Free lesson · Charts & patterns

Quantitative recognition II: shapelets and constrained dynamic time warping

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• Template: an example sequence. Shapelet: a short sequence used as a discriminative feature. Its distance can become one column in a classifier.
• First use a common documented scale. A raw-price distance mostly measures price level; aggressive per-window normalization can discard economically useful amplitude.
• DTW fills a grid. Each cell adds its local mismatch to the cheapest reachable predecessor: diagonal, up or left.
• Limit timing distortion with a fixed band. A band of zero on equal-length inputs permits only aligned pairs. Also consider run-length or slope restrictions.
• For a shapelet bank, slide each template across the observed window and retain its minimum permitted distance. Record the matching start/end times and template length.
• Discover templates, clusters, normalization and distance thresholds inside the training fold. A prototype selected using a future price continuation leaks its outcome unless the entire example belongs to training.
• A small DTW distance may pair economically different movements. Inspect the alignment path, amplitude, duration and surrounding volatility.
• Keep length comparable or document a score normalization. This lesson uses square root of total squared path cost; it is not aligned RMSE or an average-cost-optimal DTW objective.

Use the rule

• Start with aligned distance and a fixed template bank.
• Tune a small set of bands and lengths using chronological validation.
• Inspect false matches and test whether timing-flexible matching adds value over the aligned baseline.

Before moving on

Draw a tiny cost grid, recover its path and audit the timestamps of every template.

from math import sqrt, inf

def constrained_dtw(x,s,band=1):
    if not x or not s or not isinstance(band,int) or band<0: raise ValueError("Nonempty sequences and integer band required")
    n,m=len(x),len(s)
    d=[[inf]*(m+1) for _ in range(n+1)]; d[0][0]=0
    for i in range(1,n+1):
        for j in range(max(1,i-band),min(m,i+band)+1):
            d[i][j]=(x[i-1]-s[j-1])**2+min(d[i-1][j-1],d[i-1][j],d[i][j-1])
    return sqrt(d[n][m])

def shapelet_feature(observed,template,band=1):
    m=len(template)
    if m==0 or len(observed)<m: raise ValueError("Template must fit the observed window")
    return min(constrained_dtw(observed[k:k+m],template,band) for k in range(len(observed)-m+1))

print(constrained_dtw([0,1,1],[0,0,1],1))