Under-fitting

A less tempting, but nonetheless dangerous alternative, is to under-fit a model. In this case, the model is not sufficiently complex to account for interfering effects in the analyzer data. As a result, the model can provide inaccurate results even in cases where it is applied to conditions that were used to build it Under-fitting can occur if one is particularly concerned about overfitting, and zealously avoids any added complexity to the model, even if it results in the model explaining more useful information. 

Figure 8.18 provides a graphical explanation of the phenomenon of overfitting and under-fitting, based on the explanation provided by Martens and Naes.1 It shows that the overall prediction error of a model has contributions from two sources (1) the interference error and (2) the estimation error. The interference error continually decreases as the complexity of the calibration model increases, as the added complexity enables the... 

If overfitting and under-fitting are such big problems, then how can one avoid them The most commonly used tools for combating them are called model validation techniques. There are several tools that fall under this category, but they all operate with the same objective attempt to assess the performance of the model when it is applied to data that were not used to build it ... 

In tin. ihapicr. 1 hriel ti < own tv given of the behaviour of material under fit mflurmc ft tadunon, which behaviour, when the radiation < . diort waves ( i ahl or ultta-vtold), is included in the twm As t matter of fait, however, recent advances,... 

Consider the following gaseous sample in a Under fitted with a movable piston. Initially, there... 

Unlike CLS or ILS, which calculate only one model, several PCR models can be calculated by varying the number of factors. In other words, we must decide into how many latent variables (columns of T) we will compress the original matrix R. Compressing R too much (i.e. using too few factors A) will introduce systematic error because the few latent variables will not be able to describe the main variations in the measured data this is termed under-fitting and it will also be commented on in Chapter 5. On the other hand, an unnecessarily large... 

Figure 5.14 Taguchi s loss function to show the trade-off between bias (under-fitting) and variance (over-fitting), see ref. 31 for more mathematical details. Here, k would be the optimum dimensionality of the PLS model.

Under-fitting and over-fitting were discussed in some detail in Section 5.4. There, it was explained that over-fitting is much more likely to occur than under-fitting. The same can be argued for ANN-based models, although one should keep in mind that they are very much prone to over-fitting. In effect, the intrinsic behaviour of the ANNs leads them to predict the (limited number of) calibrators as closely as possible. Therefore, if any unexpected phenomenon appears in the unknown samples we want to predict (e.g. a spectral artefact or a new component), it may well happen that the net does not predict these samples properly. This is in fact also a problem with any classical (univariate) calibration, but it is exacerbated when ANNs are used." ... 

The adsorption isotherms are often Langmuirian in type (under conditions such that multilayer formation is not likely), and in the case of zeolites, both n and b vary with the cation present. At higher pressures, capillary condensation typically occurs, as discussed in the next section. Some N2 isotherms for M41S materials are shown in Fig. XVII-27 they are Langmuirian below P/P of about 0.2. In the case of a microporous carbon (prepared by carbonizing olive pits), the isotherms for He at 4.2 K and for N2 at 77 K were similar and Langmuirlike up to P/P near unity, but were fit to a modified Dubninin-Radushkevich (DR) equation (see Eq. XVII-75) to estimate micropore sizes around 40 A [186]. 

Figure B2.4.8. Relaxation of two of tlie exchanging methyl groups in the TEMPO derivative in figure B2.4.7. The dotted lines show the relaxation of the two methyl signals after a non-selective inversion pulse (a typical experunent). The heavy solid line shows the recovery after the selective inversion of one of the methyl signals. The inverted signal (circles) recovers more quickly, under the combined influence of relaxation and exchange with the non-inverted peak. The signal that was not inverted (squares) shows a characteristic transient. The lines represent a non-linear least-squares fit to the data.

Figure Cl.5.14. Fluorescence images of tliree different single molecules observed under the imaging conditions of figure Cl.5.13. The observed dipole emission patterns (left column) are indicative of the 3D orientation of each molecule. The right-hand column shows the calculated fit to each observed intensity pattern. Molecules 1, 2 and 3 are found to have polar angles of (0,( ))=(4.5°,-24.6°), (-5.3°,51.6°) and (85.4°,-3.9°), respectively. Reprinted with pennission from Bartko and Dickson [148]. Copyright 1999 American Chemical Society.

It has already been pointed out that a liquid even when subjected to simple atmospheric distillation may become superheated and then bump violently in consequence this danger is greatly increased during distillation under reduced pressure and therefore a specially designed flask, known as a Claisen flask, is used to decrease the risk of superheating. In Fig. i2(a) a Claisen flask D is shown, fitted up as part of one of the simplest types of vacuum-distillation apparatus. ... 

For general work, a very satisfactory apparatus for collecting fractions under reduced pressure is the Perkin triangle C, which is shown in Fig. 14, together with the requisite fittings for the complete... 

Fig 23(A) shows an assembly for boiling a liquid under reflux whilst adding another liquid at a rate which can be clearly seen cf. preparation of acetophenone, p. 253). The outlet A allows expansion of the vapour content, and can be fitted with a calcium chloride or soda-lime tube. The outlet A can also be used for collecting a gas evolved during the reaction cf, preparation of acetylene,... 

Add in turn benzyl chloride (8 3 g., 8 o ml.) and powdered thiourea (5 gm.) to 10 ml. of 95% ethanol in a 100 ml. flask fitted with a reflux condenser. Warm the mixture on the water-bath with gentle shaking until the reaction occurs and the effervescence subsides then boil the mixture under reflux for 30 minutes. Cool the clear solution in ice-water, filter off the crystalline deposit of the benzylthiouronium chloride at the pump, wash it with ice-cold ethyl acetate, and dry in a desiccator. Yield, 11-12 g., m.p. 170-174°. The white product is sufficiently pure for use as a reagent. It is very soluble in cold water and ethanol, but can be recrystallised by adding ethanol dropwise to a boiling suspension in ethyl acetate or acetone until a clear solution is just obtained, and then rapidly cooling. 

Add 15 g, of chloroacetic acid to 300 ml. of aqueous ammonia solution d, o-88o) contained in a 750 ml. conical flask. (The manipulation of the concentrated ammonia should preferably be carried out in a fume-cupboard, and great care taken to avoid ammonia fumes.) Cork the flask loosely and set aside overnight at room temperature. Now concentrate the solution to about 30 ml. by distillation under reduced pressure. For this purpose, place the solution in a suitable distilling-flask with some fragments of unglazed porcelain, fit a capillary tube to the neck of the flask, and connect the flask through a water-condenser and receiver to a water-pump then heat the flask carefully on a water-bath. Make the concentrated solution up to 40 ml. by the addition of water, filter, and then add 250 ml. of methanol. Cool the solution in ice-water, stir well, and set aside for ca. I hour, when the precipitation of the glycine will be complete. 

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