Statistics for stochastic processes

Statistics for stochastic processes

Course details

Material on computer: 26 hours of classroom + 6 hours of lab (R) + Rimoldi lectures

Schedule: Thursday 9:00 - 10:30

Exam

• Analysis of a data set (1h)
• Essay on one of the arguments by Prof. Rimoldi right after the first part (30 min)
• Oral examination (one question) a couple of days later

After the written part: ottimo - distinto - buono - sufficiente

Resources

• Books + web page
• Install packages "astsa" (from the web page), package "TSA"

^{12 Marzo 2018}

What is a time series?

Statistics are computed on observations (data): x₁, ..., x_n

Simple model: realizations of iid random variables X₁, ..., X_n

There are correlations among data, particularly if time is involved as ordering usually through time. Data are observations at discrete times of stochastic dynamical systems usually described with a stochastic process: discrete time series.

Course repetition

Material on compunet: 26 hours of classroom + 6 hours of lab (R) + Rimoltto lectures

Schedule: Thursday 9:00 - 10:30

Exam repetition

• Analysis of a data set (1h)
• Essay on one of the arguments by Prof. Rimott right after the first part (30 min)
• Oral examination (one question) a couple of days later

After the written part: ottimo - distinto - buono - sufficiente

Resources repetition

• Books + web page
• Install packages "forecast" (from the web page), package "TSA"

^{12 Marzo 2018}

Time series analysis

Statistics are computed on observations (data): x₁, ..., x_n

Simple model: realizations of iid random variables X₁, ..., X_n

There are correlations among data, particularly if time is involved in ordering usually through time. Data are observations at discrete times, of stochastic dynamical systems usually described with a stochastic process: discrete time series.

Notation and modeling

Notation: 3X_t, Y_t + Z

We can't assume longer independence as time series analysis.

Goals:

• Description of data (plots, descriptive statistics, _)
• Looking at trend, seasonality, random terms.

Time series (2 & ?); deterministic function (exponential) + stochastic process whose variance grows with time.

Correlation as scatter diagram with log plot.

Global temperature; trend + high seasonality.

Speech; periodic time series.

Post Riemot; spectral analysis to understand how to compute periodicity, Fourier analysis.

We're going to study linear time series sampled with time.

Difference between periodicity and seasonality.

Another tool is seasonal and trend decomposition, the command is "stl". Seasonal and trend creates smoother functions.

Remainder is a stochastic error.

Additive (linear) model = adding seasonal trend and remainder gives us the original data plot.

We will focus on finding a statistical model for the remainder term, for example with descriptive statistics (tests for normality, qq-plots, histograms).

We'll work on stationary, time series data (it's easier). Mean and variance are constant over time.

Non-stationary time series could be transformed into stationary ones working with increments −₋₁logarithm: → log (diff. common) diff + e_t estimate.

Model forecasting

After all those things, we have to diochronicate the model. The last part will be on forecasting: how the model behaves in the future.

Spectral analysis works with sine and cosine with different frequencies. Sum of them can represent time series. We will use spectral analysis only with the seasonal component.

Revise graphical test but we remind that we are refusing the zero hypothesis if the value we get is less than 0.05 (p-value remarkable). We are working at a discrete time.

Modeling time series

A time series is a collection of random variables {}_∈ = {_{−1, 0, 1, 2, …}}

How to model {}

1. Joint distribution of (₁ , …, _m) ∀ ∈ ℕ and ₁, …, _m ∈, that is to say (₁ ≤ ₁, …, _m ≤ _m) ∀ (₁, …, _m) ∈ ℜ functions of {}

Note that 1 is a complicated way ⇒ a simplified model is a Gaussian time series.