Understanding Confidence Intervals

Every Eaternity Forecast prediction includes a confidence interval that helps you understand the range of likely outcomes. This guide explains how to interpret and use these intervals effectively.

What is a Confidence Interval?

Basic Concept

A confidence interval shows the range of possible values for a prediction, not just a single number.

Example:

Pasta Carbonara - Saturday, January 20
  Point Estimate: 52 portions
  Confidence Interval: 45-59 portions (80% confidence)

Interpretation:

• Most likely: 52 portions will be sold
• Lower bound: 45 portions (10% chance of selling fewer)
• Upper bound: 59 portions (10% chance of selling more)
• Probability: 80% chance actual sales fall between 45-59

Why Confidence Intervals Matter

Single-Point Predictions Are Insufficient:

Consider these two scenarios with same point estimate:

Scenario A: High Confidence

Pasta Carbonara (stable menu item, 2 years history):
  Prediction: 52 portions
  Range: 48-56 portions (narrow, ±8% variance)
  Confidence: High

Scenario B: Low Confidence

New Mushroom Risotto (launched 1 week ago):
  Prediction: 52 portions
  Range: 35-69 portions (wide, ±33% variance)
  Confidence: Low

Decision Impact:

• Scenario A: Prepare 52-54 portions (high confidence in accuracy)
• Scenario B: Prepare 40-45 portions initially, keep ingredients ready for more (low confidence, high uncertainty)

Same point estimate, very different planning strategies.

How Confidence Intervals Are Calculated

Quantile Regression

Eaternity Forecast uses quantile regression to predict three values simultaneously:

1. 10th Percentile (Lower Bound)

   • 10% of the time, actual sales will be below this
   • 90% of the time, actual sales will be at or above this

2. 50th Percentile (Median/Point Estimate)

   • Half the time, actual sales will be below this
   • Half the time, actual sales will be above this
   • This is our "best guess"

3. 90th Percentile (Upper Bound)

   • 90% of the time, actual sales will be at or below this
   • 10% of the time, actual sales will exceed this

80% Confidence Interval = Space between 10th and 90th percentiles

Example Calculation

Historical Data for Pasta Carbonara on Saturdays (last 20 weeks):

Sorted sales: [42, 44, 46, 47, 48, 49, 50, 51, 51, 52, 53, 53, 54, 55, 56, 57, 58, 60, 62, 65]

10th percentile (2nd value):  45 portions  (lower bound)
50th percentile (10th value): 52 portions  (point estimate)
90th percentile (18th value): 59 portions  (upper bound)

Confidence interval: 45-59 portions

Neural Network Learning:

Instead of manually calculating from historical data, the neural network learns to predict these quantiles directly based on:

• Historical sales patterns
• Day of week
• Weather conditions
• Seasonal trends
• Recent trajectory
• Menu dynamics

Factors Affecting Confidence

What Makes Confidence High or Low?

High Confidence (Narrow Intervals)

Characteristics:

• ✅ Stable menu item (months/years of history)
• ✅ Consistent demand pattern
• ✅ Predictable influencing factors
• ✅ Low natural variability
• ✅ Clear seasonal patterns (if applicable)

Example:

House Bread (served daily for 3 years):
  Monday-Friday: 85 portions (range: 82-88, ±3.5%)
  Saturday-Sunday: 95 portions (range: 91-99, ±4.2%)

Why high confidence?
  - Thousands of historical data points
  - Very consistent demand
  - Minimal external factor influence
  - Predictable weekly pattern

Low Confidence (Wide Intervals)

Characteristics:

• ⚠️ New menu item (days/weeks of history)
• ⚠️ High demand variability
• ⚠️ Unpredictable influencing factors
• ⚠️ Event-driven or promotional
• ⚠️ Seasonal item at season start/end

Example:

New Seasonal Special (launched 1 week ago):
  Prediction: 35 portions
  Range: 22-48 portions (±37%)

Why low confidence?
  - Only 5-7 days of sales data
  - Unknown demand pattern
  - Unclear guest acceptance
  - Limited seasonal history

Confidence Levels by Item Age

Typical Confidence Interval Width:

Item Age	Data Points	Typical CI Width	MAPE
Week 1 (New)	5-7 days	±30-40%	18-25%
Week 2-3	10-20 days	±20-30%	14-18%
Month 2-3	30-60 days	±15-20%	11-14%
6+ months	100+ days	±10-15%	9-12%
2+ years	500+ days	±8-12%	8-10%

Learning Curve: Confidence improves rapidly in first month, stabilizes after 3-6 months

External Factors

Weather Sensitivity

Temperature-Sensitive Items:

Caesar Salad (weather-dependent):

Warm day (20°C):
  Prediction: 45 portions
  Range: 42-48 portions (narrow, weather predictable)

Cold day (5°C):
  Prediction: 28 portions
  Range: 22-34 portions (wider, more variability)

Why wider in cold? Fewer people order salads when cold, but variability is higher (some still order, some switch to soup).

Event Influence

Known Events (higher confidence):

Regular monthly staff meeting (200 attendees):
  Prediction: +180 lunch covers
  Range: +170 to +190 (narrow, event is predictable)

Unknown Events (lower confidence):

Unannounced nearby conference:
  Prediction: Normal day volume
  Range: Wider than usual (model senses uncertainty)

Note: If event announced and added to system, confidence improves

Using Confidence Intervals for Decision-Making

Decision Frameworks

Conservative Strategy (Minimize Waste)

When to use:

• Expensive ingredients
• Short shelf-life items
• High waste disposal costs
• Acceptable to occasionally stock out

Preparation Rule: Prepare to lower bound or slightly above

Example:

Fresh Fish Special (€18 cost, 1-day shelf life):
  Prediction: 28 portions
  Range: 23-33 portions

Decision: Prepare 25 portions (between lower and point estimate)
  - 90% confident we'll sell at least 23
  - Reserve fresh fish for 8 more if needed (supplier delivers 2x daily)
  - Minimal waste risk
  - Small stock-out risk acceptable

Risk Profile:

• Waste risk: Low (5-10%)
• Stock-out risk: Medium (15-20%)
• Best for: Perishables, high-cost items

Balanced Strategy (Service Quality Focus)

When to use:

• Standard menu items
• Moderate costs
• Some prep flexibility
• Stock-outs undesirable but manageable

Preparation Rule: Prepare to point estimate with slight buffer

Example:

Pasta Carbonara (€3.50 cost, easy to prepare more):
  Prediction: 52 portions
  Range: 48-56 portions

Decision: Prepare 52 portions initially
  - Keep ingredients ready for 6-8 more
  - Can prep additional in 15 minutes if needed
  - Balanced waste vs service quality

Risk Profile:

• Waste risk: Medium (10-15%)
• Stock-out risk: Low (5-10%)
• Best for: Core menu items, moderate costs

Aggressive Strategy (Never Stock Out)

When to use:

• Signature dishes
• Low-cost ingredients
• Critical guest experience items
• Leftovers easily repurposed

Preparation Rule: Prepare to upper bound

Example:

House Bread (€0.50 cost, signature item, 3-day storage):
  Prediction: 85 portions
  Range: 82-88 portions

Decision: Prepare 88 portions (upper bound)
  - Zero stock-out risk
  - Minimal waste cost (€1.50-2.00)
  - Critical to guest experience
  - Leftovers for staff meal or next-day breadcrumbs

Risk Profile:

• Waste risk: Higher (20-25%)
• Stock-out risk: Very low (less than 2%)
• Best for: Low-cost, signature items

Buffer Planning

Fixed Buffer Strategy

Add constant buffer to point estimate:

Preparation = Point Estimate + Fixed Buffer

Example (10% buffer):
  - Pasta Carbonara: 52 + 5 = 57 portions
  - Caesar Salad: 31 + 3 = 34 portions
  - Grilled Salmon: 28 + 3 = 31 portions

Pros: Simple, consistent Cons: Ignores confidence variations between items

Confidence-Based Buffer Strategy

Buffer proportional to confidence interval width:

Preparation = Point Estimate + (Interval Width × Buffer Factor)

High confidence item (narrow interval):
  - Point: 52, Range: 48-56 (width: 8)
  - Buffer: 8 × 0.25 = 2
  - Prepare: 52 + 2 = 54 portions

Low confidence item (wide interval):
  - Point: 35, Range: 22-48 (width: 26)
  - Buffer: 26 × 0.25 = 6.5
  - Prepare: 35 + 7 = 42 portions

Pros: Adapts to prediction certainty Cons: More complex calculation

Upper Bound Percentage Strategy

Prepare between point estimate and upper bound:

Preparation = Point Estimate + (Upper - Point) × Percentage

50% strategy (halfway between point and upper):
  - Pasta: 52 + (56 - 52) × 0.5 = 54 portions
  - Risotto: 35 + (48 - 35) × 0.5 = 41.5 = 42 portions

75% strategy (closer to upper bound):
  - Pasta: 52 + (56 - 52) × 0.75 = 55 portions
  - Risotto: 35 + (48 - 35) × 0.75 = 44.75 = 45 portions

Pros: Intuitively adjusts for confidence Cons: May over-prepare for low-confidence items

Combining Multiple Factors

Decision Matrix Example:

Item: Grilled Salmon
  - Prediction: 28 portions (Range: 23-33)
  - Cost: High (€6 ingredient cost)
  - Shelf life: 1 day
  - Prep flexibility: Low (1-hour advance prep needed)
  - Guest importance: Medium

Decision Process:
  1. High cost → lean toward lower bound
  2. Short shelf life → minimize waste risk
  3. Low prep flexibility → can't easily make more
  4. Medium importance → some stock-out risk acceptable

Final Decision: Prepare 26 portions (slightly above lower bound)
  - Accept 10% stock-out risk
  - Minimize expensive waste
  - Consider offering alternative if stock out

Item: House Pasta

  - Prediction: 52 portions (Range: 48-56)
  - Cost: Low (€1.50 ingredient cost)
  - Shelf life: 2 days (sauce), fresh pasta 3 days
  - Prep flexibility: High (15-min prep time)
  - Guest importance: High (signature dish)

Decision Process:
  1. Low cost → can afford some waste
  2. Longer shelf life → waste less problematic
  3. High prep flexibility → can make more if needed
  4. High importance → avoid stock-outs

Final Decision: Prepare 54 portions (above point estimate)
  - Start with 52, keep ingredients for 6-8 more
  - Zero stock-out tolerance for signature
  - Minimal financial risk

Interpreting Confidence Trends

Improving Confidence (Narrowing Intervals)

Good Signs:

New Menu Item - Mushroom Risotto:

Week 1: Prediction 22 (Range: 12-32, ±45%)
Week 2: Prediction 25 (Range: 18-32, ±28%)
Week 3: Prediction 27 (Range: 22-32, ±19%)
Week 4: Prediction 28 (Range: 24-32, ±14%)

What it means:

• Model is learning the demand pattern
• Guest acceptance stabilizing
• Prediction becoming more reliable
• Can increase preparation confidence

Declining Confidence (Widening Intervals)

Warning Signs:

Established Item - Caesar Salad:

January:  Prediction 42 (Range: 39-45, ±7%)
February: Prediction 38 (Range: 32-44, ±16%)
March:    Prediction 35 (Range: 26-44, ±26%)

Possible causes:

• Seasonal transition (winter→spring salad demand variable)
• Menu changes affecting complementary items
• New competition nearby
• Quality or recipe changes
• Promotional activities

Actions:

• Investigate operational changes
• Check for external market factors
• Provide feedback to model
• Use more conservative preparation strategy temporarily

Stable Confidence

Ideal State:

Core Menu Item - Pasta Carbonara (2 years on menu):

Consistent pattern: Prediction ±10-12% width
  - Mondays: 45 (Range: 41-49)
  - Wednesdays: 52 (Range: 48-56)
  - Fridays: 68 (Range: 62-74)
  - Saturdays: 73 (Range: 67-79)

What it means:

• Well-established demand pattern
• Predictable guest behavior
• Reliable for planning
• Minimal surprises

Common Misconceptions

Misconception 1: "Narrow intervals mean perfect predictions"

Reality: Narrow intervals mean consistent patterns, not guaranteed accuracy.

Example:

Systematic shift not yet detected:

Historical pattern: 50 portions/day (Range: 48-52)
New competition opened → Actual demand now: 42 portions/day

Week 1 after competition: Still predicts 50 (48-52)
  - Confidence high, but prediction wrong
  - Model hasn't learned new pattern yet

Week 3 after competition: Predicts 43 (40-46)
  - Adjusted to new baseline
  - Confidence restored

Lesson: High confidence reflects historical consistency, not immunity to change

Misconception 2: "Wide intervals mean the model is guessing"

Reality: Wide intervals honestly reflect genuine uncertainty.

Example:

New item, highly variable demand:
  - Day 1: 15 sold
  - Day 2: 32 sold
  - Day 3: 21 sold
  - Day 4: 28 sold
  - Day 5: 19 sold

Day 6 prediction: 23 (Range: 15-31)
  - Wide range reflects real variability
  - Point estimate (23) is average of data
  - Interval honestly communicates uncertainty

Lesson: Wide intervals are valuable information, not model failure

Misconception 3: "I should always prepare to the point estimate"

Reality: Optimal preparation depends on costs, risks, and business priorities.

Example:

Two items, same prediction:

Item A: Expensive seafood (€12 cost, 1-day shelf life)
Item B: House pasta (€1.50 cost, 2-day shelf life)

Both: Prediction 30 (Range: 25-35)

Optimal preparation:
  - Item A: 27-28 portions (minimize expensive waste)
  - Item B: 32-33 portions (avoid stock-outs, low waste cost)

Lesson: Use point estimate + confidence interval + business context together

Misconception 4: "80% confidence means 80% accuracy"

Reality: 80% confidence means 80% of actuals fall within the range.

Example:

100 predictions with 80% confidence intervals:

Expected outcome:
  - 80 predictions: Actual within [lower, upper]
  - 10 predictions: Actual below lower bound
  - 10 predictions: Actual above upper bound

This does NOT mean:
  - 80 predictions are exactly correct ❌
  - 20 predictions are totally wrong ❌

What it DOES mean:
  - 80% of the time, preparing within the range is sufficient ✅
  - 10% of the time, demand unexpectedly low (less waste)
  - 10% of the time, demand unexpectedly high (potential stock-out)

Lesson: Confidence interval is about range coverage, not point accuracy

Practical Examples

Example 1: Weekend Brunch Planning

Saturday Brunch - January 20, 2024

Eggs Benedict:
  Prediction: 42 (Range: 36-48, ±14%)
  Cost: €3.20 per portion
  Prep time: 20 min advance
  Shelf life: Same day only

Decision Analysis:
  - Moderate confidence (±14%)
  - Moderate cost
  - Some prep flexibility (can make more mid-service)
  - Short shelf life (waste is total loss)

Strategy: Balanced with slight conservatism
  - Prepare: 40 portions initially
  - Reserve: Ingredients for 10 more portions
  - Monitor: Sales rate first hour, prep more if needed

Result: Sold 44
  - Stock-out risk managed: 4 more prepared mid-service
  - Waste: 0 portions
  - Guest satisfaction: Maintained

Example 2: Weekly Planning

Monday-Friday Lunch - Pasta Carbonara

Historical Confidence Pattern:
  Monday:    45 (42-48, high confidence)
  Tuesday:   52 (49-55, high confidence)
  Wednesday: 52 (48-56, high confidence)
  Thursday:  55 (51-59, high confidence)
  Friday:    68 (62-74, moderate confidence)

Weekly Prep Plan:
  Monday: 45 (trust narrow interval)
  Tuesday: 53 (slight buffer, midweek stability)
  Wednesday: 53 (same as Tuesday)
  Thursday: 56 (slight buffer for trend)
  Friday: 66 (conservative, wider interval + end of week)

Total Weekly: 273 portions (vs pure point estimate: 272)
  - Minimal over-prep (1 portion)
  - Confidence-informed daily adjustments

Example 3: Special Event Handling

Valentine's Day - Known Event

Romantic Dinner Special (2-person dish):
  Normal Saturday: 15 orders (Range: 13-17)
  Valentine's Saturday: 32 orders (Range: 26-38)

Confidence Analysis:
  - Historical Valentine's data: 3 previous years
  - Consistent pattern: 2.0-2.2× normal Saturday
  - Moderate interval width (±19%) due to variability

Decision:
  - Prepare: 34 orders (68 portions)
  - Above point estimate due to:
    • High guest disappointment risk (romantic occasion)
    • Low waste cost (can offer post-Valentine's discount)
    • Historical tendency to exceed prediction on Valentine's

Result: Sold 36 orders
  - Slight stock-out last 30 minutes (2 orders missed)
  - Learned: Need 36-38 for future Valentine's
  - Feedback provided to model for next year

Advanced: Reading the Distribution

Understanding Skewness

Symmetric Distribution:

Pasta Carbonara - Wednesday:
  Point Estimate: 52
  Lower Bound: 48 (difference: -4)
  Upper Bound: 56 (difference: +4)

Distribution: Symmetric (equal distance from median)
Meaning: Equally likely to over/under-perform

Right-Skewed Distribution (Positive Skew):

Friday Fish Special (weather-dependent):
  Point Estimate: 28
  Lower Bound: 23 (difference: -5)
  Upper Bound: 37 (difference: +9)

Distribution: Right-skewed (longer tail upward)
Meaning: Occasionally very high demand (warm weather days)
Planning: Consider upper range more than lower

Left-Skewed Distribution (Negative Skew):

End-of-Season Salad:
  Point Estimate: 35
  Lower Bound: 24 (difference: -11)
  Upper Bound: 40 (difference: +5)

Distribution: Left-skewed (longer tail downward)
Meaning: Occasionally very low demand (decline starting)
Planning: Prepare conservatively, demand trending down

Distribution Width Changes

Stable Narrow (Ideal):

Weeks 1-4: Range width 8-10 portions
  → Reliable demand, plan confidently

Narrowing (Good):

Week 1: Width 20 portions
Week 4: Width 12 portions
  → Model learning, increase trust

Widening (Investigate):

Week 1: Width 8 portions
Week 4: Width 16 portions
  → Something changing, more cautious planning

Volatile (Caution):

Week 1: Width 12
Week 2: Width 22
Week 3: Width 10
Week 4: Width 18
  → Inconsistent patterns, use conservative approach

See Also

• Performance Study — Validation and calibration results
• Implementation Guide — Daily workflow strategies
• AI Architecture — How quantile regression works
• Features — Using confidence in dashboard and API