Recurrent Neural Networks (RNN)

1. Time Series DataÂ¶

1.1. So FarÂ¶

• 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

1.2. (Determinstic) Sequences and Difference EquationsÂ¶

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,

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

or with a difference equation and an initial condition,

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

High order homogeneous LDE

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

1.3. (Stochastic) Time Series AnalysisÂ¶

1.3.1. Stationarity and Non-Stationary SeriesÂ¶

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

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

• Example: GDP, population, weather, etc.

1.3.2. Dealing with Non-StationarityÂ¶

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*}

1.4. Time-Series DataÂ¶

(almost) all the data coming from manufacturing environment are time-series data

• sensor data,
• process times,
• material measurement,
• equipment maintenance history,
• image data, etc.

Manufacturing application is about one of the following:

• prediction of time-series values
• anomaly detection on time-series data
• classification of time-series values
• metrology and inspection

1.4.1. Definition of time-seriesÂ¶

$$x: T \rightarrow \mathbb{R}^n \;\; \text{where}\;\; T=\{\cdots, t_{-2},t_{-1},t_0,t_1,t_2, \cdots \}$$

Example: material measurements: when $n=3$

$$x(t) = \begin{bmatrix} \text{average thickness}(t)\\ \text{thickness variance}(t)\\ \text{resistivity}(t) \end{bmatrix}$$

1.4.2. Supervised and Unsupervised Learning for Time-seriesÂ¶

For supervised learning, we define two time series

$$x: T \rightarrow \mathbb{R}^n \;\; \text{and} \;\; y: T \rightarrow \mathbb{R}^m$$

Supervised time-series learning:

\begin{align*} \text{predict} \quad &y(t_k) \\ \text{given} \quad & x(t_k), x(t_{k-1}), \cdots \;\, \text{and} \;\, y(t_{k-1}), y(t_{k-2}), \cdots \end{align*}

Unsupervised time-series anomaly detection

• Find time segment that is considerably differnt from the rest

\begin{align*} \text{find} \quad & k^* \\ \text{such that} \quad & x(t_k) |_{k=k^*}^{k^*+s} \;\; \text{is significantly different from} \;\, x(t_k) |_{k=-\infty}^{\infty} \end{align*}

1.5. Markov ProcessÂ¶

1.5.1. Sequential ProcessesÂ¶

• 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 !!

1.5.2. Markov ProcessÂ¶

• (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

1.5.3. State Transition MatrixÂ¶

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 $s'$.