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9.5 Multivariate analysis
9.5 Multivariate analysis
To further enhance the differences found in the preliminary spectra comparison, in-order to provide an assessment of the frying treatment, various multivariate analyses are applied.
9.5.1 Principal Component Analysis
Recalling the principal component analysis (PCA) it will extract the patterns in the images, accounting for the largest part of the variation. The PCA was performed on the pre-processed images, to insure that it only takes the variation introduced by the meat applied with different heat treatment into account.
It was found that the first two PCA components accounts for 96.7% of the total variance (85,26% and 11,67% respectively), examining the lower components, accounting for very small amounts of variation, they mainly show noise and are therefore not examined further. To examine the first two components further, histograms of the pre-processed and transformed images are plotted.
a) b)
Figure 9.6 – a) Histogram curves PCA component 1, b) Histogram curves PCA component 2
From the histograms it is observed that different heat treatments results in different displacements of the top of the histogram curves. This displacement is generally more apparent in first component than in the second, but cannot be used directly from any of the components since the variation is very small. Instead one can use the combination of the two components to investigate the results further. To examine the combination the mean of the first and second component for each frying degree is plotted in a scatter plot.
The scatter plot is given in Figure 9.7. The plot shows two groupings of observations, which almost corresponds to the under- and adequately-processed division of meat samples however with some exceptions. To further enhance these groupings and their similarity to the under/adequate treatment classes, the border line between the under- and adequately processed observations is calculated and plotted using their classes discriminant functions. The
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border line shows that the top right grouping, corresponds to the under-processed meat, with the exceptions of two measurements namely 250oC 120[s] and 225oC 240[s], and the bottom left grouping corresponds to the adequately-processed meat.
Figure 9.7 - PCA1 and PCA2 scatter plot
Generally it seams like it is possible to do an assessment of the heat treatment using PCA, however it does not seam completely accurate. The inaccuracy is not only observed in the scatter plot, but also the histograms plotted since they show little division between the different frying degrees.
9.5.2 Canonical Discriminant Analysis
In addition to the PCA, a canonical discriminant analysis is also applied to the images to see if it is able to separate the classes better than the PCA.
The images were preprocessed as described in 9.2, and divided into the classes described in Table 9.1. The canonical discriminant analysis was then performed, deriving the optimal linear combination of the 18 bands separating the data into the two processing classes.
To examine the separation of the data, a histogram curve for a transformed image from each frying degree is derived and plotted in Figure 9.8.
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Figure 9.8 - Histogram – CDA
The histogram curves show a good separation of the different frying degrees, based on the top of the histogram curves. The under-processed samples seam to have their tops from 1 and down, whereas the adequately-processed samples seam to have their tops from 1 and up. The curves however seams to be somewhat wider, than the ones derived from the PCA. The wider curves indicate the image contains a variety of different frying degree, having such a range of different frying degrees seams inevitable in a process like this.
From the projections of the first CDF it seams like CDA is able to separate the frying degree using only one projection, and therefore it is decided to continue the heat treatment assessment using CDA.
Examining the derived linear combination, also called the canonical discriminant function, gives an impression of which bands are the most important in separating the frying degrees.
Figure 9.9 - Loadings Canonical Discriminant Function
The loadings of the CDF show that the most influential bands to the CDF seams to be 3, 10, 17 and 18. This is in accordance with the preliminary spectrum analysis, in which it was
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concluded that higher values in the upper bands implies longer heat treatment, also it was found that most of the lower bands had little or no effect on the heat treatment.
9.5.3 The Frying-Treatment Score
As concluded in the prior section, the CDF derived from the CDA can be used to give an assessment for the frying degree in the images. The next step it to define a measure of the frying treatment based on that linear combination.
The measure of frying treatment will be denoted the Frying-Treatment Score and abbreviated FTS. Recalling the CDF function, the results of applying it to a multi-spectral image is a projection of the 18 bands, thereby essentially creating a grayscale image. The grayscale images can be compared, and one will find an intensity difference between the meats at different frying degrees. However since we, for now, are not interested in a visual inspection of the meat, but rather a measure for the entire image, it is decided that the FTS for minced beef is to be defined as:
The Frying-Treatment Score (FTS) for a multi-spectral image containing minced beef, is the mean value of the pixels in the pre-processed image, containing only meat, projected with the CDF derived in 9.5.2
Having this definition of the FTS for minced beef images, it is now possible to plot the scale of the FTS. Meaning plotting the FTS for the various images, thus giving an impression of how the FTS is distributed. Furthermore using the values from the images, it is possible to examine from which FTS value the images are categorized as adequately-processed, this is simply the mean value of the two groups (under and adequately processed) mean values. This cut-off point is found to be at a FTS value of 0.95. The mean of the three sub-sample images for each frying degree, and the cut-off point is plotted on the Frying-Treatment Score scale in Figure 9.10.
Figure 9.10 - Frying-Treatment Score - Minced Meat
From the points on the scale, one observes that the no samples seams to be placed on the wrong side of the cut-off point. Furthermore it is observed that the Frying-Treatment Score seams to be increasing along with the frying treatment. Having defined the FTS the next parts will show some applications of use.
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9.5.4 Regression analysis
To investigate the relation between temperature, frying time and the FTS factor, the relation is fitted using least square regression. The model is created such that it gives an estimated time based on the measured FTS and frying temperature, this is done since this will give an estimation of a value for which the ground truth is known.
To find the optimal degree of the regression model a 3-fold cross validation is used, dividing the dataset in three subsets. This is done by having one value for each combination of time and temperature in each subset; this is possible since triple determination was used when acquiring the images.
From the cross validation the root mean square error and the R2 value is determined; this can be used to select the appropriate model. The results are shown in Table 9.4.
Polynomial degree RMSETest RMSETrain R2
1 44.78 44.71 0.00
2 39.58 32.97 0.43
3 30.97 26.85 0.62
4 120.34 24.10 0.67
5 122.51 23.57 0.68
Table 9.4 - Cross Validation Results
From the validation results of the various models, it can be concluded that the optimal relationships is the cubic relationship. Furthermore the results show that the cubic relation accounts for 62% of the variance in time. From the cubic relation contours are drawn as shown in Figure 9.11.
Figure 9.11 – ISO lines frying time
The inaccuracy which occurs in regression clearly shows in the contours drawn. For example are the contours suggesting that a high frying treatment score can be achieved at 200oC using a
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relatively short frying time, this is of-course incorrect and suggest that the top-left-most part of the plot is invalid.
As the modeling of frying time suggests the optimal relation is cubic, it can be assumed that this is also the case for modeling the FTS based on time and temperature. By doing so the parameters are estimated with a goodness of fit of R2 = 0.65, meaning 65 percent of the variance in FTS is accounted for by the temperature and frying time. This is an acceptable result, but it also shows that factors beyond the time and temperature have a significant impact on the FTS. Some of these effects can be the known varying quality parameters of minced beef, an example is the fat percent in minced meat, in [22] it is found that in one batch (from the same wholesale supplier as used for this experiment) the fat percentage can vary from 9%
to 14% in meat said to contain 15-18%.
Having estimated the parameters for the polynomial using regression, it can now be used to further model the relationship between frying time, temperature and frying-treatment. This is done by deriving the contour lines for the FTS at various interesting FTS values. The contours are plotted in Figure 9.12.
Figure 9.12 - FTS Contours, Time vs. Temperature
The model of FTS values implies that meat prepared at 120[sec] or less regardless of temperature (from 200oC to 250oC) does not seam to reach the frying degree of adequate-processed meat, which to some extend can be a fair approximation for the range plotted. It further shows that meat prepared at 200oC regardless of frying time, does not reach the adequate-processed frying degree either.