This thesis is about the study of relationships between experimental dynamical systems. The basic approach is to fit radial basis function maps between time delay embeddings of manifolds. We have shown that under certain conditions these maps are generically diffeomorphisms, and can be analysed to determine whether or not the manifolds in question are diffeomorphically related to each other. If not, a study of the distribution of errors may provide information about the lack of equivalence between the two. The method has applications wherever two or more sensors are used to measure a single system, or where a single sensor can respond on more than one time scale: their respective time series can be tested to determine whether or not they are coupled, and to what degree. One application which we have explored is the determination of a minimum embedding dimension for dynamical system reconstruction. In this special case the diffeomorphism in question is closely related to the predictor for the time series itself. Linear transformations of delay embedded manifolds can also be shown to have nonlinear inverses under the right conditions, and we have used radial basis functions to approximate these inverse maps in a variety of contexts. This method is particularly useful when the linear transformation corresponds to the delay embedding of a finite impulse response filtered time series. One application of fitting an inverse to this linear map is the detection of periodic orbits in chaotic attractors, using suitably tuned filters. This method has also been used to separate signals with known bandwidths from deterministic noise, by tuning a filter to stop the signal and then recovering the chaos with the nonlinear inverse. The method may have applications to the cancellation of noise generated by mechanical or electrical systems. In the course of this research a sophisticated piece of software has been developed. The program allows the construction of a hierarchy of delay embeddings from scalar and multi-valued time series. The embedded objects can be analysed graphically, and radial basis function maps can be fitted between them asynchronously, in parallel, on a multi-processor machine. In addition to a graphical user interface, the program can be driven by a batch mode command language, incorporating the concept of parallel and sequential instruction groups and enabling complex sequences of experiments to be performed in parallel in a resource-efficient manner.

Cloud Computing sees increased adoption as companies seek to increase flexibility and reduce cost. Although the large cloud service providers employ a pay-as-you-go pricing model and enable customers to scale up and down quickly, there is still room for improvement. Workload in the form of CPU utilization often fluctuates which leads to unnecessary cost and environmental impact for companies. To help mitigate this issue, the aim of this paper is to predict future CPU utilization using a long short-term memory (LSTM) machine learning model. By predicting utilization up to 30 minutes into the future, companies are able to scale their capacity just in time and avoid unnecessary cost and damage to the environment. The study is divided into two parts. The first part analyses how well the LSTM model performs when predicting one step at a time compared with a state-of-the-art model. The second part analyses the accuracy of the LSTM when making predictions up to 30 minutes into the future. To allow for an objective analysis of results, the LSTM is compared with a standard RNN, which is similar to the LSTM in its inherit algorithmic structure. To conclude, the results suggest that LSTM may be a useful tool for reducing cost and unnecessary environmental impact for business applications hosted on a public cloud. / Användandet av molntjänster ökar bland företag som önskar förbättrad flexibilitet och sänkta kostnader. De stora molntjänstleverantörerna använder en prismodell där kostnaden är direkt kopplad till användningen, och låter kunderna snabbt ställa om sin kapacitet, men det finns ändå förbättringsmöjligheter. CPU-behoven fluktuerar ofta vilket leder till meningslösa kostnader och onödig påverkan på klimatet när kapacitet är outnyttjad. För att lindra detta problem används i denna rapport en LSTM maskininlärningsmodell för att förutspå framtida CPU-utnyttjande. Genom att förutspå utnyttjandet upp till 30 minuter in i framtiden hinner företag ställa om sin kapacitet och undvika onödig kostnad och klimatpåverkan. Arbetet ¨ar uppdelat i två delar. Först en del där LSTM-modellen förutspår ett tidssteg åt gången. Därefter en del som analyserar träffsäkerheten för LSTM flera tidssteg in i framtiden, upp till 30 tidssteg. För att möjliggöra en objektiv utvärdering så jämfördes LSTM-modellen med ett standard recurrent neural network (RNN) vilken liknar LSTM i sin struktur. Resultaten i denna studie visar att LSTM verkar vara ¨överlägsen RNN, både när det gäller att förutspå ett tidssteg in i framtiden och när det gäller flera tidssteg in i framtiden. LSTM-modellen var kapabel att förutspå CPU-utnyttjandet 30 minuter in i framtiden med i hög grad bibehållen träffsäkerhet, vilket också var målet med studien. Sammanfattningsvis tyder resultaten på att denna LSTM-modell, och möjligen liknande LSTM-modeller, har potential att användas i samband med företagsapplikationer då man önskar att reducera onödig kostnad och klimatpåverkan.