Introduction to Time Series Analysis

What is Time Series Data?

Time series data is a sequence of observations collected over time, typically at regular intervals.

Key Characteristics

Feature	Description
Temporal ordering	Observations have a natural time sequence
Dependence	Values often depend on previous values
Regular intervals	Usually measured at fixed time periods
Single variable	Focus on how one variable changes over time

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Components of Time Series

Most time series can be decomposed into four components:

1. Trend (T)

The long-term movement in the data—overall increase or decrease over time.

2. Seasonality (S)

Regular, predictable patterns that repeat at fixed intervals (daily, weekly, yearly).

3. Cyclical (C)

Longer-term fluctuations that don’t have fixed periods (economic cycles, business cycles).

Unlike seasonality, cyclical patterns:

• Don’t have fixed period length
• Often related to economic conditions
• Can last years

4. Irregular/Random (I)

Unpredictable, random variation that can’t be attributed to other components.

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Stationarity

Why Stationarity Matters

• Non-stationary data can lead to spurious correlations
• Statistical tests assume stationarity
• Forecasting models work better with stationary data

Testing for Stationarity

Visual inspection: Does the series look stable over time?

Statistical tests:

• Augmented Dickey-Fuller (ADF) test
• KPSS test
• Phillips-Perron test

Making Data Stationary

Technique	When to Use
Differencing	Remove trend
Log transformation	Stabilize variance
Seasonal differencing	Remove seasonality
Detrending	Remove linear trend

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Autocorrelation

Autocorrelation measures how a series correlates with its past values.

Interpreting ACF Plots

Pattern	Interpretation
Slow decay	Non-stationary (needs differencing)
Cut-off after lag k	MA(k) process
Gradual decay	AR process
Seasonal spikes	Seasonal component present

Partial Autocorrelation (PACF)

PACF measures correlation with lag k after removing effects of intermediate lags.

ACF Pattern	PACF Pattern	Suggested Model
Tails off	Cuts off at p	AR(p)
Cuts off at q	Tails off	MA(q)
Tails off	Tails off	ARMA(p,q)

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Basic Forecasting Methods

1. Simple Moving Average

2. Exponential Smoothing

α Value	Behavior
Close to 1	React quickly to changes
Close to 0	Smooth, slow to react

3. Linear Trend Model

4. Seasonal Naive

Use the value from the same season last cycle:

$\hat{Y}_{t+1} = Y_{t+1-s}$

Where s is the seasonal period (e.g., 12 for monthly data with yearly seasons).

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ARIMA Models

ARIMA(p, d, q) Components

Parameter	Meaning
p	AR order (autoregressive terms)
d	Differencing order (integration)
q	MA order (moving average terms)

Common ARIMA Models

Model	Name	Use Case
ARIMA(0,1,0)	Random walk	Stock prices
ARIMA(1,0,0)	AR(1)	Simple autocorrelation
ARIMA(0,0,1)	MA(1)	Shock effects
ARIMA(1,1,1)	Common mixed	General time series

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Evaluating Forecasts

Accuracy Metrics

Metric	Advantages	Disadvantages
MAE	Easy to interpret	Scale-dependent
RMSE	Penalizes large errors	Scale-dependent
MAPE	Scale-independent	Undefined when Y=0

Train-Test Split

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Practical Example

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Summary

In this lesson, you learned:

• Time series has four components: Trend, Seasonality, Cyclical, Irregular
• Stationarity is required for most methods (constant mean/variance)
• Autocorrelation (ACF) measures correlation with past values
• Differencing removes trend to achieve stationarity
• Moving averages and exponential smoothing are simple forecasting methods
• ARIMA(p,d,q) combines AR, differencing, and MA components
• Evaluate forecasts with MAE, RMSE, or MAPE
• Always use chronological train-test splits

Practice Problems

1. A time series has these values: 10, 12, 11, 13, 12, 14 Calculate the first differences.

2. Which component explains: a) Steady increase in temperature over decades? b) Higher ice cream sales every summer? c) Economic recession effects?

3. You have monthly data with a yearly seasonal pattern. What lag would you expect to show high autocorrelation?

4. A forecast has MAE = 5 and the average actual value is 100. Is this forecast accurate?

Click to see answers

1. First differences (Yₜ - Yₜ₋₁):

• 12 - 10 = 2
• 11 - 12 = -1
• 13 - 11 = 2
• 12 - 13 = -1
• 14 - 12 = 2

First differences: 2, -1, 2, -1, 2

2. a) Trend - long-term directional movement b) Seasonality - regular yearly pattern c) Cyclical - irregular longer-term fluctuations

3. Lag 12 (12 months = 1 year)

The autocorrelation at lag 12 would be high because January this year relates to January last year, etc.

4. MAE of 5 with average value of 100 is relatively small (5% relative error).

This is reasonably accurate, though “good” depends on the application. For many business forecasts, less than 10% error is acceptable.

Next Steps

Continue your statistics journey:

• Linear Regression - Foundation for time series regression
• Correlation - Understanding relationships
• Bayesian Statistics - Alternative inference framework