1. Machine Learning
Week 4

2. Basic Probability
Envision an experiment for which the result is unknown. The collection of all possible outcomes is called the sample space. A set of outcomes, or subset of the sample space, is called an event.
A probability space is a three-tuple (W ,, Pr) where W is a sample space,  is a collection of events from the sample space and Pr is a probability law that assigns a number to each event in . For any events A and B, Pr must satsify:
Pr() = 1
Pr(A)  0
Pr(AC) = 1 – Pr(A)
Pr(A  B) = Pr(A) + Pr(B), if A  B = .
If A and B are events in  with Pr(B)  0, the conditional probability of A given B is

3. Random Variables
A random variable is “a number that you don’t know… yet”
Discrete vs. Continuous
Cumulative distribution function
Density function
Probability distribution (mass) function
Joint distributions
Conditional distributions
Functions of random variables
Moments of random variables
Transforms and generating functions

4. Conditioning
Frequently, the conditional distribution of Y given X is easier to find than the distribution of Y alone. If so, evaluate probabilities about Y using the conditional distribution along with the marginal distribution of X:
Example: Draw 2 balls simultaneously from a jar containing four balls numbered 1, 2, 3 and 4. X = number on the first ball, Y = number on the second ball, Z = XY. What is Pr(Z > 5)?
Key: Maybe easier to evaluate Z if X is known

5. Moments of Random Variables
Expectation = “average”
Variance = “volatility”
Standard Deviation
Coefficient of Variation

6. Linear Functions of Random Variables
Covariance
Correlation
If X and Y are independent then

7. Bernoulli Distribution
“Single coin flip” p = Pr(success)
N = 1 if success, 0 otherwise
Chapter 0

8. Binomial Distribution
“n independent coin flips” p = Pr(success)
N = # of successes
Chapter 0

9. Geometric Distribution
“independent coin flips” p = Pr(success)
N = # of flips until (including) first success
Chapter 0

10. Poisson Distribution
“Occurrence of rare events”  = average rate of occurrence per period;
N = # of events in an arbitrary period
Chapter 0

11. Uniform Distribution
X is equally likely to fall anywhere within interval (a,b) a b
Chapter 0

12. Exponential Distribution
X is nonnegative and it is most likely to fall near 0
Also memoryless; more on this later…
Chapter 0

13. Normal Distribution X follows a “bell-shaped” density function
From the central limit theorem, the distribution of the sum of independent and identically distributed random variables approaches a normal distribution as the number of summed random variables goes to infinity.
Chapter 0

14. Stochastic Processes
A stochastic process is a random variable that changes over time,
or a sequence of numbers that you don’t know yet.
Poisson process
Continuous time Markov chains
Chapter 0

15. Time Series
Autoregressive (AR) and Moving Average (MA) models

16. Time Series
A time series is a sequence of numerical data in which each item is associated with a particular instant in time
In fact with the current progress in computer technology we have daily series on interest rates, the hourly "telerate" interest rate index, and stock prices by the minute (or even second).

17. An analysis of a single sequence of data is called univariate time-series analysis
An analysis of several sets of data for the same sequence of time periods is called multivariatetime-series analysis or, more simply, multiple time-series analysis

18. Stochastic processes
Time series are an example of a stochastic or random process
A stochastic process is 'a statistical phenomenen that evolves in time according to probabilistic laws'
Mathematically, a stochastic process is an indexed collection of random variables

19. Stochastic processes
We are concerned only with processes indexed by time, either discrete time or continuous time processes such as

20. Continuous vs. Discrete
We base our inference usually on a single observation or realization of the process over some period of time, say [0, T] (a continuous interval of time) or at a sequence of time points {0, 1, 2, T}

21. Specification of a process
A simpler approach is to only specify the moments—this is sufficient if all the joint distributions are normal
The mean and variance functions are given by

22. Autocovariance
Because the random variables comprising the process are not independent, we must also specify their covariance

23. Autocorrelation
It is useful to standardize the autocovariance function (acvf)
Consider stationary case only
Use the autocorrelation function (acf)

24. Stationarity
Inference is most easy, when a process is stationary—its distribution does not change over time
This is strict stationarity
A process is weakly stationary if its mean and autocovariance functions do not change over time

25. Weak stationarity
The autocovariance depends only on the time difference or lag between the two time points involved

26. White noise
This is a purely random process, a sequence of independent and identically distributed random variables
Has constant mean and variance
Also

27. Several Models for Time Series
(1) a purely random process,
(2) a random walk,
(3) a movingaverage (MA) process,
(4) an autoregressive (AR) process,
(5) an autoregressive movingaverage (ARMA) process, and
(6) an autoregressive integrated moving average (ARIMA)process.

28. Purely Random Process Auto-covariance function
Auto-correlation function

29. Random Walk

30. Moving average processes
Start with {Zt} being white noise or purely random, mean zero, s.d. Z
{Xt} is a moving average process of order q (written MA(q)) if for some constants 0, 1, q we have
Usually 0 =1

31. Moving average processes
The mean and variance are given by
The process is weakly stationary because the mean is constant and the covariance does not depend on t

32. Moving average processes
If the Zt's are normal then so is the process, and it is then strictly stationary
The autocorrelation is

33. Moving average processes
Note the autocorrelation cuts off at lag q
For the MA(1) process with 0 = 1

34. Moving average processes
In order to ensure there is a unique MA process for a given acf, we impose the condition of invertibility
This ensures that when the process is written in series form, the series converges
For the MA(1) process Xt = Zt + Zt - 1, the condition is ||< 1

35. Moving average processes
For general processes introduce the backward shift operator B
Then the MA(q) process is given by

36. Moving average processes
The general condition for invertibility is that all the roots of the equation  lie outside the unit circle (have modulus less than one)

37. Autoregressive processes
Assume {Zt} is purely random with mean zero and s.d. z
Then the autoregressive process of order p or AR(p) process is

38. Autoregressive processes
The first order autoregression is
Xt = Xt Zt
Provided ||<1 it may be written as an infinite order MA process
Using the backshift operator we have
(1 – B)Xt = Zt

39. Autoregressive processes
From the previous equation we have

40. Autoregressive processes
Then E(Xt) = 0, and if ||<1

41. Autoregressive processes
The AR(p) process can be written as

42. Autoregressive processes
This is for
for some 1, 2, . . .
This gives Xt as an infinite MA process, so it has mean zero

43. Autoregressive processes
Conditions are needed to ensure that various series converge, and hence that the variance exists, and the autocovariance can be defined
Essentially these are requirements that the i become small quickly enough, for large i

44. Autoregressive processes
The i may not be able to be found however.
The alternative is to work with the i
The acf is expressible in terms of the roots i, i=1,2, ...p of the auxiliary equation

45. Autoregressive processes
Then a necessary and sufficient condition for stationarity is that for every i, |i|<1
An equivalent way of expressing this is that the roots of the equation
must lie outside the unit circle

46. ARMA processes Combine AR and MA processes
An ARMA process of order (p,q) is given by

47. ARMA processes
Alternative expressions are possible using the backshift operator

48. ARMA processes
An ARMA process can be written in pure MA or pure AR forms, the operators being possibly of infinite order
Usually the mixed form requires fewer parameters

49. ARIMA processes
General autoregressive integrated moving average processes are called ARIMA processes
When differenced say d times, the process is an ARMA process
Call the differenced process Wt. Then Wt is an ARMA process and

50. ARIMA processes Alternatively specify the process as
This is an ARIMA process of order (p,d,q)

51. ARIMA processes
The model for Xt is non-stationary because the AR operator on the left hand side has d roots on the unit circle
d is often 1
Random walk is ARIMA(0,1,0)
Can include seasonal terms—see later

52. Non-zero mean
We have assumed that the mean is zero in the ARIMA models
There are two alternatives
mean correct all the Wt terms in the model
incorporate a constant term in the model