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Exploring Data Measures of Centrality and Dispersion	Graphical Procedures Histograms, Boxplots, Ogives	Probability and Random Variables	Probability Distributions Discrete and Continuous probability Distributions. Compound and Mixed Probability Distributions
Testing Distributional Assumptions Testing Normality and Outlier Detection	Inference Procedures Confidence Intervals and Hypothesis Testing	Bivariate Data Correlation Coefficients & Hypothesis Tests, Chi Square Test for Independence	Linear Models Simple and Multiple Linear Regression Models, Residual Analysis, Regression ANOVA
Discrete Random Variables Expectations and variance of discrete random variables, Joint Distribution of discrete random variables	Continuous Random Variables Expectations and variance of continuous random variables	More on Random Variables Moment Generating Functions, Cumulant Generating Functions	Random Number Generation
Game Theory and Decision Theory Matrix Games	Experimental Design ANOVA Procedures, Testing Model Assumptions	Survival Analysis Non parametric approaches, Cox proportional hazard models	Statistical Process Control
Time Series Analysis Time Series

Defintions

Autocorrelation function

The autocovariance function is measured in squared units, so that the values obtained depend on the absolute size of the measurements. We can make this quantity independent of the absolute sizes of xn by defining a dimensionless quantity, the autocorrelation function.

The autocorrelation function (ACF) of a stationary process is defined by: \(k=corr(xt,xt+k)=k0\) The ACF of a purely indeterministic process satisfies k as k .

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Autoregressive model

The notation AR(p) refers to the autoregressive model of order p. The AR(p) model is defined as

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Backwards shift operator and difference operator

The backwards shift operator, B, acts on the process X to give a process BX such that: (BX)t=Xt-1

If we apply the backwards shift operator to a constant, then it doesn’t change it: B=

The difference operator, , is defined as = 1- B , or in other words: (X)t=Xt-Xt-1

Both operators can be applied repeatedly. For example:

(B2X)t=(B(BX))t=(BX)t-1=Xt-2

(2X)t= (X)t- (X)t-1=Xt-2Xt-1+Xt-2

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Lag Operator

In time series analysis, the lag operator or backshift operator operates on an element of a time series to produce the previous element.

For example, given some time series X={X1,X2,} then LXt=Xt-1 for all where L is the lag operator

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Ljung-Box test statistic

\[Q=T(T+2)k=1srk2/(T-k)\]

T = number of observations s = number of coefficients to test autocorrelation rk = autocorrelation coefficient (for lag k) Q = portmanteau test statistic.

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Moving Average Models

The moving average (MA) model is common approach for modeling univariate time series models. \[Xt=+t+1t-1++qt-q\]

Stationary Processes

Stationary Processes

A stationary process (or strict(ly) stationary process or strong(ly) stationary process) is a stochastic process whose joint probability distribution does not change when shifted in time. Consequently, parameters such as the mean and variance, if they are present, also do not change over time and do not follow any trends.

Stationarity is used as a tool in time series analysis, where the raw data is often transformed to become stationary; for example, economic data are often seasonal and/or dependent on a non-stationary price level. An important type of non-stationary process that does not include a trend-like behavior is the cyclostationary process.

Note that a “stationary process” is not the same thing as a “process with a stationary distribution”. Indeed there are further possibilities for confusion with the use of “stationary” in the context of stochastic processes; for example a “time-homogeneous” Markov chain is sometimes said to have “stationary transition probabilities”. Besides, all stationary Markov random processes are time-homogeneous.