Carl Boettiger

Yes­ter­day I derived the lin­earized SDEs for the sad­dle node bifur­ca­tion : \[ dX = 2\sqrt{r(t)} (\sqrt{r(t)} + \theta –X) dt + \sigma \sqrt{\sqrt{r(t)}+\theta} dB_t \] and the tran­s­crit­i­cal bifur­ca­tion \[ dX = r(t) (K — X) dt + \sqrt{K} dB_t \] Now to imple­ment them for fit­ting by like­li­hood using generic \(r(t) \).  I …

Have spent the last three days build­ing and test­ing the infra­struc­ture to fit the canon­i­cal form of a saddle-node (fold) bifur­ca­tion by like­li­hood. Actu­ally I don’t use the canon­i­cal form straight up: \( dx/dt = x^2 +r\) can­not be fit directly in this man­ner, since it needs a cou­ple scale trans­for­ma­tions on the vari­ables to …

Con­sid­er­ing the fits of the qua­dratic mod­els from yes­ter­day. Analy­sis of fits Of course these fits rely entirely on map­ping the loca­tion of the sta­ble point and the slope through that point, some­thing they still can­not do all that well. In this para­me­ter­i­za­tion the the sta­ble inter­cept is \[ \hat x = \sqrt{r}+\theta \] and slope …

Meet­ing with Alan this morn­ing.  I haven’t solved like­li­hood of power spec­tra (pre­vi­ous cou­ple entries in Stoch. Pop / warn­ing sig­nals), though after meet­ing with Sebas­t­ian (11/9) prob­a­bly the best method works from the cor­re­la­tion func­tion.  Define a new sto­chas­tic process Y \[ Y_1 = X_1 X_2 + X_2 X_3 + X_3 X_4 + \ldots …