Bitcoin Logarithmic Growth Curve 2024

9월 21일

The Bitcoin logarithmic growth curve is a concept used to analyze Bitcoin's price movements over time. The idea is based on the observation that Bitcoin's price tends to grow exponentially, particularly during bull markets. It attempts to give a long-term perspective on the Bitcoin price movements.

The curve includes an upper and lower band. These bands often represent zones where Bitcoin's price is overextended (upper band) or undervalued (lower band) relative to its historical growth trajectory. When the price touches or exceeds the upper band, it may indicate a speculative bubble, while prices near the lower band may suggest a buying opportunity.

Unlike most Bitcoin growth curve indicators, this one includes a logarithmic growth curve optimized using the latest 2024 price data, making it, in our view, superior to previous models. Additionally, it features statistical confidence intervals derived from linear regression, compatible across all timeframes, and extrapolates the data far into the future. Finally, this model allows users the flexibility to manually adjust the function parameters to suit their preferences.

The Bitcoin logarithmic growth curve has the following function:

y = 10^(a * log10(x) - b)

In the context of this formula, the y value represents the Bitcoin price, while the x value corresponds to the time, specifically indicated by the weekly bar number on the chart.

How is it made (You can skip this section if you’re not a fan of math):
To optimize the fit of this function and determine the optimal values of a and b, the previous weekly cycle peak values were analyzed. The corresponding x and y values were recorded as follows:

• 113, 18.55
  
• 240, 1004.42
  
• 451, 19128.27
  
• 655, 65502.47
  

The same process was applied to the bear market low values:

• 103, 2.48
  
• 267, 211.03
  
• 471, 3192.87
  
• 676, 16255.15
  

Next, these values were converted to their linear form by applying the base-10 logarithm. This transformation allows the function to be expressed in a linear state: y = a * x − b. This step is essential for enabling linear regression on these values.

For the cycle peak (x,y) values:

• 2.053, 1.268
  
• 2.380, 3.002
  
• 2.654, 4.282
  
• 2.816, 4.816
  

And for the bear market low (x,y) values:

• 2.013, 0.394
  
• 2.427, 2.324
  
• 2.673, 3.504
  
• 2.830, 4.211
  

Next, linear regression was performed on both these datasets. (Numerous tools are available online for linear regression calculations, making manual computations unnecessary).
Linear regression is a method used to find a straight line that best represents the relationship between two variables. It looks at how changes in one variable affect another and tries to predict values based on that relationship.

The goal is to minimize the differences between the actual data points and the points predicted by the line. Essentially, it aims to optimize for the highest R-Square value.
Below are the results:


It is important to note that both the slope (a-value) and the y-intercept (b-value) have associated standard errors. These standard errors can be used to calculate confidence intervals by multiplying them by the t-values (two degrees of freedom) from the linear regression.

These t-values can be found in a t-distribution table. For the top cycle confidence intervals, we used t10% (0.133), t25% (0.323), and t33% (0.414). For the bottom cycle confidence intervals, the t-values used were t10% (0.133), t25% (0.323), t33% (0.414), t50% (0.765), and t67% (1.063).

The final bull cycle function is:

y = 10^(4.058 ± 0.133 * log10(x) – 6.44 ± 0.324)

The final bear cycle function is:

y = 10^(4.684 ± 0.025 * log10(x) – -9.034 ± 0.063)

The main Criticisms of growth curve models:
The Bitcoin logarithmic growth curve model faces several general criticisms that we’d like to highlight briefly. The most significant, in our view, is its heavy reliance on past price data, which may not accurately forecast future trends. For instance, previous growth curve models from 2020 on TradingView were overly optimistic in predicting the last cycle’s peak.

This is why we aimed to present our process for deriving the final functions in a transparent, step-by-step scientific manner, including statistical confidence intervals. It's important to note that the bull cycle function is less reliable than the bear cycle function, as the top band is significantly wider than the bottom band.

Even so, we still believe that the Bitcoin logarithmic growth curve presented in this script is overly optimistic since it goes parly against the concept of diminishing returns which we discussed in this post:

This is why we also propose alternative parameter settings that align more closely with the theory of diminishing returns.

Our recommendations:
Drawing on the concept of diminishing returns, we propose alternative settings for this model that we believe provide a more realistic forecast aligned with this theory. The adjusted parameters apply only to the top band: a-value: 3.637 ± 0.2343 and b-parameter: -5.369 ± 0.6264. However, please note that these values are highly subjective, and you should be aware of the model's limitations.

Conservative bull cycle model:
y = 10^(3.637 ± 0.2343 * log10(x) - 5.369 ± 0.6264)