Forecast Accuracy Measures

Mean Absolute Error (MAE): Average of absolute errors between predicted and actual values.
Mean Squared Error (MSE): Average of squared errors; emphasizes larger errors.
Root Mean Squared Error (RMSE): Square root of MSE; provides error scale in original units.
Mean Absolute Percentage Error (MAPE): Average of absolute percentage errors; useful for interpretability.
R-squared: Proportion of variance explained by the model; ranges from 0 to 1.

Components:
- AR (AutoRegressive): Uses past values to predict future values.
- I (Integrated): Differencing the data to make it stationary.
- MA (Moving Average): Uses past forecast errors to improve predictions.
Model Notation: ARIMA(p,d,q) where:
- p: order of autoregressive part
- d: degree of differencing
- q: order of moving average part
Stationarity: Required for ARIMA; data must have constant mean and variance.
Model Selection: Use ACF/PACF plots and information criteria like AIC/BIC.

Purpose: Separate time series data into trend, seasonal, and residual components.
Methods:
- Additive Decomposition: Useful when seasonal fluctuations are constant.
- Multiplicative Decomposition: Useful when seasonal fluctuations increase over time.
Components:
- Trend: Long-term progression of the series.
- Seasonal: Regular pattern of fluctuations.
- Residual: Noise or irregular variations.
Tools: Can be performed using statistical software like R or Python.

Basic Concept: Weighted averages of past observations, with exponentially decreasing weights.
Types:
- Simple Exponential Smoothing: For data without trend or seasonality.
- Holt’s Linear Trend Model: For data with trends.
- Holt-Winters Seasonal Model: For data with trends and seasonality.
Smoothing Parameters:
- Alpha (α): Smoothing factor for level.
- Beta (β): Smoothing factor for trend (in Holt's).
- Gamma (γ): Smoothing factor for seasonality (in Holt-Winters).
Applications: Useful for real-time forecasting and adjusting predictions on-the-fly.

Line Graphs: Commonly used to show trends over time.
Seasonal Plots: Compare different seasons or time periods to identify patterns.
Autocorrelation Function (ACF): Visualize correlations of the time series with its lagged values; helps identify seasonality and lags.
Partial Autocorrelation Function (PACF): Shows the correlation of the time series with its lagged values after removing effects of intervening lags.
Histogram and Density Plots: Assess the distribution of residuals for normality.
Box Plots: Useful for visualizing the distribution of data across different seasons or groups.

Mean Absolute Error (MAE): Measures average magnitude of errors; simpler interpretation due to absolute values.
Mean Squared Error (MSE): Highlights significant errors more than smaller ones through squaring; used for performance evaluation.
Root Mean Squared Error (RMSE): Translates MSE back to original units, facilitating easier understanding of prediction error scale.
Mean Absolute Percentage Error (MAPE): Provides error as a percentage, enhancing interpretability across different datasets.
R-squared: Indicates how well the model explains variability in the response variable; values range from 0 (no explanatory power) to 1 (perfect explanation).

AR Component: Involves using previous time points to aid in predicting future outcomes.
Integrated Component: Focuses on differencing a dataset to achieve stationarity, ensuring stable mean and variance.
MA Component: Incorporates history of past forecast errors to refine future predictions.
Model Notation: ARIMA(p,d,q) where:
- p: reflects count of autoregressive terms.
- d: denotes levels of differencing applied.
- q: indicates count of moving average terms used.
Stationarity Requirement: ARIMA models necessitate that data exhibit constant statistical properties over time.
Model Selection Techniques: ACF/PACF plots alongside information criteria like AIC and BIC guide the selection of appropriate models.

Objective: Decomposes time series into three elements: trend, seasonal, and residual for clearer insights.
Additive Decomposition: Suitable for seasonal data exhibiting consistent fluctuations.
Multiplicative Decomposition: Applied when seasonal variations grow proportionally to the level of the series.
Components of Decomposition:
- Trend: Captures long-term directional movement.
- Seasonal: Identifies repeatable patterns across periods.
- Residual: Accounts for noise or irregular variations not captured by the other two components.
Software Utilization: Statistical tools like R or Python facilitate decomposition processes.

Core Principle: Averages past observations with weights that decrease exponentially; recent data has more influence.
Types of Models:
- Simple Exponential Smoothing: For datasets devoid of trends or seasonal patterns.
- Holt’s Linear Model: Adapts to trends present in data.
- Holt-Winters: Addresses both trends and seasonal variations.
Smoothing Parameters:
- Alpha (α): Controls level of smoothing for current data.
- Beta (β): Modulates trend smoothing in Holt’s model.
- Gamma (γ): Manages seasonal smoothing in Holt-Winters model.
Practical Uses: Effective for real-time forecasting, adjusting predictions responsive to new data.

Line Graphs: Illustrate trends over time; simple yet powerful visualization method.
Seasonal Plots: Allow for direct comparison across different conditions or time points to uncover seasonal patterns.
Autocorrelation Function (ACF): Evaluates correlations with lagged values, helping to identify underlying patterns and seasonality.
Partial Autocorrelation Function (PACF): Shows relationships of the time series with its lags after accounting for previous lags' effects.
Histogram and Density Plots: Assess residual distribution characteristics, crucial for checking normality assumptions.
Box Plots: Visual representation of data distribution segmented by seasons or categories, highlighting medians and variability.