QuanTAlib/lib/errors/mape/Mape.md

MAPE: Mean Absolute Percentage Error

The metric that lets you compare apples to oranges, as long as you don't have any zeros.

Property	Value
Category	Error Metric
Inputs	Actual, Predicted (dual series)
Parameters	period
Outputs	Single series (MAPE)
Output range	\geq 0
Warmup	period bars
PineScript	mape.pine

• Mean Absolute Percentage Error (MAPE) measures the average absolute percentage difference between actual and predicted values.
• Similar: SMAPE, MAAPE | Trading note: Mean Absolute Percentage Error; scale-independent accuracy. Undefined when actuals are zero.
• Validated against TA-Lib, Skender, and Tulip reference implementations where available.

Mean Absolute Percentage Error (MAPE) measures the average absolute percentage difference between actual and predicted values. It expresses accuracy as a percentage, making it scale-independent and easy to interpret.

Historical Context

MAPE has been widely used in forecasting and operations research since the mid-20th century. Its intuitive percentage-based interpretation makes it a favorite in business contexts where stakeholders need to understand prediction accuracy without domain expertise.

Architecture & Physics

MAPE divides each absolute error by the actual value, converting errors to percentages. This makes it independent of the scale of the data but introduces asymmetry and problems with zero values.

Properties

• Scale-independent: Expressed as percentage
• Asymmetric: Penalizes over-prediction more than under-prediction
• Undefined at zero: Cannot compute when actual value is zero
• Non-negative: MAPE ≥ 0, with 0 indicating perfect prediction
• No upper bound: Can exceed 100% for large errors

Mathematical Foundation

1. Percentage Error

For each observation, calculate the absolute percentage error:

APE_i = 100 \times \left| \frac{y_i - \hat{y}_i}{y_i} \right|

Where:

• y_i = actual value
• \hat{y}_i = predicted value

2. Mean Calculation

Average the absolute percentage errors over the period:

MAPE = \frac{100}{n} \sum_{i=1}^{n} \left| \frac{y_i - \hat{y}_i}{y_i} \right|

3. Running Update (O(1))

QuanTAlib uses a ring buffer with running sum for O(1) updates:

S_{new} = S_{old} - APE_{oldest} + APE_{newest} MAPE = \frac{S_{new}}{n}

Implementation Details

Usage Patterns

// Streaming mode - update with each new observation
var mape = new Mape(period: 20);
var result = mape.Update(actualValue, predictedValue);

// Batch mode - calculate for entire series
var results = Mape.Calculate(actualSeries, predictedSeries, period: 20);

// Span mode - zero-allocation for high performance
Mape.Batch(actualSpan, predictedSpan, outputSpan, period: 20);

Parameters

Parameter	Type	Description
period	int	Lookback window for averaging (must be > 0)

Properties

Property	Type	Description
Last	TValue	Most recent MAPE value (as percentage)
IsHot	bool	True when buffer is full
Name	string	Indicator name (e.g., "Mape(20)")
WarmupPeriod	int	Number of periods before valid output

Performance Profile

Operation Count (Streaming Mode)

O(1) per bar. Single-pass scalar transformation of (actual, forecast) pair; no lookback window required.

Operation	Count	Cost (cycles)	Subtotal
Error computation (subtract, abs/square/log)	1-3	~3-8 cy	~5-15 cy
Running accumulator update (EMA or sum)	1	~4 cy	~4 cy
Total	2-4	—	~9-19 cycles

Streaming update requires only the current actual/forecast pair and running state. ~10-15 cycles/bar typical.

Batch Mode (SIMD Analysis)

Operation	Vectorizable?	Notes
Element-wise error computation	Yes	Independent per bar; fully vectorizable with Vector<double>
Reduction (sum/mean)	Yes	Parallel reduction; AVX2 gives 4x speedup
Log/exp components	Partial	Transcendental ops; polynomial approx for SIMD

Batch SIMD: 4x-8x speedup for large windows. ~3-5 cy/bar amortized in vectorized batch mode.

Metric	Score	Notes
Throughput	~12 ns/bar	O(1) update complexity
Allocations	0	Uses pre-allocated ring buffer
Complexity	O(1)	Constant time per update
Accuracy	10/10	Exact calculation
Timeliness	9/10	No lag beyond the period
Smoothness	7/10	Moderate smoothing

Interpretation

MAPE Range	Interpretation
< 10%	Highly accurate
10-20%	Good accuracy
20-50%	Reasonable accuracy
> 50%	Poor accuracy

The Asymmetry Problem

MAPE is asymmetric because it divides by the actual value:

var mape1 = new Mape(1);
var mape2 = new Mape(1);

// Under-prediction: actual=100, predicted=50
// |100-50|/100 = 50%
mape1.Update(100, 50);  // Returns 50%

// Over-prediction: actual=50, predicted=100  
// |50-100|/50 = 100%
mape2.Update(50, 100);  // Returns 100%

Same absolute error (50), but over-prediction shows higher MAPE.

Comparison with Other Metrics

Metric	Scale	Handles Zero	Symmetric
MAPE	Percentage	No	No
MAPD	Percentage	No	No
SMAPE	Percentage	Partially	Yes
MAE	Original units	Yes	Yes
MPE	Percentage	No	Yes (signed)

Common Use Cases

1. Demand Forecasting: Inventory and supply chain planning
2. Sales Prediction: Revenue forecasting accuracy
3. Financial Modeling: Investment return predictions
4. Operations: Capacity planning and scheduling

Limitations

1. Zero Values: Undefined when actual = 0 (QuanTAlib uses epsilon fallback)
2. Asymmetry: Biases toward under-prediction
3. Scale Sensitivity: Low values inflate MAPE disproportionately
4. Outlier Impact: Single large percentage error can dominate

Edge Cases

• Identical Values: Returns 0% when actual equals predicted
• Zero Actual: Uses epsilon (1e-10) to avoid division by zero
• NaN Handling: Uses last valid value substitution
• Single Input: Not supported (requires two series)
• Period = 1: Returns current percentage error

Related Indicators

• MAPD - Mean Absolute Percentage Deviation (divides by predicted)
• SMAPE - Symmetric Mean Absolute Percentage Error
• MPE - Mean Percentage Error (signed)
• MAE - Mean Absolute Error (same units)