Intelligenta Transportsystem (ITS) utgör idag en central del i arbetet att försöka höja kvaliteten i transportnätverken, genom att exempelvis ge stöd i arbetet att leda trafik i realtid och att ge trafikanter större möjlighet att ta informerade beslut gällandes sin körning. Kortsiktig prediktion av trafikdata, däribland trafikvolym, spelar en central roll för de tjänster ITS-systemen levererar. Den starka teknologiska utvecklingen de senaste decennierna har bidragit till en ökad möjlighet till att använda datadriven modellering för att utföra kortsiktiga prediktioner av trafikdata. Säsongsbaserad ARIMA (SARIMA) är en av de vanligaste datadrivna modellerna för modellering och predicering av trafikdata, vilken använder mönster i historisk data för att predicera framtida värden. Vid modellering med SARIMA behöver en mängd beslut tas gällandes de data som används till modelleringen. Exempel på sådana beslut är hur stor mängd träningsdata som ska användas, vilka dagar som ska ingå i träningsmängden och vilket aggregationsintervall som ska användas. Därtill utförs nästintill enbart enstegsprediktioner i tidigare studier av SARIMA-modellering av trafikdata, trots att modellen stödjer predicering av flera steg in i framtiden. Besluten gällandes de parametrar som nämnts saknar ofta teoretisk motivering i tidigare studier, samtidigt som det är högst troligt att dessa beslut påverkar träffsäkerheten i prediktionerna. Därför syftar den här studien till att utföra en känslighetsanalys av dessa parametrar, för att undersöka hur olika värden påverkar precisionen vid prediktion av trafikvolym. I studien utvecklades en modell, med vilken data kunde importeras, preprocesseras och sedan modelleras med hjälp av SARIMA. Studien använde trafikvolymdata som insamlats under januari och februari 2014, med hjälp av kameror placerade på riksväg 40 i utkanten av Göteborg. Efter differentiering av data används såväl autokorrelations- och partiell autokorrelationsgrafer som informationskriterier för att definiera lämpliga SARIMA-modeller, med vilka prediktioner kunde göras. Med definierade modeller genomfördes ett experiment, där åtta unika scenarion testades för att undersöka hur prediktionsprecisionen av trafikvolym påverkades av olika mängder träningsdata, vilka dagar som ingick i träningsdata, längden på aggregationsintervallen och hur många tidssteg in i framtiden som predicerades. För utvärdering av träffsäkerheten i prediktionerna användes MAPE, RMSE och MAE. Resultaten som experimentet visar är att definierade SARIMA-modeller klarar att predicera aktuell data med god precision oavsett vilka värden som sattes för de variabler som studerades. Resultaten visade dock indikationer på att en träningsvolym omfattande fem dagar kan generera en modell som ger mer träffsäkra prediktioner än när volymer om 15 eller 30 dagar används, något som kan ha stor praktisk betydelse vid realtidsanalys. Därtill indikerar resultaten att samtliga veckodagar bör ingå i träningsdatasetet när dygnsvis säsongslängd används, att SARIMA-modelleringen hanterar aggregationsintervall om 60 minuter bättre än 30 eller 15 minuter samt att enstegsprediktioner är mer träffsäkra än när horisonter om en eller två dagar används. Studien har enbart fokuserat på inverkan av de fyra parametrarna var för sig och inte om en kombinerad effekt finns att hitta. Det är något som föreslås för framtida studier, liksom att vidare utreda huruvida en mindre träningsvolym kan fortsätta att generera mer träffsäkra prediktioner även för andra perioder under året. / Intelligent Transport Systems (ITS) today are a key part of the effort to try to improve the quality of transport networks, for example by supporting the real-time traffic management and giving road users greater opportunity to take informed decisions regarding their driving. Short-term prediction of traffic data, including traffic volume, plays a central role in the services delivered by ITS systems. The strong technological development has contributed to an increased opportunity to use data-driven modeling to perform short-term predictions of traffic data. Seasonal ARIMA (SARIMA) is one of the most common models for modeling and predicting traffic data, which uses patterns in historical data to predict future values. When modeling with SARIMA, a variety of decisions are required regarding he data used. Examples of such decisions are the amount of training data to be used, the days to be included in training data and the aggregation interval to be used. In addition, one-step predictions are performed most often in previous studies of SARIMA modeling of traffic data, although the model supports multi-step prediction into the future. Often, in previous studies, decisions are made concerning mentioned variables without theoretical motivation, while it is highly probable that these decisions affect the accuracy of the predictions. Therefore, this study aims at performing a sensitivity analysis of these parameters to investigate how different values affect the accuracy of traffic volume prediction. The study developed a model with which data could be imported, preprocessed and then modeled using a SARIMA model. Traffic volume data was used, which was collected during January and February 2014, using cameras located on highway 40 on the outskirts of Gothenburg. After differentiation of data, autocorrelation and partial autocorrelation graphs as well as information criteria are used to define appropriate SARIMA models, with which predictions could be made. With defined models, an experiment was conducted in which eight unique scenarios were tested to investigate how the prediction accuracy of traffic volume was influenced by different amount of exercise data, what days was included in training data, length of aggregation intervals, and how many steps into the future were predicted. To evaluate the accuracy of the predictions, MAPE, RMSE and MAE were used. The results of the experiment show that developed SARIMA models are able to predict current data with good precision no matter what values were set for the variables studied. However, the results showed indications that a training volume of five days can generate a model that provides more accurate predictions than when using 15 or 30-day volumes, which can be of great practical importance in real-time analysis. In addition, the results indicate that all weekdays should be included in the training data set when daily seasonality is used, SARIMA modeling handles aggregation intervals of 60 minutes better than 30 or 15 minutes, and that one-step predictions are more accurate than when one or two days horizons are used. The study has focused only on the impact of the four parameters separately and not if a combined effect could be found. Further research is proposed for investigating if combined effects could be found, as well as further investigating whether a lesser training volume can continue to generate more accurate predictions even for other periods of the year.

Predicting future sales can bring many advantages to retailers with regards to organizational performance. Using big data to make accurate forecasts can enable retailer to improve their operational performance and profitability substantially by reducing lost sales, inventory levels and labor costs. Previous research within the field of retail forecasting has mostly been dedicated to forecasting on lower time granularities such as weekly and monthly. However, despite the high practicality for retailers, forecasts on higher frequencies have not been properly covered by the current literature. This study aims to investigate how to forecast future sales using high-frequency data for a Swedish pharmacy retail chain. The forecasts are made on a daily and sub-daily time granularity using time series models SARIMA, Holt-Winter’s method and Facebook Prophet. The results show that Facebook Prophet was the most practical model and had the highest forecasting accuracy both on a daily and sub-daily frequency according to the error metrics MAPE, MAE and RMSE. / Att förutsäga framtida försäljning kan medföra många fördelar för detaljis-ter när det gäller organisationens prestanda. Att använda big data för att göra korrekta prognoser kan göra det möjligt för återförsäljare att förbättra sin lönsamhet avsevärt genom att minska förlorad försäljning, lagernivåer och arbetskostnader. Tidigare forskning inom området prognoser inom de-taljhandeln har mestadels ägnat sig åt prognoser på lägre tidsgranulariteter såsom veckovis och månadsvis. Trots att prognoser är mycket praktiska för detaljister så har prognoser på högre frekvenser inte täckts ordentligt av den aktuella litteraturen. Denna masteruppsats syftar till att undersöka hur man kan prognostisera framtida försäljning med hjälp av högfrekvent data för en svensk apotekskedja .Prognoserna görs på en daglig och sub-daglig tidsgranularitet medt idsseriemodellerna SARIMA, Holt-Winters metod och Facebook Prophet. Resultaten visar att Facebook Prophet var den mest praktiska tidsseriemodellen och hade den högsta träffsäkerheten både på en daglig och sub-daglig frekvens enligt felmåtten MAPE, MAE och RMSE.

The accurate estimation of travel time is valuable for a variety of transportation applications such as freeway performance evaluation and real-time traveler information. Given the extensive availability of traffic data collected by intelligent transportation systems, a variety of travel time estimation methods have been developed. Despite limited success under light traffic conditions, traditional corridor travel time prediction methods have suffered various drawbacks. First, most of these methods are developed based on data generated by dual-loop detectors that contain average spot speeds. However, single-loop detectors (and other devices that emulate its operation) are the most commonly used devices in traffic monitoring systems. There has not been a reliable methodology for travel time prediction based on data generated by such devices due to the lack of speed measurements. Moreover, the majority of existing studies focus on travel time estimation. Secondly, the effect of traffic progression along the freeway has not been considered in the travel time prediction process. Moreover, the impact of incidents on travel time estimates has not been effectively accounted for in existing studies.The objective of this dissertation is to develop a methodology for dynamic travel time prediction based on continuous data generated by single-loop detectors (and similar devices) and incident reports generated by the traffic monitoring system. This method involves multiple-step-ahead prediction for flow rate and occupancy in real time. A seasonal autoregressive integrated moving average (SARIMA) model is developed with an embedded adaptive predictor. This predictor adjusts the prediction error based on traffic data that becomes available every five minutes at each station. The impact of incidents is evaluated based on estimates of incident duration and the queue incurred.Tests and comparative analyses show that this method is able to capture the real-time characteristics of the traffic and provide more accurate travel time estimates particularly when incidents occur. The sensitivities of the models to the variations of the flow and occupancy data are analyzed and future research has been identified.The potential of this methodology in dealing with less than perfect data sources has been demonstrated. This provides good opportunity for the wide application of the proposed method since single-loop type detectors are most extensively installed in various intelligent transportation system deployments.

n the electrical power grid, the power load is not constant but continuouslychanging. This depends on many different factors, among which the habits of theconsumers, the yearly seasons and the hour of the day. The continuous change inenergy consumption requires the power grid to be flexible. If the energy provided bygenerators is lower than the demand, this is usually compensated by using renewablepower sources or stored energy until the power generators have adapted to the newdemand. However, if buffers are depleted the output may not meet the demandedpower and could cause power outages. The currently adopted practice in the indus-try is based on configuring the grid depending on some expected power draw. Thisanalysis is usually performed at a high level and provide only some basic load aggre-gate as an output. In this thesis, we aim at investigating techniques that are able topredict the behaviour of loads with fine-grained precision. These techniques couldbe used as predictors to dynamically adapt the grid at run time. We have investigatedthe field of time series forecasting and evaluated and compared different techniquesusing a real data set of the load of the Swedish power grid recorded hourly throughyears. In particular, we have compared the traditional ARIMA models to a neuralnetwork and a long short-term memory (LSTM) model to see which of these tech-niques had the lowest forecasting error in our scenario. Our results show that theLSTM model outperformed the other tested models with an average error of 6,1%.

Os processos espaço-temporais vêm ganhando destaque nos últimos anos, em razão do aumento de estudos compreendendo variáveis que apresentam interação entre as dimensões espacial e temporal. Com o objetivo de modelar esses processos, Pfeifer e Deutsch (1980a) propuseram uma extensão da classe de modelos univariados de Box-Jenkins, denominada por modelo espaço-temporal autoregressivo de média móvel (STARMA). Essa classe de modelos é utilizada para descrever dados de séries temporais espacialmente localizadas. Os processos passíveis de modelagem via classe de modelos STARMA são caracterizados por observações de variáveis aleatórias, em que os locais a serem incorporados no modelo são fixos no espaço. A dependência entre as n séries temporais é modelada por meio da matriz de ponderação, de modo que os modelos da classe STARMA expressem cada observação no tempo t e na localização i como uma média ponderada de combinações lineares das observações anteriores e a inovação defasada no espaço e no tempo conjuntamente. Dada a nova classe de modelos, os objetivos deste estudo foram apresentar a classe de modelos STARMA, implentar computacionalmente, no software R, rotinas que permitam a análise de dados espaço-temporais, com as rotinas implementadas estabelecer e testar modelos de séries temporais aos dados de temperaturas mínimas médias mensais de 8 estações meteorológicas situadas no Paraná e comparar a classe de modelos STARMA com a classe de modelos univariados proposta por Box e Jenkins (1970). Com este estudo verificou-se que na apresentação da classe de modelos STARMA há complexidade no conceito de ordens de vizinhança e na identificação dos modelos espaço-temporais. Em relação a criação de rotinas responsáveis pelas análises de dados espaço-temporais observou-se dificuldades em sua implementação, principalmente no momento de estimação dos parâmetros. Na comparação da classe de modelos STARMA, multivariada, com a classe de modelos SARIMA, univariada, constatou-se que ambos os modelos se ajustaram de maneira satisfatória aos dados, produzindo previsões acuradas. / Spatio-temporal processes have been highlighted lately, due to the increase of studies approaching variables that present interactions between the spatial and temporal dimensions. In order to model these processes, Pfeifer e Deutsch (1980a) have suggested an extension of the Box-Jenkins univariate model class, named spatio-temporal autoregressive moving-average model (STARMA). This model class is used to describe spatially located time series data. The processes prone to be modeled via the STARMA model class are characterized by observations of random variables, in which the locations to be incorporated in the model are spatially fixed. The dependence between the n time series is modeled through the weighing matrix. So STARMA models express each observation at time t and location i as a weighed mean of linear combinations of the previous observations and the jointly lagged innovation in space and time. Given the new class models, the objectives of this study were to present a class of models STARMA, implentar computationally, in textit R software, routines that allow the analysis of spatio-temporal data with the routines implemented to establish and test models time series data of monthly average minimum temperatures of 8 meteorological stations located in Paraná and compare the class of models STARMA with the class of univariate models proposed by Box e Jenkins (1970). With this study it was found that the presentation of the class of models STARMA no complexity in the concept of ordered neighborhood and identification of spatio-temporal models. Regarding the creation of routines responsible for the analysis of spatio-temporal observed difficulties in its implementation, especially at the time of estimation of parameters. In comparison class STARMA models, multivariate, with the class of SARIMA models, univariate, it was found that both models were adjusted satisfactorily to the data, producing accurate forecasts.

Inflation is one of the most important macroeconomic variables. It is vital that policy makers receive accurate forecasts of inflation so that they can adjust their monetary policy to attain stability in the economy which has been shown to lead to economic growth. The purpose of this study is to model inflation and evaluate if applying the Kalman filter to SARIMA models lead to higher forecast accuracy compared to just using the SARIMA model. The Box-Jenkins approach to SARIMA modelling is used to obtain well-fitted SARIMA models and then to use a subset of observations to estimate a SARIMA model on which the Kalman filter is applied for the rest of the observations. These models are identified and then estimated with the use of monthly inflation for Luxembourg, Mexico, Portugal and Switzerland with the target to use them for forecasting. The accuracy of the forecasts are then evaluated with the error measures mean squared error (MSE), mean average deviation (MAD), mean average percentage error (MAPE) and the statistic Theil's U. For all countries these measures indicate that the Kalman filtered model yield more accurate forecasts. The significance of these differences are then evaluated with the Diebold-Mariano test for which only the difference in forecast accuracy of Swiss inflation is proven significant. Thus, applying the Kalman filter to SARIMA models with the target to obtain forecasts of monthly inflation seem to lead to higher or at least not lower predictive accuracy for the monthly inflation of these countries.

I denna uppsats har olika metoder för att göra prognoser för ekonomiska tidsserier med säsongsmönster jämförts och utvärderats. Frågan som undersökningen har kretsat kring är: Vilken metod är bäst lämpad för att göra prognoser av tidsserier med säsongsmönster? De metoder som jämförs är säsongsrensningsmetoderna Census II och TRAMO/SEATS, säsongsmodellerna SARIMA och ARIMA med dummyvariabler för säsong samt en metod där medelvärdena från de fyra första metoderna används som prognoser. För att genomföra undersökningen har dessa metoder tillämpats på fyra ekonomiska tidsserier, nämligen: konsumtion, BNP, export samt byggstarter. Resultatet från undersökningen är att säsongsmodellerna är bäst för konsumtionsserien, säsongsrensningsmetoderna är bäst för BNP- och exportserien och den ena säsongsmodellen (SARIMA) är bäst för byggstartsserien medan den andra (ARIMA-dummy) är den sämsta. Val av prognosmetod beror med andra ord på vilken serie som ska prognostiseras.

In this report six different models for predicting the electrical spot price on the Nordic power exchange, Nord Pool, are developed and compared. They are evaluated against the already existing model as well as the naive test, which is a forecast where the last week’s observations are used as a prognosis for the coming week. The models developed are constructed so that the models for different time resolutions are combined to create a full model. Harmonic regression with a linear trend are used to identify a yearly trend while SARIMAX/SARIMA time series models are used on a daily and hourly basis to reveal dependencies in the data.   The model with the best prediction performance is shown to be a SARIMAX model with temperature as exogenous variable on a daily resolution, together with a SARIMA model on an hourly resolution. With an average MAPE of 12.69% and a MAPE2 of 6.90% it has the smallest prediction error out of all of the competing models when doing one week forecasts on the whole year 2009.

Os processos espaço-temporais vêm ganhando destaque nos últimos anos, em razão do aumento de estudos compreendendo variáveis que apresentam interação entre as dimensões espacial e temporal. Com o objetivo de modelar esses processos, Pfeifer e Deutsch (1980a) propuseram uma extensão da classe de modelos univariados de Box-Jenkins, denominada por modelo espaço-temporal autoregressivo de média móvel (STARMA). Essa classe de modelos é utilizada para descrever dados de séries temporais espacialmente localizadas. Os processos passíveis de modelagem via classe de modelos STARMA são caracterizados por observações de variáveis aleatórias, em que os locais a serem incorporados no modelo são fixos no espaço. A dependência entre as n séries temporais é modelada por meio da matriz de ponderação, de modo que os modelos da classe STARMA expressem cada observação no tempo t e na localização i como uma média ponderada de combinações lineares das observações anteriores e a inovação defasada no espaço e no tempo conjuntamente. Dada a nova classe de modelos, os objetivos deste estudo foram apresentar a classe de modelos STARMA, implentar computacionalmente, no software R, rotinas que permitam a análise de dados espaço-temporais, com as rotinas implementadas estabelecer e testar modelos de séries temporais aos dados de temperaturas mínimas médias mensais de 8 estações meteorológicas situadas no Paraná e comparar a classe de modelos STARMA com a classe de modelos univariados proposta por Box e Jenkins (1970). Com este estudo verificou-se que na apresentação da classe de modelos STARMA há complexidade no conceito de ordens de vizinhança e na identificação dos modelos espaço-temporais. Em relação a criação de rotinas responsáveis pelas análises de dados espaço-temporais observou-se dificuldades em sua implementação, principalmente no momento de estimação dos parâmetros. Na comparação da classe de modelos STARMA, multivariada, com a classe de modelos SARIMA, univariada, constatou-se que ambos os modelos se ajustaram de maneira satisfatória aos dados, produzindo previsões acuradas. / Spatio-temporal processes have been highlighted lately, due to the increase of studies approaching variables that present interactions between the spatial and temporal dimensions. In order to model these processes, Pfeifer e Deutsch (1980a) have suggested an extension of the Box-Jenkins univariate model class, named spatio-temporal autoregressive moving-average model (STARMA). This model class is used to describe spatially located time series data. The processes prone to be modeled via the STARMA model class are characterized by observations of random variables, in which the locations to be incorporated in the model are spatially fixed. The dependence between the n time series is modeled through the weighing matrix. So STARMA models express each observation at time t and location i as a weighed mean of linear combinations of the previous observations and the jointly lagged innovation in space and time. Given the new class models, the objectives of this study were to present a class of models STARMA, implentar computationally, in textit R software, routines that allow the analysis of spatio-temporal data with the routines implemented to establish and test models time series data of monthly average minimum temperatures of 8 meteorological stations located in Paraná and compare the class of models STARMA with the class of univariate models proposed by Box e Jenkins (1970). With this study it was found that the presentation of the class of models STARMA no complexity in the concept of ordered neighborhood and identification of spatio-temporal models. Regarding the creation of routines responsible for the analysis of spatio-temporal observed difficulties in its implementation, especially at the time of estimation of parameters. In comparison class STARMA models, multivariate, with the class of SARIMA models, univariate, it was found that both models were adjusted satisfactorily to the data, producing accurate forecasts.

El aumento del aporte energético por parte de las centrales eólicas dentro de la matriz de energías renovables no convencionales de Chile, crea la imperiosa necesidad de desarrollar herramientas que ayuden a gestionar el funcionamiento de parques eólicos, y en particular de los aerogeneradores que lo componen, con el fin de hacer más eficiente la integración y manejo en el sistema interconectado. En esta línea, el propósito de este trabajo es desarrollar un modelo predictivo para la potencia generada en un aerogenerador en base a series de tiempo históricas de variables atmosféricas del lugar donde éste se encuentra. El trabajo de memoria presenta los resultados de la implementación de un modelo SARIMA (siglas en inglés de Seasonal Auto Regressive Integrated Moving Average) y un modelo de persistencia, para predicción de velocidad de viento a horizontes de tiempo de uno y cinco pasos en una escala de tiempo de una hora por cada paso, resultados que luego son transformados a potencia eléctrica por medio de la curva de potencia del aerogenerador considerado. La investigación conecta los campos de la física, generación de energía y de teoría de estimación. Mientras que el primero aporta las ecuaciones con las cuales se describe el viento en la atmósfera y el segundo aporta la base técnica con la cual se relaciona la velocidad del viento con la potencia generada por un aerogenerador, el tercero entrega las herramientas para poder realizar predicción a distintos horizontes por medio de series de tiempo. Por ello, el reporte comienza por los fundamentos físicos que describen la velocidad del viento en la atmósfera, para seguir con los principios técnicos de un aerogenerador y continúa mencionando técnicas utilizadas en el ámbito de la predicción. Además, se trabaja con datos muestreados durante el año 1990 en la localidad de Punta Lengua de Vaca y que fueron obtenidos por el proyecto EOLO del Departamento de Geofísica de la Universidad de Chile. Los resultados de este trabajo permitieron conocer las limitaciones, ventajas y desventajas que poseen tanto el modelo de persistencia como los modelos SARIMA en el ámbito de predicción. En la misma línea, se cuantificó por medio de indicadores de desempeño la exactitud en las predicciones realizadas usando ambos modelos, para finalmente compararlos bajo distintos horizontes de predicción y usando datos de distintos lugares. Finalmente se concluye que el modelo SARIMA puede ser utilizado para predicción de potencia generada en un aerogenerador y que, en comparación con el modelo de persistencia, presenta mejores resultados en predicción a cinco pasos, pero no así en el caso de predicción a un paso, donde la relación se invierte.