A "New" Use for Kalman Filter on Price Time Series?

During the course of writing this blog I have visited the idea of using Kalman filters several times, most recently in this February 2023 post. My motivation in these previous posts could best be described as trying to smooth price data with as little lag as possible, i.e. create a zero-lag indicator. In doing so, the model most often used for the Kalman filter was a physical motion model with position, velocity and acceleration components. Whilst these "worked" in the sense of smoothing the underlying data, it is not necessarily a good model to use on financial data because, obviously, financial data is not a physical system and so I thought I would apply the Kalman filter to something that is ubiquitously used on financial data - the Exponential moving average.

Below is an Octave function to calculate a "Kalman_ema" where the prediction part of the filter is just a linear extrapolation of an exponential moving average and then this extrapolated value is used to calculate the projected price, with the measurement being the real price and its ema value.

## Copyright (C) 2024 dekalog
##
## This program is free software: you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation, either version 3 of the License, or
## (at your option) any later version.
##
## This program is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with this program.  If not, see .

## -*- texinfo -*-
## @deftypefn {} {@var{retval} =} kalman_ema (@var{price}, @var{lookback})
##
## @seealso{}
## @end deftypefn

## Author: dekalog 
## Created: 2024-04-08

function [ filter_out , P_out ] = kalman_ema ( price , lookback )

## check price is row vector
if ( size( price , 1 ) > 1 && size( price , 2 ) == 1 )
 price = price' ;
endif

P_out = zeros( numel( price ) , 2 ) ;

ema_price = ema( price , lookback )' ;
alpha = 2 / ( lookback + 1 ) ;

## initial covariance matrix
P = eye( 3 ) ;

## transistion matrix
A = [ 0 , ( 1 + alpha ) / alpha , -( 1 / alpha ) ; ...
       0 , 2 , -1 ; ...
       0 , 1 , 0 ] ;

## initial Q
Q = eye( 3 ) ;

## measurement vector
Y = [ price ; ...
       ema_price ; ...
       shift( ema_price , 1 ) ] ;
Y( 3 , 1 ) = Y( 2 , 1 ) ;

## measurement matrix
H = eye( 3 ) ;

## measurement noise covariance
R = eye( 3 ) ;

## container for Kalman filter output
filter_out = zeros( size( Y ) ) ;
errors = ones( 3 , 1 ) ;

for ii = 2 : size( Y , 2 )

  X = A * filter_out( : , ii - 1 ) ;
  P = A * P * A' + Q ;

  errors = alpha .* abs( X - Y( : , ii ) ) + ( 1 - alpha ) .* errors ;

  IM = H * X ; ## Mean of predictive distribution
  IS = ( R + H * P * H' ) ; ## Covariance of predictive mean
  K = P * H' / IS ; ## Computed Kalman gain
  X = X + K * ( Y( : , ii ) - IM ) ; ## Updated state mean
  P = P - K * IS * K' ; ## Updated state covariance

  filter_out( : , ii ) = X ;
  P_out( ii , 1 ) = P( 1 , 1 ) ;
  P_out( ii , 2 ) = P( 2 , 2 ) ;

  ## update Q and R
  Q( 1 , 1 ) = errors( 1 ) ; Q( 2 , 2 ) = errors( 2 ) ; Q( 3 , 3 ) = errors( 3 ) ;
  R = Q ;

endfor

filter_out = filter_out' ;

endfunction

Using this for smoothing either the price or the ema has no utility, but a by-product of the filter is the Covariance matrix from which it is possible to plot bands around the price series. The following chart shows the bands for various ema alpha values corresponding to various Fibonacci sequence length look backs for the ema alpha value.  

This next chart, for purposes of clarity, shows the "Golden Cross" lengths of 50 and 200

and this final chart shows an adaptive look back length which is a function of the instantaneous measured period (see here or here) of the underlying data.

Despite the wide range of the input look back lengths for the ema, it can be seen that the covariance bands around price are broadly similar. I can think of many uses for such a data driven but basically parameter insensitive measure of price variance, e.g. entry/exit levels, stop levels, position sizing etc.

More in due course.

Standard "Volume based" Indicators Replaced with PositionBook Data

In my previous post I suggested three different approaches to using PositionBook data other than directly using this data to create new, unique indicators. This post is about the first of the aforementioned ideas: modifying existing indicators that somehow incorporate volume in their construction.

The indicators I've chosen to look at are the Accumulation and Distribution index, On Balance Volume, Money Flow index, Price Volume Trend and, for comparative purposes, an indicator similar to these utilising PositionBook data. For illustrative purposes these indicators have been calculated on this OHLC data,

which shows a 20 minute bar chart of the EUR_USD forex pair. The chart starts at the New York close of 4 January 2024 and ends at the New York close on 5 January 2024. The green vertical lines span 7am to 9am London time and the red lines are 7am to 9am New York time. This second chart shows the indicators individually max-min scaled from zero to one so that they can be more easily visually compared.

 

As in the OHLC chart, the vertical lines are the London and New York opening sessions. The four "traditional" indicators more or less tell the same story, which is not surprising since their calculations involve bar to bar price changes or open to close intra-bar changes which are then multiplied by bar volume. Effectively they are all just differently scaled versions of the underlying price movement, or alternatively, just accumulated sums of different fractions of the bar volume. The PositionBook data version, called Pos Change Ind, does not use any OHLC information at all but rather uses the accumulated difference(s) between position changes multiplied by volume. For most of the day the general story told by the Pos Change Ind indicator agrees with the other indicators; however during the big run up which started just about 9am New York time there is a significant difference between Pos Change Ind and the others.

In hindsight, by looking at my order levels chart

and volume profile chart

 

it is easy to speculate about what market participants were thinking during this trading day, especially if the following PositionBook chart is taken into account.

 

For the purpose of the following brief "stream of consciousness" narrative imagine it's 7am New York time and looking back at the day's action so far it can be seen that the downward drift of the day seems to have halted with a mini double bottom, and we are now moving up with some new heavy tick volumes having accumulated over the last hour or so, forming a new volume profile point of control (POC). Over the next hour prices continue the new slight drift up with accumulating long positions and at about 8.30am we see a doji bar form on the 10 minute chart at the level of the rolling vwap for the day. Suddenly there is the big down bar, which could conceivably be a shake-out of the recently added longs, targeting the stop orders below, which finishes with an extended lower wick. This seems to be an ideal set-up for a long trade targeting either the old POC, which also happens to be the current high of the day, or the accumulated orders which happen to coincide with the level at which, currently, the greatest proportion of long positions have been entered. 

 

Of course it can seen, in hindsight, that this was a great set-up for an intra-day trade that could have caught almost the entire high-low range of the day as a profitable trade, dependent of course on exact entry and exit levels. This set-up is a synthesis of observations from the volume profile chart, the order levels chart and the position levels chart, along with the vwap indicator. The Pos Change Ind indicator does not seem to add much value over that provided by the more traditional, volume based indicators in the set-up phase.

This is not necessarily the case for the exit. It can be seen that the Pos Change Ind indicator turns down sharply several bars before all the other indicators, and this movement in the indicator is evident by the close of the bar with the long upper wick which makes the high of the day. This sharp downturn in the indicator shows that there was a mass exit of longs during the formation of this bar, made clearer by the following chart which shows the two components of the Pos Change Indicator, namely the

 

"Outside Change" and the "Inside Change." The outside change shows the total net position changes for the price levels that lie outside the range of the bar and the inside change is the net change for price levels that lie within the bar range. The greater change of the two is obviously the (red) inside change, and looking at the position levels plot we can see why. The previously mentioned level of "greatest proportion of long positions" suddenly loses that distinction - a large number of the longs at this level obviously liquidated their positions. This is important information, which shows that sentiment favouring long positions obviously changed, and it can be surmised that many long position holders were happy to get out of their trade at more or less break even prices. Also noticeable in the position levels chart is the change in blue shade from darker to lighter at the price levels within the range of the large price run-up. This reduction in colour intensity shows that those traders who entered during the run-up also exited near the top of the move. Taken together these observations could have been used as a nice short set-up targeting, for example, the then currently lower level of vwap, which in fact was subsequently hit with the day closing at this level.

As I had previously suspected, there is value in PositionBook data but it is perhaps tricky to operationalise or to easily automate within a trading system. It can be used to indicate a directional bias, or as above to show when traders exit positions. Again, as shown above, it can be used to put a new, useful twist on existing indicators, but in general it appears that use of this data is primarily visual by way of my PositionBook chart and subsequent, subjective evaluation. Whilst I am pleased with the potential insights provided, I would prefer a more structured, algorithmic use of this data, as in the third point of my previous post.

 

More in due course.

Indicator(s) Derived from PositionBook Data

Since my last post I have been trying to create new indicators from PositionBook data but unfortunately I have had no luck in doing so. I have have tried differences, ratios, cumulative sums, logs and control charts to no avail and I have decided to discontinue this line of investigation because it doesn't seem to hold much promise. The only other direct uses I can think of for this data are:

• modifying existing indicators that use volume such as Accumulation/distribution index, On balance volume and Money flow index. The idea would be to use the PositionBook data relationships instead of the price bar relationships to calculate the intermediate steps of CLV and Money Ratio.
• create a sort of PositionBook profile chart, similar to a volume profile chart. However, I suspect that this would be redundant as the high/low PositionBook nodes would be almost identical to the high/low volume nodes.
• directly use the PositionBook data as input to a machine learning model, either as a stand alone "indicator" or in a Meta learning paradigm as outlined in Advances in Financial Machine Learning.

I am not yet sure which of the above I will look at next, but whichever it is will be the subject of a future post.

Judging the Quality of Indicators.

In my previous post I said I was trying to develop new indicators from the results of my new PositionBook optimisation routine. In doing so, I need to have a methodology for judging the quality of the indicator(s). In the past I created a Data-Snooping-Tests-GitHub which contains some tests for statistical significance testing and which, of course, can be used on these new indicators. Additionally, for many years I have had a link to tssb on this blog from where a free testing program, VarScreen, and its associated manual are available. Timothy Masters also has a book, Testing and Tuning Market Trading Systems, wherein there is C++ code for an Entropy test, an Octave compiled .oct version of which is shown in the following code box.

#include "octave oct.h"
#include "octave dcolvector.h"
#include "cmath"
#include "algorithm"

DEFUN_DLD ( entropy, args, nargout,
"-*- texinfo -*-\n\
@deftypefn {Function File} {entropy_value =} entropy (@var{input_vector,nbins})\n\
This function takes an input vector and nbins and calculates\n\
the entropy of the input_vector. This input_vector will usually\n\
be an indicator for which we want the entropy value. This value ranges\n\
from 0 to 1 and a minimum value of 0.5 is recommended. Less than 0.1 is\n\
serious and should be addressed. If nbins is not supplied, a default value\n\
of 20 is used. If the input_vector length is < 50, an error will be thrown.\n\
@end deftypefn" )

{
octave_value_list retval_list ;
int nargin = args.length () ;
int nbins , k ;
double entropy , factor , p , sum ;

// check the input arguments
if ( args(0).length () < 50 )
   {
   error ("Invalid 1st argument length. Input is a vector of length >= 50.") ;
   return retval_list ;
   }

if ( nargin == 1 )
   {
   nbins = 20 ;
   }

if ( nargin == 2 )
   {
   nbins = args(1).int_value() ;
   }
// end of input checking

ColumnVector input = args(0).column_vector_value () ;
ColumnVector count( nbins ) ;
double max_val = *std::max_element( &input(0), &input( args(0).length () - 1 ) ) ;
double min_val = *std::min_element( &input(0), &input( args(0).length () - 1 ) ) ;
factor = ( nbins - 1.e-10 ) / ( max_val - min_val + 1.e-60 ) ;

for ( octave_idx_type ii ( 0 ) ; ii < args(0).length () ; ii++ ) {
      k = ( int ) ( factor * ( input( ii ) - min_val ) ) ;
      ++count( k ) ; }

sum = 0.0 ;
for ( octave_idx_type ii ( 0 ) ; ii < nbins ; ii++ ) {
      if ( count( ii ) ) {
         p = ( double ) count( ii ) / args(0).length () ;
        sum += p * log ( p ) ; }
        }

entropy = -sum / log ( (double) nbins ) ;

retval_list( 0 ) = entropy ;

return retval_list ;

} // end of function

This calculates the information content, Entropy_(information_theory), of any indicator, the value for which ranges from 0 to 1, with a value of 1 being ideal. Masters suggests that a minimum value of 0.5 is acceptable for indicators and also suggests ways in which the calculation of any indicator can be adjusted to improve its entropy value. By way of example, below is a plot of an "ideal" (blue) indicator, which has values uniformly spread across its range

with an entropy value of 0.9998. This second plot shows a "good" indicator, which has an

entropy value of 0.7781 and is in fact just random, normally distributed values with a mean of 0 and variance 1. In both plots, the red indicators fail to meet the recommended minimum value, both having entropy values of 0.2314.

It is visually intuitive that in both plots the blue indicators convey more information than the red ones. In creating my new PositionBook indicators I intend to construct them in such a way as to maximise their entropy before I progress to some of the above mentioned tests.

Update to PositionBook Chart - Revised Optimisation Method

Just over a year ago I previewed a new chart type which I called a "PositionBook Chart" and gave examples in this post and this one. These first examples were based on an optimisation routine over 6 variables using Octave's fminunc function, an unconstrained minimisation routine. However, I was not 100% convinced that the model I was using for the loss/cost function was realistic, and so since the above posts I have been further testing different models to see if I could come up with a more satisfactory model and optimisation routine. The comparison between the original model and the better, newer model I have selected is indicated in the following animated GIF, which shows the last few day's action in the GBPUSD forex pair. 

The old model is figure(200), with the darker blue "blob" of positions accumulated at the lower, beginning of the chart, and the newer model, figure(900), shows accumulation throughout the uptrend. The reasons I prefer this newer model are:

• 4 of the 6 variables mentioned above (longs above and below price bar range, and shorts above and below price bar range) are theoretically linked to each other to preserve their mutual relationships and jointly minimised over a single input to the loss/cost function, which has a bounded upper and lower limit. This means I can use Octave's fminbnd function instead of fminunc. The minimisation objective is the minimum absolute change in positions outside the price bar range, which has a real world relevance as compared to the mean squared error of the fminunc cost function.
• because fminunc is "unconstrained" occasionally it would converge to unrealistic solutions with respect to position changes outside the price bar range. This does not happen with the new routine.
• once the results of fminbnd are obtained, it is possible to mathematically calculate the position changes within the price bar range exactly, without needing to resort to any optimisation routine. This gives a zero error for the change which is arguably the most important.
• the results from the new routine seem to be more stable in that indicators I am trying to create from them are noticeably less erratic and confusing than those created from fminunc results.
• finally, fminbnd over 1 variable is much quicker to converge than fminunc over 6 variables.
  

The second last mentioned point, derived indicators, will be the subject of my next post.

Currency Strength Revisited

Recently I responded to a Quantitative Finance forum question here, where I invited the questioner to peruse certain posts on this blog. Apparently the posts do not provide enough information to fully answer the question (my bad) and therefore this post provides what I think will suffice as a full and complete reply, although perhaps not scientifically rigorous.

The original question asked was "Is it possible to separate or decouple the two currencies in a trading pair?" and I believe what I have previously described as a "currency strength indicator" does precisely this (blog search term ---> https://dekalogblog.blogspot.com/search?q=currency+strength+indicator). This post outlines the rationale behind my approach.

Take, for example, the GBPUSD forex pair, and further give it a current (imaginary) value of 1.2500. What does this mean? Of course it means 1 GBP will currently buy you 1.25 USD, or alternatively 1 USD will buy you 1/1.25 = 0.8 GBP. Now rather than write GBPUSD let's express GBPUSD as a ratio thus:- GBP/USD, which expresses the idea of "how many USD are there in a GBP?" in the same way that 9/3 shows how many 3s there are in 9. Now let's imagine at some time period later there is a new pair value, a lower case "gbp/usd" where we can write the relationship

                    (1)     ( GBP / USD ) * ( G / U ) = gbp / usd

to show the change over the time period in question. The ( G / U ) term is a multiplicative term to show the change in value from old GBP/USD 1.2500 to say new value gbp/usd of 1.2600, 

e.g.                ( G / U ) == ( gbp / usd ) / ( GBP / USD ) == 1.26 / 1.25 == 1.008

from which it is clear that the forex pair has increased by 0.8% in value over this time period. Now, if we imagine that over this time period the underlying, real value of USD has remained unchanged this is equivalent to setting the value U in ( G / U ) to exactly 1, thereby implying that the 0.8% increase in the forex pair value is entirely attributable to a 0.8% increase in the underlying, real value of GBP, i.e. G == 1.008. Alternatively, we can assume that the value of GBP remains unchanged,

 e.g.                G == 1, which means that U == 1 / 1.008 == 0.9921

which implies that a ( 1 - 0.9921 ) == 0.79% decrease in USD value is responsible for the 0.8% increase in the pair quote.

Of course, given only equation (1) it is impossible to solve for G and U as either can be arbitrarily set to any number greater than zero and then be compensated for by setting the other number such that the constant ( G / U ) will match the required constant to account for the change in the pair value.

However, now let's introduce two other forex pairs (2) and (3) and thus we have:-

                    (1)     ( GBP / USD ) * ( G / U ) = gbp / usd

                    (2)     ( EUR / USD ) * ( E / U ) = eur / usd

                    (3)     ( EUR / GBP ) * ( E / G ) = eur / gbp

We now have three equations and three unknowns, namely G, E and U, and so this system of equations could be laboriously, mathematically solved by substitution. 

However, in my currency strength indicator I have taken a different approach. Instead of solving mathematically I have written an error function which takes as arguments a list of G, E, U, ... etc. for all currency multipliers relevant to all the forex quotes I have access to, approximately 47 various crosses which themselves are inputs to the error function, and this function is supplied to Octave's fminunc function to simultaneously solve for all G, E, U, ... etc. given all forex market quotes. The initial starting values for all G, E, U, ... etc. are 1, implying no change in values across the market. These starting values consistently converge to the same final values for G, E, U, ... etc for each separate period's optimisation iterations.

Having got all G, E, U, ... etc. what can be done? Well, taking G for example, we can write

                    (4)     GBP * G = gbp

for the underlying, real change in the value of GBP. Dividing each side of (4) by GBP and taking logs we get

                    (5)     log( G ) = log( gbp / GBP )

i.e. the log of the fminunc returned value for the multiplicative constant G is the equivalent of the log return of GBP independent of all other currencies, or as the original forum question asked, the (change in) value of GBP separated or decoupled the from the pair in which it is quoted.

Of course, having the individual log returns of separated or decoupled currencies, there are many things that can be done with them, such as:-

• create indices for each currency
• apply technical analysis to these separate indices
• intermarket currency analysis
• input to machine learning (ML) models
• possibly create new and unique currency indicators

Examples of the creation of "alternative price charts" and indices are shown below

where the black line is the actual 10 minute closing prices of GBPUSD over the last week (13th to 18th August) with the corresponding GBP price (blue line) being the "alternative" GBPUSD chart if U is held at 1 in the ( G / U ) term and G allowed to be its derived, optimised value, and the USD price (red line) being the alternative chart if G is held at 1 and U allowed to be its derived, optimised value.

This second chart shows a more "traditional" index like chart

where the starting values are 1 and both the G and U values take their derived values. As can be seen, over the week there was upwards momentum in both the GBP and USD, with the greater momentum being in the GBP resulting in a higher GBPUSD quote at the end of the week. If, in the second chart the blue GBP line had been flat at a value of 1 all week, the upwards momentum in USD would have resulted in a lower week ending quoted value of GBPUSD, as seen in the red USD line in the first chart. Having access to these real, decoupled returns allows one to see through the given, quoted forex prices in the manner of viewing the market as though through X-ray vision. 

I hope readers find this post enlightening, and if you find some other uses for this idea, I would be interested in hearing how you use it.
 

Quick Update on Kalman Filter and Sensor Fusion

Managed to code it up and get it working, but at the end of the day I couldn't see any value added over just averaging the output of the indicators I was trying to fuse together via Kalman filtering. As a result, I'm giving up on this for now and looking at other things.

More in due course.