Geostatistical analysis of disease data: estimation of cancer mortality risk from empirical frequencies using Poisson kriging

BACKGROUND: Cancer mortality maps are used by public health officials to identify areas of excess and to guide surveillance and control activities. Quality of decision-making thus relies on an accurate quantification of risks from observed rates which can be very unreliable when computed from sparsely populated geographical units or recorded for minority populations. This paper presents a geostatistical methodology that accounts for spatially varying population sizes and spatial patterns in the processing of cancer mortality data. Simulation studies are conducted to compare the performances of Poisson kriging to a few simple smoothers (i.e. population-weighted estimators and empirical Bayes smoothers) under different scenarios for the disease frequency, the population size, and the spatial pattern of risk. A public-domain executable with example datasets is provided. RESULTS: The analysis of age-adjusted mortality rates for breast and cervix cancers illustrated some key features of commonly used smoothing techniques. Because of the small weight assigned to the rate observed over the entity being smoothed (kernel weight), the population-weighted average leads to risk maps that show little variability. Other techniques assign larger and similar kernel weights but they use a different piece of auxiliary information in the prediction: global or local means for global or local empirical Bayes smoothers, and spatial combination of surrounding rates for the geostatistical estimator. Simulation studies indicated that Poisson kriging outperforms other approaches for most scenarios, with a clear benefit when the risk values are spatially correlated. Global empirical Bayes smoothers provide more accurate predictions under the least frequent scenario of spatially random risk. CONCLUSION: The approach presented in this paper enables researchers to incorporate the pattern of spatial dependence of mortality rates into the mapping of risk values and the quantification of the associated uncertainty, while being easier to implement than a full Bayesian model. The availability of a public-domain executable makes the geostatistical analysis of health data, and its comparison to traditional smoothers, more accessible to common users. In future papers this methodology will be generalized to the simulation of the spatial distribution of risk values and the propagation of the uncertainty attached to predicted risks in local cluster analysis

Glosarium Matematika

273 p.; 24 cm

Using Mean Embeddings for State Estimation and Reinforcement Learning

To act in complex, high-dimensional environments, autonomous systems require versatile state estimation techniques and compact state representations. State estimation is crucial when the system only has access to stochastic measurements or partial observations. Furthermore, in combination with models of the system such techniques allow to predict the future which enables the system to asses the outcome of possible decisions. Compact state representations alleviate the curse of dimensionality by distilling the important information from high-dimensional observations. Due to noisy sensory information and non-perfect models of the system, estimates of the state never reflect the true state perfectly but are always subject to errors. The natural choice to incorporate the uncertainty about the state estimate is to use a probability distribution as representation. This results in the so called belief state. High-dimensional observations, for example images, often contain much less information than conveyed by their dimensionality. But also if all the information is necessary to describe the state of the system—for example, think of the state of a swarm with the positions of all agents—a less complex description might be a sufficient representation. In such situations, finding the generative distribution that explains the state would give a much more compact while informative representation. Traditionally, parametric distributions have been used as state representations such as most prevalently the Gaussian distribution. However, in many cases a unimodal distribution might not be sufficient to represent the belief state. Using multi-modal probability distributions, instead, requires more advanced approaches such as mixture models or particle-based Monte Carlo methods. Learning mixture models is however not straight-forward and often results in locally optimal solutions. Similarly, maintaining a good population of particles during inference is a complicated and cumbersome process. A third approach is kernel density estimation which is located at the intersection of mixture models and particle-based approaches. Still, performing inference with any of these approaches requires heuristics that lead to poor performance and a limited scalability to higher dimensional spaces. A recent technique that alleviates this problem are the embeddings of probability distributions into reproducing kernel Hilbert spaces (RKHS). Conditional distributions can be embedded as operators based on which a framework for inference has been presented that allows to apply the sum rule, the product rule and Bayes’ rule entirely in Hilbert space. Using sample based estimators and the kernel-trick of the representer theorem allows to represent the operations as vector-matrix manipulations. The contributions of this thesis are based on or inspired by the embeddings of distributions into reproducing kernel Hilbert spaces. In the first part of this thesis, I propose additions to the framework for nonparametric inference that allow the inference operators to scale more gracefully with the number of samples in the training set. The first contribution is an alternative approach to the conditional embedding operator formulated as a least-squares problem i which allows to use only a subset of the data as representation while using the full data set to learn the conditional operator. I call this operator the subspace conditional embedding operator. Inspired by the least-squares derivations of the Kalman filter, I furthermore propose an alternative operator for Bayesian updates in Hilbert space, the kernel Kalman rule. This alternative approach is numerically more robust than the kernel Bayes rule presented in the framework for non-parametric inference and scales better with the number of samples. Based on the kernel Kalman rule, I derive the kernel Kalman filter and the kernel forward-backward smoother to perform state estimation, prediction and smoothing based on Hilbert space embeddings of the belief state. This representation is able to capture multi-modal distributions and inference resolves--due to the kernel trick--into easy matrix manipulations. In the second part of this thesis, I propose a representation for large sets of homogeneous observations. Specifically, I consider the problem of learning a controller for object assembly and object manipulation with a robotic swarm. I assume a swarm of homogeneous robots that are controlled by a common input signal, e.g., the gradient of a light source or a magnetic field. Learning policies for swarms is a challenging problem since the state space grows with the number of agents and becomes quickly very high dimensional. Furthermore, the exact number of agents and the order of the agents in the observation is not important to solve the task. To approach this issue, I propose the swarm kernel which uses a Hilbert space embedding to represent the swarm. Instead of the exact positions of the agents in the swarm, the embedding estimates the generative distribution behind the swarm configuration. The specific agent positions are regarded as samples of this distribution. Since the swarm kernel compares the embeddings of distributions, it can compare swarm configurations with varying numbers of individuals and is invariant to the permutation of the agents. I present a hierarchical approach for solving the object manipulation task where I assume a high-level object assembly policy as given. To learn the low-level object pushing policy, I use the swarm kernel with an actor-critic policy search method. The policies which I learn in simulation can be directly transferred to a real robotic system. In the last part of this thesis, I investigate how we can employ the idea of kernel mean embeddings to deep reinforcement learning. As in the previous part, I consider a variable number of homogeneous observations—such as robot swarms where the number of agents can change. Another example is the representation of 3D structures as point clouds. The number of points in such clouds can vary strongly and the order of the points in a vectorized representation is arbitrary. The common architectures for neural networks have a fixed structure that requires that the dimensionality of inputs and outputs is known in advance. A variable number of inputs can only be processed by applying tricks. To approach this problem, I propose the deep M-embeddings which are inspired by the kernel mean embeddings. The deep M-embeddings provide a network structure to compute a fixed length representation from a variable number of inputs. Additionally, the deep M-embeddings exploit the homogeneous nature of the inputs to reduce the number of parameters in the network and, thus, make the learning easier. Similar to the swarm kernel, the policies learned with the deep M-embeddings can be transferred to different swarm sizes and different number of objects in the environment without further learning

Метод сглаживания вероятностей n-грамм на основе моделирования математического ожидания их встречаемости

It is shown that expectation of n-gram frequency of occurrence depends on the size of the training set and the size of the dictionary, which has been formed on the basis of this set. A method for smoothing of n-gram language model regarding probabilities of n-grams of lower order is proposed. This approach is based on the modeling of expectation function of n-gram occurrence probability. We suggest enlarging the size of the training set on the expected number of unseen n-grams instead of discounting maximum n-gram probability. To model the number of unseen n-grams expectation function of n-gram frequency of occurrence is extrapolated to zero frequency. Expectation function is modeled by the statistical analysis of occurrences of words in texts.В работе предлагается метод сглаживания n-граммной модели языка, в основе которого лежит моделирование функции математического ожидания вероятности встречаемости n-грамм. Вместо дисконтирования максимальной вероятности n-грамм предлагается увеличение мощности обучающего множества на ожидаемое число n-грамм, отсутствующих в обучающей базе текстов. Для моделирования этого числа функция математического ожидания вероятности встречаемости экстраполируется к нулевой частоте. На основе статистического анализа текстов построена модель функции математического ожидания встречаемости

Exploiting prior knowledge during automatic key and chord estimation from musical audio

Chords and keys are two ways of describing music. They are exemplary of a general class of symbolic notations that musicians use to exchange information about a music piece. This information can range from simple tempo indications such as “allegro” to precise instructions for a performer of the music. Concretely, both keys and chords are timed labels that describe the harmony during certain time intervals, where harmony refers to the way music notes sound together. Chords describe the local harmony, whereas keys offer a more global overview and consequently cover a sequence of multiple chords. Common to all music notations is that certain characteristics of the music are described while others are ignored. The adopted level of detail depends on the purpose of the intended information exchange. A simple description such as “menuet”, for example, only serves to roughly describe the character of a music piece. Sheet music on the other hand contains precise information about the pitch, discretised information pertaining to timing and limited information about the timbre. Its goal is to permit a performer to recreate the music piece. Even so, the information about timing and timbre still leaves some space for interpretation by the performer. The opposite of a symbolic notation is a music recording. It stores the music in a way that allows for a perfect reproduction. The disadvantage of a music recording is that it does not allow to manipulate a single aspect of a music piece in isolation, or at least not without degrading the quality of the reproduction. For instance, it is not possible to change the instrumentation in a music recording, even though this would only require the simple change of a few symbols in a symbolic notation. Despite the fundamental differences between a music recording and a symbolic notation, the two are of course intertwined. Trained musicians can listen to a music recording (or live music) and write down a symbolic notation of the played piece. This skill allows one, in theory, to create a symbolic notation for each recording in a music collection. In practice however, this would be too labour intensive for the large collections that are available these days through online stores or streaming services. Automating the notation process is therefore a necessity, and this is exactly the subject of this thesis. More specifically, this thesis deals with the extraction of keys and chords from a music recording. A database with keys and chords opens up applications that are not possible with a database of music recordings alone. On one hand, chords can be used on their own as a compact representation of a music piece, for example to learn how to play an accompaniment for singing. On the other hand, keys and chords can also be used indirectly to accomplish another goal, such as finding similar pieces. Because music theory has been studied for centuries, a great body of knowledge about keys and chords is available. It is known that consecutive keys and chords form sequences that are all but random. People happen to have certain expectations that must be fulfilled in order to experience music as pleasant. Keys and chords are also strongly intertwined, as a given key implies that certain chords will likely occur and a set of given chords implies an encompassing key in return. Consequently, a substantial part of this thesis is concerned with the question whether musicological knowledge can be embedded in a technical framework in such a way that it helps to improve the automatic recognition of keys and chords. The technical framework adopted in this thesis is built around a hidden Markov model (HMM). This facilitates an easy separation of the different aspects involved in the automatic recognition of keys and chords. Most experiments reviewed in the thesis focus on taking into account musicological knowledge about the musical context and about the expected chord duration. Technically speaking, this involves a manipulation of the transition probabilities in the HMMs. To account for the interaction between keys and chords, every HMM state is actually representing the combination of a key and a chord label. In the first part of the thesis, a number of alternatives for modelling the context are proposed. In particular, separate key change and chord change models are defined such that they closely mirror the way musicians conceive harmony. Multiple variants are considered that differ in the size of the context that is accounted for and in the knowledge source from which they were compiled. Some models are derived from a music corpus with key and chord notations whereas others follow directly from music theory. In the second part of the thesis, the contextual models are embedded in a system for automatic key and chord estimation. The features used in that system are so-called chroma profiles, which represent the saliences of the pitch classes in the audio signal. These chroma profiles are acoustically modelled by means of templates (idealised profiles) and a distance measure. In addition to these acoustic models and the contextual models developed in the first part, durational models are also required. The latter ensure that the chord and key estimations attain specified mean durations. The resulting system is then used to conduct experiments that provide more insight into how each system component contributes to the ultimate key and chord output quality. During the experimental study, the system complexity gets gradually increased, starting from a system containing only an acoustic model of the features that gets subsequently extended, first with duration models and afterwards with contextual models. The experiments show that taking into account the mean key and mean chord duration is essential to arrive at acceptable results for both key and chord estimation. The effect of using contextual information, however, is highly variable. On one hand, the chord change model has only a limited positive impact on the chord estimation accuracy (two to three percentage points), but this impact is fairly stable across different model variants. On the other hand, the chord change model has a much larger potential to improve the key output quality (up to seventeen percentage points), but only on the condition that the variant of the model is well adapted to the tested music material. Lastly, the key change model has only a negligible influence on the system performance. In the final part of this thesis, a couple of extensions to the formerly presented system are proposed and assessed. First, the global mean chord duration is replaced by key-chord specific values, which has a positive effect on the key estimation performance. Next, the HMM system is modified such that the prior chord duration distribution is no longer a geometric distribution but one that better approximates the observed durations in an appropriate data set. This modification leads to a small improvement of the chord estimation performance, but of course, it requires the availability of a suitable data set with chord notations from which to retrieve a target durational distribution. A final experiment demonstrates that increasing the scope of the contextual model only leads to statistically insignificant improvements. On top of that, the required computational load increases greatly

Word alignment and smoothing methods in statistical machine translation: Noise, prior knowledge and overfitting

This thesis discusses how to incorporate linguistic knowledge into an SMT system. Although one important category of linguistic knowledge is that obtained by a constituent / dependency parser, a POS / super tagger, and a morphological analyser, linguistic knowledge here includes larger domains than this: Multi-Word Expressions, Out-Of-Vocabulary words, paraphrases, lexical semantics (or non-literal translations), named-entities, coreferences, and transliterations. The first discussion is about word alignment where we propose a MWE-sensitive word aligner. The second discussion is about the smoothing methods for a language model and a translation model where we propose a hierarchical Pitman-Yor process-based smoothing method. The common grounds for these discussion are the examination of three exceptional cases from real-world data: the presence of noise, the availability of prior knowledge, and the problem of underfitting. Notable characteristics of this design are the careful usage of (Bayesian) priors in order that it can capture both frequent and linguistically important phenomena. This can be considered to provide one example to solve the problems of statistical models which often aim to learn from frequent examples only, and often overlook less frequent but linguistically important phenomena