Time Series Smoothing (Moving Average, EWMA, Holt-Winters)

This repo is a hands-on practice project for smoothing and forecasting time series using:

• Simple baseline: mean (simple average)
• Moving Average (MA) smoothing
• Exponentially Weighted Moving Average (EWMA) (custom weighted version)
• Single Exponential Smoothing (SES) via statsmodels
• Double Exponential Smoothing (Holt) via statsmodels
• Triple Exponential Smoothing (Holt–Winters) via statsmodels

The main work lives in the notebook:

• smoothing.ipynb

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What this project does

1. Generates synthetic time series (stationary noise, trend, seasonality, and combinations).
2. Visualizes raw vs. smoothed series.
3. Splits a series into train/test (last 5 points in one example).
4. Forecasts the test window using multiple smoothing methods.
5. Evaluates models using Sum of Squared Error (named mse in the notebook).

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Methods + formulas

1) Mean baseline (Simple Average)

Forecast every future value using the historical mean:

$$ \hat{y}_t = \bar{y} = \frac{1}{T}\sum_{i=1}^{T} y_i $$

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2) Error metric used in the notebook

The notebook function mse(observations, estimates) computes sum of squared errors (SSE):

$$ \text{SSE} = \sum_{t=1}^{n}(y_t - \hat{y}_t)^2 $$

Note: Standard MSE usually means average squared error: $$ \text{MSE} = \frac{1}{n}\sum_{t=1}^{n}(y_t - \hat{y}_t)^2 $$

If you want true MSE, divide the current output by n.

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3) Moving Average (MA) smoothing

For window size (k), the moving average at time (t) is:

$$ \hat{y}_t = \frac{1}{k}\sum_{i=0}^{k-1}y_{t-i} $$

In the notebook:

• A fast rolling sum is computed using cumulative sums.
• With forecast=True, the function pads the beginning with zeros for alignment.

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4) Custom EWMA (weighted 3-point smoother)

The notebook’s ewma() uses fixed weights on the last 3 observations:

Let weights be: $$ w = (w_1, w_2, w_3) = (0.160, 0.294, 0.543) $$

Then for (t \ge 3):

$$ \hat{y}_t = w_1 y_{t-2} + w_2 y_{t-1} + w_3 y_t $$

This is a finite weighted moving average (not the classic recursive EMA form).

Classic recursive EMA (common definition) is: $$ S_t = \alpha y_t + (1-\alpha)S_{t-1} $$

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5) Single Exponential Smoothing (SES)

SES is useful when the series has no trend and no seasonality.

Recursive form: $$ S_t = \alpha y_t + (1-\alpha)S_{t-1} $$

Forecast: $$ \hat{y}_{t+h} = S_t $$

6) Holt’s Linear Trend (Double Exponential Smoothing)

Holt adds a trend component.

7) Holt–Winters (Triple Exponential Smoothing)

Adds seasonality (additive form)

What you should expect to see in outputs

Running the notebook produces:

1. Plots of raw series vs smoothed series
2. Forecast plots over the test window
3. A small summary table comparing error across methods (baseline vs SES vs Holt vs Holt–Winters)

This comparison highlights a key lesson:

• If the data has trend, SES will typically underperform Holt.
• If the data has seasonality, Holt–Winters should outperform non-seasonal methods.

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Requirements

Python packages used:

• numpy
• pandas
• matplotlib
• seaborn
• statsmodels