Time Series

🎬 Time Series in R: Your Data’s Diary Through Time

The Story of Time: A Simple Analogy

Imagine you have a diary 📔 where you write down the temperature every single day for a whole year. Each page has a date and a number. That’s a time series—a collection of measurements taken at regular intervals over time!

Think of it like a movie 🎥: instead of a single photograph (one data point), you have frames playing in sequence, telling a story of how things change.

📦 Time Series Objects: Building Your Data Diary

What is a Time Series Object?

In R, a time series object is like a special notebook that knows:

• What values you recorded
• When each value was recorded
• How often you took notes (daily? monthly? yearly?)

Creating Your First Time Series

# Monthly ice cream sales for one year
sales <- c(100, 120, 150, 200,
           250, 300, 280, 260,
           200, 150, 110, 105)

# Turn it into a time series
ice_cream_ts <- ts(sales,
                   start = c(2024, 1),
                   frequency = 12)

print(ice_cream_ts)

What does this mean?

• start = c(2024, 1) → Starts in January 2024
• frequency = 12 → 12 observations per year (monthly!)

Common Frequencies

Checking Your Time Series

# When does it start?
start(ice_cream_ts)

# When does it end?
end(ice_cream_ts)

# How often?
frequency(ice_cream_ts)

📊 Time Series Plotting: Drawing the Story

Why Plotting Matters

Imagine trying to describe a roller coaster 🎢 with just numbers vs. actually seeing its picture. The picture tells you everything instantly!

Your First Time Series Plot

# Simple plot
plot(ice_cream_ts,
     main = "Ice Cream Sales 2024",
     xlab = "Month",
     ylab = "Sales",
     col = "blue",
     lwd = 2)

What you’ll see:

• X-axis shows time (months)
• Y-axis shows your values (sales)
• A line connects all the dots!

Adding Style to Your Plot

# Fancy plot with grid
plot(ice_cream_ts,
     main = "🍦 Ice Cream Sales",
     col = "darkorange",
     type = "o",    # o = both lines AND points
     pch = 19)      # solid circles

# Add a helpful grid
grid(col = "lightgray")

Reading the Plot

When you look at a time series plot, ask yourself:

• 📈 Is it going up? (Upward trend)
• 📉 Is it going down? (Downward trend)
• 🔄 Does it repeat a pattern? (Seasonality)
• 📍 Are there sudden jumps? (Anomalies)

🔗 Autocorrelation: Does Today Remember Yesterday?

The Memory of Data

Autocorrelation answers a simple question: “If today is hot, is tomorrow likely to be hot too?”

Think of it like momentum 🏃: if you’re running fast right now, you’ll probably still be running fast in the next second.

What Does “Auto” Mean?

• Auto = Self
• Correlation = Relationship

So autocorrelation means: How related is the data to itself at different time gaps?

Calculating Autocorrelation

# Check autocorrelation
acf(ice_cream_ts,
    main = "How Much Does
    Sales Remember Itself?")

Reading the ACF Plot

The plot shows vertical bars:

• Bar at lag 1: “How similar is this month to last month?”
• Bar at lag 2: “How similar is this month to 2 months ago?”
• Bars inside blue lines: No significant memory
• Bars outside blue lines: Strong memory! 📢

Real-Life Example

# Temperature autocorrelation
# If Monday is cold, Tuesday
# is probably cold too!

temps <- c(32, 33, 31, 28, 25,
           24, 26, 30, 35, 38)
acf(temps)

High autocorrelation at lag 1 = Data has short-term memory (yesterday affects today)

📐 Moving Averages: Smoothing the Bumps

The Bumpy Road Problem

Imagine your daily step count: 8000, 12000, 6000, 15000, 7000…

It’s all over the place! 🎢 Hard to see if you’re actually walking more over time.

The Solution: Take an Average

A moving average is like wearing glasses that blur out the tiny details so you can see the big picture.

graph TD
    A["Raw Data: Bumpy 📈📉"] --> B["Moving Average Filter"]
    B --> C["Smooth Data: Clear Trend ➡️"]

Calculating Moving Average in R

# Install if needed
# install.packages("zoo")
library(zoo)

# 3-month moving average
smooth_sales <- rollmean(ice_cream_ts,
                         k = 3,
                         align = "center")

# Compare: original vs smooth
plot(ice_cream_ts,
     col = "gray", lty = 2)
lines(smooth_sales,
      col = "red", lwd = 2)
legend("topright",
       c("Original", "Smoothed"),
       col = c("gray", "red"),
       lty = c(2, 1))

How It Works

For a 3-point moving average:

• Point 1: Average of values 1, 2, 3
• Point 2: Average of values 2, 3, 4
• Point 3: Average of values 3, 4, 5
• …and so on!

Original: 100, 120, 150, 200, 250
Average:  (100+120+150)/3 = 123
          (120+150+200)/3 = 157
          (150+200+250)/3 = 200

Choosing Window Size

✨ Exponential Smoothing: Smart Forgetting

The Memory Fade

Exponential smoothing is like human memory: recent events are vivid, but old memories fade.

• Yesterday’s weather? Remember clearly! 🌞
• Last week’s weather? Kind of remember…
• Last year’s weather? Barely remember! 😶‍🌫️

The Magic Number: Alpha (α)

Alpha controls how much we trust new data vs old data:

• α close to 1: “I mostly trust recent data!” (jumpy)
• α close to 0: “I mostly trust old patterns!” (smooth)

Simple Exponential Smoothing in R

# Simple exponential smoothing
library(forecast)

# Let R pick the best alpha
smooth_fit <- ses(ice_cream_ts, h = 3)

# See the smoothed values
print(smooth_fit)

# Plot it
plot(smooth_fit,
     main = "Exponential Smoothing")

The Formula (Don’t Panic!)

New Estimate = α × (New Data) + (1-α) × (Old Estimate)

Example with α = 0.3:

• Old estimate: 100
• New data: 120
• New estimate: 0.3 × 120 + 0.7 × 100 = 36 + 70 = 106

See? It moved a little toward 120, but not all the way!

🧩 Time Series Decomposition: Taking Apart the Puzzle

Every Time Series Has Hidden Layers

Like a cake 🎂, your time series has layers:

1. Trend: The long-term direction (going up or down over years)
2. Seasonal: Patterns that repeat (every summer, every December)
3. Remainder/Random: Unpredictable surprises (a random spike)

graph TD
    A["Time Series Data"] --> B{Decompose}
    B --> C["🔼 Trend"]
    B --> D["🔄 Seasonal"]
    B --> E["❓ Random"]

Decomposing in R

# Decompose our ice cream sales
parts <- decompose(ice_cream_ts)

# See all the pieces
plot(parts)

Reading the Decomposition

You’ll see 4 plots stacked:

1. Observed: Your original data
2. Trend: Is business growing or shrinking?
3. Seasonal: What months always spike?
4. Random: What’s left unexplained?

Additive vs Multiplicative

Additive (default): Seasonal swings stay the same size

• Winter: always -20 sales
• Summer: always +50 sales

decompose(ice_cream_ts,
          type = "additive")

Multiplicative: Seasonal swings grow with the trend

• When sales are low: small swings
• When sales are high: big swings

decompose(ice_cream_ts,
          type = "multiplicative")

🔮 Forecasting Basics: Predicting Tomorrow

The Crystal Ball of Data

Forecasting is using patterns from the past to guess what happens next. Like predicting tomorrow’s weather based on today! ⛅

Simple Forecast Methods

Method 1: Naive (“Tomorrow = Today”)

# Simple: next value = last value
naive_forecast <- naive(ice_cream_ts, h = 3)
plot(naive_forecast)

Method 2: Mean (“Tomorrow = Average”)

# Use the average of all past data
mean_forecast <- meanf(ice_cream_ts, h = 3)
plot(mean_forecast)

Method 3: Seasonal Naive

# Same as this month last year
snaive_forecast <- snaive(ice_cream_ts, h = 12)
plot(snaive_forecast)

The Forecast Package Magic

library(forecast)

# Let R figure out the best method!
auto_forecast <- forecast(ice_cream_ts, h = 6)

# Beautiful plot with uncertainty
plot(auto_forecast,
     main = "6-Month Ice Cream Forecast")

Understanding Forecast Output

Point Forecast: Our best guess
80% Confidence: Pretty sure it's in this range
95% Confidence: Almost certain it's in this range

The blue shaded areas show uncertainty—the further into the future, the wider they get!

Checking Forecast Accuracy

# Split data: train on past, test on future
train <- window(ice_cream_ts, end = c(2024, 9))
test <- window(ice_cream_ts, start = c(2024, 10))

# Forecast from training data
my_forecast <- forecast(train, h = 3)

# Compare to actual
accuracy(my_forecast, test)

Key Metrics:

• MAE (Mean Absolute Error): Average mistake size
• RMSE: Penalizes big mistakes more
• MAPE: Mistake as a percentage

🎯 Putting It All Together

Here’s the complete workflow for time series analysis:

graph TD
    A["1. Create ts Object"] --> B["2. Plot & Explore"]
    B --> C["3. Check Autocorrelation"]
    C --> D["4. Smooth if Needed"]
    D --> E["5. Decompose"]
    E --> F["6. Forecast!"]

Complete Example

# 1. Create time series
my_data <- ts(c(100, 120, 150, 200, 250,
                300, 280, 260, 200, 150,
                110, 105),
              start = c(2024, 1),
              frequency = 12)

# 2. Plot it
plot(my_data, main = "My Data Story")

# 3. Check memory (autocorrelation)
acf(my_data)

# 4. Smooth it
library(zoo)
smooth <- rollmean(my_data, k = 3)

# 5. Decompose it
parts <- decompose(my_data)
plot(parts)

# 6. Forecast next 6 months!
library(forecast)
future <- forecast(my_data, h = 6)
plot(future)

🌟 Key Takeaways

🚀 You Did It!

You now understand how R handles time—from creating time-aware data objects to peering into the future with forecasts. Like learning to read a diary that spans years, you can now uncover stories hidden in sequential data!

Remember: Every time series is just a story waiting to be told. Now you have the tools to read it, smooth it, break it apart, and even predict the next chapter! 📖✨