Module: Quant::Mixins::UniversalFilters

Included in:
Filters
Defined in:
lib/quant/mixins/universal_filters.rb

Overview

Source: Ehlers, J. F. (2013). Cycle Analytics for Traders:

Advanced Technical Trading Concepts. John Wiley & Sons.

Universal Filters

Ehlers devoted a chapter in his book to generalizing the algorithms for the most common filters used in trading. The universal filters makes an attempt to translate that work into working code. However, Ehler write up contained some typos and incomplete treatment of the subject matter, and as a non-mathematician, I am not sure if I have translated the formulas correctly. So far, I have not been able to prove correctness of the universal EMA vs. the optimzed EMA, but the others are still unproven and Ehlers’ many papers over the year tend to change implementation details, too.

Experimental!

The main goal with the universal filters is to provide a means to compare the optimized filters with the generalized filters and generally show correctness of the solutions. However, that also means validating the outputs of those computations, which is not my forté. My idea of validating is if I have two or more implementations that produce identical (or nearly identical) results, then I consider the implementation sound and doing what it is supposed to do.

Several are marked “experimental” because I have not been able to prove their correctness. Those that are proven correct are not marked as experimental and you’ll find their outputs show up in other specs where they’re used alongside the optimized versions of those filters.

Ehlers’ Notes on Generalized Filters

  1. All the common filters useful for traders have a transfer response that can be written as a ratio of two polynomials.

  2. Lag is very important to traders. More complex filters can be created using more input data, but more input data increases lag. Sophisticated filters are not very useful for trading because they incur too much lag.

  3. Filter transfer response can be viewed in the time domain and the frequency domain with equal validity.

  4. Nonrecursive filters can have zeros in the transfer response, enabling the complete cancellation of some selected frequency components.

  5. Nonrecursive filters having coefficients symmetrical about the center of the filter will have a delay of half the degree of the transfer response polynomial at all frequencies.

  6. Low-pass filters are smoothers because they attenuate the high-frequency components of the input data.

  7. High-pass filters are detrenders because they attenuate the low-frequency components of trends.

  8. Band-pass filters are both detrenders and smoothers because they attenuate all but the desired frequency components.

  9. Filters provide an output only through their transfer response. The transfer response is strictly a mathematical function, and interpretations such as overbought, oversold, convergence, divergence, and so on are not implied. The validity of such interpretations must be made on the basis of statistics apart from the filter.

  10. The critical period of a filter output is the frequency at which the output power of the filter is half the power of the input wave at that frequency.

  11. A WMA has little or no redeeming virtue.

  12. A median filter is best used when the data contain impulsive noise or when there are wild variations in the data. Smoothing volume data is one example of a good application for a median filter.

Filter Coefficients forVariousTypes of Filters

Filter Type           b0          b1              b2          a1        a2
EMA                   α           0               0           −(1−α)    0
Two-pole low-pass     α**2        0               0           −2*(1-α)  (1-α)**2
High-pass             (1-α/2)     -(1-α/2)        0           −(1−α)    0
Two-pole high-pass    (1-α/2)**2  -2*(1-α/2)**2   (1-α/2)**2  -2*(1-α)  (1-α)**2
Band-pass             (1-σ)/2     0               -(1-σ)/2    -λ*(1+σ)  σ
Band-stop             (1+σ)/2     -2λ*(1+σ)/2     (1+σ)/2     -λ*(1+σ)  σ

Constant Summary collapse

K = 
{
  single_pole: 1.0,
  two_pole_high_pass: Math.sqrt(2) * 0.5,
  two_pole_low_pass: Math.sqrt(2)
}.freeze

Instance Method Summary collapse

Instance Method Details

#universal_band_pass(source, previous:, period:) ⇒ Object

Band-pass filters are both detrenders and smoothers because they attenuate all but the desired frequency components. NOTE: Ehlers’ book contains a typo in the formula for the band-pass filter. I am not sure what the correct formulation is, so this is a best guess for how, left for further investigation.

Raises:

  • (ArgumentError) 


307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
 
# File 'lib/quant/mixins/universal_filters.rb', line 307

def universal_band_pass(source, previous:, period:)
  Quant.experimental("This method is unproven and may be incorrect.")
  raise ArgumentError, "source must be a Symbol" unless source.is_a?(Symbol)
  raise ArgumentError, ":previous must be a Symbol" unless previous.is_a?(Symbol)

  lambda = deg2rad(360.0 / period)
  gamma = Math.cos(lambda)
  sigma = 1 / gamma - Math.sqrt(1 / gamma**2 - 1)

  b0 = (1 - sigma) * 0.5
  b1 = 0
  b2 = -(1 - sigma) * 0.5
  a1 = -lambda * (1 + sigma)
  a2 = sigma

  universal_filter(source, previous:, b0:, b1:, b2:, a1:, a2:)
end

#universal_ema(source, previous:, period:) ⇒ Object

The EMA is optimized in the ExponentialMovingAverage module and its correctness is proven with this particular implementation.

Raises:

  • (ArgumentError) 


95
96
97
98
99
100
101
102
103
104
105
106
107
 
# File 'lib/quant/mixins/universal_filters.rb', line 95

def universal_ema(source, previous:, period:)
  raise ArgumentError, "source must be a Symbol" unless source.is_a?(Symbol)
  raise ArgumentError, ":previous must be a Symbol" unless previous.is_a?(Symbol)

  alpha = bars_to_alpha(period)
  b0 = alpha
  b1 = 0
  b2 = 0
  a1 = -(1 - alpha)
  a2 = 0

  universal_filter(source, previous:, b0:, b1:, b2:, a1:, a2:)
end

#universal_filter(source, previous:, b0:, b1:, b2:, a1:, a2:) ⇒ Object

The universal filter is a generalization of the common filters. The other algorithms in this module are derived from this one.



85
86
87
88
89
90
91
 
# File 'lib/quant/mixins/universal_filters.rb', line 85

def universal_filter(source, previous:, b0:, b1:, b2:, a1:, a2:)
  b0 * p0.send(source) +
    b1 * p1.send(source) +
    b2 * p2.send(source) -
    a1 * p1.send(previous) -
    a2 * p2.send(previous)
end

#universal_one_pole_high_pass(source, previous:, period:) ⇒ Object

A single-pole high-pass filter, also known as a first-order high-pass filter, attenuates low-frequency components of a signal while allowing higher-frequency components to pass through. In the context of processing stock price action data, applying a single-pole high-pass filter has several potential effects:

  1. Removal of Low-Frequency Trends: Similar to higher-order high-pass filters, a single-pole high-pass filter removes or attenuates slow-moving components of the price action data, such as long-term trends. This can help in focusing on shorter-term fluctuations and identifying short-term trading opportunities.

  2. Noise Amplification: As with higher-order high-pass filters, a single-pole high-pass filter can amplify high-frequency noise present in the data, especially if the cutoff frequency is set too low. This noise amplification can make it challenging to analyze the data accurately, particularly if the noise overwhelms the signal of interest.

  3. Emphasis on Short-Term Variations: By removing low-frequency components, a single-pole high-pass filter highlights short-term variations and rapid movements in the price action data. This can be beneficial for traders or analysts who are primarily interested in short-term price dynamics or intraday trading opportunities.

  4. Simple Filtering: Single-pole filters are computationally efficient and straightforward to implement, making them suitable for real-time processing and applications where simplicity is preferred. However, they may not offer the same level of noise attenuation and signal preservation as higher-order filters.

  5. Enhanced Responsiveness: Single-pole high-pass filters offer relatively high responsiveness to changes in the input signal, reflecting recent price movements quickly. This responsiveness can be advantageous for certain trading strategies that rely on timely identification of market events or short-term trends.

Overall, applying a single-pole high-pass filter to stock price action data can help in removing low-frequency trends and focusing on short-term variations and rapid movements. However, it’s essential to carefully select the cutoff frequency to balance noise attenuation with signal preservation and to consider the potential trade-offs between simplicity and filtering effectiveness.

Raises:

  • (ArgumentError) 


231
232
233
234
235
236
237
238
239
240
241
242
243
244
 
# File 'lib/quant/mixins/universal_filters.rb', line 231

def universal_one_pole_high_pass(source, previous:, period:)
  Quant.experimental("This method is unproven and may be incorrect.")
  raise ArgumentError, "source must be a Symbol" unless source.is_a?(Symbol)
  raise ArgumentError, ":previous must be a Symbol" unless previous.is_a?(Symbol)

  alpha = period_to_alpha(period, k: K[:single_pole])
  b0 = (1 - alpha / 2)
  b1 = -(1 - alpha / 2)
  b2 = 0
  a1 = -(1 - alpha)
  a2 = 0

  universal_filter(source, previous:, b0:, b1:, b2:, a1:, a2:)
end

#universal_one_pole_low_pass(source, previous:, period:) ⇒ Object

A single-pole low-pass filter, also known as a first-order low-pass filter, has a simpler response compared to a two-pole low-pass filter. It attenuates higher-frequency components of the signal while allowing lower-frequency components to pass through, but it does so with a gentler roll-off compared to higher-order filters.

Here’s what a single-pole low-pass filter typically does to stock price action data:

  1. Noise Reduction: Similar to a two-pole low-pass filter, a single-pole filter can help in reducing high-frequency noise in stock price data, smoothing out rapid fluctuations caused by market volatility or other factors.

  2. Trend Identification: It can aid in identifying trends by smoothing out short-term fluctuations, making the underlying trend more apparent. However, compared to higher-order filters, it may not provide as sharp or accurate trend signals.

  3. Signal Smoothing: Single-pole filters provide a basic level of signal smoothing, which can help in removing minor fluctuations and emphasizing larger movements in the price action. This can make the data easier to interpret and analyze.

  4. Delay and Responsiveness: Single-pole filters introduce less delay compared to higher-order filters, making them more responsive to changes in the input signal. However, this responsiveness comes at the cost of a less aggressive attenuation of high-frequency noise.

  5. Simple Filtering: Single-pole filters are computationally efficient and easy to implement, making them suitable for real-time processing and applications where simplicity is preferred.

Overall, while a single-pole low-pass filter can still be effective for noise reduction and basic trend identification in stock price action data, it may not offer the same level of precision and robustness as higher-order filters. The choice between single-pole and higher-order filters depends on the specific requirements of the analysis and the trade-offs between responsiveness and noise attenuation.

Raises:

  • (ArgumentError) 


180
181
182
183
184
185
186
187
188
189
190
191
192
193
 
# File 'lib/quant/mixins/universal_filters.rb', line 180

def universal_one_pole_low_pass(source, previous:, period:)
  Quant.experimental("This method is unproven and may be incorrect.")
  raise ArgumentError, "source must be a Symbol" unless source.is_a?(Symbol)
  raise ArgumentError, ":previous must be a Symbol" unless previous.is_a?(Symbol)

  alpha = period_to_alpha(period, k: K[:single_pole])
  b0 = alpha
  b1 = 0
  b2 = 0
  a1 = -(1 - alpha)
  a2 = 0

  universal_filter(source, previous:, b0:, b1:, b2:, a1:, a2:)
end

#universal_two_pole_high_pass(source, previous:, period:) ⇒ Object

A two-pole high-pass filter, also known as a second-order high-pass filter, attenuates low-frequency components of a signal while allowing higher-frequency components to pass through. In the context of processing stock price action data, applying a two-pole high-pass filter has several potential effects:

  1. Removal of Low-Frequency Trends: High-pass filtering can remove or attenuate long-term trends or slow-moving components of the price action data. This can be useful for focusing on shorter-term fluctuations or identifying short-term trading opportunities.

  2. Noise Attenuation: By suppressing low-frequency components, a high-pass filter can help in reducing the impact of slow-moving noise or irrelevant signals in the data. This can improve the clarity and interpretability of the price action data.

  3. Noise Amplification: High-pass filters can amplify high-frequency noise present in the data, particularly if the cutoff frequency is set too low. This noise amplification can make it challenging to analyze the data accurately, especially if the noise overwhelms the signal of interest.

  4. Emphasis on Short-Term Variations: By removing low-frequency components, a high-pass filter highlights short-term variations and rapid movements in the price action data. This can be beneficial for traders or analysts who are primarily interested in short-term price dynamics.

  5. Enhanced Responsiveness: Compared to low-pass filters, high-pass filters typically offer greater responsiveness to changes in the input signal. This means that high-pass filtered data can reflect recent price movements more quickly, which may be advantageous for certain trading strategies.

  6. Identification of Market Events: High-pass filtering can help in identifying specific market events or anomalies that occur on shorter time scales, such as intraday price spikes or volatility clusters.

Overall, applying a two-pole high-pass filter to stock price action data can help in focusing on short-term variations and removing long-term trends or slow-moving components. However, it’s essential to carefully select the cutoff frequency to balance noise attenuation with signal preservation, as excessive noise amplification can degrade the quality of the analysis. Additionally, high-pass filtering may not be suitable for all trading or analysis purposes, and its effects should be evaluated in the context of specific goals and strategies.

Raises:

  • (ArgumentError) 


288
289
290
291
292
293
294
295
296
297
298
299
300
 
# File 'lib/quant/mixins/universal_filters.rb', line 288

def universal_two_pole_high_pass(source, previous:, period:)
  raise ArgumentError, "source must be a Symbol" unless source.is_a?(Symbol)
  raise ArgumentError, ":previous must be a Symbol" unless previous.is_a?(Symbol)

  alpha = period_to_alpha(period, k: K[:two_pole_high_pass])
  b0 = (1 - alpha / 2)**2
  b1 = -2 * (1 - alpha / 2)**2
  b2 = (1 - alpha / 2)**2
  a1 = -2 * (1 - alpha)
  a2 = (1 - alpha)**2

  universal_filter(source, previous:, b0:, b1:, b2:, a1:, a2:)
end

#universal_two_pole_low_pass(source, previous:, period:) ⇒ Object

The two-pole low-pass filter can serve several purposes:

  1. Noise Reduction: Stock price data often contains high-frequency fluctuations or noise due to market volatility, algorithmic trading, or other factors. By applying a low-pass filter, you can smooth out these fluctuations, making it easier to identify underlying trends or patterns in the price action.

  2. Trend Identification: Low-pass filtering can help in identifying the underlying trend in stock price movements by removing short-term fluctuations and emphasizing longer-term movements. This can be useful for trend-following strategies or identifying potential trend reversals.

  3. Signal Smoothing: Filtering can help in removing erratic or spurious movements in the price data, providing a clearer and more consistent representation of the overall price action.

  4. Highlighting Structural Changes: Filtering can make structural changes in the price action more apparent by reducing noise and focusing on significant movements. This can be useful for detecting shifts in market sentiment or the emergence of new trends.

  5. Trade Signal Generation: Smoothed price data from a low-pass filter can be used as input for trading strategies, such as moving average-based strategies or momentum strategies, where identifying trends or momentum is crucial.

Raises:

  • (ArgumentError) 


132
133
134
135
136
137
138
139
140
141
142
143
144
145
 
# File 'lib/quant/mixins/universal_filters.rb', line 132

def universal_two_pole_low_pass(source, previous:, period:)
  Quant.experimental("This method is unproven and may be incorrect.")
  raise ArgumentError, "source must be a Symbol" unless source.is_a?(Symbol)
  raise ArgumentError, ":previous must be a Symbol" unless previous.is_a?(Symbol)

  alpha = period_to_alpha(period, k: K[:two_pole_low_pass] )
  b0 = alpha**2
  b1 = 0
  b2 = 0
  a1 = -2.0 * (1.0 - alpha)
  a2 = (1.0 - alpha)**2

  universal_filter(source, previous:, b0:, b1:, b2:, a1:, a2:)
end