Chapter 26 Key Takeaways: Business Forecasting and Trend Analysis

The Core Philosophy

Forecasting is estimation, not prediction. Every forecast involves uncertainty. The difference between a useful forecast and a misleading one is whether the uncertainty is communicated honestly.

Box's Principle, Applied: All models are wrong. The question is whether they are useful enough despite their limitations. A linear trend model that explains 80% of historical variation and provides a defensible range for planning is useful. Pretending that model provides certainty is misleading.

The Four Time Series Components

When building a forecast, identify which components are present before choosing a method.

Moving Average Quick Reference

import pandas as pd

# Simple Moving Average — equal weights, requires window observations
df["sma_6"] = df["revenue"].rolling(window=6).mean()
# Returns NaN for first window-1 periods

# Exponential Moving Average — all observations weighted, no NaN
df["ema_6"] = df["revenue"].ewm(span=6, adjust=False).mean()

# Weighted Moving Average — linear weights, recent observations count more
def wma(series: pd.Series, window: int) -> pd.Series:
    weights = range(1, window + 1)
    return series.rolling(window).apply(
        lambda x: sum(x[i] * weights[i] for i in range(window)) / sum(weights),
        raw=True,
    )

df["wma_6"] = wma(df["revenue"], 6)

Choosing a window: Small window = responds quickly to changes, noisier. Large window = smoother, slower to reflect changes. For quarterly reporting, 3–6 period windows are typically appropriate.

Linear Trend Analysis Quick Reference

import numpy as np
from scipy import stats
import pandas as pd

# Convert dates to numbers for regression
ordinals = df["date"].map(pd.Timestamp.toordinal).values
revenues = df["revenue"].values

# scipy.stats.linregress: more statistical output than polyfit
slope, intercept, r_value, p_value, std_err = stats.linregress(ordinals, revenues)

# Key outputs
r_squared = r_value ** 2   # 0 to 1: how well trend explains data
# p < 0.05: trend is statistically significant

# Forecast n periods ahead
def forecast_value(n_periods_ahead, slope, intercept, last_ordinal, period_days=90):
    future_ordinal = last_ordinal + (n_periods_ahead * period_days)
    return slope * future_ordinal + intercept

# Monthly growth rate from slope
avg_revenue = revenues.mean()
monthly_growth_pct = (slope * 30.44 / avg_revenue) * 100

Confidence Interval Formula

import numpy as np
from scipy.stats import norm

# 1. Get residuals from historical fit
fitted = slope * ordinals + intercept
residuals = revenues - fitted
residual_std = np.std(residuals, ddof=1)

# 2. Z-score for desired confidence level
z_90 = norm.ppf(0.95)   # 1.645 for 90% CI
z_95 = norm.ppf(0.975)  # 1.960 for 95% CI

# 3. Confidence band widens with forecast horizon
for horizon in range(1, 4):
    point = forecast_value(horizon, slope, intercept, last_ordinal)
    margin = z_95 * residual_std * np.sqrt(horizon)
    lower = max(0, point - margin)
    upper = point + margin
    print(f"Period +{horizon}: ${point:,.0f} ({lower:,.0f} – {upper:,.0f})")

Why confidence bands widen: Forecast errors compound over time. The further ahead you forecast, the more opportunity there is for the real trajectory to diverge from the model.

R-Squared Interpretation Guide

Seasonality Analysis Quick Reference

# Monthly seasonality profile
df["month_num"] = df["date"].dt.month
monthly_avg = df.groupby("month_num")["revenue"].mean()
seasonality_index = (monthly_avg / monthly_avg.mean() * 100).round(1)

# Index interpretation:
# 118 = this month is typically 18% ABOVE annual average
# 82  = this month is typically 18% BELOW annual average

# Applying seasonal adjustment to a linear forecast
q3_months = [7, 8, 9]
q3_index = seasonality_index[q3_months].mean() / 100  # Convert to ratio

adjusted_forecast = point_forecast * q3_index

statsmodels Methods Reference

from statsmodels.tsa.holtwinters import SimpleExpSmoothing, Holt

# Simple Exponential Smoothing — for data with no trend
model_ses = SimpleExpSmoothing(revenue_series, initialization_method="estimated")
result_ses = model_ses.fit(optimized=True)  # auto-optimizes alpha
forecast_ses = result_ses.forecast(3)       # 3 periods ahead

# Holt's Linear Trend — for data with consistent trend
model_holt = Holt(revenue_series, initialization_method="estimated")
result_holt = model_holt.fit(optimized=True)  # auto-optimizes alpha and beta
forecast_holt = result_holt.forecast(3)

# Check optimal parameters
print(f"SES alpha: {result_ses.params['smoothing_level']:.3f}")
print(f"Holt alpha: {result_holt.params['smoothing_level']:.3f}")
print(f"Holt beta:  {result_holt.params['smoothing_trend']:.3f}")

Forecast Chart Blueprint

The standard forecast visualization for business use:

|  Actual (solid line)                    |
|  ____________________________           |
| /          Trend (dashed line)    o--o  | <- Forecast points
|/                                /  /   |
|                               (shaded) | <- Confidence band
|________________________|_______________|
   Historical period    ^ End of data

Key elements: 1. Solid line: actual historical values 2. Dashed line: trend (if using linear regression) 3. Vertical separator: between historical and forecast 4. Open markers: forecast point estimates 5. Shaded band: confidence interval 6. Labels on last actual point and first forecast point

Percent Change Quick Reference

# Month-over-month (periods=1)
df["mom_pct"] = df["revenue"].pct_change(periods=1) * 100

# Year-over-year for monthly data (periods=12)
df["yoy_pct"] = df["revenue"].pct_change(periods=12) * 100

# Cumulative growth over N periods
start_value = df["revenue"].iloc[0]
end_value = df["revenue"].iloc[-1]
total_growth_pct = ((end_value - start_value) / start_value) * 100

# Compound monthly growth rate (CMGR)
n_months = len(df) - 1
cmgr = ((end_value / start_value) ** (1 / n_months) - 1) * 100

Method Selection Cheat Sheet

The Business Communication Checklist

Before presenting any forecast to a non-technical audience, verify:

[ ] Point estimate accompanied by a range (confidence interval)
[ ] Range explained in plain English ("our revenue is likely between X and Y")
[ ] R-squared mentioned and explained ("the trend explains 80% of historical variation")
[ ] Assumptions stated explicitly ("this assumes recent growth rate continues")
[ ] Limitations stated honestly ("this model won't capture a major competitive disruption")
[ ] Forecast horizon acknowledged ("confidence decreases for periods further out")
[ ] Seasonal adjustments noted if applied ("Q3 adjusted downward 6% for typical summer slowdown")

Common Forecasting Mistakes

Key Functions from This Chapter

End of Chapter 26 Key Takeaways