The following tables are **Correlation** tests of the
**Default Adaptive Filter** output with **N-day 'future'
returns**, and also the slope of the **2048-day Long-Term
Trend** considered as a **Price Projection**. The
row and column labels in the tables are two different kinds of
**Time Horizon**. The row labels in the first column are
the **Time Horizon** that is set in the **Trading &
Portfolio Parameters** dialog. This is set before each filter
calculation and determines the **Time Horizon** for which
the filter is optimized. The column labels in the first row are the
**Time Horizon** setting in the **Correlation Test
of Adaptive Filter** dialog. These set the **Time Horizon**
of the 'future' returns against which the **correlation**
of the **Adaptive filter** output is tested. We expect
the filter performance to be best for the **Time Horizon**
of 'future' returns for which it has been optimized.

All percentages are **theoretical N-day returns (annualized)**
based on the measured **correlation** and **average
N-day (log) volatility**. The **correlation** is
multiplied by the **average N-day (log) volatility**, then
converted to an actual percentage gain/loss and annualized. In other
words, the **correlation** represents the percent of the
**N-day volatility** that can be *predicted* by
the output of the **Adaptive Filter**, so this predicted
percent of the **volatility** represents a (theoretical) *gain*,
assuming N-day trading. It is just an measure of how well the
**Adaptive Filter** is performing across a variety of time
scales. Note that we define the **volatility** to be
the **average absolute deviation** of the prices.

Note that we are using a *non-standard* definition of
**correlation**, which is better suited to represent
actual gains from trading and investing. Our definition of **
correlation** between the **Adaptive Filter**
output and the **N-day future returns**, where **
N** is the **time Horizon**, over a time period
of **(Correlation Scale)** days, is as follows: The
daily product of the two data sets is taken, *without*
detrending, and then this is divided by the **average absolute
deviation** of the two data sets over the time period
(instead of the **standard deviation**)**.**
This gives a better representation of the gains from using the
**Adaptive Filter** output as a trading indicator, than
the ordinary definition of **correlation** (if the
position is varied in proportion to the **Adaptive Filter**
output).

The **Long-Term Trend** is the return from the
**2048-day Trend Line**, used as a future **Price
Projection**. It does not depend at all on the **Adaptive
Filter**. The **Long-Term Trend** varies slowly
with time, so it is not quite the same as a **Buy & Hold**
strategy. But it serves as a useful benchmark for comparison with the
**Adaptive Filter**.

The **Long-Term Trend** is independent of the
**Adaptive Filter**. It is a **2048-day **
(8 years) trend line. Evidently there has been a **regime change**
in the market for **AAPL **over the past year (256 days)
or so, which has caused the **Long-Term Trend** to lose
predictive power, as can be seen below. The results improve dramatically
when the correlation is taken over a **512-day** (2-year)
**Correlation Scale**. It appears that in general, the
longer time scales give better performance.

The explanation for the negative returns on the **128-day**
(6 months) **Correlation Scale** is the recent downturn
in **AAPL** over this time scale. The overall returns over
this time scale were negative, while the **long-term (2048-day)
returns** were of course positive, hence the negative correlation
over the past **128** days. When the **Time Horizon**
gets up to **64** days or so, it gets past the time period
of this downturn, so the **correlation** becomes positive.
Likewise, on the **512** and **1024** day
**Correlation Scale**, the **correlation**
measurement extends back **512** days (2 years) and
**1024** days (4 years), so this includes a period of a
strong uptrend in **AAPL**. Over those longer periods,
the **correlation** between the **2048-day**
return and the **(Time Horizon)-day** return is positive.
This study also proves that the **long-term (2048-day) returns**,
extended into the future as a **Price Projection**, give
varying degrees of performance, depending on the particular **
regime** or situation for **AAPL** at any given
time.

It seems that the **Adaptive Filter** beats the** Long-Term Trend**
on almost all time scales. The higher the **Correlation Scale**,
which is the time period over which the **correlation**
is measured, the more marked the difference between the two. This
difference is particularly pronounced on the two highest scales, the
**512-day** (2-year) and the **1024-day**
(4-year) scales. This is most encouraging, because it implies that
the **Adaptive Filter** is working over a variety of
different **regimes** of the **AAPL**
price action (on the average).

Note that the **Step Size** parameter was tweaked
precisely for **AAPL**, as shown here. It remains to be
seen whether this exact value for the **Step Size**
parameter is also optimal for other securities, although it should
be, at least approximately. The value of the **Step Size**
parameter that is exactly optimal might depend on the **
volatility** of the individual security or its exact **
correlation** structure, and may also vary with the **
regime**. So this provides future opportunities for tweaking
the **Least Mean Square (LMS)** filter, by implementing
a variable **Step Size** parameter.

AAPL |
128 |
64 |
32 |
16 |
8 |
4 |
2 |
1 |

2048 |
37.8670% | 37.4516% | 13.4451% | -8.9075% | -11.5343% | -13.1639% | -21.0444% | -22.2196% |

AAPL |
128 |
64 |
32 |
16 |
8 |
4 |
2 |
1 |

128 |
36.9862% | 37.1732% | 19.2053% | 2.6884% | 2.6035% | 1.6427% | -3.9170% | -4.3549% |

64 |
37.0775% | 36.3042% | 16.3239% | -1.6098% | -2.2607% | -3.3088% | -9.4199% | -9.9405% |

32 |
37.2337% | 36.4671% | 15.7581% | -3.1671% | -4.3222% | -5.6134% | -12.3234% | -12.9426% |

16 |
37.1088% | 36.1810% | 15.5808% | -3.4056% | -4.5150% | -5.8437% | -12.6507% | -13.2743% |

8 |
37.1282% | 36.2785% | 15.4366% | -4.0032% | -5.1172% | -6.5112% | -13.2068% | -13.6921% |

4 |
37.2107% | 36.3890% | 15.3794% | -4.4143% | -5.5506% | -7.1325% | -14.0367% | -14.3485% |

2 |
37.2327% | 36.3777% | 15.3632% | -4.3691% | -5.5419% | -7.2825% | -14.1199% | -13.9863% |

1 |
37.1608% | 36.2196% | 15.3391% | -3.9096% | -5.0026% | -7.1019% | -13.6639% | -12.4934% |

AAPL |
128 |
64 |
32 |
16 |
8 |
4 |
2 |
1 |

2048 |
31.3510% | 28.7990% | 16.0886% | 4.5429% | 3.0757% | 2.3991% | 1.0495% | 1.9209% |

AAPL |
128 |
64 |
32 |
16 |
8 |
4 |
2 |
1 |

128 |
30.5633% | 28.9386% | 19.2200% | 10.1431% | 9.6281% | 9.1997% | 8.7407% | 10.2151% |

64 |
30.5763% | 28.6442% | 17.8170% | 7.5726% | 6.7570% | 6.2682% | 5.4792% | 6.8107% |

32 |
30.6006% | 28.8037% | 17.6503% | 6.9122% | 5.8611% | 5.2724% | 4.2421% | 5.5070% |

16 |
30.5134% | 28.6207% | 17.4735% | 6.5503% | 5.4196% | 4.7415% | 3.6821% | 4.9563% |

8 |
30.5186% | 28.6968% | 17.4452% | 6.3357% | 5.2724% | 4.4578% | 3.3626% | 4.6452% |

4 |
30.5556% | 28.7988% | 17.4795% | 6.2756% | 5.3114% | 4.4764% | 3.3081% | 4.6885% |

2 |
30.5652% | 28.8083% | 17.4748% | 6.3330% | 5.4156% | 4.6445% | 3.4989% | 5.2141% |

1 |
30.5296% | 28.7123% | 17.3867% | 6.4427% | 5.6635% | 5.0631% | 4.0935% | 6.5378% |

AAPL |
128 |
64 |
32 |
16 |
8 |
4 |
2 |
1 |

2048 |
39.9896% | 38.9814% | 33.9749% | 31.8198% | 31.4070% | 30.4848% | 28.7416% | 28.4944% |

AAPL |
128 |
64 |
32 |
16 |
8 |
4 |
2 |
1 |

128 |
45.6965% | 45.3490% | 44.5642% | 48.6506% | 50.1331% | 48.3311% | 46.8217% | 46.4596% |

64 |
45.7512% | 45.4830% | 44.3930% | 48.0281% | 49.3483% | 47.4710% | 45.8351% | 45.4363% |

32 |
45.7082% | 45.5881% | 44.4527% | 47.8482% | 49.0354% | 47.1156% | 45.3897% | 44.9769% |

16 |
45.5953% | 45.5021% | 44.2886% | 47.5808% | 48.7786% | 46.8786% | 45.1575% | 44.7645% |

8 |
45.5592% | 45.4965% | 44.2414% | 47.3625% | 48.5851% | 46.6506% | 44.9133% | 44.5334% |

4 |
45.5585% | 45.5226% | 44.2722% | 47.3141% | 48.5624% | 46.6212% | 44.8324% | 44.5001% |

2 |
45.5557% | 45.5226% | 44.2776% | 47.3413% | 48.5971% | 46.6973% | 44.9113% | 44.7370% |

1 |
45.5339% | 45.4798% | 44.2214% | 47.3857% | 48.6868% | 46.8975% | 45.1688% | 45.3481% |

AAPL |
128 |
64 |
32 |
16 |
8 |
4 |
2 |
1 |

2048 |
22.5742% | 22.1985% | 19.6553% | 16.8546% | 15.9833% | 15.9585% | 15.3753% | 15.5428% |

AAPL |
128 |
64 |
32 |
16 |
8 |
4 |
2 |
1 |

128 |
29.7380% | 28.1708% | 26.3393% | 24.6365% | 24.0083% | 23.9953% | 23.7699% | 23.9623% |

64 |
30.2877% | 28.8497% | 26.9007% | 24.9940% | 24.3052% | 24.2724% | 23.9960% | 24.1781% |

32 |
30.4020% | 29.0488% | 27.0350% | 25.0414% | 24.3237% | 24.2899% | 23.9877% | 24.1732% |

16 |
30.4361% | 29.1172% | 27.0861% | 25.0207% | 24.3275% | 24.3226% | 24.0364% | 24.2382% |

8 |
30.4252% | 29.1679% | 27.1269% | 24.9380% | 24.2036% | 24.1834% | 23.9197% | 24.1387% |

4 |
30.4156% | 29.2211% | 27.1959% | 24.9556% | 24.1886% | 24.1307% | 23.8633% | 24.1315% |

2 |
30.4147% | 29.2378% | 27.2198% | 24.9951% | 24.2137% | 24.1334% | 23.8820% | 24.2632% |

1 |
30.4152% | 29.2177% | 27.1948% | 25.0350% | 24.2817% | 24.1872% | 24.0057% | 24.6244% |

These are some blank tables for use in future Correlation Tests:

SYMB |
128 |
64 |
32 |
16 |
8 |
4 |
2 |
1 |

2048 |

Default Adaptive Filter (Correl. Scale = 512, Step Size = 0.2)

SYMB |
128 |
64 |
32 |
16 |
8 |
4 |
2 |
1 |

128 |
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64 |
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32 |
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16 |
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8 |
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4 |
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2 |
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1 |

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