Time-Series Analysis


By Prof. Seungchul Lee
http://iai.postech.ac.kr/
Industrial AI Lab at POSTECH

Table of Contents

1. So Far

  • Regression, Classification, Dimension Reduction,

  • Based on snapshot-type data



  • Sequence matters



  • What is a sequence?
    • sentence
    • medical signals
    • speech waveform
    • vibration measurement
  • Sequence Modeling
    • Most of the real-world data is time-series
    • There are important bits to be considered
      • Past events
      • Relationship between events
        • Causality
        • Credit assignment
  • Learning the structure and hierarchy
  • Use the past and present observations to predict the future

2. (Determinstic) Sequences and Difference Equations

We will focus on linear difference equations (LDE), a surprisingly rich topic both theoretically and practivally.

For example,

$$ y[0]=1,\quad y[1]=\frac{1}{2},\quad y[2]=\frac{1}{4},\quad \cdots $$

or by closed-form expression,

$$y[n]=\left(\frac{1}{2}\right)^n,\quad n≥0 $$

or with a difference equation and an initial condition,

$$y[n]=\frac{1}{2}y[n−1],\quad y[0]=1$$

High order homogeneous LDE

$$y[n]=\alpha_1 y[n−1] + \alpha_2 y [n−2] + \cdots + \alpha_k y[n-k]$$

3. (Stochastic) Time Series Analysis

3.1. Stationarity and Non-Stationary Series

  • A series is stationary if there is no systematic change in mean and variance over time

    • Example: radio static

  • A series is non-stationary if mean and variance change over time

    • Example: GDP, population, weather, etc.

3.2. Testing for Non-Stationarity

Formally

  • Augmented DickeyFuller test

Informally

  • Auto-Correlation Function (ACF)
  • Normal Quantile Plot (Q-Q plot)

Q-Q Plot

  • Compare distribution of the residuals to normal
  • Scatter plot of residual quantiles against normal
    • Stationary data: quantiles match normal ($45^o$ line)
    • Non-stationary data: quantiles do not match (points off $45^o$ line)

3.3. Dealing with Non-Stationarity


Linear trends




Non-linear trends

  • For example, population may grow exponentially





Seasonal trends

  • Some series may exhibit seasonal trends

  • For example, weather pattern, employment, inflation, etc.




Combining Linear, Quadratic, and Seasonal Trends

  • Some data may have a combintation of trends




  • One solution is to apply repeated differencing to the series

  • For example, first remove seasonal trend. Then remove linear trend

  • Inspect model fit by examining residuals Q-Q plot

  • Anternatively, include both linear and cyclical trend terms into the model
\begin{align*} Y_t &= \beta_1 + \beta_2 Y_{t-1} \\ &+ \beta_3 t + \beta_4 t^{\beta_5} \\ &+ \beta_6 \sin \frac{2\pi}{s}t + \beta_7 \cos \frac{2\pi}{s}t \\ &+ u_t \end{align*}

4. Markov Process

4.1. Sequential Processes

  • Most classifiers ignored the sequential aspects of data
  • Consider a system which can occupy one of $N$ discrete states or categories $q_t \in \{S_1,S_2,\cdots,S_N\}$
  • We are interested in stochastic systems, in which state evolution is random
  • Any joint distribution can be factored into a series of conditional distributions


$$p(q_0,q_1,\cdots,q_T ) = p(q_0) \; p(q_1 \mid q_0) \; p( q_2 \mid q_1 q_0 ) \; p( q_3 \mid q_2 q_1 q_0 ) \cdots$$

Amost impossible to compute !!


$$p(q_0,q_1,\cdots,q_T ) = p(q_0) \; p(q_1 \mid q_0) \; p( q_2 \mid q_1 ) \; p( q_3 \mid q_2 ) \cdots$$

Possible and tractable !!

4.2. Markov Process

  • (Assumption) for a Markov process, the next state depends only on the current state:


$$ p(q_{t+1} \mid q_t,\cdots,q_0) = p(q_{t+1} \mid q_t)$$
  • More clearly


$$ p(q_{t+1} = s_j \mid q_t = s_i) = p(q_{t+1} = s_j \mid q_t = s_i,\; \text{any earlier history})$$


  • Given current state, the past does not matter
  • The state captures all relevant information from the history
  • The state is a sufficient statistic of the future

4.3. State Transition Matrix

For a Markov state 𝑠 and successor state 𝑠′, the state transition probability is defined by


$$ P_{ss'} = P\left[S_{t+1} = s' \mid S_t = s \right] $$


State transition matrix $P$ defines transition probabilities from all states $s$ to all successor states $s'$.





Example: MC episodes

  • sample episodes starting from $S_1$
In [16]:
import numpy as np

P = [[0, 0, 1],
    [1/2, 1/2, 0],
    [1/3, 2/3, 0]]
In [17]:
print(P[1][:])

[0.5, 0.5, 0]
In [18]:
a = np.random.choice(3,1,p = P[1][:])
print(a)

[1]
In [19]:
# sequential processes
# sequence generated by Markov chain
# S1 = 0, S2 = 1, S3 = 2

# starting from 0
x = 0
S = []
S.append(x)

for i in range(50):    
    x = np.random.choice(3,1,p = P[x][:])[0]    
    S.append(x)
     
print(S)    

[0, 2, 1, 0, 2, 1, 1, 0, 2, 1, 1, 1, 0, 2, 1, 0, 2, 1, 0, 2, 0, 2, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 0, 2, 1, 0, 2, 1, 1, 1, 1, 0, 2, 1, 1, 0, 2, 1, 0, 2, 1]

5. Hidden Markov Models

  • Discrete state-space model
    • Used in speech recognition
    • State representation is simple
    • Hard to scale-up the training
  • Assumption
    • We can observe something that is affected by the true state
    • Natual way of thinking
  • Limited sensors (incomplete state information)
    • But still partially related
  • Noisy senors
    • Unreliable




  • True state (or hidden variable) follows Markov chain

  • Observation emitted from state

    • $Y_t$ is noisily determined depending on the current state $X_t$

  • Forward: sequence of observations can be generated

  • Question: state estimation
$$P(X_T = s_i \mid Y_1 Y_2 \cdots Y_T)$$
  • HMM can do this, but with many difficulties

6. Kalman Filter

  • Linear dynamical system of motion



$$ \begin{align*} x_{t+1} &= A x_t + B u_t \\ z_t &= Cx_t \end{align*} $$

  • A, B, C ?

  • Continuous State space model

    • For filtering and control applications
    • Linear-Gaussian state space model
    • Widely used in many applications:
      • GPS, weather systems, etc.

  • Weakness

    • Linear state space model assumed
    • Difficult to apply to highly non-linear domains
In [20]:
%%javascript
$.getScript('https://kmahelona.github.io/ipython_notebook_goodies/ipython_notebook_toc.js')