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‎Introduction_TSA.md

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# Introduction to Time Series Analysis
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Contents
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---
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--
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- [Taxonomy of Time Series Analysis Domain](#taxonomy-of-time-series-analysis-domain)
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- [Best Practices for Forecasting Model Development](#best-practices-for-forecasting-model-development)
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- [Simple and Classical Forecasting Methods](#simple-and-classical-forecasting-methods)
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8. *Contiguous vs. Discontiguous*: Are your observations contiguous or discontiguous?
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Some useful tools to help get answers include:
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- Data visualizations (e.g. line plots, etc.).
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- Statistical analysis (e.g. ACF/PACF plots, etc.). ˆ
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- Domain experts.
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- Project stakeholders.
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- Data visualizations (e.g. line plots, etc.)
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- Statistical analysis (e.g. ACF/PACF plots, etc.)
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- Domain experts
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- Project stakeholders
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#### Step 2: Design Test Harness
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Some data preparation schemes to consider include:
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- Differencing to remove a trend.
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- Seasonal differencing to remove seasonality. ˆ
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- Standardize to center.
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- Normalize to rescale.
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- Power Transform to make normal.
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- Differencing to remove a trend
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- Seasonal differencing to remove seasonality
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- Standardize to center
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- Normalize to rescale
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- Power Transform to make normal
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This large amount of systematic searching can be slow to execute. Some ideas to speed up the evaluation of models include:
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- Use multiple machines in parallel via cloud hardware (such as Amazon EC2).
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- Reduce the size of the train or test dataset to make the evaluation process faster. ˆ
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- Use a more coarse grid of hyperparameters and circle back if you have time later. ˆ
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- Perhaps do not refit a model for each step in walk-forward validation.
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- Use multiple machines in parallel via cloud hardware (such as Amazon EC2)
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- Reduce the size of the train or test dataset to make the evaluation process faster
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- Use a more coarse grid of hyperparameters and circle back if you have time later
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- Perhaps do not refit a model for each step in walk-forward validation
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#### Step 4: Finalize Model
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There are three trend elements that require configuration. They are the same as the ARIMA model; specifically:
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- p: Trend autoregression order. ˆ
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- d: Trend difference order.
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- q: Trend moving average order.
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- p: Trend autoregression order
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- d: Trend difference order
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- q: Trend moving average order
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**Seasonal Elements**
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Hyperparameters:
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- Alpha (α): Smoothing factor for the level. ˆ
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- Beta (β): Smoothing factor for the trend.
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- Trend Type: Additive or multiplicative.
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- Dampen Type: Additive or multiplicative. ˆ
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- Phi (φ): Damping coefficient.
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- Alpha (α): Smoothing factor for the level
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- Beta (β): Smoothing factor for the trend
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- Trend Type: Additive or multiplicative
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- Dampen Type: Additive or multiplicative
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- Phi (φ): Damping coefficient
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3. Triple Exponential Smoothing
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Hyperparameters:
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- Alpha (α): Smoothing factor for the level. ˆ
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- Beta (β): Smoothing factor for the trend.
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- Trend Type: Additive or multiplicative.
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- Dampen Type: Additive or multiplicative. ˆ
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- Phi (φ): Damping coefficient.
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- Gamma (γ): Smoothing factor for the seasonality. ˆ
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- Seasonality Type: Additive or multiplicative.
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- Period: Time steps in seasonal period.
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- Alpha (α): Smoothing factor for the level
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- Beta (β): Smoothing factor for the trend
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- Trend Type: Additive or multiplicative
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- Dampen Type: Additive or multiplicative
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- Phi (φ): Damping coefficient
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- Gamma (γ): Smoothing factor for the seasonality
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- Seasonality Type: Additive or multiplicative
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- Period: Time steps in seasonal period
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---
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Below is a contrived example of a supervised learning dataset where each row is an observation comprised of one input variable (X) and one output variable to be predicted (y).
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| X | y |
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|---|-----|
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| 5 | 0.9 |
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| 4 | 0.8 |
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| 5 | 1.0 |
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| 3 | 0.7 |
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| 4 | 0.9 |
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| X | y |
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|---|-----|
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| 5 | 0.9 |
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| 4 | 0.8 |
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| 5 | 1.0 |
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| 3 | 0.7 |
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| 4 | 0.9 |
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Supervised learning problems can be further grouped into regression and classification problems.
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Time series data can be phrased as supervised learning. Given a sequence of numbers for a time series dataset, we can restructure the data to look like a supervised learning problem.
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| time | measure |
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|------|---------|
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| 1 | 100 |
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| 2 | 110 |
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| 3 | 108 |
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| 4 | 115 |
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| 5 | 120 |
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| time | measure |
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|------|---------|
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| 1 | 100 |
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| 2 | 110 |
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| 3 | 108 |
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| 4 | 115 |
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| 5 | 120 |
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We can restructure this time series dataset as a supervised learning problem by using the value at the previous time step to predict the value at the next time step. Re-organizing the time series dataset this way, the data would look as follows:
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| X | y |
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|-----|-----|
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| ? | 100 |
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| 100 | 110 |
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| 110 | 108 |
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| 108 | 115 |
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| 115 | 120 |
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| 120 | ? |
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| X | y |
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|-----|-----|
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| ? | 100 |
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| 100 | 110 |
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| 110 | 108 |
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| 108 | 115 |
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| 115 | 120 |
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| 120 | ? |
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- We can delete 1st and last row since they have missing value before training a supervised model.
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- The use of prior time steps to predict the next time step is called the sliding window method.
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For example suppose we have following dataset:
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| time | measure1 | measure2 |
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|------|----------|----------|
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| 1 | 0.2 | 88 |
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| 2 | 0.5 | 89 |
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| 3 | 0.7 | 87 |
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| 4 | 0.4 | 88 |
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| 5 | 1.0 | 90 |
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| time | measure1 | measure2 |
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|------|----------|----------|
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| 1 | 0.2 | 88 |
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| 2 | 0.5 | 89 |
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| 3 | 0.7 | 87 |
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| 4 | 0.4 | 88 |
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| 5 | 1.0 | 90 |
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Let’s also assume that we are only concerned with predicting measure2. We can re-frame this time series dataset as a supervised learning problem with a window width of one.
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| X1 | X2 | X3 | y |
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|-----|-----|-----|-----|
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| ? | ? | 0.2 | 88 |
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| 0.2 | 88 | 0.5 | 89 |
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| 0.5 | 89 | 0.7 | 87 |
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| 0.7 | 87 | 0.4 | 88 |
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| 0.4 | 88 | 1.0 | 90 |
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| 1.0 | 90 | ? | ? |
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| X1 | X2 | X3 | y |
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|-----|-----|-----|-----|
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| ? | ? | 0.2 | 88 |
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| 0.2 | 88 | 0.5 | 89 |
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| 0.5 | 89 | 0.7 | 87 |
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| 0.7 | 87 | 0.4 | 88 |
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| 0.4 | 88 | 1.0 | 90 |
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| 1.0 | 90 | ? | ? |
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We can see that as in the univariate time series example above, we may need to remove the first and last rows in order to train our supervised learning model.
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If we need to predict both `measure1` and `measure2` for the next time step. We can transform the data as follows:
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| X1 | X2 | y1 | y2 |
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|-----|-----|-----|-----|
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| ? | ? | 0.2 | 88 |
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| 0.2 | 88 | 0.5 | 89 |
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| 0.5 | 89 | 0.7 | 87 |
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| 0.7 | 87 | 0.4 | 88 |
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| 0.4 | 88 | 1.0 | 90 |
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| 1.0 | 90 | ? | ? |
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| X1 | X2 | y1 | y2 |
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|-----|-----|-----|-----|
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| ? | ? | 0.2 | 88 |
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| 0.2 | 88 | 0.5 | 89 |
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| 0.5 | 89 | 0.7 | 87 |
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| 0.7 | 87 | 0.4 | 88 |
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| 0.4 | 88 | 1.0 | 90 |
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| 1.0 | 90 | ? | ? |
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### Sliding Window With Multiple Steps
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Consider this univariate time series dataset:
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| time | measure |
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|------|---------|
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| 1 | 100 |
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| 2 | 110 |
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| 3 | 108 |
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| 4 | 115 |
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| 5 | 120 |
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| time | measure |
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|------|---------|
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| 1 | 100 |
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| 2 | 110 |
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| 3 | 108 |
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| 4 | 115 |
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| 5 | 120 |
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We can frame this time series as a two-step forecasting dataset for supervised learning with a window width of one, as follows:
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| X1 | y1 | y2 |
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|-----|-----|-----|
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| ? | 100 | 110 |
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| 100 | 110 | 108 |
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| 110 | 108 | 115 |
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| 108 | 115 | 120 |
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| 115 | 120 | ? |
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| 120 | ? | ? |
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| X1 | y1 | y2 |
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|-----|-----|-----|
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| ? | 100 | 110 |
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| 100 | 110 | 108 |
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| 110 | 108 | 115 |
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| 108 | 115 | 120 |
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| 115 | 120 | ? |
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| 120 | ? | ? |
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Specifically, that a supervised model only has X1 to work with in order to predict both y1 and y2.
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