First, we will decompose the data then re-constructed without the seasonal component — seasadj function: Returns seasonally adjusted data:. Clearly, we can see that seasonal adjusted data is not stationary. Let us double check with KPSS unit root test:. Which means that the signal is not stationary. To make it so, we will apply a useful transformation called differencing. But in general, the signal looks stationary, let us confirm again with KPSS unit root test:.
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Which means that one Differencing Transformation is sufficient enough to get the stationary data. ARIMA 0,1,2 is the best because the square errors are much lower than coefficient values. Let us fit the chosen model, check the residuals and finally forecast for 12 months:. The Residuals are almost normally distributed, slightly skewed. ACF shows no significant values except at lag Ljung-Box test with significant p-values concludes that residuals are random white noise. Forecasting for 12 months, starting from Aug till Jul After many iterations, I had estimated the values of seasonal part P,D,Q as 1,1,1.
All square errors are much lower than the absolute value of coefficients. Again, we need to make sure that residuals are random and they are unstructured:. Histogram distribution is skewed a little but still, model passes the Ljung-Box test. Forecasts from the model for the next one year are shown below:. On the other hand, We can just use auto. It will give the following result:. Actually, I have read many articles to practice time series analysis. In addition, the course and certificate I had from Microsoft Applied Machine Learning were very beneficial as well.
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Course contents Time series analysis concerns the mathematical modeling of time varying phenomena, e. Higher education credits : 7,5 Level: A Language of instruction : The course will be offered in English if non-Swedish speaking students are attending. Course material General material: Course program All the below slides, as well as the pdf and matlab files, etc can be downloaded here.
Analysis of Financial Time Series, Third Edition [Book]
An errata for the textbook is available here. Scalable learning videos. Lecture notes and schedule: Week 1: L1: Introduction and overview. Multivariate random variables. Stochastic processes. Model order selection. Reading instructions: Ch.
EC 572/472 Time Series Analysis and Forecasting
Week 5 L9: Prediction. Multivariate time series. Week 6 L Recursive estimation. State space models. Project discussion. Indeed, having a finite second moment is a necessary and sufficient condition for the weak stationarity of a strongly stationary process.
White Noise Process : A white noise process is a serially uncorrelated stochastic process with a mean of zero and a constant and finite variance. Note that this implies that every white noise process is a weak stationary process. Very close to the definition of strong stationarity, N -th order stationarity demands the shift-invariance in time of the distribution of any n samples of the stochastic process, for all n up to order N.
Thus, the same condition is required:. Naturally, stationarity to a certain order N does not imply stationarity of any higher order but the inverse is true.
Time series analysis (FMSN45/MASM17)
An interesting thread in mathoverflow showcases both an example of a 1st order stationary process that is not 2nd order stationary, and an example for a 2nd order stationary process that is not 3rd order stationary. And similarly, having a finite second moment is a sufficient and necessary condition for a 2nd order stationary process to also be a weakly stationary process. The term first-order stationarity is sometimes used to describe a series that has means that never changes with time, but for which any other moment like variance can change. Cyclostationarity is prominent in signal processing.
A stochastic process is trend stationary if an underlying trend function solely of time can be removed, leaving a stationary process. In the presence of a shock a significant and rapid one-off change to the value of the series , trend-stationary processes are mean-reverting; i. Intuitive extensions exist of all of the above types of stationarity for pairs of stochastic processes.
Weak stationarity and N -th order stationarity can be extended in the same way the latter to M - N -th order joint stationarity. A weaker form of weak stationarity, prominent in geostatistical literature see [Myers ] and [Fischer et al. An important class of non-stationary processes are locally stationary LS processes.
Alternatively, [Dahlhaus, ] defines them informally as processes which locally at each time point are close to a stationary process but whose characteristics covariances, parameters, etc. A formal definition can be found in [Vogt, ], and [Dahlhaus, ] provides a rigorous review of the subject. LS processes are of importance because they somewhat bridge the gap between the thoroughly explored sub-class of parametric non-stationary processes see the following section and the uncharted waters of the wider family of non-parametric processes, in that they have received rigorous treatment and a corresponding set of analysis tools akin to those enjoyed by parametric processes.
A great online resource on the topic is the home page of Prof. Guy Nason , who names LS processes as his main research interest. The following typology figure, partial as it may be, can help understand the relations between the different notions of stationarity we just went over:. The definitions of stationarity presented so far have been non-parametric; i. The related concept of a difference stationarity and unit root processes, however, requires a brief introduction to stochastic process modeling.
The topic of stochastic modeling is also relevant insofar as various simple models can be used to create stochastic processes see figure 5. The forecasting of future values is a common task in the study of time series data. To make forecasts, some assumptions need to be made regarding the Data Generating Process DGP , the mechanism generating the data. These assumptions often take the form of an explicit model of the process, and are also often used when modeling stochastic processes for other tasks, such as anomaly detection or causal inference. We will go over the three most common such models.
This is a memory-based model, in the sense that each value is correlated with the p preceding values; an AR model with lag p is denoted with AR p. The vector autoregressive VAR model generalizes the univariate case of the AR model to the multivariate case; now each element of the vector x[t] of length k can be modeled as a linear function of all the elements of the past p vectors:.
Like for autoregressive models, a vector generalization, VMA, exists. With a basic understanding of common stochastic process models, we can now discuss the related concept of difference stationary processes and unit roots. This concept relies on the assumption that the stochastic process in question can be written as an autoregressive process of order p, denoted as AR p :.
We can write the same process as:. The part inside the parenthesis on the left is called the characteristic equation of the process. We can consider the roots of this equation:. This means that the process can be transformed into a weakly-stationary process by applying a certain type of transformation to it, called differencing. Difference stationary processes have an order of integration , which is the number of times the differencing operator must be applied to it in order to achieve weak stationarity.
A process that has to be differenced r times is said to be integrated of order r, denoted by I r. A common sub-type of difference stationary process are processes integrated of order 1, also called unit root process. The simplest example for such a process is the following autoregressive model:. Unit root processes, and difference stationary processes generally, are interesting because they are non-stationary processes that can be easily transformed into weakly stationary processes.
As a result, while the term is not used interchangeably with non-stationarity, the questions regarding them sometimes are. I thought it worth mentioning here, as sometime tests and procedures to check whether a process has a unit root a common example is the Dickey-Fuller test are mistakenly thought of as procedures for testing non-stationarity as a latter post in this series touches upon.
Another definition of interest is a wider, and less parametric, sub-class of non-stationary processes, which can be referred to as semi-parametric unit root processes. The definition was introduced in [Davidson, ], but a concise overview of it can be found [Breitung, ].