Forecasting

Forecast the chance of dying due to the lack of sleep and overconsumption of caffeine of a student desperately trying to cram this material in 2 days.

Polynomial Trend

A mathematical model where the trend is represented by a polynomial function of time, typically in the form yt=β0+β1t+β2t2+⋯+βptp.

Forecasting with Polynomial Trends

To forecast beyond the sample, plug future time value T+j into the polynomial equation yT+j=β^0+β^1(T+j)+⋯+β^p(T+j)p.

Power Series Expansion

A method in mathematics that expresses functions as an infinite sum of terms involving powers of the argument; a polynomial can be a truncated version.

Exponential Growth

A process where the growth rate is proportional to the current value, modeled as yt=β0eβ1t.

Compound Interest

Exponential growth in finance, where the value of an investment grows according to the formula FV=PV(1+r)t.

Logistic Growth

A model that incorporates an upper limit to growth, represented by yt=γ/(1 + e^{β0 + β1t}).

Saturation Point

In logistic growth, the point where the growth rate decreases and the system approaches its upper bound.

Exponential vs. Logistic Growth

Exponential growth assumes no limits, while logistic growth accounts for constraints and slows as it nears an upper limit.

Holt’s Two-Parameter Model

A forecasting method that accounts for both level and trend using two smoothing parameters, α (level) and β (trend).

Holt’s Model Forecast Equation

The forecast for time t+p is yt+p=Ft+pTt.

Exponential Smoothing

A forecasting technique where weights decrease exponentially for past observations.

Level Smoothing Equation in Holt’s Model

Ft=αyt+(1−α)(Ft−1+Tt−1).

Trend Smoothing Equation in Holt’s Model

Tt=β(Ft−Ft−1)+(1−β)Tt−1.

Holt-Winters’ Seasonal Model

An extension of Holt’s method that includes a seasonal component to account for seasonality.

Additive Seasonal Model

A version of the Holt-Winters’ model where the seasonal component is added to the level and trend.

Multiplicative Seasonal Model

A version of the Holt-Winters’ model where the seasonal component is multiplied by the level and trend.

Seasonal Component in Holt-Winters

Captures the repeating patterns or fluctuations within the data that recur at regular intervals.

Holt-Winters Forecast Equation (Additive)

yt+p=Ft+pTt+Ct+p−r.

Initial Values in Holt-Winters

F1=y1, T1=y2−y1, C1=0.

Forecast Accuracy

Measures how well a forecasted value matches the actual observed value.

Mean Error (ME)

A measure of forecast bias calculated as the average of the differences between observed and forecasted values.

Mean Absolute Deviation (MAD)

A measure of forecast accuracy averaging the absolute differences between observed and forecasted values.

Root Mean Square Error (RMSE)

A measure of forecast accuracy that emphasizes larger errors by calculating the square root of the average of squared forecast errors.

Mean Absolute Percent Error (MAPE)

A relative measure of forecast accuracy expressing errors as a percentage of actual values.

Bias in Forecasting

Occurs when Mean Error (ME) is consistently different from zero, indicating over- or under-predicting.

Ex-post Forecast Errors

Errors observed after the forecast is made, used to evaluate and improve the forecasting model’s performance.

Ex-ante Forecast Errors

Forecast errors predicting future observations based on past data.

Pegel’s Classification

A framework for classifying exponential smoothing methods based on trend and seasonal components.

Additive Model

A model where the seasonal variations are constant over time.

Multiplicative Model

A model where the seasonal variations change proportionally with the level.

Trend Component

The underlying direction in the data over time, whether increasing, decreasing, or flat.

Smoothing Parameters α and β

Parameters used in exponential smoothing models; α controls level smoothing, β controls trend smoothing.

Degree of Polynomial

The highest power of t in the polynomial trend equation, determining the complexity of the trend line.

Exponential Growth Equation

yt=β0e^{β1t}, representing growth where the value increases exponentially with time.

S-Curve

A graph depicting logistic growth where the growth rate starts slowly, accelerates, and then decelerates.

Upper Bound in Logistic Growth

The maximum level that can be reached in a logistic growth model.

Forecasting Horizon

The time period over which forecasts are made, broken into short-term, medium-term, and long-term.

Model Validation

The process of assessing a forecasting model’s accuracy using historical data not part of the training set.

Holt’s Linear Exponential Smoothing

A forecasting method that accounts for both trend and level in a time series.

Seasonal Component StS_t

The repeating pattern in a time series due to factors like seasonality.

Alpha (α) in Exponential Smoothing

The smoothing parameter determining the weight given to the most recent observation.

Beta (β) in Exponential Smoothing

The smoothing parameter adjusting the trend component of the forecast.

Gamma (γ) in Holt-Winters

The smoothing parameter determining the weight given to the seasonal component.

Stationary Series

A time series whose statistical properties do not change over time.

Non-Stationary Series

A time series with changing statistical properties over time.

Time Series Decomposition

The process of breaking down a time series into components: trend, seasonal, and residual.

Moving Average (MA)

A forecasting technique where the forecast is the average of a specified number of past data points.

Autoregressive Integrated Moving Average (ARIMA)

A class of models combining autoregressive, differencing, and moving average components.

Forecast Error

The difference between the actual and forecasted values.

Confidence Interval in Forecasting

A range of values within which the true forecast is likely to fall.