text:
More than one million prostate biopsies are performed in the United States every year. A failure to find cancer is not definitive in a significant percentage of patients due to the presence of equivocal structures or continuing clinical suspicion. We have identified gene expression changes in stroma that can detect tumor nearby. We compared gene expression profiles of 13 biopsies containing stroma near tumor and 15 biopsies from volunteers without prostate cancer. About 3,800 significant expression changes were found and thereafter filtered using independent expression profiles to eliminate possible age-related genes and genes expressed at detectable levels in tumor cells. A stroma-specific classifier for nearby tumor was constructed on the basis of 114 candidate genes and tested on 364 independent samples including 243 tumor-bearing samples and 121 nontumor samples (normal biopsies, normal autopsies, remote stroma, as well as stroma within a few millimeters of tumor). The classifier predicted the tumor status of patients using tumor-free samples with an average accuracy of 97% (sensitivity = 98% and specificity = 88%) whereas classifiers trained with sets of 100 randomly generated genes had no diagnostic value. These results indicate that the prostate cancer microenvironment exhibits reproducible changes useful for categorizing the presence of tumor in patients when a prostate sample is derived from near the tumor but does not contain any recognizable tumor. Cancer Res; 71(7); 2476–87. ©2011 AACR.

There are more than 1 million prostate biopsy procedures carried out in the United States every year (1). More than 60% of the results are negative (2–4). However even the best current methods, including transrectal ultrasound (TRUS) procedures, may miss up to 30% of clinically significant prostate cancers (5). Indeed, about 20% to 30% of patients who give negative results on initial biopsy are rebiopsied in about 3 to 12 months [∼190,000 patients owing to the presence of prostatic intraepithelial neoplasia (PIN), high-grade PIN (HGPIN), atypical small acinar proliferation (ASAP), or other grounds for clinical suspicion of the presence of tumor (2–4, 6, 7)]. Many repeat biopsies give the diagnosis of adenocarcinoma. For example, 16% to 23% of HGPIN and up to 59% of ASAP cases give the diagnosis of adenocarcinoma upon repeat biopsy (2–4, 8). Patients deferred to repeat biopsy, receive little treatment or guidance during the interim—a period when tumors may continue to progress. Therefore, there is a need for methods that resolve false-negative and equivocal cases.

Equivocal and negative biopsies are, by definition, deficient in diagnostic tumor but contain ample stroma. Moreover, stroma near tumor may contain changes in gene expression that are not found in nontumor samples, which could be the basis for a clinical test. Epithelial cells of prostate cancer infiltrate and propagate in a microenvironment consisting largely of myofibroblast cells as well as inflammatory cells and other supporting cells and structures. It has long been appreciated that this mesenchymal component is not passive but responds to signals from the tumor component and, in turn, alters tumor properties, some of which are essential for tumor growth and progression (9, 10). Indeed, studies of prostate cancer were among the first to demonstrate an important role of the stroma in cancer progression. Mouse model studies showed that survival and growth of immortalized nontumorigenic human prostate epithelial cells as renal subcapsular xenografts required stroma from tumor-bearing prostate (10). Numerous studies have subsequently demonstrated large numbers of gene expression changes at the RNA level specific to the tumor microenvironment of prostate cancer (e.g., refs. 11–18). Similarly, a variety of protein expression changes have been associated with the microenvironment of prostate cancer. For example, reactive stroma, which is believed to occur in a subset of aggressive tumors, has been shown to correlate with changes in a variety of proteins including FGF2 (fibroblast growth factor); connective tissue growth factor; vimentin; actin, alpha, skeletal muscle; collagen, type I, alpha; and tenascin; some of which have been attributed to epithelial-derived TGF-β (12, 19).

Here, we investigate whether RNA expression changes may be identified that are sufficiently reliable to distinguish normal stroma from stroma near tumor. We have previously developed linear regression method for the identification of cell-type–specific expression of RNA from array data of prostate tumor samples (20). The method was validated using immunohistochemistry and using quantitative PCR applied to laser capture microdissection samples of tumor, stroma, and epithelia of benign prostate hyperplasia for 28 genes involving more than 400 measurements (20). Here, we have extended this approach to identify differentially expressed genes between normal volunteer prostate biopsy samples versus stroma from near tumors. More than a thousand gene expression changes were observed. A subset of stroma-specific genes was used to derive a classifier of 114 genes which accurately identifies tumor or nontumor status of a large number of independent testcases. The classifier may be useful in the diagnosis of stroma-rich biopsies from patients with equivocal pathology.

### Prostate cancer patient samples and expression analysis

Data sets 1 and 2 (Table 1) are based on postprostatectomy frozen tissue samples obtained by informed consent using Institutional Review Board (IRB)-approved and HIPPA-compliant protocols. All tissues, except where noted, were collected at surgery and escorted to pathology for expedited review, dissection, and snap freezing in liquid nitrogen. In addition, data set 1 contains 27 prostate biopsy specimens obtained as fresh snap-frozen biopsy cores from 18 normal prostates. These samples were obtained from the control untreated subjects of a clinical trial to evaluate the role of difluoromethylornithine (DFMO) to decrease the prostate size of normal men. Eighteen of these were collected before the treatment period and 9 were collected after the treatment period had ended (21). Finally, 13 samples of normal prostate tissue were obtained from the rapid autopsy program of the Sun Health Research Institute (Sun City, AZ) and were frozen within 6 hours of demise.

Table 1.

Data sets used in the studya

DataPlatformSubject no.Array no.Array: tumor/ nontumor/normalReference
1 Training + test U133Plus2 P = 87 108 68/40/0 GSE17951
B = 18 27 0/0/27
A = 13 13 0/0/13
U133A P = 82 136 65/71/0 GSE08218
U133A P = 79 79 79/0/0 Unpublished, see (22)b GSE25136
U133A P = 44 57 44/13/0 E-TABM-26 (24)c
DataPlatformSubject no.Array no.Array: tumor/ nontumor/normalReference
1 Training + test U133Plus2 P = 87 108 68/40/0 GSE17951
B = 18 27 0/0/27
A = 13 13 0/0/13
U133A P = 82 136 65/71/0 GSE08218
U133A P = 79 79 79/0/0 Unpublished, see (22)b GSE25136
U133A P = 44 57 44/13/0 E-TABM-26 (24)c

aP, samples from prostate cancer patients; B, biopsies from normal donors; A, prostate donated by rapid autopsy. Data sets 1 and 2 were collected from 5 participating institutions in San Diego County. Demographic, pathology, and clinical values are individually recorded in shadow charts and maintained in the UCI SPECS consortium database.

bData set 3 was provided by William L. Gerald (Stephenson and colleagues).

RNA for expression analysis was prepared directly from frozen tissue following dissection of OCT (optimum cutting temperature compound) blocks with the aid of a cryostat. For expression analysis, 50 μg (10 μg for biopsy tissue) of total RNA samples were processed for hybridization to Affymetrix GeneChips. Expression analysis for all samples for data set 1 were assessed using the U133 Plus 2.0 platform, whereas for data set 2, the U133A platform was used. The data have been deposited in the Gene Expression Omnibus (GEO) database with accession numbers GSE17951 (data set 1) and GSE8218 (data set 2). For data sets 1 and 2, the distributions for the 4 principal cell types (tumor epithelial cells, stroma cells, epithelial cells of benign prostatic hyperplasia, and epithelial cells of dilated cystic glands) were estimated by 3 (data set 1) or 4 pathologists (data set 2), whose estimates were averaged as described (20).

Data sets 3 and 4 were independently developed and used as test sets (Table 1). Data set 3 consists of a series of 79 samples (22, 23), whereas data set 4 (24) is composed of 57 samples from 44 patients including 13 samples of stroma near tumor and 44 tumor-bearing samples. Expression analysis of the data sets was determined using the U133A platform.

### Manual microdissection

Seventy-one of the tumor-bearing samples of data set 2 were manually microdissected to obtain tumor-adjacent stroma which was used for validation of the diagnostic classifier. For manual microdissection, the tumor-bearing tissue was embedded in an OCT block then mounted in a cryostat. Frozen sections were stained using hematoxylin and eosin (H&E) to visualize the location of the tumor. A border between tumor and adjacent stroma was marked on the glass slide using a Pilot Ultrafine Point Pen which was used as a guide to locate the border on the OCT block surface. Then the OCT-embedded block was etched with a single straight cut with a scalpel (∼1-mm deep) to divide the embedded tissue into a tumor zone and tumor-adjacent stroma. Subsequent cryosections produced 2 halves at the site of the etched cut and were separately used for H&E staining and examined to confirm their composition. Multiple subsequent frozen sections of the tumor-adjacent stroma half were then pooled and used for RNA preparation and microarray hybridization. A final frozen section was used for H&E staining and examined to confirm that the tumor-adjacent stroma remained free of tumor cells.

### Statistical tools implemented in R

The U133 Plus 2.0 platform used for data set 1 has about 55,000 probe sets whereas the U133A used for data sets 2, 3, and 4, contains 22,000 probe sets. Normalization was carried out across multiple data sets using the ∼22,000 probe sets in common to all data sets. First, data set 1 was quantile-normalized using the function “normalizeQuantiles” of LIMMA (Linear Models for Microarray Data) routine (25). Data sets 2 to 4 were then quantile-normalized by referencing normalized data set 1 using a modified function “REFnormalizeQuantiles” which was coded by Z.J. and is available at the SPECS Web site (26).

The LIMMA package from Bioconductor was used to detect differentially expressed genes.

Prediction Analysis of Microarray (PAM; ref. 27), implemented in R, was used to develop an expression-based classifier from the training sets and then applied to the test sets without further change.

A multiple linear regression (MLR) model was used to fit gene expression data and known percent cell-type composition for 4 cell types to estimate expression coefficients for each cell component (see Supplement for details). Percent cell-type distributions were estimated by 3 (data set 1) or 4 (data set 2) pathologists and exhibited an overall agreement of 4.3% SD for the 4 estimated cell types. The resulting significantly differentially expressed genes for the comparison of normal prostate biopsies to tumor-bearing prostate tissue were used for development of the diagnostic classifier.

### Development of a stroma-derived diagnostic classifier

We hypothesized that stroma within and directly adjacent to prostate cancer epithelial cells exhibits significant RNA expression changes compared with normal prostate stroma. To generate candidate biomarkers, we developed a 3-step strategy. First, we identified genes that are differentially expressed between tumor-adjacent stroma and normal stroma. Second, these differences were filtered by removing the age-related genes and removing the genes that are also expressed in tumor cells to create a stroma-specific set of differentially expressed genes. Finally, owing to the limited number of normal biopsies, we repeated steps 1 and 2 using a permutation procedure which greatly enhanced the extraction of information in the normal biopsies.

In step 1, Affymetrix gene expression data were acquired from normal frozen biopsies from each of the 15 subjects that were judged to be free of cancer by histologic examination of the 6 cores of the volunteer biopsies (21). Data from 13 of these samples (with 2 held in reserve as explained later) were compared with the gene expression data for 13 tumor-bearing patient cases from data set 1 selected with tumor cell content (T) greater than 0% but less than 10% (the average stroma content is ∼80%). These criteria ensured that the majority of stroma tissues included from the cancer-positive patients was close to tumor, whereas T less than 10% ensured that the impact from tumor cells is minimal to allow capture of altered expression signals from stroma cells rather than tumor cells. Using a moderated t test implemented in the LIMMA package of R (25), this comparison yielded 3,888 significant expression changes between these 2 groups with a value of P < 0.05. We used a relatively relaxed P value cutoff for the first step of feature selection to allow more genes to enter subsequent screening steps. The 3,888 probe sets were composed of a nearly equal number of up- and downregulated genes.

There was a substantial difference in age between the normal stroma group (average age = 51.9 years) and the near-tumor stroma group (average age = 60.6 years). In step 2, we compared the overall gene expression of the 13 normal stroma samples used for training versus 13 normal prostate specimens obtained by rapid autopsy (see “Materials and Methods”) with an average age of 82 years. Prostate glands from the rapid autopsy series with an average age of 84 years exhibited a markedly increased heterogeneity of gland shapes with stroma containing increased fibroblast and myofibroblast-like cells. The comparison revealed 8,898 significant expression changes (P < 0.05). Of these probe sets, 1,678 were also detected in the comparison of normal stroma samples to stroma near tumor. After eliminating all of these potential aging-related genes, the remaining 2,210 probe sets consisted of nearly equal numbers of up- and downregulated genes.

It remained likely that some differential expression in this comparison included expression changes specific to the residual tumor cells or epithelium cells in some samples, rather than changes between more than one type of stromal cell. To reduce the possibility that epithelial cell–derived expression changes might influence subsequent results, we removed genes that appeared to be expressed in tumor at 10% or more of the expression in stroma. However, even “pure” tumor samples are contaminated with stroma thereby risking the elimination of genes expressed only in stroma. So, identification of genes expressed in tumor was achieved using MLR analysis (described in “Materials and Methods” and Supplement). The percent cell composition of 108 samples from 87 patients in data set 1 intentionally encompassing a wide range of tissue percentages was determined by a panel of 3 pathologists (20). The distribution is shown in Figure 1A. Model diagnostics showed that the fitted model for genes significantly expressed in tumor or stroma accounted for more than 70% of the total variation (i.e., the variation of error was less than 30% of the total variation), indicating a plausible modeling scheme.

Figure 1.

Histogram of tumor percentage for data sets 1 to 4. The tumor percentage data of (A) and (B) were provided by SPECS pathologists, whereas the tumor percentage data of (C) and (D) were estimated by CellPred program (29). The stars in (A) mark the tumor percentages of the misclassified tumor-bearing cases in data set 1, which CellPred indicates may actually be nontumor samples.

Figure 1.

Histogram of tumor percentage for data sets 1 to 4. The tumor percentage data of (A) and (B) were provided by SPECS pathologists, whereas the tumor percentage data of (C) and (D) were estimated by CellPred program (29). The stars in (A) mark the tumor percentages of the misclassified tumor-bearing cases in data set 1, which CellPred indicates may actually be nontumor samples.

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Of the 2,210 probe sets, derived above, we obtained 160 probe sets that were predominantly expressed in stroma cells and also show differential expression between near-tumor stroma and normal stroma. The average expression of these 160 probe sets was estimated to be more than 2-fold greater than the average of all genes expressed in stroma, which is a consequence for the filtering steps for robustness, and also favors good sensitivity.

Finally in step 3, a permutation analysis was performed. The above procedure for the generation of differentially expressed genes between 13 of the 15 normal stroma biopsies and the 13 biopsies of stroma near tumor was repeated using a different selection of 13 biopsy samples from 15, until all 105 possible combinations of 13 normal biopsy samples drawn from 15 samples (|$C_{15}^{13} = 105$|, where |$C_n^m$| is the number of combinations of m elements chosen from a total of n elements) was complete. After filtering for genes associated with aging (discussed earlier), a total of 339 probe sets that were differentially expressed between stroma near tumor and normal stroma were generated by the 105-fold gene selection procedure (the frequency of selection is summarized in Supplementary Fig. S1). Thus, the permutation increased the basis set by 339/160 or more than 2-fold. One hundred forty-six probe sets with at least 50 occurrences in the 105-fold permutation were selected for classifier construction (listed in Table 3)

PAM (28) was used to build a diagnostic classifier. The training set (Table 2, line 1) included all the 15 normal biopsies and the initial 13 samples of stroma near tumor. Of the 146 PAM input probe sets, 131 probe sets–corresponding to 114 genes—were retained following the 10-fold cross-validation procedure of PAM (28) leading to a prediction accuracy of 96% (Table 2). Supplementary Figure S2 presents a “heatmap” of the relative expression of the 131 probe sets among all training samples. The separation of normal and near-tumor stroma samples of the training set by the classifier is illustrated by the 2 distinct populations shown in Figure 2.

Figure 2.

Plot of 2 principal components of training cases using the 131 probe set diagnostic classifier.

Figure 2.

Plot of 2 principal components of training cases using the 131 probe set diagnostic classifier.

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Table 2.

Operating characteristics for training and testing

LineData setSample no.Accuracy, %
1. Training set128 (15 + 13)96.4
1. Test set
Tumor
2. Tumor-bearing 55a 96.4
3. Tumor-bearing 65 100
4. Tumor-bearing 79 100
5. Tumor-bearing 44 100
Normal
6. Biopsies (1) 100
7. Biopsies (2) 60
8. Rapid autopsies 13 92.3
Microdissected
9. Stroma adjacent to tumor 71 97.1
10. Stroma adjacent to tumor 13 100
11. Stroma close to tumor 12 75
12. Stroma > 15 mm from tumor 28 35.7
LineData setSample no.Accuracy, %
1. Training set128 (15 + 13)96.4
1. Test set
Tumor
2. Tumor-bearing 55a 96.4
3. Tumor-bearing 65 100
4. Tumor-bearing 79 100
5. Tumor-bearing 44 100
Normal
6. Biopsies (1) 100
7. Biopsies (2) 60
8. Rapid autopsies 13 92.3
Microdissected
9. Stroma adjacent to tumor 71 97.1
10. Stroma adjacent to tumor 13 100
11. Stroma close to tumor 12 75
12. Stroma > 15 mm from tumor 28 35.7

aFifty-five test samples is less than the potential 68 of Table 1 owing to the use of 15 samples for training (line 1).

### Testing with independent data sets

The 131 probe set classifier was then tested on 243 samples that had not been used for training, and that all contained tumor, though usually very little tumor (Table 2, lines 2–5). Almost all the 243 samples were recognized as being from cancer patients with high average accuracy of about 99% (see Supplementary Table S1 for derived operating characteristics). Only 2 cases were misclassified. In Figure 1A, the 2 misclassified test are marked with asterisks. Although these samples are ostensibly given tumor percentages of 20% and 25% by pathologists, they are predicted to possibly contain little or no tumor using the CellPred program which estimates the tissue components using an in silico multiple-variate linear regression model (29). It is possible that these 2 exceptions were archived incorrectly and are not from patients with cancer or are from a very distant location relative to the tumor.

We examined whether the PAM classification results correlated with cell composition (Fig. 1). For the test cases of data sets 1 and 2, these values are known from the pathologists estimates, whereas for data sets 3 and 4 (Fig. 1C and D, respectively), these tumor cell contents were estimated using the CellPRed program (29). Examining the tumor cell percentages in all the samples in Figure 1, it is clear that the PAM classification is successful on independent test samples with a broad range of tumor epithelial cells including samples with just a few percent of epithelial cells. These observations argue that the classifier is accurate in the categorization of prostate cancer cases independent of the presence or amount of the tumor epithelial component.

The classifier was then tested using specimens composed of normal prostate stroma and epithelium. Twelve biopsies from the DFMO study, all of them different from the 15 samples used earlier for training, were separated into 2 groups. In group 1, there were 7 second biopsies from the same participants whose first biopsy samples were included in the training set, taken 12 months later. These were accurately (100%) identified as nontumor (Table 2, line 6). In group 2, there were 5 biopsy samples not from subjects previously used for training. Two of these 5 biopsy samples were categorized as being from cancer patients (Table 2, line 7). When the histories for these volunteers were investigated, it was found that both donors had consistently exhibited elevated prostate-specific antigen (PSA) levels of 6.1 and 8 ng/mL (normal values < 3 ng/mL), respectively, although no tumor was observed in either of 2 sets of sextant biopsies obtained from these volunteers. The volunteers also had a history of prostate cancer in the family. All other donors of the normal biopsy volunteers exhibited normal PSA values. The IRB-approved protocol precluded following up further to establish that these patients had cancer that had been missed in the biopsies.

The classifier was then tested on 13 specimens obtained by rapid autopsy of individuals dying of unrelated causes (Table 2, line 8). Twelve of 13 of these samples, 92% accuracy, were classified as nontumor. Histologic examination of all embedded tissue of the one “misclassified” case revealed multiple foci of small “latent” tumors.

In summary, 25 nominally normal samples were classified as being from donors without prostate cancer or were classified in accordance with abnormal features that were subsequently uncovered. These results provide further support for the ability of the classifier to discriminate among normal and abnormal prostate tissue in the absence of histologically recognizable tumor cells in the samples studied.

### Validation by manual microdissection, random classifiers, and the published literature

We sought to validate the classifier by developing histologic confirmed samples of stroma adjacent to tumor. An etching procedure was used to prepare 71 samples of tumor-adjacent stroma from patient tissues of data set 2, and 13 samples from data set 4. An additional 12 samples from data set 1 were obtained from OCT blocks entirely by manual microdissection, that is, without etching but leaving a margin of tissue between tumor and stroma, followed by histologically examining frozen section analysis of the OCT surface and bottom side of the pieces, to ensure the absence of tumor. These 12 manually excised pieces are termed “close stroma” (∼3 mm). The expression values for all 96 samples were used to test the 131 probe set classifier using the PAM procedure. The accuracy in classifying that the samples were from patients with tumor was 97% for the 71 adjacent stroma samples from data set 2, 100% for 13 adjacent stroma samples from data set 4, and 75% for the 12 “close” stroma samples from data set 1 (Table 2, lines 9–11). This is an overall accuracy of 95% for the 96 independent samples.

Five of the 96 samples appeared “misclassified” as normal. Three of these misclassifications were among the 12 “close” stroma samples in data set 1. These 12 samples were obtained by manual excision and therefore some of the samples may not have been as near to tumor as the samples obtained by the etching method. Therefore, we examined how far the expression changes characteristic of tumor stroma may extend away from the tumor; we obtained 28 samples greater than 15 mm from any known tumor and generally from the contralateral lobe (Table 2, line 12). Only 10 of the 28 samples (36%) were categorized as tumor-associated stroma. Using the Fisher's exact test, the distribution for the 28 “remote” samples was significantly different from the 12 stroma samples from “close” to tumor of the same patient tissues (P = 0.038). This result, as well as the observation of a gradient of classification frequency values from 98%, 75%, and 36% for samples adjacent, close, and greater than 15 mm from tumor, suggests that the expression changes recognized by the classifier decline with increasing distance of stroma from tumor. Such observation bears on the likely mechanism for the production of differential gene expression in tumor-adjacent stroma which is generally believed to involve the influence of “paracrine” factors emanating from tumor foci (10, 30, 31).

We found that the normal samples and rapid autopsy samples can be easily distinguished from samples containing tumor using many of the individual genes (e.g., heatmap, Supplementary Fig. S3). However, the differences that allow near stroma to be distinguished from control stroma are more subtle and vary between patients, requiring a classifier based on a number of genes.

Further validation included a comparison with 100 random classifiers generated by arbitrarily sampling 131 probe sets for each classifier. The results (Supplementary Table S1 and Supplement) showed that these random classifiers had no diagnostic value, further indicating that the results obtained with the 131 probe set classifier cannot be attributed to chance.

Finally, we sought to validate that representative genes were in fact preferentially expressed in stroma by PCR. In addition, to test the translational relevance, we utilized independent cases from a formalin-fixed and paraffin-embedded (FFPE) clinical collection. Gene expression was assessed by a modified quantitative PCR procedure (see “Materials and Methods”). In a limited survey, 4 genes were found to have reliably preserved short amplicons. Blocks of 63 tumor cases were examined and tumor and stroma regions in H&E sections were demarcated by a pathologist (D.A.M.). Punches were removed from adjacent unstained sections and used for PCR for 63 tumor portions and 38 stroma portions. For all 4 genes, highly significant preferential expression in stroma was observed (Supplementary Table S4). These results for independent cases and by an independent method further support the preferential expression of these genes in tumor stroma and further argue that the classifier may be adapted to clinical biopsies preserved in FFPE, the standard method of archiving patient biopsies.

Finally, we also reviewed 2 recent studies describing expression analysis results for subclasses of the stroma of prostate cancer (16, 17), which showed consistent findings (see Supplement). In particular, the 339 probe sets (Affymetrix arrays) we identified map to 557 genes on Agilent arrays which have been used for deriving profiles for “reactive” stroma, a special case of adjacent stroma associated with poor outcome disease (17). A total of 31 genes or probe sets appeared to be concordant (in terms of gene identity and the direction of expression alteration) between the 339 probe sets (Affymetrix arrays) we identified in this study and the 557 mapped genes (Agilent arrays) in the “reactive” stroma study (17) with P value of 0.0001 (Supplementary Table S2). The formation of reactive stroma has been associated with poor prognosis (32). Thus, it is possible that some diagnostic markers in stroma could also be of prognostic interest.

We compared the expression profiles of 15 normal biopsy samples and 13 tumor-adjacent stroma samples from prostatectomies using a permutation strategy to enhance detection of significant differences. About 3,800 significant gene expression changes were observed, which were then filtered to exclude genes known to be expressed at similar levels in epithelial tumor cells and to remove genes that change with age. The top-ranked 146 probe sets remaining after applying these filters were used for the 10-fold cross-validation procedure of PAM using the same 28 samples used for the initial training. The PAM procedure led to a 131 probe set classifier, which had a training accuracy of 96%. We then tested the classifier on a number of independent expression microarray data sets of tumor-bearing tissue including data from 110 samples generated by us (Table 2, data sets 1 and 2) and data from 123 samples generated elsewhere (Table 2, data sets 3 and 4). These samples were classified as being from cancer patients with an overall accuracy of 98%, a value compares favorably with the diagnostic accuracy of PSA-based methods of about 70% (33). Only 2 samples recorded as containing tumor cells were misclassified. Upon further investigation of these 2 samples using CellPred, a method to determine the tumor percentage of samples based solely on their expression profile (29), these samples were predicted to have little or no tumor, although they had been recorded as having more than 20% tumor, indicating their assignment as tumor may have been a bookkeeping error. Similarly, we generated data from 25 samples of normal prostate, which were recognized as nontumor with an accuracy of 92%. Only 3 samples were “misclassified”. Two of these samples were biopsies donated by men with abnormally high PSA levels and a family history of prostate cancer, although no tumor was recognized in any of the sextant biopsies taken at the beginning and end of the study period for which these volunteers were controls. In addition, one sample derived from the rapid autopsy donors was potentially “misclassified” as cancerous. Examination of multiple blocks of the glands taken from both lobes and all zones revealed tumor foci in the misclassified case. Thus, the “misclassifications” correlate well with the unusual clinical and pathologic features of the cases. In summary, the handful of misclassifications of tumor and normal each had evidence that they had been mislabeled before the test, potentially raising the actual sensitivity and specificity for classifying these samples to 100%.

Finally, for validation, we used 153 samples from data sets 1 and 4 to prepare “pure” stroma adjacent, close, and far (>15 mm) from known tumor foci. These data sets were able to detect the presence of tumor in the prostate with a decreasing accuracy of 98%, 75%, and 36%, respectively. The observation of a gradual reduction in the sensitivity of the classifier as the distance increases bears on the likely mechanism for the production of differential gene expression in tumor-adjacent stroma which is generally believed to involve the influence of “paracrine” factors emanating from tumor foci (10, 30, 31). Indeed, the tumor microenvironment is likely the source of factors that are required for tumor formation by the epithelial component (10). The amount of diffusible paracrine factors of this complex interaction mechanism likely declines with separation of target cells from the secreting cells. Indeed, a simple radial dilution model would predict a decline of effects of tumor-derived factors by at least the square of the distance of target stroma cells from a tumor focus. On the basis of this simple model, the decrease in the frequency of categorization stroma taken from more than 15 mm from a known tumor focus to 36% suggests a 50% recognition distance of about 13 mm in fresh frozen tissue. In view of the modest average fold-change of the 131 probe sets of the classifier (Table 3), the distance at which “presence of tumor” is recognized suggests a surprisingly large range of “influence” of tumor more than steady-state gene expression changes in nearby stroma. Systematic studies of differential expression as a function of known distances will be required to confirm and refine this inference.

Table 3.

One hundred forty-six diagnostic probe sets with incidence number greater than 50 for 105-fold gene selection procedure

Probe setGene symbolGene titleLogFCa
213764_s_at MFAP5 Microfibrillar associated protein 5 −1.73
209758_s_at MFAP5 Microfibrillar associated protein 5 −1.48
213765_at MFAP5 Microfibrillar associated protein 5 −1.36
210280_at MPZ Myelin protein zero (Charcot-Marie-Tooth neuropathy 1B) −1.20
210198_s_at PLP1 Proteolipid protein 1 (Pelizaeus-Merzbacher disease, spastic paraplegia 2, uncomplicated) −1.18
215104_at NRIP2 Nuclear receptor interacting protein 2 −0.94
213847_at PRPH Peripherin −0.93
214767_s_at HSPB6 Heat shock protein, alpha-crystallin-related, B6 −0.88
209843_s_at SOX10 SRY (sex determining region Y)-box 10 −0.61
209686_at S100B S100 calcium binding protein B −0.94
209915_s_at NRXN1 Neurexin 1 −0.80
214023_x_at TUBB2B Tubulin, beta 2B −0.75
214954_at SUSD5 Sushi domain containing 5 −0.98
204584_at L1CAM L1 cell adhesion molecule −1.20
204777_s_at MAL Mal, T-cell differentiation protein −0.99
205132_at ACTC1 Actin, alpha, cardiac muscle 1 −0.99
203151_at MAP1A Microtubule-associated protein 1A −0.69
210869_s_at MCAM Melanoma cell adhesion molecule −0.71
204627_s_at ITGB3 Integrin, beta 3 (platelet glycoprotein IIIa, antigen CD61) −0.82
209086_x_at MCAM Melanoma cell adhesion molecule −0.61
219314_s_at ZNF219 Zinc finger protein 219 −0.51
221204_s_at CRTAC1 Cartilage acidic protein 1 −0.56
212886_at CCDC69 Coiled-coil domain containing 69 −0.59
210814_at TRPC3 Transient receptor potential cation channel, subfamily C, member 3 −0.75
212793_at DAAM2 Dishevelled associated activator of morphogenesis 2 −0.56
212565_at STK38L Serine/threonine kinase 38 like −0.58
214606_at TSPAN2 Tetraspanin 2 −0.54
336_at TBXA2R Thromboxane A2 receptor −0.65
218660_at DYSF Dysferlin, limb girdle muscular dystrophy 2B (autosomal recessive) −0.55
214434_at HSPA12A Heat shock 70-kDa protein 12A −0.57
212274_at LPIN1 Lipin 1 −0.48
206874_s_at  – −0.44
203939_at NT5E 5′-nucleotidase, ecto (CD73) −0.49
205954_at RXRG Retinoid X receptor, gamma −0.53
219909_at MMP28 Matrix metallopeptidase 28 −0.54
206425_s_at TRPC3 Transient receptor potential cation channel, subfamily C, member 3 −0.57
205433_at BCHE Butyrylcholinesterase −0.93
35846_at THRA Thyroid hormone receptor, alpha (erythroblastic leukemia viral (v-erb-a) oncogene homolog, avian) −0.46
204736_s_at CSPG4 Chondroitin sulfate proteoglycan 4 −0.55
202806_at DBN1 Drebrin 1 −0.43
212097_at CAV1 Caveolin 1, caveolae protein, 22kDa −0.38
201841_s_at HSPB1 Heat shock 27-kDa protein 1 −0.44
206382_s_at BDNF Brain-derived neurotrophic factor −0.62
219091_s_at MMRN2 Multimerin 2 −0.44
205076_s_at MTMR11 Myotubularin-related protein 11 −0.57
204159_at CDKN2C Cyclin-dependent kinase inhibitor 2C (p18, inhibits CDK4) −0.46
212992_at AHNAK2 AHNAK nucleoprotein 2 −0.60
206024_at HPD 4-hydroxyphenylpyruvate dioxygenase −0.57
218094_s_at DBNDD2 /// SYS1-DBNDD2 Dysbindin (dystrobrevin binding protein 1) domain containing 2 /// SYS1-DBNDD2 −0.41
211276_at TCEAL2 Transcription elongation factor A (SII)-like 2 −0.52
209191_at TUBB6 Tubulin, beta 6 −0.51
213675_at  CDNA FLJ25106 fis, clone CBR01467 −0.44
211340_s_at MCAM Melanoma cell adhesion molecule −0.46
210632_s_at SGCA Sarcoglycan, alpha (50-kDa dystrophin- associated glycoprotein) −0.58
218651_s_at LARP6 La ribonucleoprotein domain family, member 6 −0.34
207876_s_at FLNC Filamin C, gamma (actin binding protein 280) −0.45
218877_s_at TRMT11 tRNA methyltransferase 11 homolog (S. cerevisiae+0.44
219416_at SCARA3 Scavenger receptor class A, member 3 −0.57
209981_at CSDC2 Cold shock domain containing C2, RNA binding −0.56
214212_x_at FERMT2 Fermitin family homolog 2 (Drosophila) −0.42
207554_x_at TBXA2R Thromboxane A2 receptor −0.44
205231_s_at EPM2A Epilepsy, progressive myoclonus type 2A, Lafora disease (laforin) −0.42
215306_at  mRNA; cDNA DKFZp586N2020 (from clone DKFZp586N2020) −0.48
218435_at DNAJC15 DNAJ (Hsp40) homolog, subfamily C, member 15 −0.49
203597_s_at WBP4 WW domain binding protein 4 (formin binding protein 21) −0.34
205303_at KCNJ8 Potassium inwardly-rectifying channel, subfamily J, member 8 −0.42
201389_at ITGA5 Integrin, alpha 5 (fibronectin receptor, alpha polypeptide) −0.50
204940_at PLN Phospholamban −0.49
220765_s_at LIMS2 LIM and senescent cell antigen-like domains 2 −0.41
203299_s_at AP1S2 Adaptor-related protein complex 1, sigma 2 subunit −0.41
201344_at UBE2D2 Ubiquitin-conjugating enzyme E2D 2 (UBC4/5 homolog, yeast) −0.38
218648_at CRTC3 CREB-regulated transcription coactivator 3 −0.33
204939_s_at PLN Phospholamban −0.45
201431_s_at DPYSL3 Dihydropyrimidinase-like 3 −0.40
215534_at  mRNA; cDNA DKFZp586C1923 (from clone DKFZp586C1923) −0.46
209169_at GPM6B Glycoprotein M6B −0.34
209651_at TGF-B1I1 Transforming growth factor beta 1 induced transcript 1 −0.42
218711_s_at SDPR Serum deprivation response (phosphatidylserine binding protein) +0.41
212358_at CLIP3 CAP-GLY domain containing linker protein 3 −0.47
218691_s_at PDLIM4 PDZ and LIM domain 4 −0.42
218266_s_at FREQ Frequenin homolog (Drosophila) −0.46
210319_x_at MSX2 Msh homeobox 2 +0.45
218545_at CCDC91 Coiled-coil domain containing 91 −0.31
44702_at SYDE1 Synapse defective 1, Rho GTPase, homolog 1 (C. elegans) −0.38
221014_s_at RAB33B RAB33B, member RAS oncogene family −0.38
221246_x_at TNS1 Tensin 1 −0.27
208789_at PTRF Polymerase I and transcript release factor −0.42
220722_s_at SLC5A7 Solute carrier family 5 (choline transporter), member 7 −0.41
209087_x_at MCAM Melanoma cell adhesion molecule −0.40
221667_s_at HSPB8 Heat shock 22kDa protein 8 −0.40
205561_at KCTD17 Potassium channel tetramerisation domain containing 17 −0.32
213808_at  Clone 23688 mRNA sequence −0.43
202565_s_at SVIL Supervillin −0.36
211964_at COL4A2 Collagen, type IV, alpha 2 −0.39
219563_at C14orf139 Chromosome 14 open reading frame 139 −0.38
214122_at PDLIM7 PDZ and LIM domain 7 (enigma) −0.30
212589_at RRAS2 Related RAS viral (r-ras) oncogene homolog 2 −0.29
205973_at FEZ1 Fasciculation and elongation protein zeta 1 (zygin I) −0.35
218818_at FHL3 Four and a half LIM domains 3 −0.36
212120_at RHOQ Ras homolog gene family, member Q −0.31
219073_s_at OSBPL10 Oxysterol binding protein-like 10 −0.37
221480_at HNRNPD Heterogeneous nuclear ribonucleoprotein D (AU-rich element RNA binding protein 1, 37kDa) −0.36
207071_s_at ACO1 Aconitase 1, soluble −0.27
211717_at ANKRD40 Ankyrin repeat domain 40 −0.28
201313_at ENO2 Enolase 2 (gamma, neuronal) −0.36
204628_s_at ITGB3 Integrin, beta 3 (platelet glycoprotein IIIa, antigen CD61) −0.31
204303_s_at KIAA0427 KIAA0427 −0.35
214439_x_at BIN1 Bridging integrator 1 −0.29
209015_s_at DNAJB6 DNAJ (Hsp40) homolog, subfamily B, member 6 −0.29
213547_at CAND2 Cullin-associated and neddylation-dissociated 2 (putative) −0.31
204058_at ME1 Malic enzyme 1, NADP(+)-dependent, cytosolic −0.34
219902_at BHMT2 Betaine-homocysteine methyltransferase 2 −0.33
214306_at OPA1 Optic atrophy 1 (autosomal dominant) −0.27
210201_x_at BIN1 Bridging integrator 1 −0.29
212509_s_at MXRA7 Matrix-remodelling associated 7 −0.27
213231_at DMWD Dystrophia myotonica, WD repeat containing −0.30
201843_s_at EFEMP1 EGF-containing fibulin-like extracellular matrix protein 1 −0.32
206289_at HOXA4 Homeobox A4 −0.29
203501_at PGCP Plasma glutamate carboxypeptidase −0.30
216894_x_at CDKN1C Cyclin-dependent kinase inhibitor 1C (p57, Kip2) −0.27
216500_at  HL14 gene encoding beta-galactoside-binding lectin, 3′ end, clone 2 −0.29
220050_at C9orf9 Chromosome 9 open reading frame 9 −0.32
209362_at MED21 Mediator complex subunit 21 −0.26
202931_x_at BIN1 Bridging integrator 1 −0.27
213480_at VAMP4 Vesicle-associated membrane protein 4 −0.24
205611_at TNFSF12 Tumor necrosis factor (ligand) superfamily, member 12 −0.29
204365_s_at REEP1 Receptor accessory protein 1 −0.29
203389_at KIF3C Kinesin family member 3C −0.26
205368_at FAM131B Family with sequence similarity 131, member B −0.27
217066_s_at DMPK Dystrophia myotonica-protein kinase −0.29
212457_at TFE3 Transcription factor binding to IGHM enhancer 3 −0.25
200685_at SFRS11 Splicing factor, arginine/serine-rich 11 −0.16
200788_s_at PEA15 Phosphoprotein enriched in astrocytes 15 −0.22
202522_at PITPNB Phosphatidylinositol transfer protein, beta −0.16
208869_s_at GABARAPL1 GABA(A) receptor-associated protein like 1 −0.19
209524_at HDGFRP3 Hepatoma-derived growth factor, related protein 3 −0.14
211347_at CDC14B CDC14 cell division cycle 14 homolog B (S. cerevisiae−0.21
212610_at PTPN11 Protein tyrosine phosphatase, non-receptor type 11 (Noonan syndrome 1) −0.23
212848_s_at C9orf3 Chromosome 9 open reading frame 3 −0.27
214643_x_at BIN1 Bridging integrator 1 −0.23
217820_s_at ENAH Enabled homolog (Drosophila) −0.19
218597_s_at CISD1 CDGSH iron sulfur domain 1 −0.18
221502_at KPNA3 Karyopherin alpha 3 (importin alpha 4) −0.20
222221_x_at EHD1 EH-domain containing 1 −0.20
32625_at NPR1 Natriuretic peptide receptor A/guanylate cyclase A (atrionatriuretic peptide receptor A) −0.22
Probe setGene symbolGene titleLogFCa
213764_s_at MFAP5 Microfibrillar associated protein 5 −1.73
209758_s_at MFAP5 Microfibrillar associated protein 5 −1.48
213765_at MFAP5 Microfibrillar associated protein 5 −1.36
210280_at MPZ Myelin protein zero (Charcot-Marie-Tooth neuropathy 1B) −1.20
210198_s_at PLP1 Proteolipid protein 1 (Pelizaeus-Merzbacher disease, spastic paraplegia 2, uncomplicated) −1.18
215104_at NRIP2 Nuclear receptor interacting protein 2 −0.94
213847_at PRPH Peripherin −0.93
214767_s_at HSPB6 Heat shock protein, alpha-crystallin-related, B6 −0.88
209843_s_at SOX10 SRY (sex determining region Y)-box 10 −0.61
209686_at S100B S100 calcium binding protein B −0.94
209915_s_at NRXN1 Neurexin 1 −0.80
214023_x_at TUBB2B Tubulin, beta 2B −0.75
214954_at SUSD5 Sushi domain containing 5 −0.98
204584_at L1CAM L1 cell adhesion molecule −1.20
204777_s_at MAL Mal, T-cell differentiation protein −0.99
205132_at ACTC1 Actin, alpha, cardiac muscle 1 −0.99
203151_at MAP1A Microtubule-associated protein 1A −0.69
210869_s_at MCAM Melanoma cell adhesion molecule −0.71
204627_s_at ITGB3 Integrin, beta 3 (platelet glycoprotein IIIa, antigen CD61) −0.82
209086_x_at MCAM Melanoma cell adhesion molecule −0.61
219314_s_at ZNF219 Zinc finger protein 219 −0.51
221204_s_at CRTAC1 Cartilage acidic protein 1 −0.56
212886_at CCDC69 Coiled-coil domain containing 69 −0.59
210814_at TRPC3 Transient receptor potential cation channel, subfamily C, member 3 −0.75
212793_at DAAM2 Dishevelled associated activator of morphogenesis 2 −0.56
212565_at STK38L Serine/threonine kinase 38 like −0.58
214606_at TSPAN2 Tetraspanin 2 −0.54
336_at TBXA2R Thromboxane A2 receptor −0.65
218660_at DYSF Dysferlin, limb girdle muscular dystrophy 2B (autosomal recessive) −0.55
214434_at HSPA12A Heat shock 70-kDa protein 12A −0.57
212274_at LPIN1 Lipin 1 −0.48
206874_s_at  – −0.44
203939_at NT5E 5′-nucleotidase, ecto (CD73) −0.49
205954_at RXRG Retinoid X receptor, gamma −0.53
219909_at MMP28 Matrix metallopeptidase 28 −0.54
206425_s_at TRPC3 Transient receptor potential cation channel, subfamily C, member 3 −0.57
205433_at BCHE Butyrylcholinesterase −0.93
35846_at THRA Thyroid hormone receptor, alpha (erythroblastic leukemia viral (v-erb-a) oncogene homolog, avian) −0.46
204736_s_at CSPG4 Chondroitin sulfate proteoglycan 4 −0.55
202806_at DBN1 Drebrin 1 −0.43
212097_at CAV1 Caveolin 1, caveolae protein, 22kDa −0.38
201841_s_at HSPB1 Heat shock 27-kDa protein 1 −0.44
206382_s_at BDNF Brain-derived neurotrophic factor −0.62
219091_s_at MMRN2 Multimerin 2 −0.44
205076_s_at MTMR11 Myotubularin-related protein 11 −0.57
204159_at CDKN2C Cyclin-dependent kinase inhibitor 2C (p18, inhibits CDK4) −0.46
212992_at AHNAK2 AHNAK nucleoprotein 2 −0.60
206024_at HPD 4-hydroxyphenylpyruvate dioxygenase −0.57
218094_s_at DBNDD2 /// SYS1-DBNDD2 Dysbindin (dystrobrevin binding protein 1) domain containing 2 /// SYS1-DBNDD2 −0.41
211276_at TCEAL2 Transcription elongation factor A (SII)-like 2 −0.52
209191_at TUBB6 Tubulin, beta 6 −0.51
213675_at  CDNA FLJ25106 fis, clone CBR01467 −0.44
211340_s_at MCAM Melanoma cell adhesion molecule −0.46
210632_s_at SGCA Sarcoglycan, alpha (50-kDa dystrophin- associated glycoprotein) −0.58
218651_s_at LARP6 La ribonucleoprotein domain family, member 6 −0.34
207876_s_at FLNC Filamin C, gamma (actin binding protein 280) −0.45
218877_s_at TRMT11 tRNA methyltransferase 11 homolog (S. cerevisiae+0.44
219416_at SCARA3 Scavenger receptor class A, member 3 −0.57
209981_at CSDC2 Cold shock domain containing C2, RNA binding −0.56
214212_x_at FERMT2 Fermitin family homolog 2 (Drosophila) −0.42
207554_x_at TBXA2R Thromboxane A2 receptor −0.44
205231_s_at EPM2A Epilepsy, progressive myoclonus type 2A, Lafora disease (laforin) −0.42
215306_at  mRNA; cDNA DKFZp586N2020 (from clone DKFZp586N2020) −0.48
218435_at DNAJC15 DNAJ (Hsp40) homolog, subfamily C, member 15 −0.49
203597_s_at WBP4 WW domain binding protein 4 (formin binding protein 21) −0.34
205303_at KCNJ8 Potassium inwardly-rectifying channel, subfamily J, member 8 −0.42
201389_at ITGA5 Integrin, alpha 5 (fibronectin receptor, alpha polypeptide) −0.50
204940_at PLN Phospholamban −0.49
220765_s_at LIMS2 LIM and senescent cell antigen-like domains 2 −0.41
203299_s_at AP1S2 Adaptor-related protein complex 1, sigma 2 subunit −0.41
201344_at UBE2D2 Ubiquitin-conjugating enzyme E2D 2 (UBC4/5 homolog, yeast) −0.38
218648_at CRTC3 CREB-regulated transcription coactivator 3 −0.33
204939_s_at PLN Phospholamban −0.45
201431_s_at DPYSL3 Dihydropyrimidinase-like 3 −0.40
215534_at  mRNA; cDNA DKFZp586C1923 (from clone DKFZp586C1923) −0.46
209169_at GPM6B Glycoprotein M6B −0.34
209651_at TGF-B1I1 Transforming growth factor beta 1 induced transcript 1 −0.42
218711_s_at SDPR Serum deprivation response (phosphatidylserine binding protein) +0.41
212358_at CLIP3 CAP-GLY domain containing linker protein 3 −0.47
218691_s_at PDLIM4 PDZ and LIM domain 4 −0.42
218266_s_at FREQ Frequenin homolog (Drosophila) −0.46
210319_x_at MSX2 Msh homeobox 2 +0.45
218545_at CCDC91 Coiled-coil domain containing 91 −0.31
44702_at SYDE1 Synapse defective 1, Rho GTPase, homolog 1 (C. elegans) −0.38
221014_s_at RAB33B RAB33B, member RAS oncogene family −0.38
221246_x_at TNS1 Tensin 1 −0.27
208789_at PTRF Polymerase I and transcript release factor −0.42
220722_s_at SLC5A7 Solute carrier family 5 (choline transporter), member 7 −0.41
209087_x_at MCAM Melanoma cell adhesion molecule −0.40
221667_s_at HSPB8 Heat shock 22kDa protein 8 −0.40
205561_at KCTD17 Potassium channel tetramerisation domain containing 17 −0.32
213808_at  Clone 23688 mRNA sequence −0.43
202565_s_at SVIL Supervillin −0.36
211964_at COL4A2 Collagen, type IV, alpha 2 −0.39
219563_at C14orf139 Chromosome 14 open reading frame 139 −0.38
214122_at PDLIM7 PDZ and LIM domain 7 (enigma) −0.30
212589_at RRAS2 Related RAS viral (r-ras) oncogene homolog 2 −0.29
205973_at FEZ1 Fasciculation and elongation protein zeta 1 (zygin I) −0.35
218818_at FHL3 Four and a half LIM domains 3 −0.36
212120_at RHOQ Ras homolog gene family, member Q −0.31
219073_s_at OSBPL10 Oxysterol binding protein-like 10 −0.37
221480_at HNRNPD Heterogeneous nuclear ribonucleoprotein D (AU-rich element RNA binding protein 1, 37kDa) −0.36
207071_s_at ACO1 Aconitase 1, soluble −0.27
211717_at ANKRD40 Ankyrin repeat domain 40 −0.28
201313_at ENO2 Enolase 2 (gamma, neuronal) −0.36
204628_s_at ITGB3 Integrin, beta 3 (platelet glycoprotein IIIa, antigen CD61) −0.31
204303_s_at KIAA0427 KIAA0427 −0.35
214439_x_at BIN1 Bridging integrator 1 −0.29
209015_s_at DNAJB6 DNAJ (Hsp40) homolog, subfamily B, member 6 −0.29
213547_at CAND2 Cullin-associated and neddylation-dissociated 2 (putative) −0.31
204058_at ME1 Malic enzyme 1, NADP(+)-dependent, cytosolic −0.34
219902_at BHMT2 Betaine-homocysteine methyltransferase 2 −0.33
214306_at OPA1 Optic atrophy 1 (autosomal dominant) −0.27
210201_x_at BIN1 Bridging integrator 1 −0.29
212509_s_at MXRA7 Matrix-remodelling associated 7 −0.27
213231_at DMWD Dystrophia myotonica, WD repeat containing −0.30
201843_s_at EFEMP1 EGF-containing fibulin-like extracellular matrix protein 1 −0.32
206289_at HOXA4 Homeobox A4 −0.29
203501_at PGCP Plasma glutamate carboxypeptidase −0.30
216894_x_at CDKN1C Cyclin-dependent kinase inhibitor 1C (p57, Kip2) −0.27
216500_at  HL14 gene encoding beta-galactoside-binding lectin, 3′ end, clone 2 −0.29
220050_at C9orf9 Chromosome 9 open reading frame 9 −0.32
209362_at MED21 Mediator complex subunit 21 −0.26
202931_x_at BIN1 Bridging integrator 1 −0.27
213480_at VAMP4 Vesicle-associated membrane protein 4 −0.24
205611_at TNFSF12 Tumor necrosis factor (ligand) superfamily, member 12 −0.29
204365_s_at REEP1 Receptor accessory protein 1 −0.29
203389_at KIF3C Kinesin family member 3C −0.26
205368_at FAM131B Family with sequence similarity 131, member B −0.27
217066_s_at DMPK Dystrophia myotonica-protein kinase −0.29
212457_at TFE3 Transcription factor binding to IGHM enhancer 3 −0.25
200685_at SFRS11 Splicing factor, arginine/serine-rich 11 −0.16
200788_s_at PEA15 Phosphoprotein enriched in astrocytes 15 −0.22
202522_at PITPNB Phosphatidylinositol transfer protein, beta −0.16
208869_s_at GABARAPL1 GABA(A) receptor-associated protein like 1 −0.19
209524_at HDGFRP3 Hepatoma-derived growth factor, related protein 3 −0.14
211347_at CDC14B CDC14 cell division cycle 14 homolog B (S. cerevisiae−0.21
212610_at PTPN11 Protein tyrosine phosphatase, non-receptor type 11 (Noonan syndrome 1) −0.23
212848_s_at C9orf3 Chromosome 9 open reading frame 3 −0.27
214643_x_at BIN1 Bridging integrator 1 −0.23
217820_s_at ENAH Enabled homolog (Drosophila) −0.19
218597_s_at CISD1 CDGSH iron sulfur domain 1 −0.18
221502_at KPNA3 Karyopherin alpha 3 (importin alpha 4) −0.20
222221_x_at EHD1 EH-domain containing 1 −0.20
32625_at NPR1 Natriuretic peptide receptor A/guanylate cyclase A (atrionatriuretic peptide receptor A) −0.22

alogFC is the logarithm fold change as tumoros stroma being compared with normal stroma. +/− represents up- or downregulated expression level in tumoros stroma.

The classifier developed here used highly selective methods to enrich for mesodermal and ectodermal derivatives compared with endoderm/epithelial derivatives. Computer-assisted gene enrichment analysis classification using DAVID (34) identified a number of statistically significant gene enrichment categories. The 10 most significant categories are summarized in Supplementary Table S3. Numerous genes associated with expression in nerve and muscle are apparent, such as the nine genes of the actin cytoskeleton enrichment category, and in the disease mutation category including MPZ (Charcot-Maire-Tooth neuropathy 1b), optic atrophy 1, EPM2a (Lafora disease), BDGF, PLN (phospholamban), SGCA (dystophin-associated glycoprotein), and EFEMP. Biochemical associations include genes related to the TGF-β pathway (SMAD3, TGFIT, ID4, and CKDN1C/p57), the Wnt pathway (FZD7, SMAD3, DAAM1, and WISP2) and interacting genes (PCH12, PCDH7, and CDH19). These pathways are associated with tumor–stroma paracrine interactions (16, 17, 32, 35, 36). Given that reactive stroma has been associated with poor prognosis (32), it is possible that some of the 131 diagnostic markers identified in stroma could also be of prognostic interest. Nevertheless, we have not ruled that classifier developed here can distinguish other prostate conditions such as acute and chronic inflammation of the prostate and, therefore, stroma near these lesions may conceivably be misdiagnosed. Additional work with samples containing such lesions could identify genes that distinguish inflammation from cancer.

Our preclinical results suggest practical applications. Assessment of suspicious initial biopsies for expression of the classifier genes was identified here by microarray but could also potentially by any number of other biomarker methods, including those available for assessment of RNA, protein, or epigenetic markers in FFPE samples. Such quantitation may have use in defining “presence of tumor” based solely on the detection of changes in the microenvironment near a focus of tumor by quantitative criteria. Such a method would be applicable to cases with an initial negative biopsy results that would otherwise be referred for rebiopsy owing to the presence of ASAP or PIN. The determination of “presence of tumor” may strengthen guidance for neoadjuvant therapy or prevention therapy or an accelerated scheduling of rebiopsy. Finally, because stroma facilitates tumor growth (10), the expression changes that occur in stroma indicating the presence of tumor might be targets for therapeutic intervention that could leave normal stroma relatively unaffected.

M. McClelland and D. Mercola are cofounders and W. Lernhardt is CEO of Proveri Inc., which is engaged in translational development of aspects of the subject matter. The other authors disclosed no potential conflicts of interest.

Average cell distribution for samples of data set 1 deposited in GEO (GSE17951) were based in part on readings by David Tarin, MD, and Linda Wasserman, MD, PhD. The authors thank Dr. Eileen Adamson for her effort in proofreading the manuscript.

This research was supported by the NIH SPECS Consortium grant U01 CA1148102 and National Cancer Institute Early Detection Research Network (EDRN) Consortium grant U01 CA152738 and the UCI Faculty Career Development Award to Z. Jia. M. McClelland was supported in part by DOD 08-1-0720.

The costs of publication of this article were defrayed in part by the payment of page charges. This article must therefore be hereby marked advertisement in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

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Please use this identifier to cite or link to this item: http://arks.princeton.edu/ark:/88435/dsp01bv73c2942
 Title: Racial Bias in Bail Decisions Authors: Arnold, DavidDobbie, WillYang, Crystal S. Issue Date: May-2017 Series/Report no.: 611 Abstract: This paper develops a new test for identifying racial bias in the context of bail decisions –a high-stakes setting with large disparities between white and black defendants. We motivate our analysis using Becker’s (1957) model of racial bias, which predicts that rates of pre-trial misconduct will be identical for marginal white and marginal black defendants if bail judges are racially unbiased. In contrast, marginal white defendants will have a higher probability of misconduct than marginal black defendants if bail judges are racially biased against blacks. To test the model, we develop a new estimator that uses the release tendencies of quasi-randomly assigned bail judges to identify the relevant race-specific misconduct rates. Estimates from Miami and Philadelphia show that bail judges are racially biased against black defendants, with substantially more racial bias among both inexperienced and part-time judges. We also find that both black and white judges are biased against black defendants. We argue that these results are consistent with bail judges making racially biased prediction errors, rather than being racially prejudiced per se. URI: http://arks.princeton.edu/ark:/88435/dsp01bv73c2942 Appears in Collections: IRS Working Papers

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# Mixture Models

Statistical models that allow for heterogeneity in the population by represeting differences between observations using probability distributions.

## Also known as

• Hierarchical models (although this can have different connotations).
• Random effects models


text:
# (Zeta function)Regularization of a divergent integral

Consider the following integral:

$$I(n)=\int_{0}^1dx\; x^{-(2n-2)} (1-x)^{n-1}$$, where $n \in \mathbb{N}$

The integral can be looked upon as a beta integral and hence can be expressed in terms of Gamma functions like so:

$$I(n)=\frac{\Gamma(3-2n) \Gamma(n)}{\Gamma(3-n)}$$

Now, for $n=1$ the integral is absolutely convergent and evaluates to 1.

But, for $n=2$ it is clearly divergent because of the Gamma function in the numerator.

The interesting situation is for $n \geq 3$. In this case both the numerator and denominator blow up but there is a limit one can take:

$$\lim_{x \to n } I(x)=\frac{(-1)^n}{2} \times \frac{(n-3)!(n-1)!}{(2n-3)!}$$

The above result can be shown by using the residue of the Gamma function at negative integers.

So, does it mean that $I(n)$ is conditionally convergent for $n \geq 3$? Also for $n$ odd like $1,3$ the result is negative even though the integrand is non-negative in the domain of integration!

In fact, for $n=3$, $$\lim_{n \to 3} \frac{\Gamma(3-2n)}{\Gamma(3-n)}=-\frac{1}{12}$$

Is $$\frac{\Gamma(-3)}{\Gamma(0)}\overset{?}{=} 1+2+3+\ldots$$

Edit 1:

I am very sorry. I had missed a minus sign in my integral. The post is corrected.

Edit 2:

This integral is clearly not convergent for $n \geq \frac{3}{2}$ as has been rightly pointed out in one of the answers, in any traditional sense like absolute or even conditional. But there seems to be some notion of the integral version of Ramanujan summation or zeta function regularization that may assign this divergent integral some finite answer. This particular integral actually came about in a quantum field theory calculation and is therefore well-motivated in some sense. It would be great if someone could throw some light on this aspect. I will also change the title of the question from "Is this integral convergent(conditionally)?" to reflect this idea.

• I don't know at the moment where the flaw is, but there have to be one. Because of the monotonicity of the integral, when it is convergent, then it has to be positive. Especially for $n=3$ you get $\int_0^1 x^4(1-x)^2 dx = \int_0^1 x^6-2x^5+x^4 dx = \frac17-\frac13+\frac15=\frac1{105}>0$. – Mundron Schmidt Jun 24 '18 at 6:00
• Isnt it simply $B(2n-1, n)$? so it converges when $2n-1>0 & n>0$ ? – markovchain Jun 24 '18 at 6:02
• @markovchain But what about the negative result? – Subho Jun 24 '18 at 6:03
• I think your $I(n)$ expression in terms of gamma fn is worng as it clearly converges for $n=2$ – markovchain Jun 24 '18 at 6:07
• I don't see how to frame it as a zeta regularisation, but it's a regularisation by analytic continuation. For $0 < n < 3/2$, the integral is obviously $\mathrm{B}(3-2n,n)$, and since that is meromorphic on the whole plane, there's your continuation. – Daniel Fischer Jun 24 '18 at 9:45

Consider that $$\int_0^{1/2}x^\alpha~dx=\begin{cases}\frac1{\alpha+1}\left(\frac12\right)^{\alpha+1} & \alpha> -1\\\infty & \alpha\leq-1\end{cases}.$$ In your case $-(2n-2)\leq-1$ holds for $n\geq \frac32$. Hence, for $n\geq \frac32$ you get $$\int_0^{1/2}x^{-(2n-2)}(1-x)^{n-1}dx\geq\left(\frac12\right)^{n-1}\int_0^{1/2}x^{-(2n-2)}~dx = \infty.$$ Therefore, your integral doesn't converge for all $n\geq \frac32$.
The same way, you can see that the integral doesn't converge for $n\leq 0$ due to the $(1-x)^{n-1}$ term.
But for $n\in\left(0,\frac32\right)$ it converges to a positive value, because of the monotonicity of the integral! This can be seen through \begin{align} \int_0^{1/2}x^{-(2n-2)}(1-x)^{n-1}~dx\leq\max\left\{1,\left(\frac12\right)^{n-1}\right\}\int_0^{1/2}x^{-(2n-2)}~dx \\=\frac1{3-2n}\max\left\{1,\left(\frac12\right)^{n-1}\right\}\left(\frac12\right)^{3-2n}<\infty \end{align} and \begin{align} \int_{1/2}^1x^{-(2n-2)}(1-x)^{n-1}~dx\leq\max\left\{1,\left(\frac12\right)^{2-2n}\right\}\int_0^{1/2}(1-x)^{n-1}~dx \\=\frac1{n}\max\left\{1,\left(\frac12\right)^{2-2n}\right\}\left(\frac12\right)^{n}<\infty. \end{align}
• I think it should be $\frac{3}{2}$ and not $\frac{1}{2}$ by your reasoning. – Subho Jun 24 '18 at 6:26


text:
# OEBitVectorXor¶

OEBitVector OEBitVectorXor(const OEBitVector &a, const OEBitVector &b)

Returns that Boolean exclusive or (or ‘xor’) of two OEBitVectors. This function returns an OEBitVector where each bit is set only if the equivalent bit positions in the two operands have different values. The result will have the same number of bits as the large of the two arguments.


text:
# American Institute of Mathematical Sciences

February  2019, 24(2): 587-613. doi: 10.3934/dcdsb.2018198

## Asymptotic boundedness and stability of solutions to hybrid stochastic differential equations with jumps and the Euler-Maruyama approximation

 1 College of Information Sciences and Technology, Donghua University, Shanghai, 201620, China 2 School of mathematics and information technology, Jiangsu Second Normal University, Nanjing 210013, China 3 Department of Applied Mathematics, Donghua University, Shanghai 201620, China 4 Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH, UK

* Corresponding author: Liangjian Hu

Received  May 2017 Revised  January 2018 Published  June 2018

Fund Project: The author Wei Mao is supported by the National Natural Science Foundation of China (11401261) and "333 High-level Personnel Training Project" of Jiangsu Province. The author Liangjian Hu is supported by the National Natural Science Foundation of China (11471071). The author Xuerong Mao is supported by the Leverhulme Trust (RF-2015-385), the Royal Society (WM160014, Royal Society Wolfson Research Merit Award), the Royal Society and the Newton Fund (NA160317, Royal Society-Newton Advanced Fellowship), the EPSRC (EP/K503174/1).

In this paper, we are concerned with the asymptotic properties and numerical analysis of the solution to hybrid stochastic differential equations with jumps. Applying the theory of M-matrices, we will study the $p$th moment asymptotic boundedness and stability of the solution. Under the non-linear growth condition, we also show the convergence in probability of the Euler-Maruyama approximate solution to the true solution. Finally, some examples are provided to illustrate our new results.

Citation: Wei Mao, Liangjian Hu, Xuerong Mao. Asymptotic boundedness and stability of solutions to hybrid stochastic differential equations with jumps and the Euler-Maruyama approximation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 587-613. doi: 10.3934/dcdsb.2018198
##### References:

show all references

##### References:
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2019 Impact Factor: 1.27


text:
# Enterprise Virtual Worlds to See Real-World Growth

3-D virtual world applications for enterprise use will grow into an industry earning $8 billion to$10 billion in annual revenue by 2014, according to a new report from GigaOM Pro (subscription required). Today, virtual worlds are primarily associated with role-playing games and avatar-based chat, but analysts Kris Tuttle and Steve Waite say they will gain widespread corporate adoption as a platform for long-distance conferencing and training, job fairs, and other business uses in the years ahead.

Companies are likely to embrace virtual worlds for such applications first, the authors argue, because they offer a cost-effective, productivity-boosting (not to mention “green”) alternative to air travel. After reaching a critical mass in the next 12-24 months and doubling year over year, the industry will soon be earning billions of dollars in annual revenue. (Anticipating skeptics, the authors note that the existing market for online enterprise training and collaboration already earns that much just in direct costs alone.)

As for the major players in this burgeoning space? Tuttle and Waite name IBM (s ibm) and Cisco (s csco), two companies that have been developing virtual world/online collaboration solutions for years, with Adobe (s ADBE), Citrix (s CTXS), Oracle (s ORCL?), and Dassault Systems (s DASTY) as other contenders to watch closely. They also note that the market’s growth will provoke fierce competition among the many enterprise virtual world solutions already out there — OLIVE, Second Life, OpenSimulator, Project Wonderland, to name just a few — leading to a shakeout in the industry. After this tumult, a few dominant behemoths and successful niche players will emerge, though it’s still unclear if the winners are already on the market, or still in development.

In any case, the report recommends that companies large and small should immediately start pilot programs based around virtual worlds. “Because the technology is delivering cost savings and improved operations,” they argue, “the time to begin is now.”

Tuttle and Waite’s full report, “Virtual Worlds for the Enterprise Market,” is now available on GigaOM Pro (subscription required).


text:
# Choose the letter of the word or phrase that best completes each statement: angle A cong angle X and angle C cong angle Z , then triangle ABC sim triangle XYZ by the _____. Choose from: a. scale factor, b. AA Similarity Postulate, c. SSS Similarity Theorem, d. SAS Similarity Theorem, e. Symmetric Property of Similarity, f. midsegment, g. dilation, h. enlargement, i. reduction, j. Transitive Property of Similarity.

Choose the letter of the word or phrase that best completes each statement: If $\mathrm{\angle }A\cong \mathrm{\angle }X$ and $\mathrm{\angle }C\cong \mathrm{\angle }Z$, then $\mathrm{△}ABC\sim \mathrm{△}XYZ$ by the _____. Choose from: a. scale factor, b. AA Similarity Postulate, c. SSS Similarity Theorem, d. SAS Similarity Theorem, e. Symmetric Property of Similarity, f. midsegment, g. dilation, h. enlargement, i. reduction, j. Transitive Property of Similarity.
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If two triangles have two pairs of congruent corresponding angles, then they are similar by the AA Similarity Postulate. The correct answer is then b. AA Similarity Postulate.
Result:
b


text:
# Find monic generators of the ideals

## Homework Statement

Let ##T## be the linear operator on ##F^4## represented in the standard basis by $$\begin{bmatrix}c & 0 & 0 & 0 \\ 1 & c & 0 & 0 \\ 0 & 1 & c &0 \\ 0 & 0 & 1 & c \end{bmatrix}.$$ Let ##W## be the null space of ##T-cI##.

a) Prove that ##W## is the subspace spanned by ##\epsilon_4##.

b) Find the monic generators of the ideals ##S(\epsilon_4;W),\,S(\epsilon_3;W),\,S(\epsilon_2;W)##, and ##S(\epsilon_1;W)##.

## The Attempt at a Solution

The first part is easy. It's trivial to see that ##T-cI## sends vectors of the form ##(0,0,0,d)## to ##0##, such that the null space is spanned by ##\epsilon_4=(0,0,0,1)##. However, I have no idea how to start the second part. I'm having some trouble understanding what is meant by ##S(\epsilon_i;W)##. Any help would be appreciated.

andrewkirk


text: 1 2 3 4 5 6 7 8 9  \textbf{Frames and iFrames now supported for Web Testing} \begin{itemize} \item \gdauts{} that contain Frames and iFrames can now be tested by \app{} \end{itemize} \textbf{Tests updated to newer versions of IE and Firefox} \begin{itemize} \item The tested versions for web \gdauts{} now include IE 9 and Firefox 10 \end{itemize}


text:
MathSciNet bibliographic data MR1044465 17B65 (17B81 22E65 81R10) Miki, Kei The representation theory of the ${\rm SO}(3)$${\rm SO}(3)$ invariant superconformal algebra. Internat. J. Modern Phys. A 5 (1990), no. 7, 1293–1318. Article

For users without a MathSciNet license , Relay Station allows linking from MR numbers in online mathematical literature directly to electronic journals and original articles. Subscribers receive the added value of full MathSciNet reviews.


text:
# Quantum Chromodynamics:

### Lecture Course by Wladimir von Schlippe

This course is an introduction to Quantum Chromodynamics (QCD), the theory of strong interactions. The theory will be developed guided by experiment:

(i) The discovery of a large number of hadrons led to the formulation of the quark model by Gell-Mann.

(ii) Deep inelastic scattering (DIS) led to the discovery of partons, which were identified with the quarks. There was indirect evidence for gluons, the carriers of the strong force.

These and other experimental results form the basis for the development of QCD. Calculations within the framework of perturbative QCD will be demonstrated and their results will be confronted with experimental data from the colliders HERA, S$\bar{p}p$S and Tevatron.

Prerequisites: Quantum Mechanics, including relativistic QM, and a basic familiarity with elementary particle physics.

To access lecture material click on one of the following links:

Lecture 1:
Lecture 2:
Lecture 3:
Lecture 4:
For further useful information click on one of the following links:

Last Revision: 21st October 1999, WvS


text:
## The $\mathbf Γ$-limit of the two-dimensional Ohta-Kawasaki energy. I. Droplet density    [PDF]

Dorian Goldman, Cyrill B. Muratov, Sylvia Serfaty
This is the first in a series of two papers in which we derive a $\Gamma$-expansion for a two-dimensional non-local Ginzburg-Landau energy with Coulomb repulsion, also known as the Ohta-Kawasaki model in connection with diblock copolymer systems. In that model, two phases appear, which interact via a nonlocal Coulomb type energy. We focus on the regime where one of the phases has very small volume fraction, thus creating small "droplets" of the minority phase in a "sea" of the majority phase. In this paper we show that an appropriate setting for $\Gamma$-convergence in the considered parameter regime is via weak convergence of the suitably normalized charge density in the sense of measures. We prove that, after a suitable rescaling, the Ohta-Kawasaki energy functional $\Gamma$-converges to a quadratic energy functional of the limit charge density generated by the screened Coulomb kernel. A consequence of our results is that minimizers (or almost minimizers) of the energy have droplets which are almost all asymptotically round, have the same radius and are uniformly distributed in the domain. The proof relies mainly on the analysis of the sharp interface version of the energy, with the connection to the original diffuse interface model obtained via matching upper and lower bounds for the energy. We thus also obtain a characterization of the limit charge density for the energy minimizers in the diffuse interface model.
View original: http://arxiv.org/abs/1201.0222


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Data Imputation in R with NAs in only one variable (categorical)

I have data frame with 44,353 entries with 17 variables (4 categorical + 13 continuous). Out of all variables only 1 categorical variable (with 52 factors) has NAs

No of factors in the categorical variables are 1601, 6, 52 and 15

When I use missforest package it throws error that it cannot handle categorical predictors with more that 53 categories.

Please suggest an imputation method in R for best accuracy. Also since the variable to be imputed is categorical I would prefer to avoid methods that use regression techniques to impute values.

• Amelia II can impute categorical values. – Sycorax Aug 2 '16 at 14:24

You could use random hot deck imputation. Roughly, this is a method where missing values are replaced with values from an observation with "similar" values in the non-missing variables. For each missing value, the algorithm generates a pool of similar observations ("donors") and randomly chooses from them. The bigger the pools of donors and the bigger the number of datasets being imputed, the better.

For example we might have data as follows

 var1  var2  var3  var4
1 B     Z     U     5.1
2 <NA>  Z     U     5.0
3 B     Z     U     4.9
4 A     W     U     5.2
5 B     W     U     4.8
6 C     U     T     6.2
7 B     C     T     5.2
8 B     T     T     6.1
9 B     S     T     6.0

Here we have one observation with a missing value in var1. The algorithm would identify observations 1 and 3 as donors since they both have the same values for the two other categorical variables and similar values for the numeric variable. Hence, B would be chosen as the imputed values. If instead row 1 was

1 A     Z     U     5.1

..then the algorithm would randomly choose either A or B as the imputed value.

The hot.deck package for R should be able to handle your situation. The only problem I see is that, since you only have ~45000 observations, it is clear that there are many combinations of the categorical variable levels that do not occur (since, if every combination occured, there would have to be a minimum of $1601 \times 6 \times 52 \times 15 = 7,492,680$ observations. So, for the observations with missing data, if the combinations of the categorical variables are not repeated at least $M$ times elsewhere in the data, and you are imputing $m$ datasets, if $m > M$ there will be insufficient donors to draw imputations from. If this occurs then the software will generate warnings.

• Yep I don't have sufficient entries for such imputation. Appreciate the efforts though. – Sagar Patel Aug 2 '16 at 18:21
• @SagarPatel it should still work. But there will just be a limit on the number of distinct imputed datasets you can create, Depending on what your final analysis is, that may not matter much. – Robert Long Aug 2 '16 at 18:49

The magic word is feature construction. Statistics and machine learning almost always take some effort to prepare data before running an algorithm. Some "tricks" with respect to your particular problem:

1. try to reduce the number of factor levels without losing too much information (e.g. combine rare levels)
2. Convert ordered input factors to numeric (tree-based methods work much faster then).
3. A possibility is also to manually dummy code an input factor which greatly reduces computational effort.

Some further hints:

• If the response variable is a factor, then a random forest does classification, not regression.
• You can try out faster algoritms like multinomial logistic regression to fill in missings.
• The categorical variables are so important that I could afford dropping rows with NA rather than imputing them. I am very naive and your other suggestions are beyond my understanding. Would really appreciate if you could break that down into simpler instructions. – Sagar Patel Jul 31 '16 at 7:44
• mention of specific functions and packages would really help – Sagar Patel Jul 31 '16 at 7:46

Do you need to impute NA's?

First I would ask if you really need to impute the missing values? If you intend to use the imputed set to train another model you might as well just add NA as a level. In my experience this is really the simplest solution when you have NA's in a categorical variable. Especially when NA's actually do mean something, which is quite common. But even if it does not it is easy, especially for random forests, to ignore that level if it is not predictive.

This will add NA as a level in the factor.

dataset$varWithNAs <- addNA(dataset$varWithNAs)

Dummy encoding large categorical features

Regarding the problem with too many levels it seems to be the factor w 1601 levels that is your main problem. This is really a lot of levels and it is hard to give you any direct usage tips as little is stated about the variable. What you always can do in the case of too many levels is to transform the variable into many boolean (true, false) variables.

I'll give you an example.

dataset <- data.frame(x1 = sample(c('a','b','c'), 10, replace=T))
#     x1
# 1   c
# 2   b
# 3   a
# 4   a
# 5   b
# 6   c
# 7   a
# 8   a
# 9   b
# 10  c

You could use the caret package to create dummy variables for your factor levels.

library(caret)
dummyObj <- dummyVars(~x1, dataset)
dummyset <- predict(dummyObj, dataset)
x1.a x1.b x1.c
# 1     0    0    1
# 2     0    1    0
# 3     1    0    0
# 4     1    0    0
# 5     0    1    0
# 6     0    0    1
# 7     1    0    0
# 8     1    0    0
# 9     0    1    0
# 10    0    0    1

In your case it will make your feature vector quite a lot wider but it is actually what is done internally in a lot of, especially linear, models before training (although not in RF which is why you get this problem). If you look at eg. the glm package it transforms the dataset into dummy variables using the model.matrix function which does the same but adds an intercept term. Removing this intercept term will give you the same answer. And as model.matrix exists in the stats package you don't need to install anything.

model.matrix(~ x1 - 1, dataset) # -1 removes the intercept
#    x1a x1b x1c
# 1    0   0   1
# 2    0   1   0
# 3    1   0   0
# 4    1   0   0
# 5    0   1   0
# 6    0   0   1
# 7    1   0   0
# 8    1   0   0
# 9    0   1   0
# 10   0   0   1

If you find that your dataset get too many features now you should resort to the options Michael M gave in his answer to reduce the feature space. Chances are you have levels that never occur or several that are very similar in meaning and can be combined etc. Of course it is tedious to do this manually when you have so many levels.


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# Topology optimization

Topology optimization may be used to facilitate the reinforcement design of concrete structures. A brief overview of the method is given and ways it can be used for reinforcement design are described, along with several test case results.
$$Three practical examples of the design of typical concrete structures based on results generated by the method are shown in the attached article.The current practice of the design and assessment of so-called discontinuity regions of concrete structures is based on either the strut and tie method or on the use of scientifically oriented programs. Both of these methods suffer from different disadvantages, but none of them can be used for a concept-decision analysis of the structure or its detail as they require accurate dimensions, location directions and amount of reinforcement in advance. The author’s experience is that even professionals have inadequate knowledge when it comes to the problem of the determination of positions and directions of the reinforcement in cases of atypical details of concrete structures.This article has been prepared for 24th Czech Concrete Days 2017. Team of authors: Michael Konecny, Jaromir Kabelac, Jaroslav Navratil Attached Downloads Topology_optimization_US.pdf (PDF, 2.2 MB)$$


text:
### Watterson estimator calculation (theta or $\theta_w$) under infinite-sites assumption

Continuing on from the previous post, I'm still working on making sense of population genetics metrics. Today's is the Watterson estimator, frequently depicted as $\theta_w$. Wikipedia defines this one as

$\hat{\theta}_w = \frac{K}{a_n}$

Where $K$ is the number of segregating sites (e.g., a SNP) and $a_n$ is the $(n-1)$th harmonic number.

To check my calculations this time around, I'm using PopGenome (whose calculations of $\pi$ also agree with the vcftools calculations from the previous post).

So, at a single SNP and a sample of 52 diploid individuals (104 chromosomes), the calculation should be

$\hat{\theta}_w = \frac{1}{a_{104}}$

To get the harmonic number, you can use a calculator or just implement it. Note that the number is calculated for $(n - 1)$. A quick example of calculating the harmonic number that I wrote in R:

harmonicNumber = 0
numChromosomes = 104
for (i in 1:(numChromosomes - 1)) {
harmonicNumber = harmonicNumber + 1.0/i
}
print(harmonicNumber)

For 104 chromosomes, the harmonic number used for the denominator is 5.216791. So for three different windows where the first window contains 0 SNPs, the second window contains 1 SNP, and the third window contains 2 SNPs, PopGenome reports the following values for $\theta_w$

> slide@theta_Watterson
pop 1
[1,]        NA
[2,] 0.1916887
[3,] 0.3833774

And we can calculate those same values from $\frac{1}{5.216791} = 0.1916887$ and $\frac{2}{5.216791} = 0.3833774$.

Dividing that by the size of the window (i.e. to get a per bp estimate) seems to be what folks do when calculating sliding window averages of $\theta_w$. Wikipedia indicates that $\theta_w$ is a measure of the "population mutation rate" which seems appropriate enough; when calculated on the basis of sliding windows, this appears to represent the density of genetic variants in a region scaled slightly for the number of individuals in the sample. This seems to indeed be corroborated by a response from the vcftools developer to a user looking for $\theta_w$, indicating that using the --SNPdensity option from vcftools provides all of the necessary information for calculating the Watterson estimator.

As a final note, this appears to be the calculation under the infinite-sites assumption. The finite-sites calculation is evidently different as indicated in the LDhat documentation, though I'm not sure what that difference is yet. It looks like McVean et al. (2002) has additional information regarding that difference.


text:
Derivation Related to Paths

1. Dec 8, 2008

MichaelT

I am trying to derivate the equation T(t) = c'(t)/||c'(t)||

||c'(t)|| being the norm of c'(t). ||T(t)||= 1 and I need to solve for T'(t)

I keep getting stuck with some nasty terms. I assume that I need to deal with the norm as the square root of the dot product of c'(t) with itself.

Any help would be much appreciated. I have a feeling there is a simpler way to solve this than the way I have been trying.

2. Dec 8, 2008

tiny-tim

Hi MichaelT!

I don't understand what you're trying to do.

If c(t) is a curve, then c'(t)/||c'(t)|| is the unit vector tangent to the curve … what is the problem?

3. Dec 8, 2008

MichaelT

So T(t) = c'(t)/||c'(t)||. We are asked to find a formula for T'(t) in terms of c.

Sorry, looking back at my original post I realize I was not so clear.

Thank you!

4. Dec 9, 2008

tiny-tim

Hi MichaelT! Thanks for your PM.

As you said:
So let's write it out in full …

||c'(t)|| = √(c'(t).c'(t)) …

use the product rule (it works for vectors just as it does for scalars) to differentiate the inside of the √, and then use the chain rule to differentiate the whole thing.

(btw … no such word as "derivate" … and "derviation" would be completely different from "differentiation" )

5. Dec 10, 2008

MichaelT

Ok, I think I have another way to approach the problem. From subsequent problems I have learned that T'(t) = c"(t)

Therefore, if I can prove that T(t) = c'(t), then I am set. Can you give me any direction on doing that? I've tried with no luck, though I do know it is true!

6. Dec 10, 2008

MichaelT

Oh wait, I was wrong about something. T(t) will only equal c'(t) if it is parametrized by the arc length.

This is what I got, if anyone cares to check (please do!)

T'(t) = [||c'(t)||(c"(t)) - c'(t)(c'(t) dot c"(t))]/||c'(t)||3

7. Dec 10, 2008

tiny-tim

Hi MichaelT!

Aren't you missing a ||c'(t)|| at the beginning?

8. Dec 10, 2008

MichaelT

yeah it looks like I did miss something! Does it look like I am approaching this the right way?

Just re-did the problem, and this is what I got

T'(t) = (c"(t)/||c't||) - [c'(t)/||c'(t)||3] (c'(t) dot c"(t))

9. Dec 11, 2008

tiny-tim

Yes, that is the right way, and the right result this time!


text:
## Reasoning ability Questions Quiz Set 9 for IBPS/SBI/SSC

Reasoning ability Questions Quiz Set 9 for IBPS/SBI/SSC

Directions (1-5): Study the following arrangement of the English alphabet and answer the questions given below: M 1 E $R B 3 A K 7 £ # 9$ U D I N 4 6 % F H 2 @ 8 W

1. How many such consonants are there in the above sequence each of which is immediately followed by a number and immediately preceded by a vowel?

1) Nil 2) One 3) Two 4) Three 5) Four

2. Four of the following five are alike in a certain way and so form a group, based on the above arrangement. Which is the one that does not belong to that group?

1) % H N 2) I 4 $3) 9 D £ 4)3 5$ 5) $B M 3. How many such vowels are there in the above arrangement, each of which is immediately preceded by a symbol and immediately followed by a consonant? 1) Nil 2) One 3) Two 4) Three 5) None of these 4. If the order of the first fifteen elements in the above arrangement is reversed, then which of the following will be the eighth to the left of the twelfth letter/number/symbol from the right end? 1) A 2) £ 3) 3 4) K 5) None of these 5. What should come in place of questions mark in the following series based on the above arrangement? E B M A £ B 9 D £? H 8 % 1) N % D 2) N F D 3) 4 F D 4) I 6 D 5) None of these Directions (6-10): Answer these questions referring to the symbol-letter number sequence given below: 2 P J @ 8$ L B 1 V # Q 6 α G W 9 K C D 3 © £ 5 F R 7 A Y 4

6. How many symbols and numbers are there in the sequence which are either immediately preceded or immediately followed by the letter which is from the first half of the English alphabet?

1) 6 2) 7 3) 8 4) 9 5) None of these

7. Four of the following five are similar in relation to their positions in the above sequence and hence form a group. Which one does not belong to that group?

1) Q K 5 2) L 6 D 3) P L δ 4) 1 G © 5) K £ 4

8. Each symbol exchanges its position with its immediate right symbol/ letter/ number. Now, how many letters are there in the sequence which are immediately followed by a number and immediately preceded by a symbol?

1) Nil 2) One 3) Two 4) Three 5) None of these

9. P @ L is to Y75 in the same way as $1 # is to 1) R £ © 2) F £ 3 3) 5 £ © 4) 5 • D 5) None of these 10. Which of the following indicates the total number of symbols, letters and number respectively, which get eliminated from the sequence when every second element of the sequence from your left is dropped from the sequence? 1) 5, 8, 2 2) 6, 9, 1 3) 5, 8, 1 4) 5, 9, 1 5) None of these Directions (11 - 15): In the following questions, the symbols @, ©, %, * and$ are used with the following meanings as illustrated below:

‘P @ Q’ means ‘P is either greater than or equal to Q’;

‘P © Q’ means ‘P is either smaller than or equal to Q’;

‘P % Q’ means ‘P is greater than Q’;

‘P * Q’ means ‘P is smaller than Q’;

‘P $Q’ means ‘P is neither greater than nor smaller than Q’. Now, in each of the following questions, assuming the given statements to be true, find which of the two conclusions I and II given below them are definitely true? Give answer 1) if only conclusion I is true; 2) if only conclusion II is true; 3) if either conclusion I or II is true; 4) if neither I nor II is true; and 5) if both conclusions I and II are true 11. Statements: M@R, R%T, T$K

Conclusions: I. K*M II. T*M

Conclusions: I. F$J II. J%F 13. Statements: D$M, M%W, W@R

 Question No Answers 1 3 2 3 3 2 4 1 5 1 6 3 7 3 8 3 9 5 10 4 11 5 12 3 13 1 14 4 15 4