Multitemporal Framing [4b]: Probabilistic Path Construction and Analysis

Formalizing intertemporal connections enables us to assess larger temporal periods for context and smaller temporal periods for developments that may confirm, clarify, or otherwise alter our outlook. Incorporating machine learning enables more robust and sophisticated probabilistic modeling and scaling across many markets. The latter establishes a foundation for structurally driven cross-market dynamics—across indices, sectors, categories, rates, commodities, etc.—as well as broader factor models.

We start with tendencies such as the number of child bars required for a parent to expand the prior parent, shown below broken down by bias type across fourteen ETFs. We then learn what upstream factors—such as regime, related states, and distance—drive different probability distributions relative to certain outcomes. Then, we assess how the expanding information provided from child bars from the in-process parent confirm and then refine that distribution. This enables us to achieve a general bias, which impacts our overall approach for the pending period, while also incorporating how such factors as early prior parent expansion impacts subsequent price movement.

This is more complex than it seems for several reasons:

1. How we interpret developments from the in-process parent is contingent on the contextual assessment.

2. We have limited training data on large temporal periods, and the data that we have is subject to issues of stationarity.

3. Optimization inherently involves sacrificing something dear, which presents a prefrontal vs limbic conflict (or system 2 vs system 1 for Kahneman devotees). This conscious tradeoff is ultimately in the hands of the practitioner. That said, our default bias with a range expansion biased model is “fuzzy” in that we are not attempting to catch every little move but rather gauge the overall direction of expansion (i.e. trend).

For long-term strategic trading

Applications for probabilistic paths vary by market and strategy (and hence target timeframes). For strategic traders factoring in structural context, the conviction and specificity regarding directionality and magnitude will naturally vary based upon the market state. In cases where we have a strong directional bias and have thus already committed to a position and/or strategy, probabilistic paths are used for intra-strategy adjustments and identification of market state transitions at variance with our outlook.

Here, we limit context to the Edge Bias (EB) derived from the relative position of the close within the just-prior parent bar. This is the most tactical contextual input in that it only provides for which boundary of the prior parent bar is likely to be expanded. While crude, it is a reliable indicator for bar/prior bar expansion. This tactical tendency becomes more strategic when incorporated into nested timeframes. It provides a means to gauge momentum (based upon the market’s response), especially when context is fully expressed.

It is not unusual for the edge bias to be misaligned with the contextual bias:

Here is an example of alignment between the prior parent bar edge bias (which is down, albeit not higher probability), the contextual framing, AND the range development signals from the in-process parent bar. We have rendered the prior parent using its child temporal component (month), enabling us to view the three key contextual factors:

1. Trend: There is an established bearish trend with steady downside progression.

2. Counter, At Resistance: The market has bounced and is now at resistance in the form of Mar/Apr support that was lost in Aug/Sept.

3. Active “Telescoping” Domain: A domain is formed when one of more sequential bars fail to expand a range formed by one of more prior bars. While most domains are transitory, larger domains are the primary vehicle for a trending market. In this case, the one bar domain formed by September has been expanded upwards. This leaves the most tactical domain as the 2-bar domain formed Aug-Sep (red rectangle), which has three captive bars (annotated in green numbers) upon December close. The 2-bar domain is a component to a 5-bar domain counting upwards and to the left (component bars annotated with circled blue and 5-bar domain outlines in dotted red line), as well as intermediate 3-bar and 4-bar domains, all of which share the September low. This level also corresponds with the prior parent low set in September (circled with magenta circle).

4. We see the development of a 9-bar overlapping price, which we refer to a rail. The narrowing of the relative range size within the context laid out above—especially the aging against the tactical 2-bar domain signals a buildup of expansion energy.

   

If the contextual bias lacks such conviction, range development on the smaller temporal periods may generate signals that enable us to extrapolate directionality. In some cases, the contextual assessment may present several scenarios that are contingent on developments of the in-process parent bar.

Tactical traders can factor contextual factors as it relates to both smaller but more certain setups as well as opportunistic intermediate terms setups.

Paths

The conventional definition of a path is a designated route through an area. Some paths may be heavily traveled, others less so. There may be intersections where the antagonist (the hiker) must decide upon a route.

In our case, the hiker is charged with traversing the area, without the ability to backtrack. Their instructions are uncertain, but we know the tendencies as well as how they vary by context.

Note that we do not have a single absorbing state—this is not like the walk/bike/train commuter problem. It is not the hiker route that we are seeking to optimize by applying stochastic decision-making to reach a given point (i.e., home).

Rather, given a contextual type, we want to understand likely routes and optimize our response at the hike begins. This accomplishes the objectives laid about earlier.

The Basic Challenge

Constructing market paths requires balancing accuracy with generalization. If we overload our path profile, the result is a unique path for almost every observation—far too rigid. And an overly simplistic approach simply does not give us enough to work with.

We opt to rely on domain expertise to provide a simple, practical approach that captures core elements of range development. The branches to not correspond to every twist and turn but rather represent key state transitions that enable us to gauge the likelihood of continuation versus reversal.

It is also readily extensible to support more data that is situational and can evolve over time. These added data take the form of higher degree paths, which we elaborate on below.

This requires conscious tradeoffs—the objective is, subjective to contextual bias, to be selective (i.e., not overly reactive) to bar-by-bar inputs from smaller timeframes. We also must be explicit with what the model says and what it does not—if the strengths are there, then we can mitigate limitations through an ensemble approach.

Our Approach

We start with the most elemental factors related to intra- and inter-period range expansion, establishing linkage between the two frames by expressing intra-range expansion in terms of inter-period edge bias follow through. This captures the arguably most important temporal dynamics while creating a readily extensible framework as intended.

Each node or branch on the prime (“0-degree”) path represents a state transition that is initiated by a concrete event in the form of a new intra-period Range Push Cycle (RPC). These are designated below with the magenta circles. Each transition rebalances the likelihood of certain outcomes over the balance of the in-process parent periods range development. Higher degree paths incorporate factors such as volatility, counters/tests, rails, and other structural input (see below).

We can visualize where two in-process parent ranges have the same intra-period range development, with both having just moved from 1 RPC to 2. What are the odds that the market continues downward versus inverts, establishing a third RPC? We expect the two scenarios to have materially different probabilities.

Constructing Path Sequences

With this framework in mind, we can now examine how these paths are structured and encoded. We start by mapping out all the possible theoretical sequences of events and outcomes into a logic tree.

Encoding Paths

Each distinct route has a corresponding unique path. As noted in the prior post, we encode this path as a variable length string with the following:

Scenario 1: “E1_S_1”

One of the most common paths is “E1_S_1”:

• the first child bar of the new in-process parent expands the prior parent’s edge bias side (up in the case)

• the first bar to expand that initial range expands in the same direction

• this expansion by definition also results in the prior parent expansion

This path is represented below:

Scenario 1: “E1_V_0_1”

Another common path is “E1_V_0_1”:

• the first child bar of the new in-process parent expands the prior parent’s edge bias side (up in this case).

• the first bar to expand that initial range expands in the opposite direction

• but it fails to expand the opposite side of the parent

• there is a second RPC upwards, which continues the prior parent expansion

This path is represented below:

Adding Historical Tendencies

The basic tree is constructed by using historical data at each branch.

Observations

First, the historical tendencies very by context. The following shows the differences for the monthly prior parent edge bias on the S&P 500:

Second, the distribution of paths varies by context, represented here with a word cloud where the color represents the four basic handoff types for a single path:

This is a vital step, but the net result is to further splinter out already meager dataset, raising questions regarding efficacy on a go-forward basis. This is a central challenge which we address with a multifaceted solution. We have mentioned several in earlier post, but most will be laid out in post 7.

An Incomplete Profile

Probabilistic paths are powerful in that they integrate concrete, quantifiably impactful events that are central to range development. However, we still have work to do as this simple state approach is simply too pixelated for our gradient world. Consider two extreme scenarios:

Identical prime paths, different movement

In this case, both scenarios have the state path and reach the same target. The difference is the depth of the counter in the first scenario. We never know with certainty what the market is going to do. Thus, a counter of this depth is problematic as it is “deep water”, representing a substantial portion of the final period range. It magnifies the risk due to a discontinuous move that does not allow room for an exit without a significant loss.

Different prime paths, similar movement

In this case, we have only a minor distinction: the slightly lower low (at “c”) in the second scenario. As mentioned above, the state of parent/prior parent expansion is material to the outcome scenarios. However, that prior parent expansion has already occurred; thus, how important is the intra-period RPC in adjusting the likely outcome between the two? Does the size of the downward expansion in the second scenario matter? What if a:b move hit the initial target—does it change our interpretation?

And in the case of a second intra-period RPC that expands away from the prior parent edge bias, the branch tells us the historical likelihood of expand the “far edge” (or FE / E2 side), which covers the span from the prior intra-period low to the prior parent low:

But by itself, the likelihood does not factor distance (below left) or time (below right):

Filling In the Gaps

Here, we fill in the obvious missing elements in the prime (0-degree) path framework: time and distance.

It’s About Time…

As a temporal model, time is integral. There is no truly static state due to each time increment altering in only by degree the situational dynamics. What are the implications of compressing this chart horizontally? Stretching it? Do we devalue early events?

Segmentation

Segmentation involves sub-dividing the child bars of the in-process parent into logical groups. This in essence establishes a “virtual” temporal period that serves as an intermediate temporal period. We addressed segmentation in an earlier post, so we will only hit the high points here.

Deciding where to sub-divide can be driven by events such as a monthly roll or (more commonly) where there is a statistically significant tendency. Several methods for sub-dividing month/day are as follows. The first keys on the fact that most months establish their first anchor within the first five days and their second anchor in the last five days. Scenarios 3 and 4 add to those with a monthly roll (e.g. WTI crude):

Segment behavior can be encoded in similar fashion to paths—indeed these are simply most abstract paths. The net effect is to dampen early signals, such as RPC count. The following represent the most common block scenarios of the over 100 potential combinations.

Dynamic Segmentation

In dynamic segmentation, we create a “completed” block that reconstitutes upon the close of each new child bar. This enables us to factor in developments as the occur to weigh the impact on the second block, which encompasses all “pending” bars. This was the traditional method for tracking pair with a low child/parent ratio, such as week/day and year/quarter (see below). However, machine learning automation and probability analysis enables us to scale this approach to high (and variable) ratio pairs as well.

Thus, in the following month/day example, we see a 3-block approach and the corresponding 2-block dynamic range.

…And Distance

Like volatility projections, range projections are a separate branch with many well-developed schools of thought. It’s complicated; indeed, it represents that same predict/react balancing act that is central to what multitemporal framing seeks to help optimize. If we rely solely on recent data, then we can be subjected to high standard deviation moves. And if we generalize over longer periods, then we potentially dilute emerging opportunities.

Simply put, if we think in terms of applied risk—probability x consequence—then range projections are risky propositions. The challenge is magnified for certain (indeed most) asset types. And it is greatly magnified on very short timeframes where we must face sigma magnitudes that are not only large but more importantly rapidly changing.

Because these projections serve a purpose, they must be performed. However, where possible we seek to rely on methods that are not contingent upon them. Among other things, this means preferring hard targets derived from a structure-based context. The implications of this approach vary by strategy, and to some extent mandates applying an ensemble of technical methods.

As is almost always the case, the S&P 500 has been one of the most straight-forward assets for the types of projections. Regime is a critical driver of these projections. Over the years, the market has seen dramatic shifts in range and volatility, especially as it relates to abrupt selloffs. Note that this evolution makes for more range surprises, but it also impacts how we interpret these outliers.

Time Based Inference

[Note: We will expand upon time-based inference is a separate post in this series, so will only touch upon it here.]

As intuition suggests, as a parent temporal period winds down, the expected range to be added diminishes. This reflects two factors: 1) there are fewer child bars left, and 2) the move over the period must cover existing parent range.

It is universal to see a reduction in the potential move as we reduce the “trading days remaining” for a month from 15 days to 5:

These percentages are subject to various identifiable factors, including the in-process parent’s period-to-date range and relative position (%R, see below), parent/prior parent expansion state, and other market states. Scheduled events can obviously distort these outcomes. Their impact is magnified on smaller temporal period, e.g. the last hour of trading a FED Day or the last day of the week for the jobs report.

Pareto Thinking

One value of a structurally driven contextual assessment is the ability to establish an array of targets using domains. As discussed in the earlier example, we do not need to predict precisely what the ultimate range of the in-process year will be—instead, we have concrete targets framing the price action.

Thus, when we discuss situations like this:

“Range Limit Out”

A “range limit out” or RLO occurs when an in-process parent period hits an extreme range. Our reaction is both content and strategy specific. When it occurs in the period is important. If earlier, we can generally assume that there is not much more to end range to obtain for that period. Consequently, we would be inclined to respond (book profits or fade) if we are using a short-term strategy. A longer-term strategy requires more consideration for whether the move implies an inflection or breakout.

The classic argument for responding to RLO is the scenario given earlier—if the in-process period approached the end target, then it is potentially a chance to book profits.

Noting Salient Cases

Factoring range size while maintaining generalization is problematic. While binning is the natural first choice, this approach is not necessarily the best suited in this case. This is due to a “wide middle” of ranges that are in most cases of little actionable value; thus, the splintering effect is a net negative.

Instead, we suggest categorizing salient cases, especially as it relates to early in the temporal period. These include:

1. Fast Intra-Range Expansion (FIRE): A wide high/low range.

2. Slow Intra-Range Expansion (SIRE): A narrow range. We also refer to the this as TUNA for “too narrow”.

3. Normal: All others.

Naturally, these thresholds must be normalized and vary by direction. Ideally, these should be based on change point analysis (meaning a change relative to a threshold binary outcome) versus simply a range above/below a certain value. Certainly, there can gradients and extreme outliers.

Breaking News!!!

News can be a determining factor in both FIRE and SIRE as it can trigger the rapid range expansion and can, in the case of scheduled events, lead to a pause in the market in the lead up. This is a complicated matter worthy of its own series if not textbook. Commodities such as oil are subject to global events, many late breaking and/or inconvenient times. Individual stocks have not only quarterly earnings but also many impactful reports, both for themselves and for related firms. The increase in regulatory and policy-related breaking news makes factoring news an evolving field. Thus, we note its importance but treat it as a placeholder for our purposes here.

Intra-Period Position

The prior parent edge bias is static in that it reflects the bias upon the close of the prior parent. Needless to say, the relative position of the in-process parent is also important. This dynamic tracking runs in parallel with the state tracking. Thus, we are tracking the state relative to parent/prior parent expansion, the original edge bias and close edge/far edge distance, the current distances relative to the prior parent targets, and the intra-parent edge bias and relative distances to its own boundaries (red lines).

As a starting point, we suggest using a simple generalized approach such as whether the in-process edge bias is aligned with the prior parent’s.

Adding Degrees

The completed prime (or 0-degree) path involved the encoding described earlier, along with time and distance factors. The latter two rely on categorization and segmentation (both static and dynamic) to maintain generalization. Higher degree paths include such factors as volatility, rolling ranges, and structural elements (e.g., domains, rails, support & resistance, exhaustion, and significant technical events).

Getting Edgy

If we can recognize when the developing range has “set an edge”, we gain a significant advantage. In the below monthly, what would an April intra-year low tell us? The only range we have for reference is the prior year—not ideal, we it suggests at least a new intra-year high. This gets us the “known known” distance from the current price to the boundary set in January. Beyond that we need more information to get into specifics. Ideally, there is a major move target hidden to the left. Assuming the prior year is indicative of a “normal” year than we can infer additional upside of around 20% of the current range.

Expanding Parent Factors

There is one additional parent dimension worth mentioning, which is the distance moved in each direction from the prior parent close. As with FRITO, we track in terms of the prior parent’s edge bias. Any move in the E1 direction is M1 and any move in the E2 direction is M2. This distance covers both the distance to the boundary as well as the expansion and is measured from the prior bar close, thus it includes discontinuities (gaps).

We see that as a practical matter there are two flavors of one-way E1 expansion: one with only a small move away from the target and one with a large move. We expect to see different proportions by edge bias. We can continue to refine this while still maintaining some degree of generalization. The point is that for each flavor of context, we can identify a range for normalized M2 for each such that if the in-process parent exceeds this value then it is likely to move to the larger range. This is what is often referred to as the barbell. This allows us to build flex to account for some counters at the open of the new parent period.

Conclusion

Probabilistic pathing is a broad topic, and we have limited ourselves here to a high-level overview. Understanding these key aspects makes the underlying probabilistic engine much more comprehensible.