DETERMINATION OF TREE SPECIES USING RANGING SCATTEROMETER DATA
Markus Törmä
[email protected] Helsinki University of Technology Institute of Photogrammetry and Remote Sensing Espoo, Finland
Abstract Economic forest inventory methods are important for a forested country like Finland. Forest inventory provides information about trees and forest stands. Remote sensing methods offer economic and efficient way to obtain information for forest inventory. One important part of information is tree species, because species can affect to determination of other informative forest inventory parameters. A special radar called scatterometer and its applicability to determine different tree species is discussed. The principle of the scatterometer and methods to normalize data and feature extraction are presented. The final determination is performed using the Bayes decision rule for minimum error.
1. INTRODUCTION Forest inventory is a procedure that provides qualitative and quantitative information about individual trees and forest stands. This information includes for example species, height and stem volume of an individual tree, or mean height, total stem volume and total biomass of forest stand. Traditionally, forest inventory has been made using a lot of field work, but this is time consuming and expensive. Remote sensing methods offer efficient and economic way to obtain information for forest inventory. The applicability of passive visible and infrared satellite sensors like Landsat for forest inventory have been studied in Finland since 1970’s (Kuusel75, Kilpel78). The problems using satellite imagery are that all needed parameters cannot be calculated and results are reliable only for large areas. Active sensors like synthetic aperture radars and scatterometers are still under study. Especially scatterometers seem to be a promising tool for the determination of information for forest inventory (Hyyppä93). The determination of qualitative and quantitative information like mean height and stem volume for forest inventory using a radar has been discussed in Hyyppä93. The determination of tree species has also been discussed in the above-mentioned publication, but in this paper a different method is discussed.
2. ACTIVE REMOTE SENSING SENSORS Active remote sensing sensors (radars) acquire information about the physical structure and electrical properties of the object by analyzing the reflected field when the sensor illuminates the object with a well defined generated field of electromagnetic waves (Elachi88). In other words, a radar sends an electromagnetic pulse to the target and measures the reflected pulse. The reflected pulse measured by the radar is used to compute the backscattering coefficient. It should be noted, that electromagnetic pulse can penetrate to the target depending on the used frequency and the properties of the target. Imaging radars form images from a wide area, where each image pixel corresponds to the reflected pulse from a specific ground area. Non-imaging radars do not form images, but they measure the distance to the target or the reflectance properties of the target. A non-imaging radar called scatterometer is used to measure very accurately the surface reflectivity as a function of frequency, polarization and illumination direction of the sensing signal (Elachi88). Ranging scatterometer measures the surface reflectivity as a function of range and other aforementioned parameters. 2.1 HUTSCAT The scatterometer used in this study is called HUTSCAT and it is developed in Helsinki University of Technology, Laboratory of Space Technology. The HUTSCAT combines the backscattering measurement accuracy of scatterometers and the range measurement accuracy of altimeters; in other words, it is a ranging scatterometer capable of measuring the backscattering behaviour of a target as a function of the range (figure 1). The objective of the HUTSCAT development was to construct a helicopter-borne radar system for remote sensing of forest canopies, sea ice and snow. The main requirements for HUTSCAT were a high measurement accuracy of the backscattering coefficient, ranging capability with a good range resolution and multichannel capability. Technical characteristics are presented in table 1 (Hyyppä93). Figure 1. The measurement principle of HUTSCAT. On the left HUTSCAT measures the backscattering behaviour of a target as a function of the range. The measured profile is on the right.
Height
Backscattered Power
The measurements are made simultaneously at eight channels (two frequencies and four polarizations). Due to the range measurement capability, HUTSCAT can identify the backscattering sources within distributed targets like forests. A more detailed discussion about HUTSCAT can be found from Hyyppä93 or Hallik93. 2.2 Measured profiles The basic output product of the HUTSCAT is a radar return spectrum of a target (backscattered power versus distance from radar, figure 3). When a forest is the target, the radar return spectrum is called Forest Canopy Profile (FCP). FCP includes information from the tree top to the ground. The backscattered power peaks at the minimum and maximum distances from the radar correspond to the tree tops and ground and the range difference can be used to determine the height of the trees. The form of FCP can be used to determine the species of the trees. Figure 2 represents an example of measured FCP (pine dominated forest). Darker areas of FCP correspond to strong backscatter from trees. Figure shows clearly the variation of the height of the trees within flight line.
Table 1. Technical characteristics of HUTSCAT. Parameter
Selected value
Centre frequency
9.8 GHz & 5.4 GHz
Modulation
FM-CW
Sweep bandwidth
300 MHz
Polarization
HH, VV, HV, VH (first transmitted and then received)
Range resolution
0.65 m
Measurement range
8 to 167 m
Antenna type
Dipole-disk paraboloid antenna
Antenna diameter
75 or 40 cm (5.4 GHz), 38 cm (9.8 GHz)
Antenna bandwidth (two-way product)
3.8° (9.8 GHz & 5.4 GHz), 5.7° (5.4 GHz, 40 cm dish), 4.7° (5.4 GHz, dual-antenna configuration)
Platform
Helicopter
Incidence angle
0 to 45° off nadir
Antenna look direction
Across flight track
Radar control
PC/AT microcomputer
Data storage
Cartridge tape unit (60 MB), Bernoulli Box disc (44 MB)
Calibration
Internal and external calibration
Add. instruments
Video camera syncronized with radar
Figure 2. An example of measured forest canopy profile (pine dominated forest). The horisontal axis represents the distance along the flight line (metres) and the vertical axis shows the range from the radar (one resolution element corresponds 0.65 m). Darker areas of the profile correspond to a strong backscatter from trees.
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2.2.1 Quality of profiles The factors that determine the quality of the measured profile are: 1. flight altitude and antenna characteristics, 2. range resolution, 3. system sensitivity, 4. flight speed and sampling rate and 5. incidence angle (Hyyppä93). 1.
2. 3.
4.
5.
The flight altitude and antenna beamwidth determine the illumination geometry of a target. The volume of the illumination is a truncated cone and it is difficult to discriminate between targets (trees in this case) within this cone. Also the antenna sidelobes should be suppressed as well as possible, because they can distort the measured profiles. The accuracy of the tree height estimate is limited by the range resolution, so it should be as high as possible. Internal reflections and their sidelobes can be interpreted as backscatter from canopy and this way they worsen the tree height estimation. In order to measure the tree height accurately, a very high system sensitivity is needed. The trees are selected with a probability proportional to the tree height and crown diameter, so trees with a wider crown have a higher probability to be sampled. The ability to determine the tree height is assumed to decrease with increasing incidence angle due to the spreading of the ground echo. When near vertical measurements are used, the ground backscatter is sensitive to variations in the incidence angle.
2.2.2 FCPs used in this study The data was measured in Ruotsinkylä test site (25 km north from Helsinki). The test site consists of 13 flight lines with a total length of about 4 km. The flight lines were divided into 20-metre-by-20-metre sample plots, and initial parameters were measured from the plots (ground truth data); tree height, different diameters, number of stems per hectare, height of crown base and tree species (Hyyppä93).
The flight lines were measured with a dual-antenna configuration (5.4 GHz) at the incidence angle 40° off nadir. The dual-antenna configuration was used to guarantee a high system sensitivity. The dataset for this study was selected from these measurements. The dataset consists of six matrices, in which each row corresponds measurements in flight direction and each column corresponds range measurements. In other words, each matrix column corresponds to one profile. Two matrices contains profiles from a pure pine forest, two from a spruce forest and last two from a birch forest. The dimensions of the matrices were different depending on the length of the flight line, tree heights and the flying altitude (Hyyppä93). The exact dimensions of the datamatrices were after removing the bare land (columns correspond to the dimension of the individual profiles and rows correspond to number of profiles in flight direction); Birch1: 72 columns and 519 rows, Birch2: 97 columns and 580 rows, Spruce1: 107 columns and 780 rows, Spruce2: 87 columns and 367 rows, Pine1: 98 columns and 387 and Pine2: 67 columns and 460 rows. The percentage of birch, spruce and pine was about 36%, 37% and 27%, respectively.
3. TRANSFORMATION OF FCPs AND FEATURE EXTRACTION Forest canopy profiles need to be transformed to a suitable form before classification. In other words, the dimensionality of each dataset should be same and errorneous profiles should be removed. 3.1 Transformation of FCPs Because the purpose of this study is to determine different tree species, nonforest areas (bare land) were removed from the dataset. Because the variations of the backscattering coefficients were quite huge, the common logarithmic transform (log10) was performed (figure 3). The flight altitudes and the height of the trees were different for each datamatrix, so the dimensions of the FCPs were different. In order to set FCPs from different datamatrices comparable and suitable for classification, FCPs were normalized. This normalization was made simply by dividing each profile to ten elements and computing a value for each element. The values of the elements were computed in three different ways; 1) mean filtering piece of original profile (called NFCP1), 2) median filtering piece of original profile (NFCP2) or 3) taking ten samples from the original profile (NFCP3). Examples about the normalized profiles of the different tree species (birch, spruce and pine) are presented in figure 4. The used normalization method is NFCP1. 3.2 Feature extraction When one FCP belongs to a class, it is quite likely that neighboring FCPs belong to the same class. In order to decrease the classification error, FCPs must include information about neighboring profiles (called context). This information can be obtained by filtering neighboring profiles or computing extra features using neighboring profiles. There are three alternatives; filtering of profiles, computing features for each element of the profile or computing features for the whole profile.
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0 0
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Figure 3. The left figure represents a measured profile from pine forest (note scale). The right figure represents same profile after common logarithmic transform. After the transform, non-zero part of the profile is normalized by dividing the profile into ten pieces and computing a value for each piece. The horisontal axes show the range from radar using the resolution elements (see table 1) and vertical axes represent the measured backscatter (on the left) and its logarithmic transform (on the right). The simplest way is to filter neighboring profiles using mean or median filtering. The filtering was performed using ±10 metres window for each profile. The motivation behind the size of the filtering window was that ground truth data was obtained using 20-metre-by-20-metre sample plots. The good point is that the dimension of profiles does not expand. The context can be taken into account by modelling the variation of the neighboring profiles. This modelling can be based on the assumption that the variation of the neighboring profiles is different for different tree species. The variation is measured by computing the mean value within the window and its deviation (this feature is called F1 afterwards), or the mean value of the gradient the within window and its deviation (F2). The mean value of the gradient is approximated by summing up the differences of the elements of the neighboring profiles and dividing them by the number of the differences. In this way, two extra features are computed for each element of the profile. The dimension of the normalized profiles is 10, so the dimension with extra features is 30. In the previous paragraph the context was modelled by computing extra features for each element of the profile. Another way to compute features is to compute them for the whole profile and this way limit the expansion of the dimension. Two different sets of features were computed; the mean of the distances between profiles and its deviation (F3), and the mean of the deviation of the difference between the elements of the profiles and its deviation (F4). The purpose of F3 is to measure the overall differences (how close to each other they are) of the profiles within the window and the purpose of F4 is to measure the differences of the form of the profiles within the window. Also a third set of features (F5), which included both earlier features (F3+F4) was computed.
Figure 4. Examples of normalized profiles. The used normalization method is NFCP1, the values for the elements of the profiles are computed by mean filtering the piece of the original profile. Horisontal axis represents the elements of the normalized profiles and vertical axis represents the corresponding logarithmically transformed and averaged backscattering.
Birch Spruce Pine
5
1 1
5
10
4. CLASSIFICATION Classifications were made using the Bayes decision rule for minimum error. The probability that X belongs to the class ωj is called a posteriori probability P(Xωj). If a posterior probability of the class j is greater than a posteriori probability of the class i (P(Xωj)>(Pωi)), pattern X is classified to the class ωj. A posteriori probability P(Xωj) can be calculated from a priori probability Pj and the conditional density function p(Xωj) using the Bayes theorem
where
is the mixture density function and c is the number of classes. When X is to be classified, a posteriori probabilities are determined for each class and X is assigned to the class with maximum a posteriori probability (Fukuna90). 4.1 Estimation of density function The values of the conditional density functions of different classes need to be estimated so that a posteriori probabilities can be computed using Bayes theorem. This estimate should also be as reliable as possible. There are two approaches to estimate them; parametric and nonparametric estimation of the density function. In parametric estimation, the estimate of density function is characterized by a set of parameters. For example, Gaussian distribution is characterized by its mean and covariance. Nonparametric estimation is based on the estimation of the density function locally using small number of neighboring samples.
Most parameter estimation methods are optimal if the conditional density functions of the classes are Gaussian. However, verifying the normality of the multivariate data is a difficult problem and if the density functions are not Gaussian, parametric estimation loses its purpose (Devijv87). Therefore, "volumetric" k-nearest neighbor density estimation method was chosen. The estimate of the conditional density function is
where k is the number of the nearest neighbors, nj is the number of known samples of the class j and v is the volume of such hypersphere, where the radius is the distance between X and its k:th nearest neighbor (Heikki92). 4.2 Error estimation The probability of misclassification or error is the most effective measure of the performance of a classification system. It is very difficult to obtain an analytic expression of the probability of error even when the conditional density functions are known. This requires the density function to have a very simple form. In practise, the probability of error must be estimated from the available samples. First, the classifier is designed using training samples and then it is tested using test samples. The percentage of misclassified test samples is taken as an estimate of the probability of the error. In order to have a reliable error estimate, the training and test samples must be statistically independent or at least different. It is important to note, that the error estimate is a random variable and has properties like bias and variance, which depend on the number of samples and the classifier used. There are two methods used in this study to determine the probability of error: 1. resubstitution and 2. leave-oneout estimation method (Devijv87). 1.
2.
Resubstitution method uses the same set of samples to train as well as test the classifier. It is well known that the resulting error estimate is optimistically biased, when a parametric classifier is used. In leave-one-out method each sample is used for design and test, although not at the same time. The classifier is designed using (n-1) samples and then tested on the remaining sample (n is the total number of samples). This is repeated n times with different design sets of size (n-1). The error estimate is the total number of misclassified samples divided by n. Leave-one-out estimator is unbiased, but it has a large variance.
5. RESULTS The classifications were made with a k varying between 1 and 10. A priori probabilities were computed from the training samples. In other words, a priori probability for the class i is computed by dividing the number of samples of class i by the number of all samples. Error estimation was performed using resubstitution (called res) and leaveone-out (loo) estimators. Estimated errors are averaged errors, the number of misclassifications is computed and divided by the number of all samples.
5.1 Results using normalized forest canopy profiles The classification of the normalized profiles (NFCP1, NFCP2 and NFCP3) were performed first. The best classification was obtained using normalized profiles with mean filtered pieces (figure 5). Using normalization method NFCP1 and k-value 6 res error was 15.32% and loo error was 18.11%. Figure 5 indicates, that the least biased errors can be obtained using k=6 or k=7. Using NFCP2 and NFCP3 the classification errors were higher; loo errors for NFCP2 varied between 20% and 24% and for NFCP3 varied between 29% and 38%. It should be noted, that all estimated errors are biased and it is difficult to determine the least biased errors. 5.2 Results using information from neighboring profiles The simplest way to take into account the information from neighboring profiles is to filter the neighboring profiles. Filtering was performed using mean and median filtering. The results were quite surprising; the errors were very close to zero and differences between different filtered normalized profiles (NFCP1, NFCP2 and NFCP3) were small. In all cases the errors increased with increasing k, so it is quite difficult to choose an optimal k. As an example, figure 6 represents the errors obtained using mean filtered profiles with normalization method NFCP1. The figure indicates, that a suitable value for k is between 2 and 4 in this case, because the error curves are quite flat. The mean filtered profiles performed better than the median filtered profiles, the estimated errors were smaller with almost all k-values. Using the mean filtered profiles, the estimated errors behaved as can be seen from table 2 (table represents the estimated errors using k-value 1, 3 and 6). The estimated errors for median filtered profiles are represented in table 3. The estimated classification errors were a little higher compared to the filtered profiles, when features F1 and F2 were used. The estimated errors of feature F1 were quite close to the filtered profiles, but a little higher and differences between res and loo errors were larger. This is due to the increased dimension of the data, the original dimension was 10 and it increased to 30. In general, increasing the dimension of the data increases the bias of the estimated error. Some of the estimated errors of F1 and F2 are represented in table 4 and table 5. At first, classifications with features F3, F4 and F5 were made without normalized profiles. The purpose was to study if these features could contain enough information for a correct classification. The answer is no. All estimated classification errors were very high. It should be noted, that if all profiles would be classified to the class with most samples, the classification error would be about 63%. In some cases, the estimated errors were larger than 63%. Using feature F3 computed from NFCP1 loo errors varied between 55% and 62%, from NFCP2 loo errors varied between 52% and 58% and from NFCP3 loo errors varied between 56% and 61%. Using feature F4 computed from NFCP1 loo errors varied between 58% and 62%, from NFCP2 loo errors varied between 52% and 59% and from NFCP3 loo errors varied between 56% and 65%. Using feature F5 computed from NFCP1 loo errors varied between 54% and 57%, from NFCP2 loo errors varied between 49% and 55% and from NFCP3 loo errors varied between 54% and 60%.
When classifications were performed with features F3, F4 and F5 including normalized profiles, the results were much better. Compared to the results of the normalized profiles, the estimated classification errors decreased a little. In all cases loo error decreased first to the minimum error and then started to increase. Using feature F3 with NFCP1 loo errors varied between 15% and 19%, with NFCP2 loo errors varied between 16% and 20% and with NFCP3 loo errors varied between 28% and 34%. Using feature F4 with NFCP1 loo errors varied between 15% and 18%, with NFCP2 loo errors varied between 15% and 19% and with NFCP3 loo errors varied between 27% and 34%. Using feature F5 with NFCP1 loo errors varied between 14% and 19%, with NFCP2 loo errors varied between 14% and 20% and with NFCP3 loo errors varied between 27% and 32%. The classifications were also performed with features F3 and F4 including mean and median filtered profiles. The mean filtered profiles with features performed better than the median filtered profiles with features and the estimated errors were really close to the mean filtered profiles. The estimated errors for the mean filtered profiles with feature F3 and F4 are represented in table 6 and table 7. Figure 7 represents the obtained errors when the mean filtered profiles with feature F4 were used. It is quite difficult to compare the results to the results of the filtered profiles, because the estimated errors are close to each other and in both cases loo error increases with increasing k. However, the form of the error curves in figure 6 and figure 7 indicate, that the mean filtered profiles perform better, because corresponding error curves in figure 6 are flatter when k is 4 or less, and start to increase rapidly. Figure 7 shows that the errors, when mean filtered normalized profiles with feature F4 are used, increase quite steadily, and it is difficult to choose an optimal k. 5.3 Examples about other studies Hyyppä (1993) used a similar data in his study, in fact the data in this study is a subset of Hyyppä’s data. The pre-processing was made using Butterworth low-pass filtering to smooth the shape of the radar return spectra. The filtering was performed in both directions (in columns and in rows). The spectra corresponding to the centre of the tree tops were selected by picking the local maxima of the height profile. Then five consecutive radar return spectras were averaged. The classification was carried out using the principal component analysis, using the first two principal components and multiple discriminant analysis. The estimated classification errors for pine, spruce and birch were 34%, 25% and 35%, respectively. Salo (1992) used completely different data in his study. The pre-processing steps included the common logarithmic transform, Butterworth low-pass filtering, picking the local maxima and normalization of the profiles using cubic splines. The Kohonen self-organizing feature map was used in classification and the average classification error was 7.8%.
6. CONCLUSIONS These experiments indicate that different tree species can be determined quite well using ranging scatterometer. The best procedure to process the data is to normalize the measured profiles with the method NFCP1 (the elements of the normalized profiles are computed by mean filtering the original profile). The classification accuracy can be increased by mean filtering the profiles in the flight direction. So, the best results were obtained with the mean filtered data, also the results of the mean filtered data with feature F4 were quite close. In both cases it is quite difficult to estimate an exact classification error due to the biased error estimates, but however, it is quite close to zero. Compared the other two studies introduced previously, the method presented here performs better. To get a better understanding about the performance of the presented method, it should be tested using more data and also try to determine the forest stand development classes (seedling stand, young thinning stand, advanced thinning stand and mature stand). Also other features like moments and their feasibility to the determination of the tree species should be investigated. Finally, I would like to thank Dr. J. Hyyppä, Dr. J. Pulliainen and Mr. M. Inkinen for datasets, help and assistance.
References: Devijv87
Elachi88
Fukuna90
Hallik93
Heikki92
Hyyppä93
Kilpel78
Kuusel75
Salo92
P. Devijver, J. Kittler (ed) Pattern Recognition Theory and Applications (Chapter 1.) Springer-Verlag, 1987. C. Elachi Spaceborne Radar Remote Sensing: Applications and Techniques IEEE Press, New York, 1988. K. Fukunaga Introduction to Statistical Pattern Recognition Academic Press, San Diego, 1990. M. Hallikainen, J. Hyyppä, J. Haapanen, T. Tares, P. Ahola, J. Pulliainen, M. Toikka A Helicopter-Borne Eight-Channel Ranging Scatterometer for Remote Sensing: Part I: System Description IEEE Transactions on Geoscience and Remote Sensing, Vol. 31, No. 1, pp.161-168, 1993. J. Heikkilä Hahmontunnistus - tilastollinen lähestymistapa Teknillinen korkeakoulu, Fotogrammetrian ja kaukokartoituksen laboratorio, 1992 (translated from P. Devivjer, J. Kittler: Pattern Recognition - Statistical Approach). J. Hyyppä Development and Feasibility of Airborne Ranging Radar for Forest Assessment Dissertation Thesis, Helsinki University of Technology, Laboratory of Space Technology, 1993. E. Kilpelä, S. Jaakkola, R. Kuittinen, J. Talvitie Automated Earth Resources Surveys Using Satellite and Aircraft Scanner Data Technical Research Centre of Finland, Building Technology and Community Development, Pub. 15, 1978. K. Kuusela, S. Poso Demontration of the applicability of satellite data to forestry Communicationes Instituti Forestalis Fenniae 83.4, reprint, 1975. T. Salo Neuroverkon hyödyntäminen tutkamittauksen perusteella suoritettavassa metsänarvioinnissa ("Utilizing Neural Networks in Forest Inventory Based on Radar Measurements") Master’s Thesis, Helsinki University of Technology, Laboratory of Space Technology, 1992.
Table 2. The classification errors when mean filtered normalized profiles were used. NFCP* correspond to the used normalization method (see 3.1), and res and loo correspond to the resubstitution and leave-one-out error estimation methods. k
NFCP1res
NFCP1loo
NFCP2res
NFCP2loo
NFCP3res
NFCP3loo
1
0.00%
0.06%
0.00%
0.10%
0.00%
0.13%
3
0.03%
0.19%
0.03%
0.29%
0.13%
0.36%
6
0.39%
1.00%
0.48%
1.13%
0.52%
0.84%
Table 3. The classification errors when median filtered normalized profiles were used. k
NFCP1res
NFCP1loo
NFCP2res
NFCP2loo
NFCP3res
NFCP3loo
1
0.00%
0.19%
0.00%
0.19%
0.00%
0.36%
3
0.26%
0.42%
0.19%
0.45%
0.32%
0.65%
6
0.71%
1.33%
0.91%
1.26%
1.16%
1.84%
Table 4. The classification errors when normalized profiles with feature F1 were used. k
NFCP1res
NFCP1loo
NFCP2res
NFCP2loo
NFCP3res
NFCP3loo
1
0.00%
0.19%
0.00%
0.81%
0.00%
2.17%
3
0.29%
0.84%
0.61%
1.33%
1.00%
2.52%
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0.94%
1.75%
2.23%
3.88%
2.68%
4.49%
Table 5. The classification errors when normalized profiles with feature F2 were used. k
NFCP1res
NFCP1loo
NFCP2res
NFCP2loo
NFCP3res
NFCP3loo
1
0.00%
1.88%
0.00%
2.00%
0.00%
4.59%
3
1.03%
2.36%
1.49%
2.88%
2.81%
5.46%
6
2.59%
4.11%
4.20%
6.21%
5.21%
7.57%
Table 6. The classification errors when mean filtered profiles with feature F3 were used. k
NFCP1res
NFCP1loo
NFCP2res
NFCP2loo
NFCP3res
NFCP3loo
1
0.00%
0.13%
0.00%
0.19%
0.00%
0.10%
3
0.03%
0.52%
0.03%
0.84%
0.13%
0.32%
6
0.71%
1.29%
0.81%
1.42%
0.48%
0.87%
Table 7. The classification errors when mean filtered profiles with feature F4 were used. k
NFCP1res
NFCP1loo
NFCP2res
NFCP2loo
NFCP3res
NFCP3loo
1
0.00%
0.03%
0.00%
0.16%
0.00%
0.10%
3
0.00%
0.32%
0.00%
0.42%
0.13%
0.26%
6
0.42%
0.97%
0.61%
1.23%
0.42%
0.87%
Figure 5. The classification errors when normalized profiles were used. Normalization was made using method NFCP1. The lower curve corresponds to the resubstitution and upper curve to the leave-one-out error estimation methods. The horisontal axis represents the k-value used and the vertical axis shows corresponding errors.
Figure 6. The classification errors when mean filtered normalized profiles were used. Normalization was made using method NFCP1. The lower curve corresponds to the resubstitution and upper curve to the leave-one-out error estimation methods. The horisontal axis represents the k-value used and the vertical axis shows corresponding errors.
Figure 7. The classification errors when mean filtered normalized profiles with feature F4 were used. Normalization was made using method NFCP1. The lower curve corresponds to the resubstitution and upper curve to the leave-one-out error estimation methods. The horisontal axis represents the k-value used and the vertical axis shows corresponding errors.
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2.5% 2.0% 1.5% 1.0% 0.5% 0.0%
2.5% 2.0% 1.5% 1.0% 0.5% 0.0%
