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9. Time Series Analysis & Forecasting

Time series data is any data collected over time at regular intervals (daily sales, monthly temperature, hourly stock prices, yearly population, etc.). Forecasting = predicting future values based on past patterns.

R has one of the strongest time series ecosystems — especially for classical statistical forecasting.

9.1 Time Series Objects – ts, xts, zoo

R has several classes for handling time series data.

ts – The classic base R time series class (regular frequency)

R

# Monthly data starting from Jan 2020 sales <- c(120, 135, 148, 162, 175, 190, 210, 225, 240, 255, 270, 290) ts_sales <- ts(sales, start = c(2020, 1), frequency = 12) print(ts_sales) plot(ts_sales, main = "Monthly Sales", ylab = "Sales", xlab = "Time")

zoo – Irregular time series (very flexible)

R

library(zoo) dates <- as.Date(c("2025-01-01", "2025-01-05", "2025-01-12", "2025-01-20")) values <- c(100, 120, 115, 140) z <- zoo(values, dates) plot(z, main = "Irregular Time Series", ylab = "Value")

xts – Modern, high-performance extension of zoo (most recommended in 2026 for financial/time series)

R

library(xts) dates <- seq(as.Date("2025-01-01"), by = "day", length.out = 30) values <- cumsum(rnorm(30)) + 100 xts_data <- xts(values, order.by = dates) plot(xts_data, main = "Daily Random Walk", ylab = "Value") # Subsetting xts_data["2025-01-10/2025-01-20"] # slice by date range

Quick recommendation (2026):

• Use ts for simple, regular, monthly/quarterly data

• Use xts for financial, daily, or irregular high-frequency data

• Use zoo only if you need very old compatibility

9.2 Decomposition – Trend, Seasonality, Remainder

Decomposition breaks a time series into:

• Trend – long-term direction

• Seasonal – repeating pattern

• Remainder (residual) – random noise

Classical decomposition (additive/multiplicative)

R

# AirPassengers dataset (built-in) data("AirPassengers") plot(AirPassengers) # Additive decomposition decomp_add <- decompose(AirPassengers, type = "additive") plot(decomp_add) # Multiplicative (better for increasing variance) decomp_mult <- decompose(AirPassengers, type = "multiplicative") plot(decomp_mult)

STL decomposition (more robust – handles changing seasonality)

R

library(forecast) stl_decomp <- stl(AirPassengers, s.window = "periodic") plot(stl_decomp)

Seasonal decomposition with X-13-ARIMA-SEATS (very advanced, used in official statistics)

R

library(seasonal) seas_decomp <- seas(AirPassengers) plot(seas_decomp)

Key takeaway: Use STL for most modern work — it’s robust and handles most real-world series well.

9.3 ARIMA & SARIMA Models

ARIMA (AutoRegressive Integrated Moving Average) is the classic statistical forecasting model.

Components

• AR(p) – autoregression (depends on past values)

• I(d) – differencing (make stationary)

• MA(q) – moving average (depends on past errors)

SARIMA adds seasonal components (P,D,Q,m)

Step-by-step ARIMA in R

R

library(forecast) # 1. Make stationary (ADF test) adf.test(AirPassengers) # p > 0.05 → non-stationary # Difference once diff_series <- diff(AirPassengers, differences = 1) adf.test(diff_series) # now stationary # 2. Auto ARIMA (best automatic choice) auto_model <- auto.arima(AirPassengers, seasonal = TRUE, stepwise = FALSE, approximation = FALSE) summary(auto_model) # 3. Forecast fc <- forecast(auto_model, h = 24) # 24 months ahead plot(fc, main = "ARIMA Forecast – Air Passengers")

Manual SARIMA (when you know parameters)

R

sarima_model <- Arima(AirPassengers, order = c(0,1,1), seasonal = c(0,1,1,12)) checkresiduals(sarima_model) # residuals should be white noise fc_manual <- forecast(sarima_model, h = 12) plot(fc_manual)

9.4 Prophet & forecast Package

Prophet (by Facebook/Meta) – Extremely easy and powerful for business time series

R

library(prophet) # Prepare data (must have ds = date, y = value) df_prophet <- AirPassengers %>% as.data.frame() %>% rownames_to_column("ds") %>% mutate(ds = as.Date(ds, format = "%b %Y")) %>% rename(y = x) # Fit model m <- prophet(df_prophet, yearly.seasonality = TRUE, weekly.seasonality = FALSE) # Future dataframe future <- make_future_dataframe(m, periods = 24, freq = "month") # Forecast forecast_prophet <- predict(m, future) # Plot plot(m, forecast_prophet) prophet_plot_components(m, forecast_prophet)

forecast package – Traditional, comprehensive, still very strong

R

library(forecast) fit <- ets(AirPassengers) # Exponential smoothing fc_ets <- forecast(fit, h = 24) plot(fc_ets) # Or auto.arima as shown earlier

When to choose (2026):

• Prophet → Business forecasting, strong seasonality, holidays, missing data

• ARIMA/SARIMA → Classical stats, high accuracy needed, research

• ETS → Exponential smoothing (good for short-term)

9.5 Real-world Forecasting Project

Project: Monthly Sales Forecasting (End-to-End)

R

library(tidyverse) library(forecast) library(prophet) # 1. Load & prepare (assume you have monthly sales data) sales <- read_csv("monthly_sales.csv") %>% mutate(ds = as.Date(month_year, format = "%Y-%m"), y = sales_amount) # 2. EDA ggplot(sales, aes(x = ds, y = y)) + geom_line(color = "steelblue", size = 1) + labs(title = "Monthly Sales Trend", x = "Date", y = "Sales") + theme_minimal() # 3. Prophet model m <- prophet(sales, yearly.seasonality = TRUE) future <- make_future_dataframe(m, periods = 12, freq = "month") fc <- predict(m, future) plot(m, fc) + labs(title = "Prophet Forecast – Monthly Sales") # 4. ARIMA alternative auto_fit <- auto.arima(sales$y, seasonal = TRUE) fc_arima <- forecast(auto_fit, h = 12) plot(fc_arima) # 5. Compare & choose best model accuracy(fc_arima) # RMSE, MAE, etc.

Key Takeaways from Project:

• Always visualize trend & seasonality first

• Prophet is easiest for business users

• ARIMA gives more control & statistical diagnostics

• Evaluate with hold-out set or cross-validation (tsCV in forecast)

This completes the full Time Series Analysis & Forecasting section — now you can confidently analyze and predict time-based data in R!

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