The time domain sample signal E_trans(t)and the reference signal E₀(t)were collected by focusing the terahertz beam with and without the sample in the optical path, respectively. After applying the fast Fourier transform(FFT), we obtained reference and sample signals in the frequency domain, E₀(ω)and E_trans(ω), respectively, which were used to calculate the absorbance spectrum. The absorbance A is a dimensionless relative quantity and is calculated as in Eq.(1)^[20]

A=-1/dln(E²_trans(ω))/(E²₀(ω))(1)

where ω is the frequency of terahertz wave, and d is the thickness of sample.

PCA is a statistical extraction method which reduces the dimensionality of a data set through mathematical transformation. The comprehensive variables derived from the transformation are the principal components which summarize the features of the data set. All the principal components are uncorrelated and ordered. Each principal component is a linear combination of the original variables. When the cumulative variance contribution rate of the first k principal components is large enough(typically P≥85%), the original data set can be replaced approximately with the first k principal components.

PCA projects the n-dimensional features to the k-dimension(k<n). The greatest variance lies on the first coordinate, called the first principal component, the second greatest variance lies on the second coordinate, and so on.Consider a p-dimensional data matrix, X=(X₁,X₂,…,X_p)^T,

PC₁=a₁'X=a₁₁X₁+a₂₁X₂+…+a_p1X_p

PC₂=a₂'X=a₁₂X₁+a₂₂X₂+…+a_p2X_p



PC_p=a_p'X=a_1pX₁+a_2pX₂+…+a_ppX_p

After the transformation, the original data of p variables X₁,X₂,…,X_p are replaced by the variables PC₁, PC₂, …, PC_p in the new coordinate system which successively inherit the maximum possible variance from X^[21].

Before PCA is carried out, the original absorbance spectra should be standardized in order to solve the comparability of data. After standardized processing, the original data are transformed into dimensionless. Standard scores are also called z-values, z-scores, normal scores, and standardized variables. The computation of the z-score are based on the mean and standard deviation of the original data.

The standard score was calculated using the sample mean and sample standard deviation as estimates of the population values^[22]. In these cases, the z-score is

z=(x-(-overx))/S(2)

where(-overx)is the mean of the sample, and S is the standard deviation of the sample.

The absolute value of z represents the distance between the raw score and the sample mean in units of the standard deviation. z is negative when the raw score is below the mean, positive when above.

The LDA classifier was applied for identification after the PCA data projection. Multivariate analysis(PCA and LDA)combined with terahertz spectroscopy is an excellent method with good potential for the identification and classification of biological materials. LDA is a classical dimension reduction linear discriminant analysis. Prior knowledge and experience of categories can be used in dimensionality reduction and classification.Both LDA and PCA are commonly used for dimensionality reduction. PCA ignores class labels and it is an unsupervised algorithm. In contrast to PCA, LDA computes the directions that will represent the axes that that maximize the separation between multiple classes and it is a supervised algorithm. It is common to use both LDA and PCA in combination: PCA for dimensionality reduction followed by an LDA.

The evaluation of the classification performance was carried out using the LOO approach^[23]. LOO is commonly used when the data set is too small to provide sufficient samples for separate, independent training and test sets.There are N samples in the whole dataset; each sample was used as the test set alone and the remaining(N-1)samples were used as the training set. The category estimation was performed on that data, and then compared to the known category. This procedure was repeated N times(the total number of the measured spectra), while each time a different datum was left out. LOO will get N models, and the average of the total prediction error of the N models is used as the performance index.

The terahertz time domain transmittance spectra were measured for 30 treated American ginseng samples and 30 untreated American ginseng samples. Then, the time domain spectra were transformed to power density spectra by fast Fourier transformation and the absorbance was calculated according to Eq.(1). The effective frequency range from 0.2 to 1.4 THz of the absorption spectra was determined according to power density spectra.

The absorbance spectra of 30 different treated American ginseng samples and 30 different untreated American ginseng samples are shown in Fig.3. From Fig.3, we can find that there are no significant characteristic peaks for treated American ginseng and untreated American ginseng. The extraction of the bioactive components has little influence on the absorbance spectra of American ginseng. The absorbance spectra were highly similar for treated American ginseng and untreated American ginseng in our frequency range, thus it is difficult to distinguish them by fingerprint analysis.

Fig.3 The terahertz absorbance spectra of treated American ginseng and untreated American ginseng

PCA was applied to reduce the dimensionality and extract features from the original absorbance spectra. Before carrying out PCA, the original absorbance spectra were standardized by Eq.(2).After performing PCA, as seen in table 1, the top three eigenvectors(principal components)extracted from the terahertz absorbance spectra data of 60 samples were 0.515 9, 0.052 9 and 0.005 9, respectively. The variance contribution rates of the top three principal components of the absorption coefficient spectra were 88.15%, 9.04% and 1.01%, respectively, which were representative of spectral features of the original absorbance spectra. As shown in table 1, the accumulated variance of the top three eigenvectors reaches 98.10%. PCA effectively reduced the number of variables of the data and maintained the spectral features of the original absorbance spectra. In the spectra around 0.7 to 1.0 THz, the number of spectral variables was reduced from 70 to 3 by using the PCA method.

Table 1 Total variance explained by each component

Fig.4 shows that the treated American ginseng samples and untreated American ginseng samples can be classified clearly by using the scoring chart of their top three principal components. The treated American ginseng and untreated American ginseng can be distinguished by the combination of the terahertz technique and the PCA method. Furthermore, this combination could be extended to other Chinese herbal medicines.

Fig.4 Principal component scores of the absorbance spectra of treated American ginseng and untreated American ginseng

These three principal component scores were input to the linear discrimination analysis LDA model for classification and discrimination. Symbols -1 and 1 represent treated American ginseng and untreated American ginseng respectively. The LOO approach was used to evaluate the LDA model classification performance. Good results were obtained as shown in table 2, where, M_i represents the number of samples identified as class i in the test. Fig.5 presents the posterior of 60 samples if they were discriminated correctly with PCA-LDA model. Table 2 shows that the PCA-LDA model can distinguish the treated American ginseng from the untreated American ginseng accurately. The recognition rate of the PCA-LDA model is 100% and 96.7% for the spectra of the treated American ginseng and untreated American ginseng acquired with terahertz spectroscopy. The total recognition rate is 98.3%.

Table 2 Results of validation of the model

Fig.5 The posterior of 60 samples with PCA-LDA model