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    Avoiding some Dangers of the "MPP 3 Parameter Fits"
    From: Antoine Couëtte
    Date: 2015 Feb 26, 06:09 -0800

    Resent with enclosure

    The "MPP 2 Parameter Fit" (MPP2) statistics (also related further down as first order statistics) are considered to be an excellent added value method to classical celnav data processing under most cases. First order statistics compute a least square position as well as the actual dispersion of the LOP's around such position.

    There also exist more refined statististics (also related further down as higher order statistics) such as the "MPP3 Parameter Fit" (MPP3) which computes a modified position through an "optimized constant offset" introduced into the initial raw data set in order to minimizethe LOP's dispersion. The new position obtained from such "offset data" might all too often be considered as being "superior" to the first position derived from the initial raw data set. This is far from being always the case. In other words, what looks like a Great Idea - as in some cases minimizing the LOP's dispersion really is - might run the Navigator into big trouble if higher order statistics results are blindly relied upon.

    If this topic has earlier been adressed on NavList, I am not aware of it and please be indulgent accordingly.

    If not, let us just consider the following Real World case. And also refer to the enclosed Drawings.

    1 - "SET#1" of 4 LOP's

    SET#1     #1-1: +3.5 NM / 159°        #1-2: +3.0 NM / 171°          #1-3: +2.4 NM / 183°       #1-4: +1.1 NM / 204°

    1.1 - Processing SET#1 through MPP2 yields "Position 1.1" lying into Azimuth 130° at 4.0 NM from DR Position with a Dispersion of Observations equal to 7.4 E-3 NM (0.0 NM actually... and the Observer from the French Merchant Navy has now become a supertanker Captain).

    1.2 - If we further process SET#1 through MPP3 we see that the previously defined "optimized constant offset" is equal to +0.15 NM. All observations subsequently reprocessed after removing such +0.15 NM offset yield a second position - "Position 1.2" - lying into Azimuth 128.16° at 3.90 NM from DR Position with a new LOP's dispersion value exactly equal to 0.0 NM. So far, so good: "Position 1.2" remains very close from "Position 1.1", less than 0.2 NM ... STILL we should be already surprised - if not shocked - at the "huge" ratio between the "optimized constant offset(0.15 NM) compared to the Dispersion of Observations (7.4 E-3 NM): this ration exceeds 20 !!!

    2 - If we introduce some "random noise" into SET#1 - only the intercepts are to be modified and not the Azimuths - we then obtain for example the following SET#2 derived from SET#1:

    SET#2    #2-1: +2.5 NM / 159°        #2-2: +3.5 NM / 171°          #2-3: +2.9 NM / 183°       #2-4: +0.4 NM / 204°

    2.1 - Processing SET#2 through MPP2 yields "Position 2.1" lying into Azimuth 128° at 3.9 NM from DR Position. The Dispersion of Observations equals 0.7 NM, a value which remains quite fair and reasonable.

    Here we can see that first order statistics cope quite well with the noise introduced into our initial data set: "Position 2.1" remains within 0.3 NM from "Position 1.1".

    2.2 - If we further process SET#2 through MPP3we see that the previously defined "optimized constant offset" is equal to ... -18.78 NM ! i.e. over 25 times bigger than the 0.7 NM Dispersion of Observations here-above. All observations reprocessed after removing such -18.78 NM offset yield a second position - "Position 2.2" - lying into Azimuth 174.01° at 22.11 NM from DR Position with a new LOP's dispersion equal to 0.13 NM. It is no longer possible to "squeeze shrink" the Dispersion value obtained here.

    3 - We could study this example in greater depth, and in particular modify SET#1 and SET#2 with one same (known) "constant bias". For example, let's increase each of the SET#2 intercepts by +1.0 NM to simulate a constant 1.0' index error. It gives us SET#3 as follows:

    SET#2    #2-1: +3.5 NM / 159°        #2-2: +4.5 NM / 171°          #2-3: +3.9 NM / 183°       #2-4: +1.4 NM / 204°

    3.1 - MPP2 now yields "Position 3.1" at 139°/DR/4.6 NM which remains quite close (0.9 NM) from "Position 1.1".

    3.2 - And MPP3 does exactly "smoke out" our +1.0' index error. And eventually "Position 3.2" falls back exactly onto "Position 2.2", i.e. at 174.01°/DR/22.11NM.

    4 - LESSONS LEARNT:

    First order statistics - e.g. MPP2 - remain very robust against noise and even against uncorrected systematic constant error/bias (such as constant index error) as long as such constant systematic errors remain within some reasonable range: let's say better than 1 arc minute for index error, a quantity which should stay within Navigator's realistic reach. In that sense Statistics are GREAT !!!

    There are cases when higher order statistics - MPP 3 or equivalent - are overly sensitive to random noise which totally ruin the end-results. In other words, the assumption governing the use of such high order statitics - minimizing the LOP's dispersion through som "optimized constant offset" into the raw data - is not always a good idea.

    Accordingly high order statistics need to be treated with [much] caution and care.

    Hence the following recommended rules:

    First order statistics (e.g. 1 Parameter MPP) should be used as frequently as possible.

    Care, caution and sound judgment is to be exercised when using higher order statistics (e.g. "3 Parameter MPP"). A good and solid rule of thumb to start trusting LOP's results requires the following 2 conditions:

    - Using a sufficient number of observations: 10 to 12 Observations should be an absolute minimum number. AND

    - Observations should be evenly distributed over the whole horizon.

    Ignorance of these simple 2 rules may hit [quite] hard.

    In other words, with higher order statistics, we need to understand what we are doing.

    In real life, I very rarely use 3 Parameter MPP fits, but [almost] every single time 2 Parameter MPP fits.

    Kermit

    Antoine M. "Kermit" Couëtte

       
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