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Analysis of Theoretical Physics SE statistics - model for estimation of Q&A site traffic trend (v1 May 26 replaced by v2 May 31)

Analysis of Theoretical Physics SE statistics - model for estimation of Q&A site traffic trend (v1 May 26 replaced by v2 May 31)

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- 1. Analysis of Theoretical Physics SE statistics Alexander Yu. Vlasov May 31, 2012 Number of questions in Q&A StackExchange site Theoretical Physics (un-til the closure) is presented on the plot below: 400 300 200 Q 100 0 0 50 100 150 200 T The plot may be approximated with a simple model. Let’s suggest: 1. Before opening of the Q&A site some number of old questions already existed. 2. After opening of the Q&A site the rate of incoming questions is the sum of two terms: it may be either an old or a new question. 3. The rate of incoming new questions is proportional to the total number of already asked questions. 1
- 2. Let us denote number of old questions as Q0 = Q0 (0) and number ofquestions in Q&A site as Q(T ). We may write system of two equations dQ0 (T ) = −λ0 Q0 (T ) (1a) dT dQ(T ) = λ0 Q0 (T ) + λQ(T ) (1b) dT Due to condition Q(0) = 0 a solution of Eqs. (1) may be written as λ0 + λ −λ0 T Q0 (T ) = C e (2a) λ0 Q(T ) = C (eλT − e−λ0 T ) (2b) Using nonlinear least square estimation (nls function from R packagewas used) it was found C = 248.1, λ = 0.002335, λ0 = 0.01322 (also, due toEq. (2a), Q0 = C λ0 +λ ≈ 292). Plot of the function Q(T ) for given model λ0together with data is represented below: Parameters Std. Error t value Pr(> |t|) C = 248.1 4.016 61.79 <2E-16 λ = 0.002335 6.817E-05 34.25 <2E-16 λ0 = 0.01322 2.61E-04 50.64 <2E-16 Residual standard error: 3.928 on 406 degrees of freedom 2
- 3. The function Q(T ) may be used for an extrapolation for the imaginaryscenario if Theoretical Physics SE site were not be closed. Derivative of the function Q(T ) dQ = C (λeλT + λ0 e−λ0 T ) (3) dTrepresents rate of the questions per unit of time (day). 3

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