提高必然导致牺牲空间分辨率为代价。然而，通过将较低空间分辨率的多光谱影像和较高空间分辨率的影像的全色波段影像的融合，可以产生多光谱和高空间分辨率的影像。因此，各种基于不同算法的融合方法得到了迅速地发展和广泛地应用。 随着遥感技术的发展，由于对图像解译和反演目标参数的需要，一些简单的融合方法在很大程度上已经无法满足对于光谱信息保持，空间纹理信息增加的迫切需求。例如，对于检测植被活力和生长状态，反演陆地生产力，进行环境评价和矿产勘测等，如果融合后的图像信息的保真度无法满足要求，将会导致错误结果的产生。 通常采用的遥感图像融合方法有IHS变换、Brovey变换、主成分变换、小波变换等。虽然，这些融合方法都能够增加多光谱影像的空间纹理信息特征。但IHS、Brovey、主成分变换等方法易使融合后的影像失真；小波变换光谱信息虽保真相对较好，但小波基选择困难，且计算相对复杂（李存军等，2004）。 基于Gram-schmidt算法的图像融合方法既能使融合影像保真度较好，计算又较为简单。本文将对该影像融合算法的原理、方法以及所具备的优势做较为详尽的介绍。 2 算法简介
二项分布、泊松分布与泊松逼近 雅各布·伯努利与二项分布公式 雅各布·伯努利（Jacob Bernoulli，1654—1705）来自数学史上的传奇家族—瑞士巴塞尔的伯努利家族，该家族的三代成员中产生了8位数学家，在17世纪和18世纪微积分理论及应用的发展中占有领先地位，雅各布·伯努利是其家族第一代数学家中的第一位，他与弟弟约翰·伯努利（Johann Bernoulli，1667—1748）、侄子丹尼尔·伯努利（Daniel Bernoulli，1700—1782）在数学史上享有声誉。 家族简介 在科学史上，父子科学家、兄弟科学家并不鲜见，然而，在一个家族跨世纪的几代人中，众多父子兄弟都是科学家的较为罕见，其中，瑞士的伯努利（也译作贝努力、伯努利）家族最为突出。 伯努利家族3代人中产生了8位科学家，出类拔萃的至少有3位；而在他们一代又一 代的众多子孙中，至少有一半相继成为杰出人物。伯努利家族的后裔有不少于120位被人们系统地追溯过，他们在数学、科学、技术、工程乃至法律、管理、文学、艺术等方面享有名望，有的甚至声名显赫。最不可思议的是这个家族中有两代人，他们中的大多数数学家，并非有意选择数学为职业，然而却忘情地沉溺于数学之中，有人调侃他们就像酒鬼碰到了烈酒。 老尼古拉·伯努利（Nicolaus Bernoulli，公元1623～1708年）生于巴塞尔，受过良好教育，曾在当地政府和司法部门任高级职务。他有3个有成就的儿子。其中长子雅各布（Jocob，公元1654～1705年）和第三个儿子约翰（Johann，公元1667～1748年）成为著名的数学家，第二个儿子小尼古拉（Nicolaus I，公元1662～1716年）在成为彼得堡科学院数学界的一员之前，是伯尔尼的第一个法律学教授。 雅各布·伯努利
ISSN 1000-0054CN 11-2223/N 清华大学学报(自然科学版)J T singh ua Un iv (Sci &Tech ),2008年第48卷第7期 2008,V o l.48,N o.7w 11 http://qhx bw.chinajo https://www.doczj.com/doc/935810046.html, 基于Poisson 重建的极化SAR 图像对比增强 邓启明, 陈亦伦, 张卫杰, 杨 健 (清华大学电子工程系,北京100084) 收稿日期:2007-04-16 基金项目:国家自然科学基金资助项目(40571099); 高校博士点基金资助项目;新世纪优秀人才支持计划 作者简介:邓启明(1982—),男(汉),湖北,博士研究生。通讯联系人:杨健,教授, E-mail:deng qm 05@mails.tsin https://www.doczj.com/doc/935810046.html, 摘 要:为了能够同时增强多类目标,提出一种基于P oisso n 重建的极化合成孔径雷达(SAR )图像对比增强方法。该方法对多类目标和对应的背景杂波分别进行广义相对最优极化增强(G OP CE ),并得到相应的最优极化状态和特征系数;以此定义图像中各像素点的最优局部梯度,并在最小二乘准则下,根据局部梯度建立离散P oisso n 方程;通过快速Fo urier 变换求解该Po isson 方程,得到最终的多目标增强图像。实验结果表明:利用极化SA R 数据,使用该方法增强后的图像的直方图保持应有的峰值,且更加均衡,能够达到增强多类目标的效果,从而有利于目标检测等后续处理。关键词:图像处理;极化;合成孔径雷达(SA R );对比增 强;Poisso n 重建 中图分类号:T P 751 文献标识码:A 文章编号:1000-0054(2008)07-1108-04 Contrast enhancement of polarimetric SAR images based on the Poisson reconstructions DE NG Qim ing ,CHEN Yilun ,ZHANG Weijie ,YA NG Jian (Department of Electronic Engineering ,Tsinghua University , Beij ing 100084,China ) Abstract :A contrast enhancement method bas ed on the Poisson reconstruction w as developed to enh an ce multiple targets in polarimetr ic s ynthetic aperture radar (SAR)images.Each kind of target is enhanced relative to its corr esp on ding clutter by us ing the gen eralized optimization of polar imetric contrast enhancemen t (GOPCE ). Th e obtained optimal polarization states and ch aracteristic coefficients define th e optimal local contras t for each pixel.A dis crete Pois son equation is th en cons tructed us ing leas t squares m inimization,w hich is s olved using fast Fourier transforms.T ests w ith polarimetric SAR data demonstrated that the meth od produces an en han ced image with a more uniform histogram wh ile keeping its peaks. Satisfactory res ults can be ob tained w hen en han cing multiple targets w ith various types of clutter for sub sequ ent applications such as target detection. Key words :image process ing;polarimetric;s ynthetic aperture radar (SA R);contrast enhancement;Poiss on recon struction 在极化合成孔径雷达(synthetic aperture r adar,SAR)数据处理应用中,对比增强是一个重要的研究课题。极化对比增强是通过融合目标多维的极化信息来增强目标和背景杂波之间的对比度,在目标检测中有着极为重要的应用。多年来,研究者提出了多种利用极化信息增强目标的方法。例如:Ko stinski 和Boerner [1] 提出了经典的相对最优极化增强(generalized o ptimization of polarimetric contrast enhancement,OPCE)技术,其主要目的是选取一种最优极化状态以增强目标回波或接收功 率,同时尽可能地抑制杂波干扰。杨健等[2] 利用交叉迭代法,得到了快速数值求解OPCE 的方法,提高了运算速度。进而杨健等[3]还通过引入包含更多极化信息的参数,提出了广义相对最优极化增强(GOPCE )方法,取得了比传统OPCE 更好的效果。但是,传统方法的模型均定义为对单一类别的目标进行对比增强,即能够对某类目标较之背景杂波进行较好地增强,但在增强多类目标时,不能取得很好的效果。也就是说,使用OPCE 或GOPCE 进行增强某类目标时,有可能将另一类目标连同杂波一起抑制。 Poisson 重建是利用图像的梯度来重构图像的 方法。陈亦伦和杨健[4] 将该方法用于极化SAR 图像的融合中,将各个通道的局部对比度融合后重建,可以在融合图像里显示各极化通道的信息。但该方法在融合过程中仅用到了各极化通道的幅度值,没有完全利用极化信息。
相关分析与回归分析 一、试验目标与要求 本试验项目的目的是学习并使用SPSS软件进行相关分析与回归分析，具体包括： (1)皮尔逊pearson简单相关系数的计算与分析 (2)学会在SPSS上实现一元及多元回归模型的计算与检验。 (3)学会回归模型的散点图与样本方程图形。 (4)学会对所计算结果进行统计分析说明。 (5)要求试验前，了解回归分析的如下内容。 参数α、β的估计 回归模型的检验方法：回归系数β的显着性检验（t－检验）； 回归方程显着性检验（F－检验）。 二、试验原理 1．相关分析的统计学原理 相关分析使用某个指标来表明现象之间相互依存关系的密切程度。用来测度简单线性相关关系的系数是Pearson简单相关系数。 2．回归分析的统计学原理 相关关系不等于因果关系，要明确因果关系必须借助于回归分析。回归分析是研究两个变量或多个变量之间因果关系的统计方法。其基本思想是，在相关分析的基础上，对具有相关关系的两个或多个变量之间数量变化的一般关系进行测定，确立一个合适的数据模型，以便从一个已知量推断另一个未知量。回归分析的主要任务就是根据样本数据估计参数，建立回归模型，对参数与模型进行检验与判断，并进行预测等。 线性回归数学模型如下： 在模型中，回归系数是未知的，可以在已有样本的基础上，使用最小二乘法对回归系数进行估计，得到如下的样本回归函数： 回归模型中的参数估计出来之后，还必须对其进行检验。如果通过检验发现模型有缺陷，则必须回到模型的设定阶段或参数估计阶段，重新选择被解释变量与解释变量及其函数形式，或者对数据进行加工整理之后再
次估计参数。回归模型的检验包括一级检验与二级检验。一级检验又叫统计学检验，它是利用统计学的抽样理论来检验样本回归方程的可靠性，具体又可以分为拟与优度评价与显着性检验；二级检验又称为经济计量学检验，它是对线性回归模型的假定条件能否得到满足进行检验，具体包括序列相关检验、异方差检验等。 三、试验演示内容与步骤 1．连续变量简单相关系数的计算与分析 在上市公司财务分析中，常常利用资产收益率、净资产收益率、每股净收益与托宾Q值4个指标来衡量公司经营绩效。本试验利用SPSS对这4个指标的相关性进行检验。操作步骤与过程： 打开数据文件“上市公司财务数据(连续变量相关分析).sav”,依次选择“【分析】→【相关】→【双变量】”打开对话框如图，将待分析的4个指标移入右边的变量列表框内。其他均可选择默认项，单击ok提交系统运行。 图5.1 Bivariate Correlations对话框 结果分析： 表给出了Pearson简单相关系数，相关检验t统计量对应的p值。相关系数右上角有两个星号表示相关系数在0.01的显着性水平下显着。从表中可以看出，每股收益、净资产收益率与总资产收益率3个指标之间的相关系数都在0.8以上，对应的p值都接近0，表示3个指标具有较强的正相关关系，而托宾Q值与其他3个变量之间的相关性较弱。 表5.1 Pearson简单相关分析 Correlations 每股收益 率净资产 收益率 资产收益 率 托宾Q 值 每股收益率Pearson Correlation 1 .877(* *) .824(**)-.073 Sig. ..000.000.199
利用全局方法进行泊松前景提取* 曹春红1,2,3+，王庆敏1,2,3 1.东北大学信息科学与工程学院，沈阳110819 2.东北大学医学影像计算教育部重点实验室，沈阳110819 3.南京大学计算机软件新技术国家重点实验室，南京210023 Using Global Method to Extract Poisson Foreground CAO Chunhong 1,2,3+,WANG Qingmin 1,2,3 1.College of Information Science and Engineering,Northeastern University,Shenyang 110819,China 2.Key Laboratory of Medical Image Computing of Ministry of Education,Northeastern University,Shenyang 110819,China 3.State Key Laboratory for Novel Software Technology,Nanjing University,Nanjing 210023,China +Corresponding author:E-mail:caochunhong@https://www.doczj.com/doc/935810046.html, CAO Chunhong,WANG https://www.doczj.com/doc/935810046.html,ing global method to extract Poisson foreground.Journal of Frontiers of Computer Science and Technology,2016,10(5)：688-698. Abstract:The image foreground extraction is using image processing algorithms to quickly and accurately extract the target image that people are interested in.The precision of image foreground extraction directly affects the subsequent processing of target image.In order to improve the accuracy of image foreground extraction,this paper proposes a new Poisson matting algorithm using global method.In order to improve the speed,this paper uses a diffusion-search method,and analyzes the effectiveness and accuracy of the algorithm.Diffusion method is to find the least cost sampling points by calculating the cost of each sampling point in a small neighborhood,it can obtain the optimal solution adjacent areas;Search method is to find the optimal sampling point by certain rules,it can acceler-ate the speed to find the optimal sampling point.Finally,the experiment proves that Poisson extraction algorithm based on global foreground will get more accurate results for foreground extraction. *The National Natural Science Foundation of China under Grant No.61300096(国家自然科学基金);the Fundamental Research Funds for the Central Universities of China under Grant No.N130404013(中央高校基本科研业务费专项基金). Received 2015-06,Accepted 2015-08. CNKI 网络优先出版:2015-09-02,https://www.doczj.com/doc/935810046.html,/kcms/detail/11.5602.TP.20150902.1103.006.html ISSN 1673-9418CODEN JKYTA8 Journal of Frontiers of Computer Science and Technology 1673-9418/2016/10(05)-0688-11 doi:10.3778/j.issn.1673-9418.1506096E-mail:fcst@https://www.doczj.com/doc/935810046.html, https://www.doczj.com/doc/935810046.html, Tel:+86-10-89056056