## How To Convert Numpy Array To A Dataframe

Now, let’s come back to the original topic of converting the MNIST data in IDX format. Visually, you can represent a NumPy array as something like this: This is a visual representation of a NumPy array that contains five values: 88, 19, 46, 74, 94. Since you have now completed an easy calculation to convert the precipitation values using numpy array calculations, you can use this numpy array to plot the precipitation data, rather than relying on Python lists. Unlike NumPy library which provides objects for multi-dimensional arrays, Pandas provides in-memory 2d table object called Dataframe. DataFrame(np. unique() array(['Oceania'], dtype=object) 6. Here is my attempt: # Create a NumPy array, which has four elements. Python | Flatten a 2d numpy array into 1d array Given a 2d numpy array, the task is to flatten a 2d numpy array into a 1d array. Because although this is a 1-dimensional array, numpy will broadcast it as a 1 x n matrix while performing matrix operations. Below are a few methods to solve the task. asarray_chkfinite Similar function which checks input for NaNs and Infs. Transpose DataFrame. I would like to convert the output numpy array to a pandas dataframe that will replicate the line segments that make up the polyline so that I land up with the following columns:. 0 (with less JSON SQL functions). How should I procede with this, should I convert the dictionary into numpy array or pandas df and HOW ? I have been trying a couple of things but no luck. Tutorial: Pandas Dataframe to Numpy Array and store in HDF5. The fundamental object of NumPy is its ndarray (or numpy. fromfunction Construct an array by executing a function on grid. In many cases, it is helpful to use a uniquely valued identifying field of the data as its index. This table lays out the different dtypes and default return types of to_numpy() for various dtypes within pandas. Numpy Adjacency Matrix To Graph. I wonder if there is a direct way to import the contents of a csv file into a record array, much in the way that R's read. howto make Python list from numpy. DataFrame(data_tuples, columns=['Month','Day']) Month Day 0 Jan 31 1 Apr 30 2 Mar 31 3 June 30 3. arange(9) array We can use NumPy’s reshape function to convert the 1d-array to 2d-array of dimension 3×3, 3 rows and 3 columns. I have two dataframes df and df2. array1: Numpy Array, original array array2: Numpy Array, To Append the original array. Columns are named, rows are numbered (but can be named) and can be easily selected and calculated upon. Begin End Duration ID 42 40680 40846 167 18 and I want to convert a numpy array in this form : array([40680 , 40860 ,167,18]) I am using for conversion as_matrix function and I used after it reshape(1,4) but it is not working!! It is getting me this format : [[40680 40846 167 18]] any suggestions please ? I need to. 5][locs]) … and that peeves me since I end up with a huge array copy in memory. Recommend：python - Convert Pandas Dataframe to numpy for sklearn. I tried the following data_np = data. Below are a few methods to solve the task. array should be used instead. How to Find the Average of a List in Python You can pass a list or array of numbers to the "numpy. fromfunction Construct an array by executing a function on grid. python,list,numpy,multidimensional-array. However, rdkit has CreateFromBitString method. To convert Pandas DataFrame to Numpy Array, use the function DataFrame. Convert input to a contiguous array. 2 Machine Learning with Tensorflow - Pandas Dataframe to Numpy Array How do I select multiple rows and columns from a pandas. Explore Popular Topics Like Government, Sports, Medicine, Fintech, Food, More. Python in combination with Numpy allows for using python to solve simultaneous equations in a few simple steps. I am interested in knowing how to convert a pandas dataframe into a numpy array, including the index, and set the dtypes. values() with the rename_axis() function and you will get the converted NumPy array from pandas dataframe. In general you can concatenate a whole sequence of arrays along any axis: numpy. table(), read. A data frame is a tabular data, with rows to store the information and columns to name the information. According to documentation of numpy. In Spark, SparkContext. In Python, data is almost universally represented as NumPy arrays. unique() array(['Oceania'], dtype=object) 6. concatenate( LIST, axis=0 ) but you do have to worry about the shape and dimensionality of each array in the list (for a 2-dimensional 5x3 output, you need to ensure that they are all 2-dimensional n-by-3 arrays already). reshape() method. Numpy arrays are great alternatives to Python Lists. 2 Answers 2. df_to_numpy(df) # Each column is strictly equal to the original. This means that you would receive one summary value for each row or each column in the two-dimensional numpy array. In this tutorial, we will see How To Create NumPy Arrays From Python Data Structures. You know you have to assign the newly created columns to the old column in pandas/numpy otherwise you changed nothing. (1 reply) I would like to convert a dataframe to an array of lists, one for every record. Once the DataFrame is returned, you can decide how to split the rows of the returned DataFrame into X and y components for supervised learning any way you wish. Now, let’s come back to the original topic of converting the MNIST data in IDX format. Python | Flatten a 2d numpy array into 1d array Given a 2d numpy array, the task is to flatten a 2d numpy array into a 1d array. How can I do this for dataframe with same datatype and different dataypes. Convert a pandas dataframe in a numpy array, store data in a. array), an n-dimensional array that is also present in some form in array-oriented languages such as Fortran 90, R, and MATLAB, as well as predecessors APL and J. When you need a no-copy reference to the underlying data, Series. I would like to put this results into a report and the best way would be to put a table w. I wish to have it transformed into a Pandas DataFrame D where each column of the 2D numpy array in S becomes a 1D numpy array in a column of D: D should look like: 0 1 a [0,2,3] [1,3,4] b [5,7,9] [6,8,10] Note, my actual data set is 1238500 arrays sized (32,8) so i was trying to avoid iterating over rows. Can you show an actual example (construct a dataframe, and then what you'd like to be able to do)? I would like to have a view on internal data already stored by dataframes as a numpy array. Learn more about python, numpy, ndarray MATLAB. dataframe:. and so can not be converted to a list. As an example, I will create a PySpark dataframe from a pandas dataframe. NumPy N-dimensional Array. Questions: I am interested in knowing how to convert a pandas dataframe into a numpy array, including the index, and set the dtypes. array1: Numpy Array, original array array2: Numpy Array, To Append the original array. Convert a 1d array (which is a result of complex computations) to a tuple to use it as a basic index in another Nd array to access particular elements. How can I convert this CSV file (with 3 columns of data) imported as a dataframe into individual columns of data? Or can I directly import each column of data into a 1d array and use it in the function kde_scipy?. Changing the Index of a DataFrame. to use for sklearn logistic prediction. Note that the same concepts would apply by using double quotes):. The first thing is the conversion of the pandas. Shape of the vector v: (3,) This way of creating a row vector is not wrong. DataFrame({'a': [1, 2, 3], 'b': [4, 5, 6]}) print(df) # a b # 0 1 4 # 1 2 5 # 2 3 6 array = np. mean" function included in the library, and it returns the mean. Python | Change column names and row indexes in Pandas DataFrame; How to get rows/index names in Pandas dataframe; Split a column in Pandas dataframe and get part of it; Getting Unique values from a column in Pandas dataframe; Apply uppercase to a column in Pandas dataframe; Adding new column to existing DataFrame in Pandas. Change lowest level of column labels into innermost row index. The main data structure in NumPy is the ndarray, which is a shorthand name for N-dimensional array. DataFrame(data=matrix, index=np. Or better yet just just translate the 2D index into a linear index mathematically. This also avoids the substantial overhead of converting the list to a numpy array. Transposing numpy array is extremely simple using np. The 1d-array starts at 0 and ends at 8. I'm using Spyder since the variable explorer lets me see matrices in a friendly way (studying density matrices in the console is a nightmare). Python has a very powerful library, numpy , that makes working with arrays simple. Previous: Write a NumPy program to convert 1-D arrays as columns into a 2-D array. In this article we will discuss how to save 1D & 2D Numpy arrays in a CSV file with or without header and footer. Once loaded, you convert the CSV data to a NumPy array and use it for machine learning. Getting into Shape: Intro to NumPy Arrays. According to documentation of numpy. Converting a PySpark dataframe to an array In order to form the building blocks of the neural network, the PySpark dataframe must be converted into an array. Python has a wonderful method that is called as slicing this can be used to access the ith column of a NumPy multidimensional array we will use this as follows:-. csv", delimiter=";", skip_header=1) the arguments inside the brackets are the file name, the delimiter, and skip_header set to 1 will make the csv load to an array without the header row. NumPy Array Object Exercises, Practice and Solution: Write a NumPy program to convert a list and tuple into arrays. To accomplish this goal, you may use the following Python code, which will allow you to convert the DataFrame into a list, where: The top part of the code, contains the syntax to create the DataFrame with our data about products and prices. Some key differences between lists include, numpy arrays are of fixed sizes, they are homogenous I,e you can only contain, floats or strings, you can easily convert a list to a numpy array, For example, if you would like to perform vector operations you can cast a list to a numpy array. asfortranarray Convert input to an ndarray with column-major memory order. One of them has data of same datatype and the other has data of different datatypes. The first thing is the conversion of the pandas. 2 Answers 2. asarray_chkfinite Similar function which checks input for NaNs and Infs. to use for sklearn logistic prediction. recarray (aka "record array") with€np. I am pretty sure it has to do something with how my "actin2. Each column is named after the same: column name in the data frame. result = array(arr2, str) and it will determine the length of the string for you. Fundamentally, transposing numpy array only make sense when you have array of 2 or more than 2 dimensions. So, first, we must import numpy as np. dataframe:. So here, we can see the dtype=np. tolist() #> [1, 'a'] To summarise, the main differences with python lists are: Arrays support vectorised operations, while lists don’t. asfortranarray Convert input to an ndarray with column-major memory order. I want to create a series in pandas using a numpy array. In fact, the time it takes to do so usually prohibits this from any data set that is at all interesting. >gapminder['continent']. Unlike NumPy library which provides objects for multi-dimensional arrays, Pandas provides in-memory 2d table object called Dataframe. Changing the Index of a DataFrame. array1: Numpy Array, original array array2: Numpy Array, To Append the original array. Whats people lookup in this blog:. Whats people lookup in this blog: Convert Dataframe Column To Array Python; Convert Dataframe Column To Series Python. So, I tried to convert numpy array to rdkit fp with the method. Return is NOT a Numpy-matrix, rather, a Numpy-array. >gapminder_Ocean. array, responseName = "value") d1 d2 d3 value 1 A1 B1 C1 0 2 A2 B1 C1 0 3 A1 B2 C1 0 4 A2 B2 C1 0 5 A1 B3 C1 0 6 A2 B3 C1 0 7 A1 B1 C2 0 8 A2 B1 C2 0 9 A1 B2 C2 0 10 A2 B2 C2 0 11 A1 B3 C2 0 12 A2 B3 C2 0 13 A1 B1 C3 0 14 A2 B1 C3 0 15 A1 B2 C3 0 16 A2 B2 C3 0 17 A1 B3 C3 0 18 A2 B3 C3 0 19 A1. I have an array of size 1801 that will be all of the column names in the dataframe. How to Create an Array in Python. > So if you can advise how to convert this data. NumPy is a commonly used Python data analysis package. Beside functions, and environments, most of the objects an R user is interacting with are vector-like. Oftentimes I find myself converting pandas. I have an array of size 1801 that will be all of the column names in the dataframe. You can access the numbers the same way you do for multidimensional arrays, and the as. I've mostly dealt with the ones that use numpy readers like loadtxt and genfromtxt. Machine learning data is represented as arrays. If none are provided, all the columns from the dataframe are extracted. NumPy Array Object Exercises, Practice and Solution: Write a NumPy program to convert a list and tuple into arrays. As numpy arrays are homogeneous i. I have this dataframe. com>, oyekomova wrote: > I would like to know how to convert a csv file with a header row into a > floating point array without the header row. You can create numpy array casting python list. Python Numpy - Duplicate or Copy Array Data to Another Array; Python NumPy Array - Iterate over all elements; Python Numpy - Add a constant to all the elements of array; Python Numpy - Multiply a constant to all the elements of array; Python Numpy - Divide all the elements of array by a constant; How to Convert Pandas DataFrame to NumPy Array. Show first n rows. Numpy Array - Multiply a constant to all elements of the array Multiplying a constant to a NumPy array is as easy as multiplying two numbers. Here's some example code on how to do this with PIL, but the general idea is the same. NumPy is a Python Library/ module which is used for scientific calculations in Python programming. How can I convert a shapely point geometry like following to numpy array? [, as. The 1d-array starts at 0 and ends at 8. As we cannot directly use Sparse Vector with scikit-learn, we need to convert the sparse vector to a numpy data structure. I want to import some data and store it in numpy array, do some operations on it and then convert it into pandas series How to do this. Numpy arrays essentially come in two flavors: Vectors and Matrics. The Dataframe will be 288 rows (289 counting the columns names) and 1801 columns. from_pandas(). fromfunction Construct an array by executing a function on grid. Those written in Python and I can outline their behavior. A pandas series is a labeled list of data. NumPy / SciPy arr = array([]) arr. N needed another overload to take arrays. Home Community Categories Machine Learning How do I convert a pandas dataframe to a numpy. In this chapter, we will discuss how to create an array from existing data. array(range(1, 36))) However, I am receiving the following error:. Shape of the vector v: (3,) This way of creating a row vector is not wrong. Web Development I want to get a 2d-numpy array from a column of a pandas dataframe df having a numpy vector in each row. Python in combination with Numpy allows for using python to solve simultaneous equations in a few simple steps. savetxt() Python's Numpy module provides a function to save numpy array to a txt file with custom delimiters and other custom options i. I recently spent a day working on the performance of a Python function and learned a bit about Pandas and NumPy array indexing. The explainer requires numpy arrays as input and h2o requires the train and test data to be in h2o frames. DataFrame(data_tuples, columns=['Month','Day']) Month Day 0 Jan 31 1 Apr 30 2 Mar 31 3 June 30 3. NumPy is, just like SciPy, Scikit-Learn, Pandas, etc. In this tutorial, we will see How To Create NumPy Arrays From Python Data Structures. When you need a no-copy reference to the underlying data, Series. Posted on sáb 06 setembro 2014 in Python. This routine is useful for converting Python sequence into ndarray. What is a Structured Numpy Array ? A Structured Numpy Array is an array of structures (Similar to C struct). array(range(1, 36))) However, I am receiving the following error:. A DataFrame is a 2D numpy array under the hood: [code]>>> import numpy as np >>> import pandas as pd >>> df = pd. Although it's very simple, but the concept behind this technique is very unique. Shape of the vector v: (3,) This way of creating a row vector is not wrong. array([7, 8, 9. You can convert a Pandas DataFrame to Numpy Array to perform some high-level mathematical functions. Since we have no idea were bayFails comes from, the only advice would be to read the Pandas docs since extracting data would be rountinely done by many programmers (I would guess by using itertuples or. I have two dataframes df and df2. Below are a few methods to solve the task. asfortranarray Convert input to an ndarray with column-major memory order. Transposing numpy array is extremely simple using np. Python : how to convert row index values of a Pandas DataFrame to a list or Numpy array +1 vote asked Oct 5 in Programming Languages by pythonuser ( 8k points). table(), read. Python Forums on Bytes. com/u/15199/vs001. Why do you need this ? Aren't dataframes based on numpy arrays anyways ? You should be able to use a dataframe where you need an a numpy array. Creating a Pandas DataFrame from a Numpy array: How do I specify the index column and column headers? Convert pandas dataframe to NumPy array. 5][locs]) … and that peeves me since I end up with a huge array copy in memory. Although it’s very simple, but the concept behind this technique is very unique. This example list is incredibly useful, and we would like to get all the good examples and comments integrated in the official numpy documentation so that they are also shipped with numpy. I have converted a feature class (polylines) to a numpy array and have expoded the polylines to vertices using the "explode_to_points" method. fromfunction Construct an array by executing a function on grid. How do I assign a vector to a subset of rows of a column in a pandas DataFrame with NaNs? This is a bug in 0. Many times you may want to do this in Python in order to work with arrays instead of lists. asfortranarray Convert input to an ndarray with column-major memory order. Python : How to find the indices of elements in a sublist in a list or numpy array +2 votes asked Oct 3, 2018 in Programming Languages by pythonuser ( 8k points). At that stackoverflow page there's also the numpy structured array. axis: It is optional default is 0. How to pretty-printing a numpy. We will learn how to change the data type of an array from float to integer. However, using Numpy arrays and functions has proven tricky, as the Numpy float dtype evidently does not match the Spark FloatType(). As numpy arrays are homogeneous i. It is a vector that contains data of the same type as linear memory. Python has a very powerful library, numpy , that makes working with arrays simple. zeros() Python's Numpy module provides a function to create a numpy array of given shape & type and all values in it initialized with 0's i. This table lays out the different dtypes and default return types of to_numpy() for various dtypes within pandas. to_pickle (self, path[, compression, protocol]) Pickle (serialize) object to file. Since I want to keep the names that were in the pandas. Find the examples here. array() method as an argument and you are done. One of the primary tools for working with numerical data is the NumPy array. python,list,numpy,multidimensional-array. The receiving DataFrame is not extended to accommodate the new series. array with chori zonVar. Can you show an actual example (construct a dataframe, and then what you'd like to be able to do)? I would like to have a view on internal data already stored by dataframes as a numpy array. array = np. To convert this NumPy multidimensional array to an MXNet NDArray, we’re going to use the mx. numpy descends into the lists even if you request a object dtype as it treats object arrays containing nested lists of equal size as ndimensional: np. > So if you can advise how to convert this data. A Pandas Index extends the functionality of NumPy arrays to allow for more versatile slicing and labeling. Simply pass the python list to np. In order to use multiple numpy arrays within the same plot, you need to make sure that the dimensions of the arrays are compatible. Visually, you can represent a NumPy array as something like this: This is a visual representation of a NumPy array that contains five values: 88, 19, 46, 74, 94. This sounds odd, I tested this and after converting to ints the csv file has also only ints. When I use MATLAB engine for python, the outputs of functions are not numpy arrays. You will have to access the data within the class. Convert input to a contiguous array. To accomplish this goal, you may use the following Python code, which will allow you to convert the DataFrame into a list, where: The top part of the code, contains the syntax to create the DataFrame with our data about products and prices. ndim-levels deep nested list of Python scalars. read_csv(filename) #convert dataframe to. Convert sympy matrix objects to numpy arrays. DataFrame(data=matrix, index=np. A dataframe object is most similar to a table. Vectors are strictly 1-d array whereas Matrices are 2-d but matrices can have only one row/column. Visually, you can represent a NumPy array as something like this: This is a visual representation of a NumPy array that contains five values: 88, 19, 46, 74, 94. In Spark, SparkContext. I have this dataframe. And we can also specify column names with the list of tuples. * Similar to a SQL table or Spreadsheet. Return the matrix obtained by converting all the variables in a data frame to numeric mode and then binding them together as the columns of a matrix. arange(9) array We can use NumPy's reshape function to convert the 1d-array to 2d-array of dimension 3×3, 3 rows and 3 columns. asfortranarray Convert input to an ndarray with column-major memory order. Converting a PySpark dataframe to an array In order to form the building blocks of the neural network, the PySpark dataframe must be converted into an array. Convert python list to numpy array. Example 1. genfromtxt:. DataFrame(data=matrix, index=np. We can use numpy ndarray tolist() function to convert the array to a list. Below is the code to create the DataFrame in Python, where the values under the ‘Price’ column are stored as strings (by using single quotes around those values. fromfunction Construct an array by executing a function on grid. I wish to have it transformed into a Pandas DataFrame D where each column of the 2D numpy array in S becomes a 1D numpy array in a column of D: D should look like: 0 1 a [0,2,3] [1,3,4] b [5,7,9] [6,8,10] Note, my actual data set is 1238500 arrays sized (32,8) so i was trying to avoid iterating over rows. This page details how to convert a matplotlib figure to a numpy 3D array of RGBA values, or directly to a PIL ( Python Imaging Library ) Image. To convert this NumPy multidimensional array to an MXNet NDArray, we’re going to use the mx. Sample NumPy array: d1 = [10, 20, 30, 40, 50]. , int64) results in an array of the same type. Add numpy array as column to Pandas data frame; How do I convert a pandas Series or index to a Numpy array? How to determine whether a column/variable is numeric or not in Pandas/NumPy? Ambiguity in Pandas Dataframe / Numpy Array "axis" definition; Convert pandas dataframe to NumPy array. This function will explain how we can convert the pandas Series to numpy Array. Changing the Index of a DataFrame. Rather, they are matlab engine variables. Convert a 1d array (which is a result of complex computations) to a tuple to use it as a basic index in another Nd array to access particular elements. The DataFrame class resembles a collection of NumPy arrays but with labeled axes and mixed data types across the columns. Convert input to a contiguous array. In this article, we show how to retrieve a subset of a pandas DataFrame object in Python. This is just memorandum for my self. Questions: I am interested in knowing how to convert a pandas dataframe into a numpy array, including the index, and set the dtypes. DataFrame objects to lists of formatted row strings, so I can print the rows into, e. Change innermost row index into lowest level of column labels. googlegroups. tif" image is acquired. Converting to NumPy Array. NumPy Array Object Exercises, Practice and Solution: Write a NumPy program to convert a list and tuple into arrays. However, as it seems, as. array), an n-dimensional array that is also present in some form in array-oriented languages such as Fortran 90, R, and MATLAB, as well as predecessors APL and J. To do this, I have been utilizing pandas. value_counts() Africa 624 Asia 396 Europe 360 Americas 300 Oceania 24 If you just want the unique values from a pandas dataframe column, it is pretty simple. Here is my attempt: # Create a NumPy array, which has four elements. dataframe: label A B C ID 1 NaN 0. The dtype will be a lower-common-denominator dtype (implicit upcasting); that is to say if the dtypes (even of numeric types) are mixed, the one that accommodates all will be chosen. For example, you can download the Pima Indians dataset into your local directory (download from here). patch file to my source, but it fails because my file index (10655) is older than the patch index (10755). ]) I wish the array to be in the form 1010, so it can be manipulated. If none are provided, all the columns from the dataframe are extracted. Transpose DataFrame. array characterizes all columns as "object". The following are code examples for showing how to use numpy. Creating a Pandas DataFrame from a Numpy array: How do I specify the index column and column headers? Convert pandas dataframe to NumPy array. In general you can simply use a library like PIL or OpenCV to open the images and convert them to array. In this tutorial, you will learn how to perform many operations on NumPy arrays such as adding, removing, sorting, and manipulating elements in many ways. How do I convert a numpy array into a pandas dataframe? Ask Question Convert pandas dataframe to NumPy array. I have an array of size 1801 that will be all of the column names in the dataframe. If I have a function that can use values from a row in the dataframe as input, then I can map it to the entire dataframe. It is like a spreadsheet with column names and row labels. Transposing numpy array is extremely simple using np. reshape() method. The benefit is fancy indexing. DataFrame(px, columns=['R', 'G', 'B']) Thank you Hugo python numpy pandas | this question asked Feb 21 '14 at 15:49 Hugo 180 1 15 34 1 you have to pass DataFrame with 2D input, not. The function is defined with default parameters so that if you call it with just your data, it will construct a DataFrame with t-1 as X and t as y. Hi, Imagine I have two numpy arrays containing zeros and ones (or bools) effectively being fingerprints: np_1, np_2 = some_fingerprints_as_np_arrays() I want to convert them both to rdkit fingerprint objects so I can use DiceSimilarity: from rdkit import DataStructs # this won't work becuse of type incompatibility DataStructs. unique() Instead, we can simply count the number of unique values in the country column and find that there are 142 countries in the data set. Sort columns. If the array is multi-dimensional, a nested list is returned. recarray (aka "record array") with€np. asfortranarray Convert input to an ndarray with column-major memory order. It is very important to reshape you numpy array, especially you are training with some deep learning network. get_dummies(). If you are new to Python, you may be confused by some of the pythonic ways of accessing data, such as negative indexing and array slicing. array([7, 8, 9. In our example, we need a two dimensional numpy array which represents the features data. In addition, pandas s a package for data manipulation that uses the DataFrame objects from R (as well as different R packages) in a Python. So here, we can see the dtype=np. Because although this is a 1-dimensional array, numpy will broadcast it as a 1 x n matrix while performing matrix operations. To convert Pandas DataFrame to Numpy Array, use the function DataFrame. 1, out soon). Here is the solution I currently use: import numpy as np def scale_array(dat, out_range=(-1,. NumPy Array Object Exercises, Practice and Solution: Write a NumPy program to convert a list and tuple into arrays. In many cases, it is helpful to use a uniquely valued identifying field of the data as its index. Here axis is not passed as an argument so, elements will append with the original array a, at the end. Now I know that I can just modify the patch's file index but I would like to know if there is any way using git to make a patch ignore the file index differences?. Using python to solve simultaneous equations relies on matrix linear algebra and can be done by using a built-in function (method 1) or manually (method 2) manually manipulating the matrices to solve. Same load via np. The 1d-array starts at 0 and ends at 8. # import numpy import numpy as np Let us create a NumPy array using arange function in NumPy. As numpy arrays are homogeneous i.