WIPFunctionLyaponovLibrary "WIPFunctionLyaponov"
Lyapunov exponents are mathematical measures used to describe the behavior of a system over
time. They are named after Russian mathematician Alexei Lyapunov, who first introduced the concept in the
late 19th century. The exponent is defined as the rate at which a particular function or variable changes
over time, and can be positive, negative, or zero.
Positive exponents indicate that a system tends to grow or expand over time, while negative exponents
indicate that a system tends to shrink or decay. Zero exponents indicate that the system does not change
significantly over time. Lyapunov exponents are used in various fields of science and engineering, including
physics, economics, and biology, to study the long-term behavior of complex systems.
~ generated description from vicuna13b
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To calculate the Lyapunov Exponent (LE) of a given Time Series, we need to follow these steps:
1. Firstly, you should have access to your data in some format like CSV or Excel file. If not, then you can collect it manually using tools such as stopwatches and measuring tapes.
2. Once the data is collected, clean it up by removing any outliers that may skew results. This step involves checking for inconsistencies within your dataset (e.g., extremely large or small values) and either discarding them entirely or replacing with more reasonable estimates based on surrounding values.
3. Next, you need to determine the dimension of your time series data. In most cases, this will be equal to the number of variables being measured in each observation period (e.g., temperature, humidity, wind speed).
4. Now that we have a clean dataset with known dimensions, we can calculate the LE for our Time Series using the following formula:
λ = log(||M^T * M - I||)/log(||v||)
where:
λ (Lyapunov Exponent) is the quantity that will be calculated.
||...|| denotes an Euclidean norm of a vector or matrix, which essentially means taking the square root of the sum of squares for each element in the vector/matrix.
M represents our Jacobian Matrix whose elements are given by:
J_ij = (∂fj / ∂xj) where fj is the jth variable and xj is the ith component of the initial condition vector x(t). In other words, each element in this matrix represents how much a small change in one variable affects another.
I denotes an identity matrix whose elements are all equal to 1 (or any constant value if you prefer). This term essentially acts as a baseline for comparison purposes since we want our Jacobian Matrix M^T * M to be close to it when the system is stable and far away from it when the system is unstable.
v represents an arbitrary vector whose Euclidean norm ||v|| will serve as a scaling factor in our calculation. The choice of this particular vector does not matter since we are only interested in its magnitude (i.e., length) for purposes of normalization. However, if you want to ensure that your results are accurate and consistent across different datasets or scenarios, it is recommended to use the same initial condition vector x(t) as used earlier when calculating our Jacobian Matrix M.
5. Finally, once we have calculated λ using the formula above, we can interpret its value in terms of stability/instability for our Time Series data:
- If λ < 0, then this indicates that the system is stable (i.e., nearby trajectories will converge towards each other over time).
- On the other hand, if λ > 0, then this implies that the system is unstable (i.e., nearby trajectories will diverge away from one another over time).
~ generated description from airoboros33b
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Reference:
en.wikipedia.org
www.collimator.ai
blog.abhranil.net
www.researchgate.net
physics.stackexchange.com
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This is a work in progress, it may contain errors so use with caution.
If you find flaws or suggest something new, please leave a comment bellow.
_measure_function(i)
helper function to get the name of distance function by a index (0 -> 13).\
Functions: SSD, Euclidean, Manhattan, Minkowski, Chebyshev, Correlation, Cosine, Camberra, MAE, MSE, Lorentzian, Intersection, Penrose Shape, Meehl.
Parameters:
i (int)
_test(L)
Helper function to test the output exponents state system and outputs description into a string.
Parameters:
L (float )
estimate(X, initial_distance, distance_function)
Estimate the Lyaponov Exponents for multiple series in a row matrix.
Parameters:
X (map)
initial_distance (float) : Initial distance limit.
distance_function (string) : Name of the distance function to be used, default:`ssd`.
Returns: List of Lyaponov exponents.
max(L)
Maximal Lyaponov Exponent.
Parameters:
L (float ) : List of Lyapunov exponents.
Returns: Highest exponent.
WIP
Realtime All-Time High and All-Time Low Tracker [WIP]This is a study intended to port the work of /u/QuantNomad's "Kozlod - All-time high/low alerts" to pine version 4 without giving any alerts. It is intended to capture the most extreme points on any given price chart in real time, the absolute high and absolute low points. Ideally, the plotted lines would strictly diverge from one another in opposite directions and widen with new highs or lows on a hypothetical "all-time" resolution.
Most of the original code was replaced and I'm trying to resolve a bug where the script fails to register new highs or lows while at the same time making sure that earlier data is plotted correctly. If applied to an asset that has not yet recovered , is just too new, or has not achieved a lower low than its opening price, this indicator ceases to function correctly. This would not be an issue if pine script was more general purpose and had something similar to Python's max(list) function.
Any and all suggestions are welcome. This is simply to serve as a springboard for any programmers trying to design algorithms or strategies that use these variables on any price chart.
Happy Hunting!
- Patch Hemlock
