Differencing: Subtracting consecutive values to remove trends. First-order differencing: Subtracting from . Second-order differencing: Differencing the differenced data. Log transformation: Applying logarithms to stabilize variance. Seasonal differencing: Subtracting values from the same season of the previous cycle. Detrending: Removing trends by fitting and subtracting a trend line. Moving average smoothing: Averaging neighboring points to smooth data. Exponential smoothing: Giving more weight to recent observations for smoothing.
MSE is the average of the squared differences between actual and predicted values. Formula: . MSE penalizes larger errors more heavily due to squaring the differences. The smaller the MSE, the better the model fits the data. MSE helps measure the accuracy of predictive models. It’s sensitive to outliers due to squaring errors, emphasizing large errors. MSE is often used to compare different models, with lower MSE preferred. Root Mean Square Error (RMSE) is the square root of MSE, used for easier interpretation.
A curve or model that uses only past or present data for prediction. One-sided curves are used for forward prediction without future data. They are simple and practical for real-time forecasting scenarios. Useful when predicting data where future values aren’t available. Examples include predicting stock prices based only on past prices. In rolling forecasts, one-sided curves update with new data as it's available. They focus on forward prediction, unlike models that fit past and future data. One-sided predictions can sometimes be biased if sudden shifts occur.
A stationary series has statistical properties (mean, variance) that don’t change over time. Stationarity simplifies the analysis, making the series more predictable. Autocovariance in stationary series depends only on the lag, not time. Strict stationarity implies that the probability distribution is time-invariant. Weak stationarity requires constant mean, variance, and autocovariance. Non-stationary series show trends, seasonality, or structural breaks. Stationarity tests include the Augmented Dickey-Fuller (ADF) test. Stationarity is crucial for many TSA models like ARIMA, which assume stationary data.
A model that expresses a time series as a function of past forecast errors. MA models use past errors (residuals) to predict future values. The model's order defines how many past errors are included. MA(1): The first-order moving average model considers the most recent error. MA(2): The second-order model includes the two most recent errors. Residuals (errors) in MA models are assumed to be white noise. Inversion can transform MA models into autoregressive models for estimation. Used when residuals of an AR model show patterns instead of being random.
Finance: Predicting stock prices, profits, and market risks. Weather forecasting: Predicting temperature and rainfall patterns. Economics: Forecasting GDP, inflation, and unemployment rates. Supply chain management: Demand forecasting and inventory optimization. Medicine: Monitoring patient health over time (e.g., heart rates). Marketing: Analyzing customer buying behavior and sales patterns. Energy: Forecasting energy consumption and optimizing distribution. Speech & signal processing: Analyzing audio and signal data patterns.
Time series are data points collected at regular intervals over time. TSA is the study of data points ordered in time to extract patterns and insights. TSA deals with temporal dependency, where current data depends on previous values. Univariate time series involves a single variable tracked over time. Multivariate time series tracks multiple variables over time. Components include trend, seasonality, cyclical, and irregular patterns. Trends reflect long-term upward or downward data movements. The goal of TSA is to model and forecast future points based on past data.
Autoregressive (AR) models predict values using previous data points in the series. Linear regression models can use other explanatory variables along with time. Autoregressive Integrated Moving Average (ARIMA) combines AR, differencing, and MA. Lagged variables (previous time points) are often used as predictors. Explanatory variables can include external factors (e.g., economic indicators). AR models predict based on past values (e.g., AR(1) uses ). Regression models are evaluated by metrics like R-squared and AIC. The residuals (errors) in time series regression should ideally be white noise.
