Markov Chain Properties Of Snowball Sampling

The classical sampling methods in the traditional sampling survey system belong to probability sampling.Researchers determine the population size by using the characteristics of the sampling box,assign the inclusion probability of a fixed value to each population unit,obtain samples by combining with random sampling,and finally make statistical inferences based on the weight of samples.With the development of network technology and the rise of the concept of big data,new elements have been added to the characteristics of sampling survey.The overall complexity of research is enhanced,and the new characteristics of data brought by the era of big data(large quantity,complex composition,etc.)make the overall characteristics more complex.On this basis,it highlights the importance of improbability sampling in practical application.There are no more than two reasons for choosing improbability sampling in this era.Big data technology and environment facilitate improbability sampling.The difficulties encountered by probability sampling are exactly the development opportunities of non-probability sampling.Snowball sampling,as the basic representative of non-probability sampling,is only used as a tool to introduce its methods in general studies,and there is no relevant theoretical research,which further illustrates the feasibility of studying snowball sampling.In this paper,the relationship between snowball sampling and markov chain is analyzed.First,it is proved that snowball sampling process is a markov chain,and there are two markov chain states in snowball sampling,namely target group and non-target group,which are interlinked and irreducible.Similarly,the snowball sampling process can reach the stable distribution state of markov chain,and the stable distribution of snowball sampling is the ideal state of sampling.A series of properties of snowball sampling can be studied through the properties of the stable distribution.Second,defines a new theorem,snowball sampling error,by showing that snowball sampling and the relationship between the markov chains,can be defined for stationary distribution method puts forward a new theorem of snowball sampling error,the theorem is proved that the stable distribution of snowball sampling process,and its error of accuracy is limited and is effective,the definition of sampling error can solve the traditional problem of non probability sampling error cannot be defined.In addition,this paper focuses on the analysis of snowball sampling process in the case of rejection,which can be divided into 8 general cases and 2 special cases.In either case,it can be further explained that snowball sampling in the case of rejection will cause unpredictable results to the whole sampling process.One of the characteristics of improbability sampling is that the sampling process is based on people's subjective will,so there are many unknown situations.Finally,this paper studies the snowball sampling combined with hidden markov model,the feasibility and effectiveness of real snowball sampling for hidden markov model,is nothing more than three factors(initial distribution,implicit state transition probability matrix and observation state transition probability matrix),these three elements combination of known and unknown have practical algorithm can solve,corresponding to the three problems in the HMM,so in reality application can use the algorithm to calculate when snowball sampling,effectively improve the efficiency of snowball sampling.The markov chain property of snowball sampling is a theoretical derivation.Among them,first of all,the assumption condition is a little harsh.In practical application,transfer probability matrix may not be obtained.Secondly,the markov chain process of inhomogeneous snowball sampling is not further discussed.In the actual snowball sampling process,the transfer probability matrix of each round is related to the number of rounds n of each round,that is,the transfer probability matrix of each round is different.Such markov chain is called inhomogeneous markov chain.Finally,no matter how changeable the situation is,in the face of the actual snowball sampling process,specific problem specific analysis is still needed.