A Forex Pair Snapshot Chart

After yesterday's Heatmap Plot of Forex Temporal Clustering post I thought I would consolidate all the chart types I have recently created into one easy, snapshot overview type of chart. Below is a typical example of such a chart, this being today's 10 minute EUR_USD forex pair chart up to a few hours after the London session close (the red vertical line).

The top left chart is a Market/Volume Profile Chart with added rolling Value Area upper and lower bounds (the cyan, red and white lines) and also rolling Volume Weighted Average Price with upper and lower standard deviation lines (magenta).

The bottom left chart is the turning point heatmap chart as described in yesterday's post.

The two rightmost charts are also Market/Volume Profile charts, but of my Currency Strength Candlestick Charts based on my Currency Strength Indicator. The upper one is the base currency, i.e. EUR, and the lower is the quote currency. 

The following charts are the same day's charts for:

GBP_USD,

USD_CHF

and finally USD_JPY

The regularity of the turning points is easily seen in the lower lefthand charts although, of course, this is to be expected as they all share the USD as a common currency. However, there are also subtle differences to be seen in the "shadows" of the lighter areas.

For the nearest future my self-assigned task will be to observe the forex pairs, in real time, through the prism of the above style of chart and do some mental paper trading, and perhaps some really small size, discretionary live trading, in additional to my normal routine of research and development.

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Heatmap Plot of Forex Temporal Clustering of Turning Points

Following up on my previous post, below is the chart of the temporal turning points that I have come up with.

This particular example happens to be 10 minute candlesticks over the last two days of the GBP_USD forex pair.

The details I have given about various turning points over the course of my last few posts have been based on identifying the "ix" centre value of turning point clusters. However, for plotting purposes I felt that just displaying these ix values wouldn't be very illuminating. Instead, I have taken the approach of displaying a sort of distribution of turning points per cluster. I would refer readers to my temporal clustering part 3 post wherein there is a coloured histogram of the R output of the clustering algorithm used. What I have done for the heatmap background of the above chart is normalise each separate, coloured histogram by the maximum value within the cluster and then plotted these normalised cluster values using Octave's pcolor function. An extra step taken was to raise the values to the power four just to increase the contrast within and between the sequential histogram backgrounds.

Each normalised histogram has a single value of one, which is shown by the bright yellow vertical lines, one per cluster. This represents the time of day at which, within the cluster window, the greatest number of turns occured in the historical lookback period. The darker green lines show other times within the cluster at which other turns occured.

The hypothesis behind this is that there are certain times of the day when price is more likely to change direction, a turning point, than at other times. Such times are market opens, closes etc. and the above chart is a convenient visual representation of these times. The lighter the backgound, the greater the probability that such a turn will occur, based upon the historical record of such turn timings.

Enjoy!
 

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Temporal Clustering Times on Forex Majors Pairs

In the following code box there are the results from the temporal clustering routine of my last few posts on the four forex majors pairs of EUR_USD, GBP_USD, USD_CHF and USD_JPY.

###### EUR_USD 10 minute bars #######
## In the following order
## Both Delta turning point filter and "normal" TPF combined ##
## Delta turning point filter only ##
## "Normal" turning point filter only

###################### Monday ##############################################
K_opt == 8, ix values == 13  38    63    89     112    135    162    186    ## averaged over all 15 n_bars 1 to 15 inclusive
                         00  4:10  8:20  12:40  16:30  20:20  00:50  4:50
                         
K_opt == 8, ix values == 13 39 64 89 112 135 161 186                        ## averaged over n_bars 1 to 6 inclusive ( upto and include 1 hour )

K_opt == 5, ix_values == 21 60 97 134 175                                   ## averaged over n_bars 7 to 15 inclusive ( over 1 hour )
K == 6,     ix values == 21 59 94 125 158 184

K_opt == 11, ix values == 9 26 43 60 78 95 113 132 151 169 185              ## averaged over all 15 n_bars 1 to 15 inclusive

K_opt == 8, ix values == 13  36  61  86 111 136 161 186                     ## averaged over n_bars 1 to 6 inclusive ( upto and include 1 hour )

K_opt == 8, ix values == 13  34  61  87 110 137 164 187                     ## averaged over n_bars 7 to 15 inclusive ( over 1 hour )

K_opt == 8, ix values == 13  38  63  88 112 137 162 186                     ## averaged over all 15 n_bars 1 to 15 inclusive

K_opt == 10, ix values == 10  31  52  72  91 112 131 150 169 188            ## averaged over n_bars 1 to 6 inclusive ( upto and include 1 hour )

K_opt == 8, ix values == 12  35  62  88 112 137 164 187                     ## averaged over n_bars 7 to 15 inclusive ( over 1 hour )

###################### Tuesday #############################################
K_opt == 6, ix values == 131   169    206   244    283    322               ## averaged over all 15 n_bars 1 to 15 inclusive
                         19:40 02:00  8:10  14:30  21:00  03:30
                         
K_opt == 6, ix values == 131 170 207 245 284 323                            ## averaged over n_bars 1 to 6 inclusive ( upto and include 1 hour )

K_opt == 7, ix values == 131 168 206 243 274 305 330                        ## averaged over n_bars 7 to 15 inclusive ( over 1 hour )

K_opt == 11, ix values == 124 143 164 184 205 226 247 268 289 310 331       ## averaged over all 15 n_bars 1 to 15 inclusive

K_opt == 11, ix values == 124 144 164 185 204 225 246 267 288 309 332       ## averaged over n_bars 1 to 6 inclusive ( upto and include 1 hour ) 

K_opt == 7, ix values = 133 169 206 241 273 304 329                         ## averaged over n_bars 7 to 15 inclusive ( over 1 hour )

K_opt == 9, ix values == 127 152 175 202 228 253 278 305 330                ## averaged over all 15 n_bars 1 to 15 inclusive

K_opt == 9, ix values == 127 152 177 202 228 253 278 304 329                ## averaged over n_bars 1 to 6 inclusive ( upto and include 1 hour )

K_opt == 7, ix values == 132 168 205 242 273 304 329                        ## averaged over n_bars 7 to 15 inclusive ( over 1 hour )

###################### Wednesday ###########################################
K_opt == 6, ix values == 275    312    351    389    426    465             ## averaged over all 15 n_bars 1 to 15 inclusive
                         19:40  01:50  08:20  14:40  20:50  03:20
                         
K_opt == 6, ix values == 275 313 352 391 428 466                            ## averaged over n_bars 1 to 6 inclusive ( upto and include 1 hour )

K_opt == 6, ix values == 274 312 350 389 424 463                            ## averaged over n_bars 7 to 15 inclusive ( over 1 hour )

K_opt == 9, ix values == 272 299 322 347 372 397 422 449 474                ## averaged over all 15 n_bars 1 to 15 inclusive

K_opt == 11, ix values == 268 288 308 329 348 369 390 411 432 453 476       ## averaged over n_bars 1 to 6 inclusive ( upto and include 1 hour )

K_opt == 6, ix values == 275 312 351 388 424 463                            ## averaged over n_bars 7 to 15 inclusive ( over 1 hour )

K_opt == 9, ix values == 272 297 322 348 373 398 423 449 474                ## averaged over all 15 n_bars 1 to 15 inclusive 

K_opt == 9, ix values == 271 297 322 348 373 398 423 448 473                ## averaged over n_bars 1 to 6 inclusive ( upto and include 1 hour )

K_opt == 6, ix values == 276 311 350 389 426 465                            ## averaged over n_bars 7 to 15 inclusive ( over 1 hour )

####################### Thursday ###########################################
K_opt == 6, ix values == 420    457    495    532    570    609             ## averaged over all 15 n_bars 1 to 15 inclusive
                         19:50  02:00  08:20  14:30  20:50  03:20
                         
K_opt == 6, ix values == 420 457 494 531 570 610                            ## averaged over n_bars 1 to 6 inclusive ( upto and include 1 hour )

K_opt == 6, ix values == 420 457 495 532 568 607                            ## averaged over n_bars 7 to 15 inclusive ( over 1 hour )

K_opt == 9, ix values == 416 443 466 492 518 543 568 593 618                ## averaged over all 15 n_bars 1 to 15 inclusive 

K_opt == 10, ix values == 414 437 460 483 506 527 550 573 596 619           ## averaged over n_bars 1 to 6 inclusive ( upto and include 1 hour )

K_opt == 9, ix values == 416 443 466 493 520 543 568 595 618                ## averaged over n_bars 7 to 15 inclusive ( over 1 hour )

K_opt == 9, ix values == 415 440 465 492 518 543 568 593 618                ## averaged over all 15 n_bars 1 to 15 inclusive 

K_opt == 9, ix values ==  415 440 465 492 518 543 568 593 618               ## averaged over n_bars 1 to 6 inclusive ( upto and include 1 hour )

K_opt == 7, ix values == 420 457 494 529 561 592 617                        ## averaged over n_bars 7 to 15 inclusive ( over 1 hour )

####################### Friday #############################################
K_opt == 5, ix values == 564    599    635    670     703                   ## averaged over all 15 n_bars 1 to 15 inclusive
                         19:50  01:40  07:40  13:30   19:00
                         
K_opt == 6, ix values == 563 596 627 654 680 707                            ## averaged over n_bars 1 to 6 inclusive ( upto and include 1 hour ) 
K == 5,     ix values == 564 599 635 668 703

K_opt == 5, ix values == 564 601 639 674 705                                ## averaged over n_bars 7 to 15 inclusive ( over 1 hour )

K_opt == 9, ix values == 556 575 595 614 633 652 672 691 711                ## averaged over all 15 n_bars 1 to 15 inclusive

K_opt == 11, ix values == 554 570 587 602 619 634 651 667 682 698 713       ## averaged over n_bars 1 to 6 inclusive ( upto and include 1 hour )

K_opt == 9, ix values == 556 575 595 614 633 652 671 691 711                ## averaged over n_bars 1 to 6 inclusive ( upto and include 1 hour )

K_opt == 9, ix values == 556 575 596 613 634 652 672 691 711                ## averaged over n_bars 7 to 15 inclusive ( over 1 hour )

K_opt == 9, ix values == 556 575 594 613 633 652 672 691 710                ## averaged over all 15 n_bars 1 to 15 inclusive 

K_opt == 9, ix values == 556 575 594 613 634 653 672 691 710                ## averaged over n_bars 1 to 6 inclusive ( upto and include 1 hour )

K_opt == 5, ix values == 564 600 637 674 705                                ## averaged over n_bars 7 to 15 inclusive ( over 1 hour )

############################################################################

###### GBP_USD 10 minute bars #######
## In the following order
## Both Delta turning point filter and "normal" TPF combined ##

###################### Monday ##############################################
K_opt = 8, ix_values = 13    36    61    86     111    136    162    186    ## averaged over all 15 n_bars 1 to 15 inclusive
                       0:00  3:50  8:00  12:10  16:20  20:30  0:50   4:50

K_opt = 9, ix_values = 12  34  56  78  99 120 141 164 187                   ## averaged over n_bars 1 to 6 inclusive ( upto and include 1 hour ) 

K_opt = 8, ix_values = 12  35  61  86 110 136 163 186                       ## averaged over n_bars 7 to 15 inclusive ( over 1 hour )

###################### Tuesday #############################################
K_opt = 12, ix_values = 124    143    162   180   199   216   235   254   274   293   312   332     ## averaged over all 15 n_bars 1 to 15 inclusive
                        18:30  21:40  0:50  3:50  7:00  9:50  13:00 16:10 19:30 22:40 1:50  5:10

K_opt = 11, ix_values = 124 143 164 185 206 227 248 269 290 311 332         ## averaged over n_bars 1 to 6 inclusive ( upto and include 1 hour )  

K_opt = 9, ix_values = 128 154 177 205 230 254 279 307 330                  ## averaged over n_bars 7 to 15 inclusive ( over 1 hour )

###################### Wednesday ###########################################
K_opt = 11, ix_values = 269   290   311  331  352  373   394   415   434   455   476   ## averaged over all 15 n_bars 1 to 15 inclusive
                        18:40 22:10 1:40 5:00 8:30 12:00 15:30 19:00 22:10 1:40  5:10

K_opt = 11, ix_values = 269 289 310 330 351 372 393 413 434 455 476         ## averaged over n_bars 1 to 6 inclusive ( upto and include 1 hour ) 

K_opt = 8, ix_values = 275 310 341 367 394 422 451 475                      ## averaged over n_bars 7 to 15 inclusive ( over 1 hour )

###################### Thursday ############################################
K_opt = 9, ix_values = 415   440   465  492  517   542   568   594   618    ## averaged over all 15 n_bars 1 to 15 inclusive
                       19:00 23:10 3:20 7:50 12:00 16:10 20:30 0:50  4:50

K_opt = 9, ix_values = 415 440 465 491 517 542 568 593 618                  ## averaged over n_bars 1 to 6 inclusive ( upto and include 1 hour )  

K_opt = 9, ix_values = 416 441 464 492 519 542 569 596 619                  ## averaged over n_bars 7 to 15 inclusive ( over 1 hour )

###################### Friday ##############################################
K_opt = 9, ix_values = 557   576   595  614  633  652   671   690   711     ## averaged over all 15 n_bars 1 to 15 inclusive
                       18:40 21:50 1:00 4:10 7:20 10:30 13:40 16:50 20:20

K_opt = 9, ix_values = 557 576 595 614 633 652 671 691 711                  ## averaged over n_bars 1 to 6 inclusive ( upto and include 1 hour ) 

K_opt = 8, ix_values = 557 576 599 621 642 665 686 709                      ## averaged over n_bars 7 to 15 inclusive ( over 1 hour )

############################################################################

###### USD_CHF 10 minute bars #######
## In the following order
## Both Delta turning point filter and "normal" TPF combined ##

###################### Monday ##############################################
K_opt = 11, ix_values = 8      25    42   61   79    96    113   131   150   169  188   ## averaged over all 15 n_bars 1 to 15 inclusive
                        23:10  2:00  4:50 8:00 11:00 13:50 16:40 19:40 22:50 2:00 5:10

K_opt = 11, ix_values =  9  26  43  60  79  96 114 133 151 170 189          ## averaged over n_bars 1 to 6 inclusive ( upto and include 1 hour ) 

K_opt = 7, ix_values =  13  38  66  99 127 157 184                          ## averaged over n_bars 7 to 15 inclusive ( over 1 hour )

###################### Tuesday #############################################
K_opt = 9, ix_values = 127   152   177  202  228   253   279   306  330     ## averaged over all 15 n_bars 1 to 15 inclusive
                       19:00 23:10 3:20 7:30 11:50 16:00 20:20 0:50 4:50

K_opt = 11, ix_values = 124 144 165 185 204 225 246 267 288 309 331         ## averaged over n_bars 1 to 6 inclusive ( upto and include 1 hour )  

K_opt = 7, ix_values = 133 170 205 240 270 301 328                          ## averaged over n_bars 7 to 15 inclusive ( over 1 hour )

###################### Wednesday ###########################################
K_opt = 10, ix_values = 270   293   316  342  365   388   411   432   454  475  ## averaged over all 15 n_bars 1 to 15 inclusive
                        18:50 22:40 2:30 6:50 10:40 14:30 18:20 21:50 1:30 5:00

K_opt = 12, ix_values = 268 287 308 327 346 365 384 401 420 439 458 477     ## averaged over n_bars 1 to 6 inclusive ( upto and include 1 hour )  

K_opt = 7, ix_values = 276 313 349 383 414 444 471                          ## averaged over n_bars 7 to 15 inclusive ( over 1 hour )

###################### Thursday ############################################
K_opt = 11, ix_values = 413   432   452  471  491  512   533   554   575   598  619  ## averaged over all 15 n_bars 1 to 15 inclusive
                        18:40 21:50 1:10 4:20 7:40 11:10 14:40 18:10 21:40 1:30 5:00

K_opt = 12, ix_values = 412 431 450 469 488 507 526 545 563 582 601 621     ## averaged over n_bars 1 to 6 inclusive ( upto and include 1 hour ) 

K_opt = 9, ix_values = 415 440 463 491 518 543 570 597 619                  ## averaged over n_bars 7 to 15 inclusive ( over 1 hour )

###################### Friday ##############################################
K_opt = 9, ix_values = 557   576   596  615  634  653   672   691   710     ## averaged over all 15 n_bars 1 to 15 inclusive
                       18:40 21:50 1:10 4:20 7:30 10:40 13:50 17:00 20:10

K_opt = 9, ix_values = 556 575 595 614 633 652 671 690 710                  ## averaged over n_bars 1 to 6 inclusive ( upto and include 1 hour ) 

K_opt = 7, ix_values = 558 579 602 629 652 677 705                          ## averaged over n_bars 7 to 15 inclusive ( over 1 hour )                

############################################################################

###### USD_JPY 10 minute bars #######
## In the following order
## Both Delta turning point filter and "normal" TPF combined ##

###################### Monday ##############################################
K_opt = 12, ix_values = 8     24   41   58   73    90    107   124   141   158  173  190  ## averaged over all 15 n_bars 1 to 15 inclusive
                        23:10 1:50 4:40 7:30 10:00 12:50 15:40 18:30 21:20 0:10 2:40 5:30

K_opt = 12, ix_values = 8  24  41  56  73  90 107 124 141 158 173 190       ## averaged over n_bars 1 to 6 inclusive ( upto and include 1 hour )

K_opt = 5, ix_values = 20  60  99 136 175                                   ## averaged over n_bars 7 to 15 inclusive ( over 1 hour )

###################### Tuesday #############################################
K_opt = 9, ix_values = 128   154   179  204  229   254   279   306  331     ## averaged over all 15 n_bars 1 to 15 inclusive
                       19:10 23:30 3:40 7:50 12:00 16:10 20:20 0:50 5:00

K_opt = 9, ix_values = 128 153 178 203 228 254 279 305 330                  ## averaged over n_bars 1 to 6 inclusive ( upto and include 1 hour )

K_opt = 7, ix_values = 133 168 205 240 271 302 329                          ## averaged over n_bars 7 to 15 inclusive ( over 1 hour )

###################### Wednesday ###########################################
K_opt = 11, ix_values = 269   289   310  331  352  373   394   414   433   454  476  ## averaged over all 15 n_bars 1 to 15 inclusive
                        18:40 22:00 1:30 5:00 8:30 12:00 15:30 18:50 22:00 1:30 5:10

K_opt = 9, ix_values = 272 297 322 348 374 399 424 449 474                  ## averaged over n_bars 1 to 6 inclusive ( upto and include 1 hour )

K_opt = 10, ix_values = 269 288 309 331 352 376 398 423 450 475             ## averaged over n_bars 7 to 15 inclusive ( over 1 hour )

###################### Thursday ############################################
K_opt = 9, ix_values = 416   442   467  492  518   543   568   593  618     ## averaged over all 15 n_bars 1 to 15 inclusive
                       19:10 23:30 3:40 7:50 12:10 16:20 20:30 0:40 4:50

K_opt = 12, ix_values = 412 431 450 469 488 507 526 545 564 583 602 621     ## averaged over n_bars 1 to 6 inclusive ( upto and include 1 hour )

K_opt = 7, ix_values = 420 455 492 527 560 591 618                          ## averaged over n_bars 7 to 15 inclusive ( over 1 hour )

###################### Friday ##############################################
K_opt = 7, 8 or 9
ix_values 7 = 561   588   613       638        663   686    709             ## averaged over all 15 n_bars 1 to 15 inclusive
ix_values 8 = 557   578   599  622  643        666   687    710
ix_values 9 = 557   576   596  616  635  653   672   691    711             ## timings are for this bottom row
              18:40 21:50 1:10 4:30 7:40 10:40 13:50 17:00  20:20

K_opt = 8, ix_values = 558 579 600 621 644 665 687 709                      ## averaged over n_bars 1 to 6 inclusive ( upto and include 1 hour )

K_opt = 6, ix_values = 563 594 621 646 676 705                              ## averaged over n_bars 7 to 15 inclusive ( over 1 hour )

############################################################################

This is based on 10 minute bars over the last year or so. Readers should read my last few previous posts for background.

The first set of results, EUR_USD, are what the charts of my previous posts were based on and include combined results of my "Delta Turning Point Filter" and "Normal Turning Point Filter" and the results for each filter separately. Since there doesn't appear to be significant differences between these, the other three pairs' results are the combined filter results only.

The K_opt variable is the optimal number of clusters (see my temporal-clustering-part-3 post for how "optimal" is decided) and the ix_values are also described in this post. For convenience the first set of ix_values per day have the relevant times anotated underneath and therefore it is a simple matter to count forwards/backwards in 10 minute increments to place times to the other ix_values. The variable n_bars is an input to the turning point filter functions and essentially indicates the lookback/lookforward period (n_bar == 2 would mean 2 x 10 minute periods) used for determining a local high/low according to each function's logic.

As to how to interpret this, a typical sequence of times per day might look like this:

18:40 22:00 1:30 5:00 8:30 12:00 15:30 18:50 22:00 1:30 5:10

where the highlighted times represent the BST times for the period covering the London session open to the New York session close for one day. The preceding and following times are the two "book-ending" Asian sessions. 

Close inspection of these results reveals some surprising regularities. In even just the above single example (an actual copy and paste of a code box example) there appear to be definite times per day at which a local high/low occurs. I hopefully will be able to incorporate this into some type of chart for a nice visual presentation of the data. 

More in due course. Enjoy.

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Temporal Clustering on Real Prices, Part 2

Below are some more out of sample plots for the Temporal Clustering solutions of the EUR_USD forex pair for the week just gone. The details of how these solutions are derived is explained in my previous post, Temporal Clustering on Real Prices. First is Tuesday's solution

where the major (blue vertical lines) turns are a combination of optimal K values of 6 and 7 (5 sets of data in total) plus 2 sets of data each for K = 9 and 11 (red and green vertical lines). The price plot is

Next up is Wednesday's solution

where the blue vertical lines represent 5 sets of data with K = 6 and the red and green vertical lines 3 sets and 1 set with K = 9 and K= 11 respectively. The price plot is

Thursday's solution is

where black/blue vertical lines are K values of 9 and 6 respectively, whilst green/red are K values 10 and 7. Thursday's price plot is

Finally, Friday's solution is

where the major blue vertical lines are K = 9 over 5 sets of data, with the remainder being K = 5, 6 and 11 over the last 4 sets of data. Friday's price plot is

The above seems to tie in nicely with my previous post about Forex Intraday Seasonality whereby the above identified turning points signify the end points of said intraday tendencies to trend. Readers might also be interested in another paper I have come across, Segmentation and Time-of-Day Patterns in Foreign Exchange Markets, which gives a possible, theoretical explanation as to why such patterns manifest themselves. In particular, for the EUR_USD pair, the paper states  

• "the US dollar appreciates significantly from 8:00 to 12:00 GMT
  and the euro appreciates significantly from 16:00 to 22:00 GMT"
  

Readers can judge for themselves whether this appears to be true, out of sample, by inspecting the above plots. Enjoy!

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Temporal Clustering on Real Prices

Having now had time to run the code shown in my previous post, Temporal Clustering, part 3, in this post I want to show the results on real prices.

Firstly, I have written two functions in Octave to identify market turning points and each function takes as input an n_bar argument which determines the lookback/lookforward length along price series to determine local relative highs and lows. I ran both these for n_bar values of 1 to 15 inclusive on EUR_USD forex 10 minute bars from July 2012 upto and including last week's set of 10 minute bars. I created 3 sets of turning point data per function by averaging the function outputs over n_bar 1 - 15, 1 - 6 and 7 - 15, and also averaged the outputs over the average of the 2 functions over the same ranges. In total this gives 9 slightly different sets of turning point data.

I then ran the optimal K clustering code, shown in previous posts, over each set of data to get the "solutions" per set of data. Six of the sets had an optimal K value of 8 and a combined plot of these is shown below.

For each "solution" turning point ix (ix ranges from 1 to 198) a turning point value of 1 is added to get a sort of spike train plot through time. The ix = 1 value is 22:00 BST on Sunday and ix = 198 is 06:50 BST on Tuesday. I chose this range so that there would be a buffer at each end of the time range I am really interested in: 7:00 BST to 22:00 BST, which covers the time from the London open to the New York close. The vertical blue lines are plotted for clarity to help identify the the turns and are plotted as 3 consecutive lines 10 minutes apart. The added text shows the time of occurence of the first bar of each triplet of lines, the time being London BST. The following second plot is the same as above but with the other 3 "solutions" of K = 5, 10 and 11 added.

For those readers who are familiar with the Delta Phenomenon the main vertical blue lines could conceptually be thought of as MTD lines with the other lines being lower timeframe ITD lines, but on an intraday scale. However, it is important to bear in mind that this is NOT a Delta solution and therefore rules about numbering, alternating highs and lows and inversions etc. do not apply. It is more helpful to think in terms of probability and see the various spikes/lines as indicating times of the day at which there is a higher probability of price making a local high or low. The size of a move after such a high or low is not indicated, and the timings are only approximate or alternatively represent the centre of a window in which the high or low might occur.

The proof of the pudding is in the eating, however, and the following plots are yesterday's (23 November 2020) out of sample EUR_USD forex pair price action with the lines of the above "solution" overlaid. The first plot is just the K = 8 solution plot

whilst this second plot has all lines shown.

Given the above caveats about caution with regards to the lines only being probabilities, it seems uncanny how accurately the major highs and lows of the day are picked out. I only wish I had done this analysis sooner as then yesterday could have been one of my best trading days ever!

More soon.

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