Constant Bias Scale

The Constant Bias Scale module applies a linear transformation to each value in a series—first multiplying by a scale factor, then adding a bias term.

$$\text{CBS}(x) = x \cdot \text{scale} + \text{bias}$$

This simple yet powerful operation is fundamental to signal processing and data normalization.

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How It Works

flowchart LR
    subgraph Input
        X["x₁, x₂, ... xₙ"]
    end
    
    subgraph Transform["Linear Transformation"]
        M["× Scale"]
        A["+ Bias"]
    end
    
    subgraph Output
        Y["y₁, y₂, ... yₙ"]
    end
    
    X --> M --> A --> Y

For each input value:

1. Multiply by the scale factor (stretches or compresses)
2. Add the bias term (shifts up or down)

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Step-by-Step Example

Input: [0, 1, 2, 3, 4]
Scale: 7
Bias: 3

Input (x)	× Scale	+ Bias	Output
0	0 × 7 = 0	0 + 3	3
1	1 × 7 = 7	7 + 3	10
2	2 × 7 = 14	14 + 3	17
3	3 × 7 = 21	21 + 3	24
4	4 × 7 = 28	28 + 3	31

Result: [3, 10, 17, 24, 31]

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Understanding the Parameters

Scale: Amplitude Control

The scale parameter controls the spread or amplitude of your data:

Scale	Effect
> 1	Amplify / stretch the range
= 1	No change to spread
0 < scale < 1	Compress the range
= 0	Flatten to constant
< 0	Invert and scale

Bias: Level Shift

The bias parameter shifts the entire output up or down:

Bias	Effect
> 0	Shift values upward
= 0	No shift
< 0	Shift values downward

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Practical Applications

1. Re-centering an Oscillator

Many oscillators (RSI, Stochastic) range from 0 to 100. To center them around zero:

Original Range	Scale	Bias	New Range
0 to 100	1	-50	-50 to +50
0 to 100	0.02	-1	-1 to +1

2. Converting Units

Transform data between different scales:

• Fahrenheit to Celsius: Scale = 5/9, Bias = -17.78 (approximate after input shift)
• Percentage to decimal: Scale = 0.01, Bias = 0

3. Normalizing Custom Ranges

Map any range [a, b] to a new center point by choosing appropriate scale and bias.

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Combining CBS Modules

Multiple CBS transformations in sequence compose into a single equivalent CBS:

$$\text{CBS}_2(\text{CBS}_1(x)) = (x \cdot s_1 + b_1) \cdot s_2 + b_2 = x \cdot (s_1 \cdot s_2) + (b_1 \cdot s_2 + b_2)$$

This means chained transformations can always be reduced to one.

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Edge Cases

Scenario	Result
Scale = 0	All outputs equal bias (constant)
Bias = 0, Scale = 1	Output equals input (identity)
Empty input	Rejected — the series must contain at least one value
Missing (null) value in input	Output is null at the same position

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Missing Data

A missing datapoint (null) — a data gap or a closed-market slot — produces null at the same position in the output. CBS never invents a value for a missing input; in particular, a missing slot is never replaced with the bias, which is what scaling a fabricated zero would produce. The math is never faked.

CBS itself is defined for every real number, so there is no invalid-input case: nulls in the output mirror missing inputs. The one exception is arithmetic with no defined result (for example an infinite value multiplied by a scale of zero), which is also null at that position.

In live execution, CBS returns exactly one datapoint: the transformed newest value, or null when that value is missing.

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Further Reading

• Wikipedia: Linear Function
• Wikipedia: Affine Transformation
• Feature Scaling in Machine Learning

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Indicator Assessment

---
config:
  xyChart:
    width: 600
    height: 400
  themeVariables:
    xyChart:
      plotColorPalette: "#1abc9c"
---
xychart-beta horizontal
    title "Constant Bias Scale Assessment (Scale: 1-10)"
    x-axis ["Popularity", "Reliability", "Complexity", "Versatility", "Recent Interest"]
    y-axis "Score" 0 --> 10
    bar [6, 10, 2, 10, 4]

Note: Single bar shown as scores are identical across asset classes for this fundamental mathematical operation.

Dimension Explanations

Dimension	Score	Rationale
Popularity	6	Linear transformations are universally used but often as building blocks within larger systems, not standalone indicators. Well understood in engineering and data science.
Reliability	10	Mathematically deterministic—identical inputs always produce identical outputs. No statistical uncertainty or market-dependent behavior.
Complexity	2	Elementary arithmetic (multiply then add). Requires only basic algebra to understand and apply correctly.
Versatility	10	Applicable to any numerical series across all asset classes, timeframes, and use cases. Fundamental building block for normalization, unit conversion, and signal processing.
Recent Interest	4	As a fundamental operation, interest is stable rather than trending. No recent surge in research—it simply works and has for centuries.