ARIMA Time Series Forecasting: Models & Examples

ARIMA may help with one series measured at set time steps. It uses past values, changes, and past errors to predict later values. This page runs a small ARIMA-style example on made-up data and checks the forecast on held-back values.

Worked Example (Simulated): Forecast Six Held-Back Values

Question: Can a model fitted to the first 42 values of a made-up series predict the last 6?

Inputs: Made-up data, seed 42. The script builds 48 values. Each change from one value to the next is linked to the change before it, with a built-in link of 0.55 plus random noise. It fits the first 42 values and saves the last 6 to check the forecast. This script fits the link by least squares on the changes. It is close to, not the same as, a packaged ARIMA fit.

import numpy as np

SEED = 42
N = 48
TRAIN_SIZE = 42
HOLDOUT_SIZE = N - TRAIN_SIZE
TRUE_PHI = 0.55  # the link built into the made-up series
rng = np.random.default_rng(SEED)

# Build the series: each change follows the one before it, plus noise.
delta = np.empty(N - 1)
delta[0] = 0.8
noise = rng.normal(loc=0.0, scale=1.5, size=N - 2)
for t in range(1, N - 1):
    delta[t] = 0.8 + TRUE_PHI * (delta[t - 1] - 0.8) + noise[t - 1]
series = np.concatenate(([100.0], 100.0 + np.cumsum(delta)))

# Fit the link between changes to the training changes by least squares.
train_delta = np.diff(series[:TRAIN_SIZE])
X = np.column_stack((np.ones(len(train_delta) - 1), train_delta[:-1]))
intercept, phi = np.linalg.lstsq(X, train_delta[1:], rcond=None)[0]

# Forecast each held-out change in turn, then add it to the last level.
forecasts = []
previous_delta = train_delta[-1]
level = series[TRAIN_SIZE - 1]
for _ in range(HOLDOUT_SIZE):
    next_delta = intercept + phi * previous_delta
    level += next_delta
    forecasts.append(level)
    previous_delta = next_delta
forecasts = np.array(forecasts)
actual = series[TRAIN_SIZE:]
miss = actual - forecasts
mae = np.mean(np.abs(miss))

print(f"seed={SEED}; observations={N}; train={TRAIN_SIZE}; holdout={HOLDOUT_SIZE}")
print("fit: AR(1) on first differences, least squares (not a packaged ARIMA fit)")
print(f"built-in phi={TRUE_PHI}; estimated intercept={intercept:.3f}; "
      f"estimated phi={phi:.3f}")
print("holdout actual=" + ", ".join(f"{x:.2f}" for x in actual))
print("holdout forecast=" + ", ".join(f"{x:.2f}" for x in forecasts))
print("holdout miss=" + ", ".join(f"{x:.2f}" for x in miss))
print(f"holdout MAE={mae:.2f}")

Printed output:

seed=42; observations=48; train=42; holdout=6
fit: AR(1) on first differences, least squares (not a packaged ARIMA fit)
built-in phi=0.55; estimated intercept=0.534; estimated phi=0.424
holdout actual=139.78, 142.02, 142.62, 143.66, 144.76, 146.06
holdout forecast=138.73, 139.65, 140.57, 141.49, 142.42, 143.34
holdout miss=1.05, 2.38, 2.05, 2.17, 2.34, 2.71
holdout MAE=2.12

Result: Every forecast was below the true value. The miss was 1.05 at the first step and 2.71 at the sixth, and the average miss (MAE) was 2.12. The fit found a link of 0.424; the series was built with 0.55.

Limit: The data are made up and the run is one series with six held-back values. It says little about forecasts for real business data.

Sources

• Hyndman R, Athanasopoulos G. Forecasting: Principles and Practice (3rd ed.). otexts.com/fpp3. Cited as a textbook on ARIMA forecasting.
• statsmodels developers. statsmodels: time series analysis. statsmodels.org/stable/tsa. Cited as documentation for packaged ARIMA fits.

The Challenge: Making Sense of Time-Based Data

Every business generates time series data - sales figures, website traffic, inventory levels, customer acquisition rates. The challenge isn't collecting this data; it's extracting actionable insights that drive strategic decisions. Traditional spreadsheet-based forecasting methods fall short when dealing with:

To see where your own measure is heading, with a likely range, try the free forecasting tool.

• Seasonal Patterns: Recurring patterns like holiday sales spikes or quarterly business cycles
• Trend Analysis: Long-term growth or decline trajectories hidden in noisy data
• Autocorrelation: How past values influence future outcomes in complex ways
• Confidence Intervals: Understanding the uncertainty in predictions for risk management

This is where ARIMA (AutoRegressive Integrated Moving Average) models may help, and MCP Analytics offers them through natural language interaction with Claude.

Understanding ARIMA: Past Values, Changes, and Past Errors

ARIMA uses past values, changes, and past errors. Its three parts are:

AR (AutoRegressive): Learning from the Past

The AutoRegressive component assumes that future values depend on recent past values. Think of it as asking: "How do the last few months of sales predict next month?" The AR(p) component uses p previous observations to predict the current value.

I (Integrated): Achieving Stability

The Integrated component addresses non-stationarity - when a time series has changing means or trends over time. Differencing transforms the data to focus on period-to-period changes rather than absolute values.

MA (Moving Average): Smoothing Random Fluctuations

The Moving Average component models the relationship between current values and past prediction errors. It helps smooth out random fluctuations and capture the impact of external shocks on the system.

ARIMA(p, d, q) where:
- p = number of autoregressive terms (how many past values to consider)
- d = degree of differencing (how many times to difference the data)
- q = number of moving average terms (how many past errors to consider)

Example: ARIMA(2,1,1) uses:
- 2 past values for autoregression
- 1 level of differencing for stationarity
- 1 past error for moving average smoothing

Seasonal ARIMA: Capturing Business Cycles

Most business data exhibits seasonal patterns - quarterly earnings cycles, monthly sales fluctuations, or weekly website traffic patterns. Seasonal ARIMA (SARIMA) extends the basic model to handle these cyclical behaviors.

SARIMA Model Structure

A SARIMA model is denoted as ARIMA(p,d,q)(P,D,Q)[s], where the uppercase letters represent seasonal components and [s] indicates the seasonal period (e.g., 12 for monthly data with yearly seasonality).

Real-World Applications: ARIMA in Action

Let's examine how different industries leverage ARIMA forecasting through MCP Analytics to solve specific business challenges:

Retail: Inventory Optimization

Retail teams forecast demand to plan inventory. The worked example near the top of this page shows the method on a made-up series, not on retail data.

SaaS: Revenue Forecasting for Fundraising

B2B SaaS companies may use ARIMA models to create revenue forecasts for investor presentations and strategic planning:

Manufacturing: Demand Planning

Manufacturing companies may use ARIMA for production planning and supply chain optimization:

• Raw Material Procurement: Forecast component demand to optimize purchasing schedules
• Capacity Planning: Predict peak production periods for workforce scheduling
• Quality Control: Monitor process variations and predict maintenance needs

The MCP Analytics Advantage: Professional Tools, Simple Interface

MCP Analytics offers ARIMA modeling through conversation. Instead of writing R or Python code, you can ask for a forecast in natural language:

Model Selection

The tool can choose ARIMA settings using:

• AIC/BIC Optimization: Statistical measures to balance model complexity and accuracy
• Out-of-Sample Testing: Out-of-sample testing on held-back data
• Stationarity Testing: Automatic differencing to achieve statistical stability
• Residual Analysis: Diagnostic checks on model assumptions

Comprehensive Diagnostics

ARIMA analyses can include diagnostic reports:

Best Practices for ARIMA Forecasting

Data Requirements

ARIMA forecasting depends on the data:

• Sufficient History: Enough history to cover several complete seasonal cycles
• Consistent Intervals: Regular time periods without gaps
• Stable Business Context: Avoid periods with major structural changes
• Clean Data: Handle outliers and missing values appropriately

Model Interpretation

Checking Forecasts

Check forecasts against business context:

• Sanity Checks: Do predictions align with business knowledge?
• External Factors: Consider upcoming market changes or campaigns
• Confidence Intervals: Use uncertainty ranges for scenario planning
• Model Monitoring: Update forecasts regularly as new data arrives

Advanced Techniques: Beyond Basic ARIMA

External Variables (ARIMAX)

ARIMAX models incorporate external variables as predictors:

• Marketing Spend: Include advertising budget as a predictor
• Economic Indicators: Incorporate GDP, unemployment, or interest rates
• Weather Data: Account for seasonal weather impacts
• Competitor Activity: Include market share or pricing data

Ensemble Forecasting

Combine ARIMA with other forecasting methods to compare forecasts:

• ARIMA + Linear Regression: Capture both temporal and explanatory patterns
• Multiple Seasonal Cycles: Handle both weekly and yearly seasonality
• Hierarchical Forecasting: Forecast multiple product lines simultaneously

Why Use ARIMA

Reasons teams use ARIMA:

• Provide Interpretable Results: Clear understanding of what drives predictions
• Handle Uncertainty: Confidence intervals for risk management
• Require Minimal Data Engineering: Work with raw time series data
• Scale Across Industries: Used in many fields

With MCP Analytics, you can ask for these forecasts by asking questions about your data.

Working on this for your business? See the time series forecasting tool.

About ARIMA Forecasting

ARIMA (AutoRegressive Integrated Moving Average) models are a common technique in time series analysis, used for business planning and financial analysis. MCP Analytics offers these models through natural language interaction.