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The EKF algorithm is summarized in Table 58 on p 209 As discussed in Section 51053, the EKF is formulated using a complementary lter framework The main assumptions of the Kalman lter derivation were that the system model was linear, driven by Gaussian white noise processes, with measurements corrupted by Gaussian white noise processes The complementary lter approach ensures that these main assumptions are satis ed because the Kalman lter is designed to estimate the error state vector The Kalman lter derivation, properties, implementation issues, and examples are presented in 5 High performance navigation systems require accurate deterministic and stochastic modeling Deterministic modeling requires careful analysis of the relationships between quantities represented in distinct reference frames to determine the proper system model as represented by the functions f and h There are a few issues that motivate the need for stochastic modeling First, the initial conditions are not exactly known, but are conveniently represented by a mean value and an error covariance matrix Second, the sensors for u and y are not perfect, but are a ected by various corrupting factors The imperfections in sensing u may be addressed by additive white process noise or by state augmentation The imperfections in sensing y may be modeled by white measurement noise or by state augmentation (with additional process noise contributions) Through linearization and the state augmentation process, the designer is able to construct a dynamic error model that is linear with white Gaussian measurement and process noise vectors Reference frames, deterministic modeling, and stochastic modeling are discussed in s 2, 3, and 4, respectively The expected system performance and the performance under various design con gurations can be analyzed by the methods in 6

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Sections 112 and 71 outlined the aided navigation system design and analysis methodology This section contains a short subsection that performs each step indicated in Sections 112 and 71 using the tools from s 2 6 The intent is to provide a rather complete presentation of the methodology, subject to design assumptions, for a very simple application This is the same methodology and presentation format that will be used in each of the aided navigation applications considered in Part II The navigation problem that is of interest in this subsection is stated as follows A point is free to translate along a line with position p, velocity v, and acceleration a Two navigation sensors are available: an accelerometer and a position sensor The accelerometer is sampled at the high rate of f1 = 100Hz and is accurately modeled as u = a a bu 1 (73)

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2 where 1 is a Gaussian, white noise process with PSD equal to 1 = 2 2 (m/s2 ) The symbol represents a constant scale factor error 10 3 Hz with (0) N (0, (001)2 ) The accelerometer output signal u contains a bias bu that is modeled as a constant plus random walk process

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What did I discover First, most backtest programs have preprogrammed strategies built into the software For example, you could quickly run tests on the moving average crossover strategy for any time period You can also run tests on many other popular strategies When my friend generated the test on the indicators, he didn t include stop losses, so the results were a bit skewed Although all of the indicators resulted in huge gains, the category Maximum Drawdown was rather interesting Maximum Drawdown refers to the point where you had your worst losses In the depths of a market crash, almost all of the indicators had rather large drawdowns My trading friend explained that because we didn t use stop losses, this was to be expected When most people look at the results of a backtest, they immediately look at the Net Profit category And if you saw the final results of our indicator backtest, you d be impressed But the pros might be more interested in Maximum Drawdown, because to them, large drawdowns are unacceptable There was another tidbit: every single indicator beat the buyand-hold strategy over the three-year period Not surprisingly, however, during the last year of the test, when the market went up almost 70 percent in a year, buy-and-hold did just as well as the indicators In addition to not using stop losses, it took some time to agree on the rules Moving Averages were simple: When the 20day crossed over the 50-day, we bought When the 20-day crossed below the 50-day, we sold It was also easy to create rules for MACD: When MACD crossed above the zero line, we bought When MACD crossed below the zero line, we sold But the rules for RSI, Stochastics, and Bollinger Bands were not as clear Just to experiment, we bought when RSI crossed below 30 and sold when it crossed above 70 For Stochastics, we used 20 and 80, although we were aware these were not designed to be actionable trades In case you were curious, Bollinger Bands had the best results, which was probably because the three-year period was so volatile and Bollinger Bands self-adapt in this type of environment.

(74)

where E bu (0) = bu , var(bu (0)) = Pbu (0) = 0012 The process noise 1 is white, Gaussian, and uncorrelated with p, v, a, and 1 The PSD of 1 3 2 2 (m/s ) 2 is bu = (001) Hz The aiding sensor provides information about the position At discretetime instants, with a sampled rate of f2 = 10 Hz, the sensor output is accurately modeled as yk = pk + by (k) + 2 (k) (75)

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