Time-dependent covariates have proven to be extremely useful in the Cox model. From a clinical point of view, when assessing the hazard for a subject why not make use of our most up-to-date information? And indeed, models with TD covariates can often have much better loglikelihood. An important complement to hazard ratios, particularly in multi-state models, are the 3 measures of absolute risk (of which the first is most familiar)
- probability in state k at time t
- expected number of visits to state k
- expected sojourn time in state k
Creating honest estimates of absolute risk, when the model has TD covariates, has been a baffling problem; with mostly bad ideas in the literature. We were motivated to address the issue by analysis of data from the Mayo Clinic Study of Aging and a simple 3 state model of cognitively unimpaired, dementia, and death. The affects of age, sex, amyloid burden (time dependent), APOE e4 genotype, and comorid conditions (time dependent) on both the rates of transition between states and on the absolute rates (lifetime risk of dementia, expected years in dementia) were of key interest. An approach based on classic observed/expected events in epidemiology studies led us to an approach takes explicit advantage of multistate models and seems to work well. Several important theory and technical issues remain open.
In 1970 Matiyasevich, building on earlier work of Davis--Putnam--Robinson, proved that every enumerable subset of Z is Diophantine, thus showing that Hilbert's 10th problem is undecidable for Z. The problem of extending this result to the ring of integers of number fields (and more generally to finitely generated infinite rings) has attracted significant attention and, thanks to the efforts of many mathematicians, the task has been reduced to the problem of constructing, for certain quadratic extension of number fields L/K, an elliptic curve E/K with rk(E(L))=rk(E(K))>0.
In this talk I will explain joint work with Peter Koymans, where we use Green--Tao to construct the desired elliptic curves, settling Hilbert 10 for every finitely generated infinite ring.
Title: Improving Probabilistic Forecastingof Extreme Wind Speeds in the Netherlands
Date: Friday, October 25th
Time: 13:00
Location: BBG 401
Supervisors: Dr. Sjoerd Dirksen, Dr.Bastien François (KNMI), Dr. Kirien Whan (KNMI)
Second Reader: Dr. Chiheb Ben Hammouda
Abstract:
Forecasting wind speeds is importantbecause of its large impact on society. The forecasts are issued by NumericalWeather Prediction (NWP) models. These NWP models often contain biases and haveerrors in dispersion, therefore they are subjugated to statisticalpost-processing. A commonly used method to perform post-processing is ensemble model output statistics (EMOS), where the goal is to fit theparameters of a probability distribution based on NWP output. In early versionsof EMOS, linear regression was employed for this task. In recent approaches,more complex models such as neural networks have been introduced. While neuralnetworks are able to significantly improve performance up to medium range windspeeds, they struggle with high wind speeds. The models are often trained usingthe continuous ranked probability score (CRPS), a proper scoring rule thatequally weighs all possible forecast values. In this work, we propose using aweighted version of the CRPS (wCRPS) to address the challenges associated withextreme wind speeds. The wCRPS is a proper scoring rule that emphasizesparticular regions of the forecast through a weight function. We also exploredifferent parametric distributions, namely the truncated normal (TN), log-normal(LN), generalized extreme value (GEV) and mixture distributions.
Our findings suggest that using the wCRPS with an appropriate weightfunction can enhance performance on extremes. However, for models using linearregression, we observed a body-tail trade-off, where increased performance onextremes came at the cost of worse predictions for average wind speeds. Wedeveloped an approach where the weight function is selected based on userpreference by selecting hyperparameters using a multi-objective optimizationalgorithm. For the convolutional neural network-based models, we found thatwith an appropriate weight function the performance on extremes could beincreased. Further investigation on the weight function of neural network-basedmodels is advised, as the best choice of weight function may not have beenincluded in our search space. Additionally, the best-performing weight functionis shown to be model-specific. Regarding the choice of distribution, nosignificant effect was observed.
With kind regards,
Simon Hakvoort
Speaker: Henk Bruin (U Vienna)
Title: Ergodic properties of $\mathbb Z^d$-extensions over translation flows
Abstract:
Studying $\mathbb Z^d$-extensions (i.e., skew-product with $\mathbb Z^d$
as fiber space) are more difficult to study than their base dynamics;
they are non-compact and their invariant measures are infinite. But they
model important systems such as Ehrenfest's wind-tree model and flows on
infinite surfaces.
In this talk I want to present some result concerning their ergodicity
and recurrence, based on joint papers with Olga Lukina, and with Charles
Fourgeron, Davide Ravotti, Dalia Terhesiu.
Location and time: JKH 15a, Room 003 at 11-12.45
Afternoon session:
Speaker: Timon Idema (TUD)
Title: The maths of membranes - how differential geometry can be useful for biology
Abstract: Membranes in living cells adapt a wide variety of continually evolving shapes closely related to their function. These shapes are regulated by curvature-inducing proteins, which also interact via the membrane deformations they impose. We study such membrane-mediated interactions in the globally curved and crowded setting of membranes inside the living cell. To do so, we rely heavily on tools from differential geometry to describe the shape and evolution of the membranes. In this talk, I will focus on the application of these mathematical tools to our biological system.
I will start with introducing the necessary framework of two-dimensional surfaces embedded in three-dimensional space. We will discuss how we can build complicated networks of membrane tubes and sheets by coupling global membrane properties to locally induced curvature. We find that the collective action of many proteins can change the overall membrane shape, and lead to the formation of many striking patterns that can be tested both in vivo and in biomimetic soft matter systems. Moreover, by coupling to active components like molecular motors or a growing and shrinking cytoskeleton, we can make the membrane dynamical, adapting its shape in response to a varying environment. These membrane dynamics are the basis of many biological functions, and by studying them we can eventually understand not only what a cell does, but also how it manages to do just that.
Location and time: JKH 15a, Room 003, 14.15-16.00