Transformations de tableaux
Restez organisé à l'aide des collections
Enregistrez et classez les contenus selon vos préférences.
Earth Engine accepte les transformations de tableaux telles que la transposition, l'inverse et la pseudo-inverse.
Prenons l'exemple d'une régression des moindres carrés ordinaires (OLS) d'une série temporelle d'images. Dans l'exemple suivant, une image avec des bandes pour les prédicteurs et une réponse est convertie en image de tableau, puis "résolue" pour obtenir des estimations des coefficients des moindres carrés de trois manières. Commencez par rassembler les données d'image et les convertir en tableaux:
Éditeur de code (JavaScript)
// Scales and masks Landsat 8 surface reflectance images.
function prepSrL8(image) {
// Develop masks for unwanted pixels (fill, cloud, cloud shadow).
var qaMask = image.select('QA_PIXEL').bitwiseAnd(parseInt('11111', 2)).eq(0);
var saturationMask = image.select('QA_RADSAT').eq(0);
// Apply the scaling factors to the appropriate bands.
var opticalBands = image.select('SR_B.').multiply(0.0000275).add(-0.2);
var thermalBands = image.select('ST_B.*').multiply(0.00341802).add(149.0);
// Replace the original bands with the scaled ones and apply the masks.
return image.addBands(opticalBands, null, true)
.addBands(thermalBands, null, true)
.updateMask(qaMask)
.updateMask(saturationMask);
}
// Load a Landsat 8 surface reflectance image collection.
var collection = ee.ImageCollection('LANDSAT/LC08/C02/T1_L2')
// Filter to get only two years of data.
.filterDate('2019-04-01', '2021-04-01')
// Filter to get only imagery at a point of interest.
.filterBounds(ee.Geometry.Point(-122.08709, 36.9732))
// Prepare images by mapping the prepSrL8 function over the collection.
.map(prepSrL8)
// Select NIR and red bands only.
.select(['SR_B5', 'SR_B4'])
// Sort the collection in chronological order.
.sort('system:time_start', true);
// This function computes the predictors and the response from the input.
var makeVariables = function(image) {
// Compute time of the image in fractional years relative to the Epoch.
var year = ee.Image(image.date().difference(ee.Date('1970-01-01'), 'year'));
// Compute the season in radians, one cycle per year.
var season = year.multiply(2 * Math.PI);
// Return an image of the predictors followed by the response.
return image.select()
.addBands(ee.Image(1)) // 0. constant
.addBands(year.rename('t')) // 1. linear trend
.addBands(season.sin().rename('sin')) // 2. seasonal
.addBands(season.cos().rename('cos')) // 3. seasonal
.addBands(image.normalizedDifference().rename('NDVI')) // 4. response
.toFloat();
};
// Define the axes of variation in the collection array.
var imageAxis = 0;
var bandAxis = 1;
// Convert the collection to an array.
var array = collection.map(makeVariables).toArray();
// Check the length of the image axis (number of images).
var arrayLength = array.arrayLength(imageAxis);
// Update the mask to ensure that the number of images is greater than or
// equal to the number of predictors (the linear model is solvable).
array = array.updateMask(arrayLength.gt(4));
// Get slices of the array according to positions along the band axis.
var predictors = array.arraySlice(bandAxis, 0, 4);
var response = array.arraySlice(bandAxis, 4);
Configuration de Python
Consultez la page
Environnement Python pour en savoir plus sur l'API Python et l'utilisation de geemap pour le développement interactif.
import ee
import geemap.core as geemap
Colab (Python)
import math
# Scales and masks Landsat 8 surface reflectance images.
def prep_sr_l8(image):
# Develop masks for unwanted pixels (fill, cloud, cloud shadow).
qa_mask = image.select('QA_PIXEL').bitwiseAnd(int('11111', 2)).eq(0)
saturation_mask = image.select('QA_RADSAT').eq(0)
# Apply the scaling factors to the appropriate bands.
optical_bands = image.select('SR_B.').multiply(0.0000275).add(-0.2)
thermal_bands = image.select('ST_B.*').multiply(0.00341802).add(149.0)
# Replace the original bands with the scaled ones and apply the masks.
return (
image.addBands(optical_bands, None, True)
.addBands(thermal_bands, None, True)
.updateMask(qa_mask)
.updateMask(saturation_mask)
)
# Load a Landsat 8 surface reflectance image collection.
collection = (
ee.ImageCollection('LANDSAT/LC08/C02/T1_L2')
# Filter to get only two years of data.
.filterDate('2019-04-01', '2021-04-01')
# Filter to get only imagery at a point of interest.
.filterBounds(ee.Geometry.Point(-122.08709, 36.9732))
# Prepare images by mapping the prep_sr_l8 function over the collection.
.map(prep_sr_l8)
# Select NIR and red bands only.
.select(['SR_B5', 'SR_B4'])
# Sort the collection in chronological order.
.sort('system:time_start', True)
)
# This function computes the predictors and the response from the input.
def make_variables(image):
# Compute time of the image in fractional years relative to the Epoch.
year = ee.Image(image.date().difference(ee.Date('1970-01-01'), 'year'))
# Compute the season in radians, one cycle per year.
season = year.multiply(2 * math.pi)
# Return an image of the predictors followed by the response.
return (
image.select()
.addBands(ee.Image(1)) # 0. constant
.addBands(year.rename('t')) # 1. linear trend
.addBands(season.sin().rename('sin')) # 2. seasonal
.addBands(season.cos().rename('cos')) # 3. seasonal
.addBands(image.normalizedDifference().rename('NDVI')) # 4. response
.toFloat()
)
# Define the axes of variation in the collection array.
image_axis = 0
band_axis = 1
# Convert the collection to an array.
array = collection.map(make_variables).toArray()
# Check the length of the image axis (number of images).
array_length = array.arrayLength(image_axis)
# Update the mask to ensure that the number of images is greater than or
# equal to the number of predictors (the linear model is solvable).
array = array.updateMask(array_length.gt(4))
# Get slices of the array according to positions along the band axis.
predictors = array.arraySlice(band_axis, 0, 4)
response = array.arraySlice(band_axis, 4)
Notez que arraySlice() renvoie toutes les images de la série temporelle pour la plage d'indices spécifiée le long de bandAxis (axe 1). À ce stade, l'algèbre matricielle peut être utilisée pour déterminer les coefficients OLS:
Éditeur de code (JavaScript)
// Compute coefficients the hard way.
var coefficients1 = predictors.arrayTranspose().matrixMultiply(predictors)
.matrixInverse().matrixMultiply(predictors.arrayTranspose())
.matrixMultiply(response);
Configuration de Python
Consultez la page
Environnement Python pour en savoir plus sur l'API Python et l'utilisation de geemap pour le développement interactif.
import ee
import geemap.core as geemap
Colab (Python)
# Compute coefficients the hard way.
coefficients_1 = (
predictors.arrayTranspose()
.matrixMultiply(predictors)
.matrixInverse()
.matrixMultiply(predictors.arrayTranspose())
.matrixMultiply(response)
)
Bien que cette méthode fonctionne, elle est inefficace et rend le code difficile à lire. Il est préférable d'utiliser la méthode pseudoInverse() (matrixPseudoInverse() pour une image de tableau):
Éditeur de code (JavaScript)
// Compute coefficients the easy way.
var coefficients2 = predictors.matrixPseudoInverse()
.matrixMultiply(response);
Configuration de Python
Consultez la page
Environnement Python pour en savoir plus sur l'API Python et l'utilisation de geemap pour le développement interactif.
import ee
import geemap.core as geemap
Colab (Python)
# Compute coefficients the easy way.
coefficients_2 = predictors.matrixPseudoInverse().matrixMultiply(response)
Du point de vue de la lisibilité et de l'efficacité de calcul, le meilleur moyen d'obtenir les coefficients OLS est solve() (matrixSolve() pour une image de tableau). La fonction solve() détermine la meilleure façon de résoudre le système à partir des caractéristiques des entrées, à l'aide du pseudo-inverse pour les systèmes surdéterminés, de l'inverse pour les matrices carrées et de techniques spéciales pour les matrices presque singulières:
Éditeur de code (JavaScript)
// Compute coefficients the easiest way.
var coefficients3 = predictors.matrixSolve(response);
Configuration de Python
Consultez la page
Environnement Python pour en savoir plus sur l'API Python et l'utilisation de geemap pour le développement interactif.
import ee
import geemap.core as geemap
Colab (Python)
# Compute coefficients the easiest way.
coefficients_3 = predictors.matrixSolve(response)
Pour obtenir une image multibande, projetez l'image du tableau dans un espace à dimension inférieure, puis aplatissez-la:
Éditeur de code (JavaScript)
// Turn the results into a multi-band image.
var coefficientsImage = coefficients3
// Get rid of the extra dimensions.
.arrayProject([0])
.arrayFlatten([
['constant', 'trend', 'sin', 'cos']
]);
Configuration de Python
Consultez la page
Environnement Python pour en savoir plus sur l'API Python et l'utilisation de geemap pour le développement interactif.
import ee
import geemap.core as geemap
Colab (Python)
# Turn the results into a multi-band image.
coefficients_image = (
coefficients_3
# Get rid of the extra dimensions.
.arrayProject([0]).arrayFlatten([['constant', 'trend', 'sin', 'cos']])
)
Examinez les résultats des trois méthodes et observez que la matrice de coefficients résultante est la même, quel que soit le solveur. La flexibilité et l'efficacité de solve() en font un bon choix pour la modélisation linéaire à usage général.
Sauf indication contraire, le contenu de cette page est régi par une licence Creative Commons Attribution 4.0, et les échantillons de code sont régis par une licence Apache 2.0. Pour en savoir plus, consultez les Règles du site Google Developers. Java est une marque déposée d'Oracle et/ou de ses sociétés affiliées.
Dernière mise à jour le 2025/07/25 (UTC).
[null,null,["Dernière mise à jour le 2025/07/25 (UTC)."],[[["\u003cp\u003eEarth Engine enables array transformations like transpose, inverse, and pseudo-inverse for advanced analysis, such as ordinary least squares (OLS) regression on image time series.\u003c/p\u003e\n"],["\u003cp\u003eUsers can convert image collections to arrays, extract predictors and responses, and apply matrix operations to derive regression coefficients.\u003c/p\u003e\n"],["\u003cp\u003eEarth Engine offers multiple methods for solving linear systems, with \u003ccode\u003esolve()\u003c/code\u003e being the most efficient and adaptable for various scenarios, including overdetermined systems and nearly singular matrices.\u003c/p\u003e\n"],["\u003cp\u003eArray images resulting from calculations can be transformed back into multi-band images for visualization and further analysis.\u003c/p\u003e\n"]]],["The content demonstrates ordinary least squares (OLS) regression on a Landsat 8 image time series using Earth Engine. Key actions include preparing images by masking and scaling, creating predictor and response variables (constant, trend, seasonal, and NDVI), and converting the collection to an array. OLS coefficients are then calculated using three methods: direct matrix operations, pseudo-inverse, and the `matrixSolve()` function. Finally, the coefficient array is projected and flattened into a multi-band image. `matrixSolve()` is highlighted as the most efficient and flexible method.\n"],null,["# Array Transformations\n\nEarth Engine supports array transformations such as transpose, inverse and pseudo-inverse.\nAs an example, consider an ordinary least squares (OLS) regression of a time series of\nimages. In the following example, an image with bands for predictors and a response is\nconverted to an array image, then \"solved\" to obtain least squares coefficients estimates\nthree ways. First, assemble the image data and convert to arrays:\n\n### Code Editor (JavaScript)\n\n```javascript\n// Scales and masks Landsat 8 surface reflectance images.\nfunction prepSrL8(image) {\n // Develop masks for unwanted pixels (fill, cloud, cloud shadow).\n var qaMask = image.select('QA_PIXEL').bitwiseAnd(parseInt('11111', 2)).eq(0);\n var saturationMask = image.select('QA_RADSAT').eq(0);\n\n // Apply the scaling factors to the appropriate bands.\n var opticalBands = image.select('SR_B.').multiply(0.0000275).add(-0.2);\n var thermalBands = image.select('ST_B.*').multiply(0.00341802).add(149.0);\n\n // Replace the original bands with the scaled ones and apply the masks.\n return image.addBands(opticalBands, null, true)\n .addBands(thermalBands, null, true)\n .updateMask(qaMask)\n .updateMask(saturationMask);\n}\n\n// Load a Landsat 8 surface reflectance image collection.\nvar collection = ee.ImageCollection('LANDSAT/LC08/C02/T1_L2')\n // Filter to get only two years of data.\n .filterDate('2019-04-01', '2021-04-01')\n // Filter to get only imagery at a point of interest.\n .filterBounds(ee.Geometry.Point(-122.08709, 36.9732))\n // Prepare images by mapping the prepSrL8 function over the collection.\n .map(prepSrL8)\n // Select NIR and red bands only.\n .select(['SR_B5', 'SR_B4'])\n // Sort the collection in chronological order.\n .sort('system:time_start', true);\n\n// This function computes the predictors and the response from the input.\nvar makeVariables = function(image) {\n // Compute time of the image in fractional years relative to the Epoch.\n var year = ee.Image(image.date().difference(ee.Date('1970-01-01'), 'year'));\n // Compute the season in radians, one cycle per year.\n var season = year.multiply(2 * Math.PI);\n // Return an image of the predictors followed by the response.\n return image.select()\n .addBands(ee.Image(1)) // 0. constant\n .addBands(year.rename('t')) // 1. linear trend\n .addBands(season.sin().rename('sin')) // 2. seasonal\n .addBands(season.cos().rename('cos')) // 3. seasonal\n .addBands(image.normalizedDifference().rename('NDVI')) // 4. response\n .toFloat();\n};\n\n// Define the axes of variation in the collection array.\nvar imageAxis = 0;\nvar bandAxis = 1;\n\n// Convert the collection to an array.\nvar array = collection.map(makeVariables).toArray();\n\n// Check the length of the image axis (number of images).\nvar arrayLength = array.arrayLength(imageAxis);\n// Update the mask to ensure that the number of images is greater than or\n// equal to the number of predictors (the linear model is solvable).\narray = array.updateMask(arrayLength.gt(4));\n\n// Get slices of the array according to positions along the band axis.\nvar predictors = array.arraySlice(bandAxis, 0, 4);\nvar response = array.arraySlice(bandAxis, 4);\n```\nPython setup\n\nSee the [Python Environment](/earth-engine/guides/python_install) page for information on the Python API and using\n`geemap` for interactive development. \n\n```python\nimport ee\nimport geemap.core as geemap\n```\n\n### Colab (Python)\n\n```python\nimport math\n\n\n# Scales and masks Landsat 8 surface reflectance images.\ndef prep_sr_l8(image):\n # Develop masks for unwanted pixels (fill, cloud, cloud shadow).\n qa_mask = image.select('QA_PIXEL').bitwiseAnd(int('11111', 2)).eq(0)\n saturation_mask = image.select('QA_RADSAT').eq(0)\n\n # Apply the scaling factors to the appropriate bands.\n optical_bands = image.select('SR_B.').multiply(0.0000275).add(-0.2)\n thermal_bands = image.select('ST_B.*').multiply(0.00341802).add(149.0)\n\n # Replace the original bands with the scaled ones and apply the masks.\n return (\n image.addBands(optical_bands, None, True)\n .addBands(thermal_bands, None, True)\n .updateMask(qa_mask)\n .updateMask(saturation_mask)\n )\n\n\n# Load a Landsat 8 surface reflectance image collection.\ncollection = (\n ee.ImageCollection('LANDSAT/LC08/C02/T1_L2')\n # Filter to get only two years of data.\n .filterDate('2019-04-01', '2021-04-01')\n # Filter to get only imagery at a point of interest.\n .filterBounds(ee.Geometry.Point(-122.08709, 36.9732))\n # Prepare images by mapping the prep_sr_l8 function over the collection.\n .map(prep_sr_l8)\n # Select NIR and red bands only.\n .select(['SR_B5', 'SR_B4'])\n # Sort the collection in chronological order.\n .sort('system:time_start', True)\n)\n\n\n# This function computes the predictors and the response from the input.\ndef make_variables(image):\n # Compute time of the image in fractional years relative to the Epoch.\n year = ee.Image(image.date().difference(ee.Date('1970-01-01'), 'year'))\n # Compute the season in radians, one cycle per year.\n season = year.multiply(2 * math.pi)\n # Return an image of the predictors followed by the response.\n return (\n image.select()\n .addBands(ee.Image(1)) # 0. constant\n .addBands(year.rename('t')) # 1. linear trend\n .addBands(season.sin().rename('sin')) # 2. seasonal\n .addBands(season.cos().rename('cos')) # 3. seasonal\n .addBands(image.normalizedDifference().rename('NDVI')) # 4. response\n .toFloat()\n )\n\n\n# Define the axes of variation in the collection array.\nimage_axis = 0\nband_axis = 1\n\n# Convert the collection to an array.\narray = collection.map(make_variables).toArray()\n\n# Check the length of the image axis (number of images).\narray_length = array.arrayLength(image_axis)\n# Update the mask to ensure that the number of images is greater than or\n# equal to the number of predictors (the linear model is solvable).\narray = array.updateMask(array_length.gt(4))\n\n# Get slices of the array according to positions along the band axis.\npredictors = array.arraySlice(band_axis, 0, 4)\nresponse = array.arraySlice(band_axis, 4)\n```\n\nNote that `arraySlice()` returns all the images in the time series for the\nrange of indices specified along the `bandAxis` (the 1-axis). At this point,\nmatrix algebra can be used to solve for the OLS coefficients:\n\n### Code Editor (JavaScript)\n\n```javascript\n// Compute coefficients the hard way.\nvar coefficients1 = predictors.arrayTranspose().matrixMultiply(predictors)\n .matrixInverse().matrixMultiply(predictors.arrayTranspose())\n .matrixMultiply(response);\n```\nPython setup\n\nSee the [Python Environment](/earth-engine/guides/python_install) page for information on the Python API and using\n`geemap` for interactive development. \n\n```python\nimport ee\nimport geemap.core as geemap\n```\n\n### Colab (Python)\n\n```python\n# Compute coefficients the hard way.\ncoefficients_1 = (\n predictors.arrayTranspose()\n .matrixMultiply(predictors)\n .matrixInverse()\n .matrixMultiply(predictors.arrayTranspose())\n .matrixMultiply(response)\n)\n```\n\nAlthough this method works, it is inefficient and makes for difficult to read code. A\nbetter way is to use the `pseudoInverse()` method\n(`matrixPseudoInverse()` for an array image):\n\n### Code Editor (JavaScript)\n\n```javascript\n// Compute coefficients the easy way.\nvar coefficients2 = predictors.matrixPseudoInverse()\n .matrixMultiply(response);\n```\nPython setup\n\nSee the [Python Environment](/earth-engine/guides/python_install) page for information on the Python API and using\n`geemap` for interactive development. \n\n```python\nimport ee\nimport geemap.core as geemap\n```\n\n### Colab (Python)\n\n```python\n# Compute coefficients the easy way.\ncoefficients_2 = predictors.matrixPseudoInverse().matrixMultiply(response)\n```\n\nFrom a readability and computational efficiency perspective, the best way to get the OLS\ncoefficients is `solve()` (`matrixSolve()` for an array image). The\n`solve()` function determines how to best solve the system from characteristics\nof the inputs, using the pseudo-inverse for overdetermined systems, the inverse for square\nmatrices and special techniques for nearly singular matrices:\n\n### Code Editor (JavaScript)\n\n```javascript\n// Compute coefficients the easiest way.\nvar coefficients3 = predictors.matrixSolve(response);\n```\nPython setup\n\nSee the [Python Environment](/earth-engine/guides/python_install) page for information on the Python API and using\n`geemap` for interactive development. \n\n```python\nimport ee\nimport geemap.core as geemap\n```\n\n### Colab (Python)\n\n```python\n# Compute coefficients the easiest way.\ncoefficients_3 = predictors.matrixSolve(response)\n```\n\nTo get a multi-band image, project the array image into a lower dimensional space, then\nflatten it:\n\n### Code Editor (JavaScript)\n\n```javascript\n// Turn the results into a multi-band image.\nvar coefficientsImage = coefficients3\n // Get rid of the extra dimensions.\n .arrayProject([0])\n .arrayFlatten([\n ['constant', 'trend', 'sin', 'cos']\n]);\n```\nPython setup\n\nSee the [Python Environment](/earth-engine/guides/python_install) page for information on the Python API and using\n`geemap` for interactive development. \n\n```python\nimport ee\nimport geemap.core as geemap\n```\n\n### Colab (Python)\n\n```python\n# Turn the results into a multi-band image.\ncoefficients_image = (\n coefficients_3\n # Get rid of the extra dimensions.\n .arrayProject([0]).arrayFlatten([['constant', 'trend', 'sin', 'cos']])\n)\n```\n\nExamine the outputs of the three methods and observe that the resultant matrix of\ncoefficients is the same regardless of the solver. That `solve()` is flexible\nand efficient makes it a good choice for general purpose linear modeling."]]
