Reducing monthly averaged MODIS data over several ROIs

Question

I want to create monthly averages for the MODIS Land Surface Temperature (LST) product for several ROIs. I am able to create an image collection containing the monthly averaged MODIS data, but I have not figured out how to reduce this data for the ROIs.

Here is the first part, with help from here and here:

var modis = ee.ImageCollection('MODIS/006/MOD11A2');
var months = ee.List.sequence(1, 12);

Map.centerObject(areas, 7);
Map.addLayer(areas, {}, 'ROI');

// First I divide the data into monthly averages
var byMonth = ee.ImageCollection.fromImages(
      months.map(function (m) {
        return modis.filter(ee.Filter.calendarRange(m, m, 'month'))
                    .select(0).mean() // selecting band 1: LST_Day_1km
                    .set('month', m); 
}));
print('byMonth:', byMonth);

var check = ee.Image(byMonth.first());
Map.addLayer(check, {}, 'check');

Then comes the next part, where I try to use .reduceRegions() to reduce over multiple regions (my FeatureCollection "areas"). The idea is to loop/map through the 12 months, reducing the monthly data from "byMonth" to a value for each ROI. Unfortunately it doesn't work.

var reduced = months.map(function(mm) { 
  return byMonth.reduceRegions({
  collection: areas, // a featureCollection containing multiple polygons
  reducer: ee.Reducer.mean(),
  scale: 30,
  maxPixels: 1e9
}).filter(ee.Filter.eq('month', mm));
});
print(reduced);

Answer 1

You had a bunch of bugs in there. This ought to do it:

var modis = ee.ImageCollection('MODIS/006/MOD11A2');
var months = ee.List.sequence(1, 12);

Map.centerObject(areas, 7);
Map.addLayer(areas, {}, 'ROI');

// First I divide the data into monthly averages
var byMonth = ee.ImageCollection.fromImages(
      months.map(function (m) {
        return modis.filter(ee.Filter.calendarRange(m, m, 'month'))
                    .select(0).mean() // selecting band 1: LST_Day_1km
                    .set('month', m); 
}));
print('byMonth:', byMonth);

var check = ee.Image(byMonth.first());
Map.addLayer(check, {}, 'check');

var reduced = months.map(function(mm) { 
  return ee.Image(byMonth
      .filter(ee.Filter.eq('month', mm))
      .first())
      .reduceRegions({
        collection: areas, // a featureCollection containing multiple polygons
        reducer: ee.Reducer.mean(),
        scale: 1000,
      });
});
print(reduced.get(0));