# Generation of count matrix (2022)

## Single-cell RNA-seq data - raw data to count matrix

Depending on the library preparation method used, the RNA sequences (also referred to as reads or tags), will be derived either from the 3’ ends (or 5’ ends) of the transcripts (10X Genomics, CEL-seq2, Drop-seq, inDrops) or from full-length transcripts (Smart-seq).

Image credit: Papalexi E and Satija R. Single-cell RNA sequencing to explore immune cell heterogeneity, Nature Reviews Immunology 2018 (https://doi.org/10.1038/nri.2017.76)

The choice of method involves the biological question of interest. The following advantages are listed below for the methods:

• 3’ (or 5’)-end sequencing:
• More accurate quantification through use of unique molecular identifiers distinguishing biological duplicates from amplification (PCR) duplicates
• Larger number of cells sequenced allows better identity of cell type populations
• Cheaper per cell cost
• Best results with > 10,000 cells
• Full length sequencing:
• Detection of isoform-level differences in expression
• Identification of allele-specific differences in expression
• Deeper sequencing of a smaller number of cells
• Best for samples with low number of cells

Many of the same analysis steps need to occur for 3’-end sequencing as for full-length, but 3’ protocols have been increasing in popularity and consist of a few more steps in the analysis. Therefore, our materials are going to detail the analysis of data from these 3’ protocols with a focus on the droplet-based methods (inDrops, Drop-seq, 10X Genomics).

## 3’-end reads (includes all droplet-based methods)

For the analysis of scRNA-seq data, it is helpful to understand what information is present in each of the reads and how we use it moving forward through the analysis.

For the 3’-end sequencing methods, reads originating from different molecules of the same transcript would have originated only from the 3’ end of the transcripts, so would have a high likelihood of having the same sequence. However, the PCR step during library preparation could also generate read duplicates. To determine whether a read is a biological or technical duplicate, these methods use unique molecular identifiers, or UMIs.

• Reads with different UMIs mapping to the same transcript were derived from different molecules and are biological duplicates - each read should be counted.
• Reads with the same UMI originated from the same molecule and are technical duplicates - the UMIs should be collapsed to be counted as a single read.
• In image below, the reads for ACTB should be collapsed and counted as a single read, while the reads for ARL1 should each be counted.

Image credit: modified from Macosko EZ et al. Highly Parallel Genome-wide Expression Profiling of Individual Cells Using Nanoliter Droplets, Cell 2015 (https://doi.org/10.1016/j.cell.2015.05.002)

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So we know that we need to keep track of the UMIs, but what other information do we need to properly quanitify the expression in each gene in each of the cells in our samples? Regardless of droplet method, the following are required for proper quantification at the cellular level:

• Sample index: determines which sample the read originated from
• Added during library preparation - needs to be documented
• Cellular barcode: determines which cell the read originated from
• Each library preparation method has a stock of cellular barcodes used during the library preparation
• Unique molecular identifier (UMI): determines which transcript molecule the read originated from
• The UMI will be used to collapse PCR duplicates

For example, when using the inDrops v3 library preparation method, the following represents how all of the information is acquired in four reads:

Image credit: Sarah Boswell, Director of the Single Cell Sequencing Core at HMS

• R1 (61 bp Read 1): sequence of the read (Red top arrow)
• R2 (8 bp Index Read 1 (i7)): cellular barcode - which cell read originated from (Purple top arrow)
• R3 (8 bp Index Read 2 (i5)): sample/library index - which sample read originated from (Red bottom arrow)
• R4 (14 bp Read 2): read 2 and remaining cellular barcode and UMI - which transcript read originated from (Purple bottom arrow)

The analysis workflow for scRNA-seq is similar for the different droplet-based scRNA-seq methods, but the parsing of the UMIs, cell IDs, and sample indices, will differ between them. For example, below is a schematic of the 10X sequence reads, where the indices, UMIs and barcodes are placed differently:

Image credit: Sarah Boswell, Director of the Single Cell Sequencing Core at HMS

## Single-cell RNA-seq workflow

The scRNA-seq method will determine how to parse the barcodes and UMIs from the sequencing reads. So, although a few of the specific steps will slightly differ, the overall workflow will generally follow the same steps regardless of method. The general workflow is shown below:

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Image credit: Luecken, MD and Theis, FJ. Current best practices in single‐cell RNA‐seq analysis: a tutorial, Mol Syst Biol 2019 (doi: https://doi.org/10.15252/msb.20188746)

The steps of the workflow are:

• Generation of the count matrix (method-specific steps): formating reads, demultiplexing samples, mapping and quantification
• Quality control of the raw counts: filtering of poor quality cells
• Clustering of filtered counts: clustering cells based on similarities in transcriptional activity (cell types = different clusters)
• Marker identification: identifying gene markers for each cluster
• Optional downstream steps

Regardless of the analysis being done, conclusions about a population based on a single sample per condition are not trustworthy. BIOLOGICAL REPLICATES ARE STILL NEEDED! That is, if you want to make conclusions that correspond to the population and not just the single sample.

## Generation of count matrix

We are going to start by discussing the first part of this workflow, which is generating the count matrix from the raw sequencing data. We will focus on the 3’ end sequencing used by droplet-based methods, such as inDrops, 10X Genomics, and Drop-seq.

After sequencing, the sequencing facility will either output the raw sequencing data as BCL or FASTQ format or will generate the count matrix. If the reads are in BCL format, then we will need to convert to FASTQ format. There is a useful command-line tool called bcl2fastq that can easily perform this conversion.

NOTE: We do not demultiplex at this step in the workflow. You may have sequenced 6 samples, but the reads for all samples may be present all in the same BCL or FASTQ file.

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The generation of the count matrix from the raw sequencing data will go through similar steps for many of the scRNA-seq methods.

umis and zUMIs are command-line tools that estimate expression of scRNA-seq data for which the 3’ ends of transcripts were sequenced. Both tools incorporate collapsing of UMIs tocorrect for amplification bias. The steps in this process include the following:

1. Formatting reads and filtering noisy cellular barcodes
2. Demultiplexing the samples
3. Mapping/pseudo-mapping to transcriptome
4. Collapsing UMIs and quantification of reads

If using 10X Genomics library preparation method, then the Cell Ranger pipeline would be used for all of the above steps.

## 1. Formatting reads and filtering noisy cellular barcodes

The FASTQ files can then be used to parse out the cell barcodes, UMIs, and sample barcodes. For droplet-based methods, many of the cellular barcodes will match a low number of reads (< 1000 reads) due to:

• encapsulation of free floating RNA from dying cells
• simple cells (RBCs, etc.) expressing few genes
• cells that failed for some reason

These excess barcodes need to be filtered out of the sequence data prior to read alignment. To do this filtering, the ‘cellular barcode’ and the ‘molecular barcode’ are extracted and saved for each cell. For example, if using ‘umis’ tools, the information is added to the header line for each read, with the following format:

@HWI-ST808:130:H0B8YADXX:1:1101:2088:2222:CELL_GGTCCA:UMI_CCCTAGGAAGATGGAGGAGAGAAGGCGGTGAAAGAGACCTGTAAAAAGCCACCGN+@@@DDBD>=AFCF+<CAFHDECII:DGGGHGIGGIIIEHGIIIGIIDHII#
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Known cellular barcodes used in the library preparation method should be known, and unknownbarcodes would be dropped, while allowing for an acceptable number of mismatches to the known cellular barcodes.

The next step of the process is to demultiplex the samples, if sequencing more than a single sample. This is the one step of this process not handled by the ‘umis’ tools, but is accomplished by ‘zUMIs’. We would need to parse the reads to determine the sample barcode associated with each cell.

## 3. Mapping/pseudo-mapping to cDNAs

To determine which gene the read originated from, the reads are aligned using traditional (STAR) or light-weight methods (Kallisto/RapMap).

## 4. Collapsing UMIs and quantification of reads

The duplicate UMIs are collapsed, and only the unique UMIs are quantified using a tool like Kallisto or featureCounts. The resulting output is a cell by gene matrix of counts:

Image credit: extracted from Lafzi et al. Tutorial: guidelines for the experimental design of single-cell RNA sequencing studies, Nature Protocols 2018 (https://doi.org/10.1038/s41596-018-0073-y)

Each value in the matrix represents the number of reads in a cell originating from the corresponding gene. Using the count matrix, we can explore and filter the data, keeping only the higher quality cells.

This lesson has been developed by members of the teaching team at the Harvard Chan Bioinformatics Core (HBC). These are open access materials distributed under the terms of the Creative Commons Attribution license (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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## FAQs

### Generation of count matrix? ›

The generation of the count matrix from the raw sequencing data will go through similar steps for many of the scRNA-seq methods. umis and zUMIs are command-line tools that estimate expression of scRNA-seq data for which the 3' ends of transcripts were sequenced.

### What is a count matrix RNA seq? ›

RNA seq data is often analyzed by creating a count matrix of gene counts per sample. This matrix is analyzed using count-based models, often built on the negative binomial distribution.

### What is matrix MTX? ›

matrix. mtx : a matrix of count values, where rows are associated with the gene IDs above and columns correspond to the cellular barcodes. Note that there are many zero values in this matrix.

### What is a gene expression matrix? ›

Gene expression matrix. The ENCODE gene expression matrix is obtained by collecting into a single file the gene quantification files produced by the ENCODE3 long RNAseq pipeline. The matrix is created using inhouse scripts, and provided in both TSV and JSON formats.

### What does featureCounts mean? ›

featureCounts can be used to quantify reads generated from either RNA or DNA sequencing technologies in terms of any type of genomic feature. It implements chromosome hashing, feature blocking and other strategies to assign reads to features with high efficiency.

### How does R calculate TPM? ›

Inspecting the salmon output
1. divide the number of reads for each transcript by it's length (reads per kilobase - RPK)
2. sum the RPK values and divide by 1 million to get a scaling factor.
3. divide the RPK values by the scaling factor to get the TPM.
Jan 14, 2019

### What does log2 fold change mean? ›

Fold change:

This value is typically reported in logarithmic scale (base 2). For example, log2 fold change of 1.5 for a specific gene in the “WT vs KO comparison” means that the expression of that gene is increased in WT relative to KO by a multiplicative factor of 2^1.5 ≈ 2.82.

### What is count matrix? ›

We will work with a count matrix, which has genes along the rows and samples along the columns. The elements in the matrix give the number of reads which could be uniquely aligned to a given gene for a given sample.

### What is matrix format? ›

matrix, a set of numbers arranged in rows and columns so as to form a rectangular array. The numbers are called the elements, or entries, of the matrix. Matrices have wide applications in engineering, physics, economics, and statistics as well as in various branches of mathematics.

### What is 10x barcode? ›

The 10X barcoded gel beads consist of a pool barcodes which are used to separately index each cell's transcriptome. The individual gel barcodes are delivered to each cell via flow-cytometry, where each cell is fed single-file along a liquid tube and tagged with a 10X gel bead.

### How do you analyze gene expression data? ›

A common approach to interpreting gene expression data is gene set enrichment analysis based on the functional annotation of the differentially expressed genes (Figure 13). This is useful for finding out if the differentially expressed genes are associated with a certain biological process or molecular function.

### What is gene expression data used for? ›

Gene expression profiling has been widely used to characterize cell status to reflect the health of the body, to diagnose genetic diseases, etc.

### What is gene expression data? ›

Gene expression is the process by which information from a gene is used in the synthesis of a functional gene product that enables it to produce end products, protein or non-coding RNA, and ultimately affect a phenotype, as the final effect.

### How do you calculate TPM from raw count? ›

Here's how you calculate TPM: Divide the read counts by the length of each gene in kilobases. This gives you reads per kilobase (RPK). Count up all the RPK values in a sample and divide this number by 1,000,000.

### What is the output of featureCounts? ›

Basically, it is a tab-separated file, and some of its columns are "comma-separated" fields, because featureCounts outputs one field per exon. Geneid is first column and counts is last (or n last columns, in case you read n bam files), you can use cut to get a file similar to HTSeq.

### What is FPKM value? ›

FPKM stands for fragments per kilobase of exon per million mapped fragments. It is analogous to RPKM and is used specifically in paired-end RNA-seq experiments [17].

### How do you convert TPM to RPKM? ›

TPM = (((mean transcript length in kilobases) x RPKM) / sum(RPKM all genes)) * 10^6. Excellent paper btw -- very clear. I used the method to convert other transcript counts measured by other instruments to TPMs -- but everyone can use the conversion from RPKMs (and FPKMs) to TPMs.

### What is the difference between CPM and RPKM? ›

Since RPKM actually builds on CPM by adding feature length normalization, edgeR's implementation calculates RPKM by simply dividing each feature's CPM (variable y in the code) by that feature's length multiplied by one thousand. This adds feature length normalization to sequencing depth-normalized counts.

### How do I convert counts to FPKM? ›

Here's how you do it for RPKM:
1. Count up the total reads in a sample and divide that number by 1,000,000 – this is our “per million” scaling factor.
2. Divide the read counts by the “per million” scaling factor. ...
3. Divide the RPM values by the length of the gene, in kilobases.
Jul 22, 2015

### Why is log2 used? ›

Log2 aids in calculating fold change, by which measure the up-regulated vs down-regulated genes between samples. Usually, Log2 measured data more close to the biologically-detectable changes.

### What is a 1 fold difference? ›

It is defined as the ratio between the two quantities; for quantities A and B the fold change of B with respect to A is B/A. In other words, a change from 30 to 60 is defined as a fold-change of 2. This is also referred to as a "one fold increase".

### What is log2 ratio? ›

The logarithms of the expression ratios are also treated symmetrically, so that a gene upregulated by a factor of 2 has a log2(ratio) of 1, a gene down- regulated by a factor of 2 has a log2(ratio) of −1, and a gene expressed at a constant level (with a ratio of 1) has a log2(ratio) equal to zero.

### What is a gene count? ›

In RNA-Seq, the abundance level of a gene is measured by the number of reads that map to that gene. Once the reads have been mapped to our reference, we must now count the number of reads that map to RNA units of interest to obtain gene/exon/transcript counts.

### What is read count? ›

The Read Count quantitation is the simplest and most commonly used quantitation. It counts up the reads within a probe and can correct this raw count according to a few different factors which might bias the result - allowing it to be compared to other data sets.

### How do you normalize data count? ›

DESeq2-normalized counts: Median of ratios method
1. Step 1: creates a pseudo-reference sample (row-wise geometric mean) ...
2. Step 2: calculates ratio of each sample to the reference. ...
3. Step 3: calculate the normalization factor for each sample (size factor)

### Why is matrix used? ›

In geometry, matrices are widely used for specifying and representing geometric transformations (for example rotations) and coordinate changes. In numerical analysis, many computational problems are solved by reducing them to a matrix computation, and this involves often to compute with matrices of huge dimension.

### What is matrix formula? ›

A matrix equation is an equation of the form Ax = b , where A is an m × n matrix, b is a vector in R m , and x is a vector whose coefficients x 1 , x 2 ,..., x n are unknown.

### What are the types of matrix? ›

The various types of matrices are row matrix, column matrix, null matrix, square matrix, diagonal matrix, upper triangular matrix, lower triangular matrix, symmetric matrix, and antisymmetric matrix.

### How many cells are in 10x? ›

The optimal cell concentration for a 10X Genomics single cell RNA sequencing experiment is 400-1200 cells/µL in a minimal volume of 30-100 µL.

### How many cells are in a 10x single cell? ›

Answer: In each standard 10x Single Cell Gene Expression, Immune Profiling, ATAC, or Multiome library, it is possible to target up to 10,000 cells. Each chip can process eight libraries in parallel, allowing users to target up to 80,000 cells per chip.

### How does 10x sequencing work? ›

10x Genomics' single-cell RNA-seq (scRNA-seq) technology, the Chromium™ Single Cell 3' Solution, allows you to analyze transcriptomes on a cell-by-cell basis through the use of microfluidic partitioning to capture single cells and prepare barcoded, next-generation sequencing (NGS) cDNA libraries.

### Which technique is used to study gene expression? ›

In addition to Northern blot tests and SAGE analyses, there are several other techniques for analyzing gene expression. Most of these techniques, including microarray analysis and reverse transcription polymerase chain reaction (RT-PCR), work by measuring mRNA levels.

### What are the methods of gene expression? ›

These methods include Northern blots, RT-PCR, macroarrays, microarrays, differential-display RT-PCR, serial analysis of gene expression (SAGE), comparative expressed sequence tag (EST) analysis, and massively parallel signature sequencing (MPSS).

### What is the study of gene expression called? ›

Gene expression analysis is most simply described as the study of the way genes are transcribed to synthesize functional gene products — functional RNA species or protein products.

### What are the two stages of gene expression? ›

It consists of two major steps: transcription and translation. Together, transcription and translation are known as gene expression. During the process of transcription, the information stored in a gene's DNA is passed to a similar molecule called RNA (ribonucleic acid) in the cell nucleus.

### What factors affect gene expression? ›

Environmental factors such as diet, temperature, oxygen levels, humidity, light cycles, and the presence of mutagens can all impact which of an animal's genes are expressed, which ultimately affects the animal's phenotype.

### Why is gene expression analysis important? ›

Gene expression profiling has been used extensively in biological research and has resulted in significant advances in the understanding of the molecular mechanisms of complex disorders, including cancer, heart disease, and metabolic disorders.

### What is a count matrix? ›

We will work with a count matrix, which has genes along the rows and samples along the columns. The elements in the matrix give the number of reads which could be uniquely aligned to a given gene for a given sample.

### What is a read count matrix? ›

A count matrix is a single table containing the counts for all samples, with the genes in rows and the samples in columns.

### What is FPKM value? ›

FPKM stands for fragments per kilobase of exon per million mapped fragments. It is analogous to RPKM and is used specifically in paired-end RNA-seq experiments [17].

### What does a Voom plot show? ›

The voom plot shows how the coefficient of variation of the counts depends on the count size.

Depending on the library preparation method used, the RNA sequences (also referred to as reads or tags), will be derived either from the 3’ ends (or 5’ ends) of the transcripts (10X Genomics, CEL-seq2, Drop-seq, inDrops) or from full-length transcripts (Smart-seq).

Many of the same analysis steps need to occur for 3’-end sequencing as for full-length, but 3’ protocols have been increasing in popularity and consist of a few more steps in the analysis.. For the 3’-end sequencing methods, reads originating from different molecules of the same transcript would have originated only from the 3’ end of the transcripts, so would have a high likelihood of having the same sequence.. However, the PCR step during library preparation could also generate read duplicates.. To determine whether a read is a biological or technical duplicate, these methods use unique molecular identifiers, or UMIs.. Reads with different UMIs mapping to the same transcript were derived from different molecules and are biological duplicates - each read should be counted.. Cellular barcode : determines which cell the read originated from (purple top arrow) Each library preparation method has a stock of cellular barcodes used during the library preparation. Unique molecular identifier (UMI) : determines which transcript molecule the read originated from The UMI will be used to collapse PCR duplicates (purple bottom arrow). The scRNA-seq method will determine how to parse the barcodes and UMIs from the sequencing reads.. After sequencing, the sequencing facility will either output the raw sequencing data as BCL or FASTQ format or will generate the count matrix .. The generation of the count matrix from the raw sequencing data will go through similar steps for many of the scRNA-seq methods.. Formatting reads and filtering noisy cellular barcodes Demultiplexing the samples Mapping/pseudo-mapping to transcriptome Collapsing UMIs and quantification of reads. The next step of the process is to demultiplex the samples, if sequencing more than a single sample.

Depending on the library preparation method used, the RNA sequences (also referred to as reads or tags), will be derived either from the 3’ ends (or 5’ ends) of the transcripts (10X Genomics, CEL-seq2, Drop-seq, inDrops) or from full-length transcripts (Smart-seq).

Reads with different UMIs mapping to the same transcript were derived from different molecules and are biological duplicates - each read should be counted.. Cellular barcode : determines which cell the read originated from (purple top arrow) Each library preparation method has a stock of cellular barcodes used during the library preparation. The scRNA-seq method will determine how to parse the barcodes and UMIs from the sequencing reads.. Formatting reads and filtering noisy cellular barcodes Demultiplexing the samples Mapping/pseudo-mapping to transcriptome Collapsing UMIs and quantification of reads. The next step of the process is to demultiplex the samples, if sequencing more than a single sample.

Approximate time: 30 minutes

The complexity of scRNA-seq data, which is generally characterized as a large volume of data , representing thousands of cells, and by a low depth of sequencing per cell , resulting in a large number of genes without any corresponding reads (zero inflation), makes analysis of the data more involved than bulk RNA-seq.. The analysis workflow for scRNA-seq is generally similar for the differing scRNA-seq methods, but some specifics, such as the parsing of the UMIs, cell IDs, and sample IDs, will differ between them.. While the 10X sequence reads have the UMI and barcodes placed differently:. The scRNA-seq method will determine the how to parse the barcodes and UMIs from the sequencing reads.. After generating the count matrix, the raw counts will be assessed to filter out poor quality cells with a low number of genes or UMIs, high mitochondrial gene expression indicative of dying cells, or low number of genes per UMI.. After removing the poor quality cells, the cells are clustered based on similarities in transcriptional activity, with the idea that the different cell types separate into the different clusters.. The generation of the count matrix from the raw sequencing data will go through the following steps for many of the scRNA-seq methods.. ’ umis provides tools for estimating expression in RNA-seq data which performs. sequencing of end tags of transcript, and incorporate molecular tags to. correct for amplification bias.’ The steps in this process include the following:. Formatting reads and filtering noisy cellular barcodes Demultiplexing the samples Pseudo-mapping to cDNAs Counting molecular identifiers. The FASTQ files can then be used to parse out the cell barcodes, UMIs, and sample barcodes.. Many of the cellular barcodes will match a low number of reads (< 1000 reads) due to encapsulation of free floating RNA from dying cells, small cells, or set of cells that failed for some reason.. Now we have our count matrix containing the counts per gene for each cell, which we can use to explore our data for quality information.

Depending on the library preparation method used, the RNA sequences (also referred to as reads or tags), will be derived either from the 3’ ends (or 5’ ends) of the transcripts (10X Genomics, CEL-seq2, Drop-seq, inDrops) or from full-length transcripts (Smart-seq).

Contribute to hbctraining/scRNA-seq_online development by creating an account on GitHub.

Cellular barcode : determines which cell the read originated from (purple top arrow) Each library preparation method has a stock of cellular barcodes used during the library preparation. The scRNA-seq method will determine how to parse the barcodes and UMIs from the sequencing reads.. The generation of the count matrix from the raw sequencing data will go through similar steps for many of the scRNA-seq methods.. Formatting reads and filtering noisy cellular barcodes Demultiplexing the samples Mapping/pseudo-mapping to transcriptome Collapsing UMIs and quantification of reads. The next step of the process is to demultiplex the samples, if sequencing more than a single sample.

Approximate time: 30 minutes

The complexity of scRNA-seq data, which is generally characterized as a large volume of data , representing thousands of cells, and by a low depth of sequencing per cell , resulting in a large number of genes without any corresponding reads (zero inflation), makes analysis of the data more involved than bulk RNA-seq.. The analysis workflow for scRNA-seq is generally similar for the differing scRNA-seq methods, but some specifics, such as the parsing of the UMIs, cell IDs, and sample IDs, will differ between them.. While the 10X sequence reads have the UMI and barcodes placed differently:. The scRNA-seq method will determine the how to parse the barcodes and UMIs from the sequencing reads.. After generating the count matrix, the raw counts will be assessed to filter out poor quality cells with a low number of genes or UMIs, high mitochondrial gene expression indicative of dying cells, or low number of genes per UMI.. The generation of the count matrix from the raw sequencing data will go through the following steps for many of the scRNA-seq methods.. ’ umis provides tools for estimating expression in RNA-seq data which performs. sequencing of end tags of transcript, and incorporate molecular tags to. correct for amplification bias.’ The steps in this process include the following:. The FASTQ files can then be used to parse out the cell barcodes, UMIs, and sample barcodes.. Many of the cellular barcodes will match a low number of reads (< 1000 reads) due to encapsulation of free floating RNA from dying cells, small cells, or set of cells that failed for some reason.

Machines cannot understand characters and words. So when dealing with text data we need to represent it in numbers to be understood by the machine. Countvectorizer is a method to convert text to…

We have 8 unique words in the text and hence 8 different columns each representing a unique word in the matrix.. Since the words ‘is’ and ‘my’ were repeated twice we have the count for those particular words as 2 and 1 for the rest.. These words could be like the word ‘the’ that occur in every document and does not provide and valuable information to our text classification or any other machine learning model and can be safely ignored.. Max_df looks at how many documents contain the word and if it exceeds the max_df threshold then it is eliminated from the sparse matrix.. The words ‘is’, ‘to’, ‘james’, ‘my’ and ‘of’ have been removed from the sparse matrix as they occur in more than 1 document.. As you can see the word ‘james’ appears in 4 out of 5 documents(85%) and hence crosses the threshold of 75% and removed from the sparse matrix. Min_df stands for minimum document frequency, as opposed to term frequency which counts the number of times the word has occurred in the entire dataset, document frequency counts the number of documents in the dataset (aka rows or entries) that have the particular word.. min_df can take absolute values(1,2,3..) or a value representing a percentage of documents(0.50, ignore words appearing in 50% of documents). Even though all the words occur twice in the above input our sparse matrix just represents it with 1. The numbers do not represent the count of the words but the position of the words in the matrix

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Formally, we can write. \begin{align}. \lambda_i =\lim_{\delta \rightarrow 0^{+}} \left[ \frac{P\big(X(\delta)\neq i | X(0)=i\big)}{\delta}\right]. \hspace{30pt} (11.7). \end{align}. Since we go from state $i$ to state $j$ with probability $p_{ij}$, we call the quantity $g_{ij}=\lambda_i p_{ij}$, the transition rate from state $i$ to state $j$ .. \end{align*}. The transition matrix for the corresponding jump chain is given by. . \nonumber P = \begin{bmatrix}. p_{00} & p_{01} \\. p_{10} & p_{11}. \end{bmatrix} =. \begin{bmatrix}. 0 & 1 \\. 1 & 0. \end{bmatrix}.. \end{align*}. Thus, the generator matrix is given by. \begin{align*}. G= \begin{bmatrix}. -\lambda & \lambda \\[5pt]. \lambda & -\lambda \\[5pt]. \end{bmatrix}.. \end{align*} We have. \begin{align*}. P'(t)= \begin{bmatrix}. -\lambda e^{-2\lambda t} & \lambda e^{-2\lambda t} \\[5pt]. \lambda e^{-2\lambda t} & -\lambda e^{-2\lambda t} \\[5pt]. \end{bmatrix},. \end{align*}. where $P'(t)$ is the derivative of $P(t)$.. We also have. . \nonumber P(t) G= \begin{bmatrix}. \frac{1}{2}+\frac{1}{2}e^{-2\lambda t} & \frac{1}{2}-\frac{1}{2}e^{-2\lambda t} \\[5pt]. \frac{1}{2}-\frac{1}{2}e^{-2\lambda t} & \frac{1}{2}+\frac{1}{2}e^{-2\lambda t} \\[5pt]. \end{bmatrix} \begin{bmatrix}. -\lambda & \lambda \\[5pt]. \lambda & -\lambda \\[5pt]. \end{bmatrix}= \begin{bmatrix}. -\lambda e^{-2\lambda t} & \lambda e^{-2\lambda t} \\[5pt]. \lambda e^{-2\lambda t} & -\lambda e^{-2\lambda t} \\[5pt]. \end{bmatrix},. . . \nonumber G P(t) = \begin{bmatrix}. -\lambda & \lambda \\[5pt]. \lambda & -\lambda \\[5pt]. \end{bmatrix} \begin{bmatrix}. \frac{1}{2}+\frac{1}{2}e^{-2\lambda t} & \frac{1}{2}-\frac{1}{2}e^{-2\lambda t} \\[5pt]. \frac{1}{2}-\frac{1}{2}e^{-2\lambda t} & \frac{1}{2}+\frac{1}{2}e^{-2\lambda t} \\[5pt]. \end{bmatrix} = \begin{bmatrix}. -\lambda e^{-2\lambda t} & \lambda e^{-2\lambda t} \\[5pt]. \lambda e^{-2\lambda t} & -\lambda e^{-2\lambda t} \\[5pt]. \end{bmatrix}.. \end{align*}. Using the Chapman-Kolmogorov equation, we can write. \begin{align*}. P_{ij}(t+\delta) &=\sum_{k \in S} P_{ik}(t)p_{kj}(\delta) \\. &=p_{ij}(t)p_{jj}(\delta)+\sum_{k \neq j} P_{ik}(t)p_{kj}(\delta)\\. &\approx p_{ij}(t)(1+g_{jj} \delta)+\sum_{k \neq j} P_{ik}(t)\delta g_{kj}\\. &= p_{ij}(t)+ \delta p_{ij}(t) g_{jj} + \delta \sum_{k \neq j} P_{ik}(t) g_{kj}\\. &= p_{ij}(t)+ \delta \sum_{k \in S} P_{ik}(t) g_{kj}.. \end{align*}. Thus,. \begin{align*}. \frac{P_{ij}(t+\delta)-p_{ij}(t)}{\delta} \approx \sum_{k \in S} P_{ik}(t) g_{kj},. \end{align*}. which is the $(i,j)$th element of $P(t)G$.. The values of $g_{ii}$'s are not usually shown because they are implied by the other values, i.e.,. $$g_{ii}=-\sum_{j \neq i} g_{ij}.$$. For example, Figure 11.24 shows the transition rate diagram for the following generator matrix. . G = \begin{bmatrix}. -5 & 5 & 0 \\[5pt]. 1 & -2 & 1\\[5pt]. 3 & 1 & -4 \\[5pt]. \end{bmatrix},. \hspace{30pt} (11.8). . Figure 11.24 - The transition rate diagram for the continuous-time Markov chain defined by Equation 11.8.The print version of the book is available through Amazon here .

The basic reproduction number, $$R_0$$, is defined as the expected number of secondary cases produced by a single (typical) infection in a completely susceptible population. It is important to note that $$R_0$$ is a dimensionless number and not a rate, which would have units of $$\mathrm{time}^{-1}$$. Some authors incorrectly call $$R_0$$ the “basic reproductive rate.”"

The basic reproduction number, $$R_0$$, is defined as the expected number of secondary cases produced by a single (typical) infection in a completely susceptible population.. where $$\tau$$ is the transmissibility (i.e., probability of infection given contact between a susceptible and infected individual), $$\bar{c}$$ is the average rate of contact between susceptible and infected individuals, and $$d$$ is the duration of infectiousness.. If $$R_0$$ is the number of secondary infections produced by a single typical infection in a rarefied population, how do we define it when there are multiple types of infected individuals.. We define the next generation matrix as the square matrix $$\mathbf{G}$$ in which the $$ij$$th element of $$\mathbf{G}$$, $$g_{ij}$$, is the expected number of secondary infections of type $$i$$ caused by a single infected individual of type $$j$$, again assuming that the population of type $$i$$ is entirely susceptible.. Define $$f$$ as the expected number of infected women and $$m$$ as the expected number of infected men given contact with a single infected member of the opposite sex in a completely susceptible population.. $$\beta$$ is the effective contact rate, $$\lambda$$ is the “birth” rate of susceptibles, $$\mu$$ is the mortality rate, $$k$$ is the progression rate from exposed (latent) to infected, $$\gamma$$ is the removal rate.. where $$\beta$$ is the effective contact rate, $$\lambda$$ is the “birth” rate of susceptibles, $$\mu$$ is the mortality rate, $$k$$ is the progression rate from exposed (latent) to infected, $$\gamma$$ is the removal rate.. To calculate the next generation matrix for the SEIR model, we need to enumerate the number of ways that (1) new infections can arise and (2) the number of ways that individuals can move between compartments.. Generations in epidemic models are the waves of secondary infection that flow from each previous infection.. So, the first generation of an epidemic is all the secondary infections that result from infectious contact with the index case, who is of generation zero.. If $$R_i$$ denotes the reproduction number of the $$i$$th generation, then $$R_0$$ is simply the number of infections generated by the index case, i.e., generation zero.. The number of secondary infections generated by the case in generation zero is $$R_0=3$$.

Create a matrix or construct one from other matrices.

A single row of data has spaces or commas in between the elements, and a semicolon separates the rows.. Now create a matrix with the same numbers, but arrange them in two rows.. The first and second arguments of these functions are the number of rows and number of columns of the matrix, respectively.. Then, create a 4-by-4 matrix whose diagonal elements are the elements of A .. You can add one or more elements to a matrix by placing them outside of the existing row and column index boundaries.. For example, create a 2-by-3 matrix and add an additional row and column to it by inserting an element in the (3,4) position.. To expand the size of a matrix repeatedly, such as within a for loop, it's usually best to preallocate space for the largest matrix you anticipate creating.. For example, preallocate a matrix that holds up to 10,000 rows and 10,000 columns by initializing its elements to zero.. If you need to preallocate additional elements later, you can expand it by assigning outside of the matrix index ranges or concatenate another preallocated matrix to A .. A matrix is a two-dimensional, rectangular array of data elements arranged in rows and columns.

Learn how to make a matrix in python? Steps to create a matrix in python using user input, how to Create an empty matrix using NumPy in python and how to create a matrix using for loop in python.

How to create a matrix in python using user input Create an empty matrix using NumPy in python How to create a matrix in python 3 How to do matrix multiplication in python How to create a matrix using for loop in python How to create a matrix in Python using a list Multiply 8-rows, 1-column matrix and an 1-row, 8-column to get an 8-rows.. After writing the above code (how to create a matrix in python using user input), Once you will print “matrix” then the output will appear as a “[[2 4] [6 3]] ” .. To create an empty matrix, we will first import NumPy as np and then we will use np.empty() for creating an empty matrix.. Here, np.empty() matrix of 0 rows and 0 columns is used for creating an empty matrix in python.. The main rule for matrix multiplication is “number of rows in first matrix must be equal to number of column s in second matrix” and in this case, that rule is satisfied so we can proceed with the multiplication now.. How to create a matrix in python using user input Create an empty matrix using NumPy in python How to create a matrix in python 3 How to do matrix multiplication in python How to create a matrix using for loop in python How to create a matrix in Python using a list Multiply 8-rows, 1-column matrix and an 1-row, 8-column to get an 8-rows.

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